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10794:{\displaystyle {\begin{aligned}f\cdot f^{-1}&=e\cdot (f\cdot f^{-1})&&{\text{(left identity)}}\\&=((f^{-1})^{-1}\cdot f^{-1})\cdot (f\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot ((f^{-1}\cdot f)\cdot f^{-1})&&{\text{(associativity)}}\\&=(f^{-1})^{-1}\cdot (e\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot f^{-1}&&{\text{(left identity)}}\\&=e&&{\text{(left inverse)}}\end{aligned}}}
40:
20741:
3968:
29728:
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16906:
9543:{\displaystyle {\begin{aligned}b&=b\cdot e&&{\text{(}}e{\text{ is the identity element)}}\\&=b\cdot (a\cdot c)&&{\text{(}}c{\text{ and }}a{\text{ are inverses of each other)}}\\&=(b\cdot a)\cdot c&&{\text{(associativity)}}\\&=e\cdot c&&{\text{(}}b{\text{ is an inverse of }}a{\text{)}}\\&=c&&{\text{(}}e{\text{ is the identity element and }}b=c{\text{)}}\end{aligned}}}
298:
10982:
20748:
8284:
27757:
10810:
21465:, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomial equations and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their
8075:
4596:. The underlying set of the group is the above set of symmetries, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first
20589:
are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois
13714:
In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the
8954:(1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of
13715:
reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup
10977:{\displaystyle {\begin{aligned}f\cdot e&=f\cdot (f^{-1}\cdot f)&&{\text{(left inverse)}}\\&=(f\cdot f^{-1})\cdot f&&{\text{(associativity)}}\\&=e\cdot f&&{\text{(right inverse)}}\\&=f&&{\text{(left identity)}}\end{aligned}}}
21612:
of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group
8070:
23373:. They can, for instance, be used to construct simple modelsâimposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example is the group of
16925:
into another, and the group operation is "concatenation" (tracing one loop then the other). For example, as shown in the figure, if the topological space is the plane with one point removed, then loops which do not wrap around the missing point (blue)
15778:
14551:
21230:
A group action gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and
17613:
17058:, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as
25602:
13253:, highlighted in red in the Cayley table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The
11199:
for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are done). However,
16800:
17648:. Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of
14781:
8279:{\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}
10200:, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker.
19245:
23377:, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of
19592:
12735:
1516:
19777:
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3135:
by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.
22998:
under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every
9139:
16061:
21878:
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4872:
13696:
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a
Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be
21383:
11758:
233:âan active mathematical disciplineâstudies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as
24109:
24061:
24013:
21225:
24334:
is any small category in which every morphism is an isomorphism. In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined:
22536:
21227:
from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.
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20689:
analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.
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2236:, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.
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11962:
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13700:. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.
19944:
18076:
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12240:
12199:
7963:
145:. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.
20951:
20869:(the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the
19191:
8588:
are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.
2250:, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.
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17142:
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12008:. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.
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if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the
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The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and â if
17644:
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An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set
22490:
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and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the
16542:
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4559:
sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the
23125:. The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other
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of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by
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3812:. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.
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is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of
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245:. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of
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containing these three elements; in other words, all relations are consequences of these three. The quotient of the free group by this normal subgroup is denoted
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10987:
These proofs require all three axioms (associativity, existence of left identity and existence of left inverse). For a structure with a looser definition (like a
3192:
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27018:
Behler, Florian; Wickleder, Mathias S.; Christoffers, Jens (2014), "Biphenyl and bimesityl tetrasulfonic acid â new linker molecules for coordination polymers",
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may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally,
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17007:
In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein,
1415:
27311:
23935:
More general structures may be defined by relaxing some of the axioms defining a group. The table gives a list of several structures generalizing groups.
16909:
The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers under addition.
9064:
7763:, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose
21608:
is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (
28187:
19198:
22959:, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids
21833:
3916:. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called
17737:â fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.
7562:
4817:
21313:
20704:
and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft
18772:, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer
1386:
1045:
28507:
26986:
19686:
16549:
7898:
27399:
23163:
19808:
19805:
of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as
12688:
9153:
The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements
28379:
18369:
17034:, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as
16005:
25163:
database of mathematics publications lists 1,779 research papers on group theory and its generalizations written in 2020 alone. See
27713:
20485:
are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to
18839:
13644:
8686:
8627:
3646:. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition,
28439:
21014:. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the
16863:
of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains
11698:
28890:
28746:
24068:
24020:
23972:
23938:
For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a
22808:
17946:, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent
9030:
603:
262:
21176:
18294:
28688:
28662:
28463:
28402:
28356:
28332:
28310:
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28173:
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28104:
28085:
28058:
27974:
27955:
27929:
27888:
27852:
27825:
27804:
27783:
27746:
27722:
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27606:
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27517:
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27194:
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27115:
27009:
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26869:
26836:
26782:
26741:
26720:
22497:
20666:
by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on.
20523:, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.
4503:
These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example,
21469:) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and
20685:. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of
22426:
22091:
21489:
17442:, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no
15785:
14846:
14609:
13178:
553:
26731:
24613:
21256:
were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the
12339:
12247:
22090:
More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups:
13258:
11564:
2570:
8789:
developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of
28346:
27554:
24287:
11903:
8534:
are their own inverse, because performing them twice brings the square back to its original orientation. The rotations
8065:{\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}
7652:). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table.
1379:
1038:
548:
17501:
17405:
17149:
3455:
29244:
28288:
28004:
27471:
27300:
27061:
23139:
17321:
137:
The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers,
17:
23251:
19902:
18034:
15174:
12206:
12165:
3815:
Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are
27400:"Herstellung von Graphen mit vorgegebener abstrakter Gruppe [Construction of graphs with prescribed group]"
22087:
is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.
20914:
20715:
has found further application in elementary particle physics, where its occurrence is related to the appearance of
19148:
11241:
75:
of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is
28434:
27462:
21492:âa precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
18575:
17109:
15924:
is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the
7399:
27759:
List of papers reviewed on MathSciNet on "Group theory and its generalizations" (MSC code 20), published in 2020
22276:
20183:
19287:
19250:
17000:), because the loop cannot be smoothly deformed across the hole, so each class of loops is characterized by its
16422:
14083:
13895:
13840:
10355:), one can show that every left inverse is also a right inverse of the same element as follows. Indeed, one has
28677:
The
Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory
28550:
27420:
25293:
24199:
23122:
20795:
9037:
and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of
964:
28541:
Traité des substitutions et des équations algébriques [Study of
Substitutions and Algebraic Equations]
23158:; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the
19966:
is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are
8591:
In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in
1651:. The following properties of integer addition serve as a model for the group axioms in the definition below.
29759:
29682:
28394:
28182:
27432:
26828:
20712:
15773:{\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}
9053:
Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under
8971:
8892:
had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas,
8798:
7619:
that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (
3283:
1372:
1031:
22897:
17620:
8876:
Geometry was a second field in which groups were used systematically, especially symmetry groups as part of
3426:
29530:
27159:
26431:
25481:
22832:
17947:
17188:
form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example,
14546:{\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}
11852:
9029:, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the
3568:
whenever the group operation is notated as multiplication; in this case, the identity is typically denoted
28520:
22796:
22459:
17855:
28620:
28161:
25075:
23000:
16895:
16515:
16231:
15634:
11593:
Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A
11112:
11066:
8913:
7483:
Given this set of symmetries and the described operation, the group axioms can be understood as follows.
1241:
648:
462:
29206:
27587:
Quaternions and
Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality
17608:{\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}
15861:
15575:
15338:
13312:
10994:
The same result can be obtained by only assuming the existence of a right identity and a right inverse.
29703:
29461:
28739:
28421:
27590:
27509:
27424:
20833:
20821:
20603:
20177:
17251:
17031:
12038:
3705:
380:
254:
24420:
Finally, it is possible to generalize any of these concepts by replacing the binary operation with an
23167:
22570:
22399:
22247:
22171:
21673:
21616:
21582:
21528:
21101:
21022:
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18003:
16646:
16263:
16178:
16149:
16093:
15931:
15898:
14667:
14558:
14314:
13584:
13000:
8745:
8596:
8479:
8446:
8413:
8380:
7735:
7658:: The associativity axiom deals with composing more than two symmetries: Starting with three elements
7624:
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7112:
7082:
7052:
7022:
6992:
6850:
6820:
6790:
6760:
6730:
6588:
6558:
6528:
6498:
6468:
6326:
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6266:
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6206:
6174:
6144:
6114:
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5884:
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5394:
5364:
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5136:
5106:
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4438:
4401:
4368:
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4191:
4152:
4114:
3933:
3249:
27072:
22862:
The gap between the classification of simple groups and the classification of all groups lies in the
21577:
20692:
Group theory helps predict the changes in physical properties that occur when a material undergoes a
20326:
16874:
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by
16204:
is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism
15961:
14026:
13467:
12491:
12101:
11664:
10314:
8955:
3493:
whenever the group operation is notated as addition; in this case, the identity is typically denoted
1728:
27552:(1937), "Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy",
23150:. Other locally compact topological groups include the group of points of an algebraic group over a
22932:
21093:
are linear representations in which the group acts on a vector space, such as the three-dimensional
20441:
Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element
17146:, has been described above. The integers, with the operation of multiplication instead of addition,
3888:
is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.
29708:
29698:
29407:
28801:
28444:
21262:
21015:
20813:
20406:
20369:
16930:
and are the identity element of the fundamental group. A loop which wraps around the missing point
14032:
13615:
12651:
9997:
9710:
8926:
8838:
8351:
8296:
7257:
7226:
6936:
6646:
6356:
6002:
5770:
5538:
5194:
5168:
4936:
3974:
3909:
3853:
846:
580:
457:
345:
270:
157:
24582:
24552:
24522:
24491:
24461:
23948:
20832:
are well-behaved. Geometric properties that remain stable under group actions are investigated in
17718:
17379:
17288:
17085:
16844:
15971:
12133:
9198:
3404:
3378:
3356:
3327:
3227:
3147:
1630:
726:
701:
664:
29713:
29417:
29307:
29201:
27947:
25388:
For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See
23189:
22821:
22813:
20591:
19802:
19321:
18968:
18748:, and therefore that the modular product is nonzero. The identity element is represented by
18157:, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on
16639:
16078:
11247:
8780:
1332:
30:
This article is about basic notions of groups in mathematics. For a more advanced treatment, see
18964:
18912:
17467:
17217:
17022:
In addition to the above theoretical applications, many practical applications of groups exist.
16489:
16390:
13042:
11600:
11203:
10228:
9963:
9674:
2935:
2901:
2762:
2726:
29595:
29520:
29441:
29431:
29397:
29347:
29086:
27245:, Lecture Notes in Mathematics (in French), vol. 1441, Berlin, New York: Springer-Verlag,
26800:
26770:
25347:
The transition from the integers to the rationals by including fractions is generalized by the
25263:
23967:(including zero) under addition form a monoid, as do the nonzero integers under multiplication
23508:
23398:
23374:
22960:
20655:
18973:
17712:
17316:
17008:
13173:
In the example of symmetries of a square, the identity and the rotations constitute a subgroup
12634:
11509:
11008:
10188:
The group axioms for identity and inverses may be "weakened" to assert only the existence of a
9058:
3913:
3905:
3857:
3816:
2246:
The axioms for a group are short and natural ... Yet somehow hidden behind these axioms is the
996:
786:
266:
91:
27411:
27051:
26726:, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article.
23024:
20654:
and the composition of these operations follows the group law. For example, an element of the
19626:
14929:
12567:
11830:
11459:
10105:
9834:
7492:
4717:
lists the results of all such compositions possible. For example, rotating by 270° clockwise (
4643:
3827:
3674:
2433:
29754:
29590:
29584:
29357:
29237:
29072:
29024:
28732:
28482:
25540:
24888:
23452:
23437:
21771:
20905:
20852:
20829:
20599:
20531:
20289:
20152:. The group operation is multiplication of complex numbers. In the picture, multiplying with
20099:
20066:
18533:
16975:
11312:
9014:
8979:
8963:
8826:
8802:
3901:
2342:
1597:
870:
246:
48:
27736:
25611:, p. 1, §1.1: "The idea of a group is one which pervades the whole of mathematics both
23393:. The full symmetry group of Minkowski space, i.e., including translations, is known as the
23050:
21953:
21885:
19654:
17004:
around the missing point. The resulting group is isomorphic to the integers under addition.
15378:
13147:
11389:
11046:
10262:
10059:
9788:
9247:
The group axioms also imply that the inverse of each element is unique: Let a group element
9142:
4915:
3871:
3617:
3066:
2322:
29554:
29477:
29435:
29378:
29300:
29048:
29042:
28806:
28628:
28604:
28584:
28555:
28259:
28114:
28068:
28034:
27984:
27898:
27862:
27835:
27677:
27650:
27616:
27594:
27563:
27450:
27387:
27334:
27260:
27233:
27167:
27151:
27098:
26919:
26891:
26846:
26812:
26792:
26762:
25573:
25397:
24410:
23457:
23442:
23354:
23350:
22964:
22817:
22674:
22599:
22543:
22432:
22200:
21975:
21806:
21700:
21090:
20901:
20866:
20848:
20760:
20755:
20590:
theory (see below). Conceptually, group theory can be thought of as the study of symmetry.
19597:
19414:
19365:
17071:
17063:
16891:
16864:
16677:
16361:
16312:
16209:
16122:
15054:
14249:
13557:
13416:
12949:
12922:
11632:
11005:
inverse (or vice versa) is not sufficient to define a group. For example, consider the set
10193:
8908:
8564:
8537:
8510:
7348:
7317:
7286:
7198:
7170:
7142:
6962:
6908:
6880:
6700:
6672:
6618:
6438:
6410:
6382:
6056:
6028:
5974:
5946:
5796:
5742:
5714:
5686:
5510:
5482:
5454:
5426:
5276:
5248:
5220:
5018:
4990:
4962:
4722:
4506:
4334:
4305:
4276:
4077:
4042:
4007:
3820:
2104:
2070:
1979:
1945:
1319:
1311:
1283:
1278:
1269:
1226:
1168:
810:
798:
416:
350:
250:
76:
28078:
The Words of
Mathematics: An Etymological Dictionary of Mathematical Terms Used in English
27906:
25576:
for an example where symmetry greatly reduces the complexity analysis of physical systems.
21388:
15088:
229:
and geometry, the group notion was generalized and firmly established around 1870. Modern
8:
29611:
29535:
29481:
29457:
29401:
29342:
29327:
29317:
28781:
28430:
28139:
Solomon, Ronald (2018), "The classification of finite simple groups: A progress report",
27545:
27369:
25616:
25505:
action on the vector space of solutions of the differential equations is considered. See
25485:
25361:
25310:
24855:
24414:
23418:
22848:
22762:
22332:
21667:
20788:
20579:
20565:
20239:
20127:
20010:
elements is isomorphic to this group. A second example for cyclic groups is the group of
19449:
19032:
18503:
18421:
18224:
18103:
17975:
17921:
17681:
17651:
17191:
16899:
15244:
15119:
14901:
9145:
can be inserted anywhere within such a series of terms, parentheses are usually omitted.
8889:
8885:
8790:
3321:
3171:
3128:
1572:
1337:
1327:
1178:
1061:
385:
280:
200:
28539:
27738:
Combinatorial Group Theory: Presentations of Groups in Terms of
Generators and Relations
27598:
27567:
26978:
25544:
24824:
24793:
24339:
21480:
generalizes the above type of Galois groups by shifting to field theory and considering
18619:
18272:
16795:{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}
15149:
15029:
15006:
14983:
14365:
3649:
3542:
3202:
2186:
29570:
29487:
29391:
29322:
29290:
29269:
29264:
28981:
28797:
28791:
28680:
28635:
28572:
28499:
28278:
28244:, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press,
28206:
28047:
27629:
27219:
27203:
26974:
26956:
25433:
25429:
25348:
25331:
25067:
24390:
24366:
24263:
24239:
24177:
24146:
24138:
24118:
23868:
23397:. By the above, it plays a pivotal role in special relativity and, by implication, for
23390:
23329:
23309:
23289:
23231:
23211:
23100:
23080:
22877:
22740:
22720:
22700:
22654:
22634:
22379:
22310:
22227:
22142:
22117:
22097:
22070:
22050:
22028:
22004:
21929:
21907:
21749:
21729:
21709:
21647:
21559:
21428:
21408:
21257:
21152:
21132:
21058:
20985:
20961:
20874:
20700:
materials, where the change from a paraelectric to a ferroelectric state occurs at the
20674:
20635:
20611:
20553:
20468:
20446:
20267:
20217:
20155:
20105:
20044:
20015:
19993:
19971:
19949:
19784:
19455:
19392:
19345:
19124:
19102:
19080:
19058:
19010:
18946:
18890:
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18797:
18775:
18753:
18729:
18707:
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18663:
18641:
18479:
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18399:
18377:
18349:
18250:
18202:
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18160:
18131:
18083:
17953:
17899:
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17793:
17773:
17756:
17746:
17445:
17353:
17059:
17016:
16953:
16933:
16883:
16824:
16339:
16292:
15925:
15612:
15550:
15529:
15508:
15485:
15464:
15443:
15420:
15399:
15328:
15300:
15276:
14963:
14824:
14802:
14776:{\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}
14589:
14432:
14412:
14390:
14345:
14283:
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14207:
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14005:
13999:
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13266:
13125:
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13081:
12978:
12900:
12878:
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12832:
12810:
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12611:
12589:
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12525:
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12391:
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12299:
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12018:
11989:
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11805:
11783:
11763:
11588:
11572:
11537:
11515:
11491:
11437:
11415:
11367:
11343:
11319:
11294:
11182:
11160:
10991:) one may have, for example, that a left identity is not necessarily a right identity.
10292:
10206:
10161:
10135:
10085:
10037:
9941:
9919:
9894:
9868:
9814:
9768:
9748:
9652:
9632:
9610:
9588:
9568:
9290:
9270:
9250:
9176:
9156:
9038:
9034:
8814:
8323:
7876:
7852:
7830:
7808:
7786:
7766:
7711:
7687:
7663:
7540:
7518:
4691:
4669:
4619:
4599:
4271:
rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by
4265:
3865:
3785:
3763:
3743:
3595:
3573:
3520:
3498:
3044:
3016:
2994:
2969:
2881:
2861:
2839:
2817:
2702:
2680:
2660:
2640:
2546:
2522:
2498:
2474:
2409:
2387:
2367:
2347:
2298:
2272:
2217:
2164:
2138:
2050:
2028:
1925:
1894:
1872:
1852:
1832:
1812:
1792:
1704:
1682:
1660:
1604:
1547:
1525:
1143:
1134:
1092:
370:
342:
218:
193:
185:
115:
28516:
27870:
8940:
The convergence of these various sources into a uniform theory of groups started with
8794:
204:
29764:
29661:
29631:
29501:
29411:
29362:
29295:
28904:
28707:
28684:
28658:
28503:
28459:
28451:
28398:
28362:
28352:
28328:
28306:
28284:
28263:
28245:
28223:
28169:
28126:
28100:
28081:
28054:
28042:
28000:
27970:
27951:
27925:
27884:
27848:
27821:
27800:
27779:
27742:
27718:
27708:
27694:
27663:
27636:
27602:
27532:
27513:
27503:
27486:
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27457:
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27373:
27347:
27296:
27277:
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27190:
27171:
27137:
27111:
27057:
27042:
27005:
26960:
26927:
26907:
26865:
26832:
26778:
26737:
26716:
25393:
25314:
25061:
24605:
23414:
23147:
23126:
23018:
23012:
22956:
22863:
22852:
21462:
21232:
21080:
20954:
20701:
20686:
16918:
16914:
16887:
16878:
groups to them and studying the properties of the corresponding groups. For example,
12434:
11289:
9018:
8918:
3861:
3132:
2267:
2255:
775:
618:
512:
64:
29143:
27794:
23394:
22197:
of symmetries of a square is a finite group of order 8. In this group, the order of
20828:
of a prescribed form, giving group-theoretic criteria for when solutions of certain
16879:
941:
177:
29731:
29616:
29285:
29230:
29211:
29148:
29123:
29115:
29107:
29099:
29091:
29078:
29060:
29054:
28564:
28528:
28491:
28473:
28196:
28148:
27921:
27902:
27571:
27365:
27207:
27084:
27037:
27027:
26899:
26879:
25612:
25532:
25306:
23464:
23447:
22856:
21515:
21458:
21450:
20856:
20836:
20772:. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.
20693:
20663:
17043:
17012:
16868:
12633:
The collection of all groups, together with the homomorphisms between them, form a
12630:, provided that any specific elements mentioned in the statement are also renamed.
9022:
8999:
8959:
8951:
8881:
8806:
3897:
2290:
2016:
1622:
1163:
926:
918:
910:
902:
894:
882:
822:
762:
752:
594:
536:
411:
173:
80:
68:
27089:
22970:
This way of defining groups lends itself to generalizations such as the notion of
1188:
29651:
29641:
29560:
29387:
29312:
29164:
28998:
28867:
28776:
28624:
28600:
28580:
28274:
28255:
28110:
28064:
28030:
27980:
27913:
27894:
27858:
27831:
27673:
27646:
27612:
27446:
27383:
27330:
27256:
27229:
27147:
27133:
27132:, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York:
27094:
27032:
26887:
26861:
26842:
26808:
26788:
26758:
25518:
This was crucial to the classification of finite simple groups, for example. See
24514:
23598:
23575:
23378:
22927:
22836:
21520:
21485:
21481:
21094:
21076:
20716:
20682:
17371:
17055:
16834:
14302:
8995:
2157:
1255:
1249:
1236:
1216:
1207:
1173:
1110:
1010:
1003:
989:
946:
834:
757:
587:
501:
441:
321:
142:
138:
84:
44:
28015:
27944:
Symmetry and the
Monster: The Story of One of the Greatest Quests of Mathematics
23405:
are central to the modern description of physical interactions with the help of
22273:
etc. is 2. Both orders divide 8, as predicted by
Lagrange's theorem. The groups
20708:
mode, a vibrational lattice mode that goes to zero frequency at the transition.
29666:
29655:
29574:
29425:
29337:
29169:
29011:
28964:
28957:
28950:
28943:
28915:
28882:
28852:
28847:
28837:
28786:
28535:
28237:
27732:
27690:
26750:
25401:
24574:
24454:
24169:
23943:
23733:
23710:
23410:
23402:
23193:
23159:
22840:
22366:
22162:
21466:
21446:
20825:
20570:
20547:
20039:
19240:{\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}
17001:
16927:
16812:
14794:
14076:
12875:, with the same operation. Concretely, this means that the identity element of
11576:
8967:
8941:
8810:
8786:
4560:
3195:
1297:
1017:
953:
643:
623:
560:
525:
446:
436:
421:
406:
360:
337:
238:
169:
161:
107:
24435:). With the proper generalization of the group axioms, this gives a notion of
20526:
The study of finitely generated abelian groups is quite mature, including the
17026:
relies on the combination of the abstract group theory approach together with
8987:
29748:
29580:
29545:
29179:
29138:
29066:
28987:
28972:
28920:
28872:
28827:
28612:
28411:
28366:
28298:
27813:
27549:
27499:
27395:
26712:
26704:
25536:
25415:
25259:
25040:
25020:
25010:
23912:
23130:
22995:
22844:
21477:
21236:
20809:
20805:
20697:
20696:, for example, from a cubic to a tetrahedral crystalline form. An example is
20035:
18154:
17315:
The desire for the existence of multiplicative inverses suggests considering
17285:, which is a rational number, but not an integer. Hence not every element of
14204:
may or may not be the same as its right cosets. If they are (that is, if all
13735:
determines left and right cosets, which can be thought of as translations of
13254:
10189:
8904:
8830:
3809:
2247:
1915:
1183:
1148:
1105:
936:
858:
692:
565:
431:
226:
164:
of the object, and the transformations of a given type form a general group.
28267:
20530:; and reflecting this state of affairs, many group-related notions, such as
14662:
and similarly for the other elements of the group. (In fact, in the case of
4530:
sends a point to its rotation 90° clockwise around the square's center, and
29564:
29497:
29471:
29352:
29133:
28933:
28895:
28842:
28832:
28822:
28763:
28711:
28672:
28650:
28458:, History of Mathematics, Providence, R.I.: American Mathematical Society,
28386:
28320:
27992:
27576:
27125:
26100:
23688:
23665:
23486:
23406:
23143:
22971:
22791:
21501:
21248:
21010:
20897:
20659:
20575:
19333:
17067:
17023:
16808:
16000:
factors canonically as a quotient homomorphism followed by an isomorphism:
12786:
10203:
In particular, assuming associativity and the existence of a left identity
9007:
9003:
8983:
8934:
8922:
8870:
8818:
4877:
4714:
3109:
3105:
1357:
1288:
1122:
791:
490:
479:
426:
401:
396:
355:
326:
289:
258:
242:
230:
222:
103:
72:
31:
27466:(2nd ed.), Reading, MA: Addison-Wesley Publishing, pp. 588â596,
26911:
26134:
24196:
in which every morphism is an isomorphism: given such a category, the set
23204:
A standard example is the general linear group introduced above: it is an
18724:
ensures that the usual product of two representatives is not divisible by
16841:
Examples and applications of groups abound. A starting point is the group
15671:. The group operation on the quotient is shown in the table. For example,
12438:
is a homomorphism that has an inverse homomorphism; equivalently, it is a
29635:
29621:
29539:
29525:
29491:
28862:
28857:
28456:
Pioneers of
Representation Theory: Frobenius, Burnside, Schur, and Brauer
28342:
27624:
26096:
26053:
25035:
25025:
24544:
24436:
23382:
23205:
23151:
22991:
20678:
20670:
20561:
20557:
16066:
9026:
8975:
8930:
8877:
8822:
3805:
3140:
1347:
1342:
1231:
1221:
1195:
189:
56:
28639:
27224:
26077:
25658:
25559:
isomorphism, there are about 49 billion groups of order up to 2000. See
22831:
groups was a major achievement in contemporary group theory. There are
20861:
20574:
The (2,3,7) triangle group, a hyperbolic reflection group, acts on this
29645:
29625:
29550:
29174:
28592:
28576:
28495:
28210:
27939:
26948:
26853:
26820:
25160:
23824:
23801:
23643:
23620:
23198:
23155:
22986:
21511:
20535:
19587:{\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}
16829:
16672:
by generators and relations, because the first isomorphism theorem for
16084:
15965:
8893:
3804:. If this additional condition holds, then the operation is said to be
1097:
958:
686:
153:
27077:
Electronic
Research Announcements of the American Mathematical Society
13554:. In the example of symmetries of a square, the subgroup generated by
11579:. These are the analogues that take the group structure into account.
9017:'s 1960â61 Group Theory Year brought together group theorists such as
2020:
of addition because adding it to any integer returns the same integer.
1511:{\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}
29034:
28716:
28153:
27435:, Readings in Mathematics, vol. 129, New York: Springer-Verlag,
27273:
27202:
26831:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
26239:
25502:
23778:
23755:
23386:
23370:
23179:
23004:
21609:
21170:
20817:
20733:
20173:
19772:{\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}
18471:
17039:
17027:
14023:
12645:
12439:
10988:
9858:
9061:
applications of the associativity axiom show that the unambiguity of
8897:
4814:). Using the above symbols, highlighted in blue in the Cayley table:
3852:; then the identity may be denoted id. In the more specific cases of
1567:
1352:
1158:
1115:
1083:
779:
181:
165:
28568:
28522:
Manuscrits de Ăvariste Galois [Ăvariste Galois' Manuscripts]
28201:
26284:
26182:
25279:
However, a group is not determined by its lattice of subgroups. See
16905:
16628:{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }
29421:
29383:
29332:
28771:
27485:(2nd ed.), West Sussex, England: John Wiley & Sons, Ltd.,
26467:
26443:
25413:
The stated property is a possible definition of prime numbers. See
24331:
23553:
23530:
23358:
22975:
21470:
20595:
19195:. Hence all group axioms are fulfilled. This example is similar to
16922:
12798:
11568:
9134:{\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}
7375:
4430:
3917:
1519:
1153:
316:
234:
149:
99:
27820:, vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag,
27270:
Universal Algebra and Applications in Theoretical Computer Science
22244:
that this element generates. The order of the reflection elements
20740:
9141:
generalizes to more than three factors. Because this implies that
7394:
yellow (last column), respectively. The result of the composition
29253:
24484:
22824:
all finite groups is a problem considered too hard to be solved.
22134:
20842:
20765:
18221:, as shown in the illustration. This is expressed by saying that
17035:
16875:
12730:{\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}
8991:
1410:
658:
572:
273:
as geometric objects, has become an active area in group theory.
95:
39:
28351:(second ed.), Princeton, N.J.: Princeton University Press,
26332:
17374:. The set of all such irreducible fractions is commonly denoted
12685:
factors canonically as an isomorphism followed by an inclusion,
25045:
25030:
25015:
25005:
25000:
23939:
23846:
20775:
20726:
20705:
19338:
16056:{\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}
14898:
denotes its set of cosets. Then there is a unique group law on
11560:
8907:. Certain abelian group structures had been used implicitly in
8835:
On the theory of groups, as depending on the symbolic equation
8793:
of degree higher than 4. The 19th-century French mathematician
4223:
4184:
4070:
4035:
3921:
3108:
of a set and a binary operation on this set that satisfies the
1087:
297:
26979:"The status of the classification of the finite simple groups"
26440:, Chapter VI (see in particular p. 273 for concrete examples).
26308:
26065:
25449:
The additive notation for elements of a cyclic group would be
23003:, such as the red arc in the figure, looks like a part of the
21873:{\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}
20630:
if every group element can be associated to some operation on
17950:. Modular addition, defined in this way for the integers from
17751:
17615:
does form an abelian group under multiplication, also denoted
16921:
of loops, where loops are considered equivalent if one can be
16867:. These groups are predecessors of important constructions in
9048:
4145:
4107:
4000:
3967:
134:(these properties characterize the integers in a unique way).
27187:
The JahnâTeller Effect in C60 and Other Icosahedral Complexes
25993:
25556:
24421:
23362:
21004:
matrices with real entries. Its subgroups are referred to as
20538:, describe the extent to which a given group is not abelian.
19174:
18595:
15327:. The quotient group can alternatively be characterized by a
13709:
7612:{\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}
7383:
4867:{\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}
4781:) is the same as performing a reflection along the diagonal (
4710:"). This is the usual notation for composition of functions.
27240:
26807:(2nd ed.), Lexington, Mass.: Xerox College Publishing,
26777:(3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc.,
26663:
26083:
25886:
25094:
Some authors include an additional axiom referred to as the
23349:
Lie groups are of fundamental importance in modern physics:
19472:. In multiplicative notation, the elements of the group are
13691:{\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}
10183:
8733:{\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}
8674:{\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}
7895:. These two ways must give always the same result, that is,
3904:
if one can be changed into the other using a combination of
27071:
Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001),
27017:
26651:
26373:
26371:
26122:
25480:
More rigorously, every group is the symmetry group of some
23366:
21670:. Parallel to the group of symmetries of the square above,
17708:, therefore the axiom of the inverse element is satisfied.
7803:
into a single symmetry, then to compose that symmetry with
4363:
reflections about the horizontal and vertical middle line (
2011:
94:
are groups endowed with other properties. For example, the
29222:
26194:
26146:
25915:
25913:
25898:
25330:
Elements which do have multiplicative inverses are called
24113:, and likewise adjoining inverses to any (abelian) monoid
22839:" that do not belong to any of the families. The largest
21378:{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
20747:
19282:
that have a multiplicative inverse. These groups, denoted
17896:. Every integer is equivalent to one of the integers from
12562:
simply by renaming its elements according to the function
11753:{\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)}
8340:
when composed with it either on the left or on the right.
28724:
28553:(1986), "The evolution of group theory: A brief survey",
28478:"Gruppentheoretische Studien (Group-theoretical studies)"
25802:
25766:
25682:
25204:
is abelian, because of the ambiguity of whether it means
24104:{\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )}
24056:{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}
24008:{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}
22802:
21699:
can also be interpreted as the group of symmetries of an
19247:
above: it consists of exactly those elements in the ring
18153:
A familiar example is addition of hours on the face of a
15003:
is a homomorphism. Explicitly, the product of two cosets
14342:, the group of symmetries of a square, with its subgroup
28391:
Essays in the History of Lie Groups and Algebraic Groups
28185:(1951), "On the lattice of subgroups of finite groups",
26455:
26407:
26368:
26296:
25944:
25942:
25940:
25622:
22371:
fundamental theorem of finitely generated abelian groups
21220:{\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )}
20528:
fundamental theorem of finitely generated abelian groups
18470:, replaces the usual product by its representative, the
17062:
and fields. Further abstract algebraic concepts such as
17011:
employs geometric concepts, for example in the study of
13144:, indeed form a group. In this case, the inclusion map
8978:'s papers. The theory of Lie groups, and more generally
3452:
whose underlying set is the set of nonzero real numbers
168:
appear in symmetry groups in geometry, and also in the
26248:
26170:
26158:
26029:
26005:
25981:
25971:
25969:
25954:
25910:
25590:
23129:, such as the field of complex numbers or the field of
22531:{\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}
22369:
of finite cyclic groups; this statement is part of the
16807:
A presentation of a group can be used to construct the
10804:
Similarly, the left identity is also a right identity:
2458:, such that the following three requirements, known as
160:: The symmetries of an object form a group, called the
28220:
Foundations of Differentiable Manifolds and Lie Groups
27631:
Galois' Dream: Group Theory and Differential Equations
26603:
26519:
26017:
25790:
25742:
25670:
25646:
17262:
14160:. Similar considerations apply to the right cosets of
14136:, i.e., when the two elements differ by an element of
3127:
A group and its underlying set are thus two different
27241:
Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990),
27070:
26615:
26041:
25937:
25560:
24891:
24858:
24827:
24796:
24616:
24585:
24555:
24525:
24494:
24464:
24393:
24369:
24342:
24290:
24266:
24242:
24202:
24180:
24149:
24121:
24071:
24023:
23975:
23951:
23332:
23312:
23292:
23254:
23234:
23214:
23166:. A generalization used in algebraic geometry is the
23103:
23083:
23053:
23027:
22935:
22900:
22880:
22765:
22743:
22723:
22703:
22677:
22657:
22637:
22602:
22573:
22546:
22500:
22462:
22435:
22402:
22382:
22335:
22313:
22279:
22250:
22230:
22203:
22174:
22145:
22120:
22100:
22073:
22053:
22031:
22007:
21978:
21956:
21932:
21910:
21888:
21836:
21809:
21774:
21752:
21732:
21712:
21676:
21650:
21619:
21585:
21562:
21531:
21431:
21411:
21391:
21316:
21265:
21179:
21155:
21135:
21104:
21061:
21025:
20988:
20964:
20917:
20877:
20638:
20614:
20493:
20471:
20449:
20409:
20372:
20329:
20292:
20270:
20242:
20220:
20186:
20158:
20130:
20108:
20069:
20047:
20018:
19996:
19974:
19952:
19905:
19811:
19787:
19689:
19657:
19629:
19600:
19478:
19458:
19417:
19395:
19368:
19348:
19290:
19253:
19201:
19151:
19127:
19105:
19083:
19061:
19035:
19013:
18976:
18949:
18915:
18893:
18842:
18822:
18800:
18778:
18756:
18732:
18710:
18688:
18666:
18644:
18622:
18578:
18536:
18506:
18482:
18454:
18424:
18402:
18380:
18352:
18297:
18275:
18253:
18227:
18205:
18183:
18163:
18134:
18106:
18086:
18037:
18006:
17978:
17956:
17924:
17902:
17858:
17836:
17816:
17796:
17776:
17721:
17684:
17654:
17623:
17549:
17504:
17470:
17448:
17408:
17382:
17356:
17324:
17291:
17254:
17220:
17214:
is an integer, but the only solution to the equation
17194:
17152:
17112:
17088:
17015:. Further branches crucially applying groups include
16978:
16956:
16936:
16847:
16704:
16680:
16649:
16552:
16518:
16492:
16425:
16393:
16364:
16342:
16315:
16295:
16266:
16234:
16212:
16181:
16152:
16125:
16096:
16008:
15974:
15934:
15901:
15864:
15851:{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}
15788:
15679:
15637:
15615:
15578:
15553:
15532:
15511:
15488:
15467:
15446:
15423:
15402:
15381:
15341:
15303:
15279:
15247:
15177:
15152:
15122:
15091:
15057:
15032:
15009:
14986:
14966:
14932:
14904:
14849:
14827:
14805:
14703:
14695:, the cosets generated by reflections are all equal:
14670:
14655:{\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}
14612:
14592:
14561:
14455:
14435:
14415:
14393:
14368:
14348:
14317:
14286:
14252:
14230:
14210:
14190:
14168:
14144:
14086:
14035:
14008:
13982:
13956:
13898:
13843:
13819:
13795:
13763:
13741:
13721:
13647:
13618:
13610:
consists of these two elements, the identity element
13587:
13560:
13538:
13516:
13496:
13476:
13450:
13428:
13399:
13377:
13357:
13315:
13293:
13269:
13244:{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}
13181:
13150:
13128:
13106:
13084:
13045:
13003:
12981:
12952:
12925:
12903:
12881:
12859:
12835:
12813:
12769:
12745:
12691:
12654:
12614:
12592:
12570:
12548:
12528:
12494:
12468:
12448:
12416:
12394:
12342:
12322:
12302:
12250:
12209:
12168:
12136:
12104:
12080:
12041:
12021:
11992:
11970:
11906:
11855:
11833:
11808:
11786:
11766:
11701:
11667:
11635:
11603:
11540:
11518:
11494:
11462:
11440:
11418:
11392:
11370:
11346:
11322:
11297:
11250:
11206:
11185:
11163:
11115:
11069:
11049:
11011:
10813:
10364:
10317:
10295:
10265:
10231:
10209:
10164:
10138:
10108:
10088:
10062:
10040:
10000:
9966:
9944:
9922:
9897:
9871:
9837:
9817:
9791:
9771:
9751:
9713:
9677:
9655:
9635:
9613:
9591:
9571:
9316:
9293:
9273:
9253:
9201:
9193:
of a group are equal, because the group axioms imply
9179:
9159:
9067:
8947:
Traité des substitutions et des équations algébriques
8841:
8748:
8689:
8630:
8599:
8567:
8540:
8513:
8482:
8449:
8416:
8383:
8354:
8326:
8299:
8078:
7966:
7901:
7879:
7855:
7833:
7811:
7789:
7769:
7738:
7714:
7690:
7666:
7627:
7565:
7543:
7521:
7495:
7448:
7402:
7351:
7320:
7289:
7260:
7229:
7201:
7173:
7145:
7115:
7085:
7055:
7025:
6995:
6965:
6939:
6911:
6883:
6853:
6823:
6793:
6763:
6733:
6703:
6675:
6649:
6621:
6591:
6561:
6531:
6501:
6471:
6441:
6413:
6385:
6359:
6329:
6299:
6269:
6239:
6209:
6177:
6147:
6117:
6087:
6059:
6031:
6005:
5977:
5949:
5917:
5887:
5857:
5827:
5799:
5773:
5745:
5717:
5689:
5657:
5627:
5597:
5567:
5541:
5513:
5485:
5457:
5429:
5397:
5367:
5337:
5307:
5279:
5251:
5223:
5197:
5171:
5139:
5109:
5079:
5049:
5021:
4993:
4965:
4939:
4918:
4886:
4820:
4789:
4756:
4725:
4694:
4672:
4646:
4622:
4602:
4571:
4536:
4509:
4474:
4441:
4404:
4371:
4337:
4308:
4279:
4233:
4194:
4155:
4117:
4080:
4045:
4010:
3977:
3936:
3874:
3830:
3788:
3766:
3746:
3708:
3677:
3652:
3620:
3598:
3576:
3545:
3523:
3501:
3458:
3429:
3407:
3381:
3359:
3330:
3286:
3252:
3230:
3205:
3174:
3150:
3069:
3047:
3019:
2997:
2972:
2938:
2904:
2884:
2864:
2842:
2820:
2765:
2729:
2705:
2683:
2663:
2643:
2573:
2549:
2525:
2501:
2477:
2436:
2412:
2390:
2370:
2350:
2325:
2301:
2275:
2220:
2189:
2167:
2141:
2107:
2073:
2053:
2031:
1982:
1948:
1928:
1897:
1875:
1855:
1835:
1815:
1795:
1731:
1707:
1685:
1663:
1633:
1607:
1575:
1550:
1528:
1418:
729:
704:
667:
118:
27812:
27731:
27342:
Eliel, Ernest; Wilen, Samuel; Mander, Lewis (1994),
26757:, American Elsevier Publishing Co., Inc., New York,
26543:
26338:
26211:
26209:
26059:
25966:
25826:
25057:
24662:{\displaystyle \mathbb {Z} /n\mathbb {Z} =\{0,1,2\}}
19342:
The 6th complex roots of unity form a cyclic group.
13419:
is important in understanding the group as a whole.
12381:{\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}
12289:{\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}
11155:. This structure does have a left identity (namely,
8809:
of a particular polynomial equation in terms of the
7953:{\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}
28380:
Historically important publications in group theory
27717:(2nd ed.), Berlin, New York: Springer-Verlag,
27662:, Universitext, Berlin, New York: Springer-Verlag,
27657:
27293:
Structure and Dynamics: An Atomic View of Materials
26579:
26473:
26356:
25925:
25874:
25706:
25694:
24017:. Adjoining inverses of all elements of the monoid
22869:
19890:{\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}
13510:and their inverses. It is the smallest subgroup of
3891:
2797:
2622:{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}
28417:The Collected Mathematical Papers of Arthur Cayley
28046:
27756:
27628:
27184:
26675:
26627:
26591:
26531:
26290:
26272:
26110:
25850:
25838:
25814:
25754:
25730:
24903:
24872:
24836:
24805:
24661:
24593:
24563:
24533:
24502:
24472:
24399:
24375:
24351:
24320:
24272:
24248:
24226:
24186:
24155:
24127:
24103:
24055:
24007:
23959:
23381:. The latter servesâin the absence of significant
23338:
23318:
23298:
23278:
23240:
23220:
23109:
23089:
23069:
23039:
22947:
22918:
22886:
22779:
22749:
22729:
22709:
22689:
22663:
22643:
22615:
22588:
22559:
22530:
22484:
22448:
22417:
22388:
22347:
22319:
22299:
22265:
22236:
22216:
22189:
22151:
22126:
22106:
22079:
22059:
22037:
22013:
21991:
21962:
21938:
21916:
21894:
21872:
21822:
21793:
21758:
21738:
21718:
21691:
21656:
21634:
21600:
21568:
21546:
21437:
21417:
21397:
21377:
21302:
21219:
21161:
21141:
21119:
21067:
21045:
20994:
20970:
20945:
20883:
20644:
20620:
20513:
20477:
20455:
20428:
20391:
20354:
20311:
20276:
20254:
20226:
20206:
20164:
20142:
20114:
20088:
20053:
20024:
20002:
19980:
19958:
19938:
19889:
19793:
19771:
19673:
19641:
19613:
19586:
19464:
19430:
19401:
19381:
19354:
19310:
19274:
19239:
19185:
19133:
19111:
19089:
19067:
19047:
19019:
18997:
18955:
18933:
18899:
18879:
18828:
18806:
18784:
18762:
18738:
18716:
18694:
18672:
18650:
18628:
18606:
18560:
18518:
18488:
18460:
18436:
18408:
18386:
18358:
18334:
18281:
18259:
18239:
18211:
18189:
18169:
18140:
18118:
18092:
18070:
18021:
17990:
17962:
17936:
17908:
17886:
17842:
17822:
17802:
17782:
17729:
17698:
17668:
17638:
17607:
17528:
17488:
17454:
17432:
17390:
17362:
17340:
17299:
17275:
17238:
17206:
17176:
17136:
17096:
17054:Many number systems, such as the integers and the
16990:
16962:
16942:
16855:
16794:
16686:
16664:
16627:
16536:
16504:
16476:
16405:
16377:
16348:
16328:
16301:
16281:
16252:
16218:
16196:
16167:
16138:
16111:
16055:
15992:
15949:
15916:
15887:
15850:
15772:
15661:
15621:
15601:
15559:
15538:
15517:
15494:
15473:
15452:
15429:
15408:
15387:
15364:
15309:
15285:
15261:
15229:
15161:
15136:
15106:
15075:
15041:
15018:
14995:
14972:
14952:
14918:
14890:
14833:
14811:
14775:
14685:
14654:
14598:
14576:
14545:
14441:
14421:
14399:
14377:
14354:
14332:
14292:
14270:
14236:
14216:
14196:
14174:
14150:
14126:
14067:
14014:
13988:
13962:
13937:
13882:
13825:
13801:
13769:
13747:
13727:
13690:
13629:
13602:
13573:
13544:
13522:
13502:
13482:
13456:
13434:
13405:
13383:
13363:
13343:
13299:
13275:
13243:
13162:
13134:
13112:
13090:
13066:
13029:
12987:
12965:
12938:
12909:
12887:
12865:
12841:
12819:
12775:
12751:
12729:
12677:
12620:
12598:
12576:
12554:
12534:
12512:
12474:
12454:
12422:
12400:
12380:
12328:
12308:
12288:
12234:
12193:
12154:
12122:
12086:
12066:
12027:
11998:
11976:
11956:
11890:
11839:
11814:
11792:
11772:
11752:
11685:
11653:
11621:
11546:
11524:
11500:
11480:
11446:
11424:
11404:
11376:
11352:
11328:
11303:
11280:
11224:
11191:
11179:), and each element has a right inverse (which is
11169:
11145:
11099:
11055:
11035:
10976:
10793:
10345:
10301:
10281:
10249:
10215:
10170:
10144:
10120:
10094:
10074:
10046:
10022:
9984:
9950:
9928:
9903:
9877:
9849:
9823:
9803:
9777:
9757:
9735:
9695:
9661:
9641:
9619:
9597:
9577:
9542:
9299:
9279:
9259:
9225:
9185:
9165:
9133:
8860:
8763:
8732:
8673:
8614:
8580:
8553:
8526:
8497:
8464:
8431:
8398:
8365:
8332:
8310:
8278:
8064:
7952:
7885:
7861:
7839:
7817:
7795:
7775:
7753:
7720:
7696:
7672:
7642:
7611:
7549:
7527:
7507:
7463:
7430:
7364:
7333:
7302:
7271:
7240:
7214:
7186:
7158:
7130:
7100:
7070:
7040:
7010:
6978:
6950:
6924:
6896:
6868:
6838:
6808:
6778:
6748:
6716:
6688:
6660:
6634:
6606:
6576:
6546:
6516:
6486:
6454:
6426:
6398:
6370:
6344:
6314:
6284:
6254:
6224:
6192:
6162:
6132:
6102:
6072:
6044:
6016:
5990:
5962:
5932:
5902:
5872:
5842:
5812:
5784:
5758:
5730:
5702:
5672:
5642:
5612:
5582:
5552:
5526:
5498:
5470:
5442:
5412:
5382:
5352:
5322:
5292:
5264:
5236:
5208:
5182:
5154:
5124:
5094:
5064:
5034:
5006:
4978:
4950:
4924:
4901:
4866:
4804:
4771:
4738:
4700:
4678:
4658:
4628:
4608:
4586:
4551:
4522:
4489:
4456:
4419:
4386:
4350:
4321:
4292:
4248:
4209:
4170:
4132:
4093:
4058:
4023:
3988:
3951:
3928:The elements of the symmetry group of the square,
3880:
3842:
3794:
3772:
3752:
3732:
3689:
3661:
3636:
3604:
3582:
3554:
3529:
3507:
3478:
3444:
3415:
3389:
3367:
3338:
3312:
3272:
3238:
3214:
3186:
3158:
3085:
3053:
3025:
3003:
2978:
2956:
2922:
2890:
2870:
2848:
2826:
2783:
2747:
2711:
2689:
2669:
2649:
2621:
2555:
2531:
2507:
2483:
2448:
2418:
2396:
2376:
2356:
2331:
2307:
2281:
2226:
2198:
2173:
2147:
2125:
2091:
2059:
2037:
2000:
1966:
1934:
1903:
1881:
1861:
1841:
1821:
1801:
1779:
1713:
1691:
1669:
1641:
1613:
1587:
1556:
1534:
1510:
737:
712:
675:
124:
28428:
28188:Transactions of the American Mathematical Society
26567:
26555:
26495:
26419:
26395:
26383:
26320:
26206:
25862:
25718:
25634:
24409:. Groupoids arise in topology (for instance, the
24321:{\displaystyle \operatorname {Hom} (x,x)\simeq G}
24258:, one can build a small category with one object
21488:of a polynomial. This theory establishesâvia the
21173:real vector space is simply a group homomorphism
19055:above, the inverse of the element represented by
16950:times cannot be deformed into a loop which wraps
13309:to be a subgroup: it is sufficient to check that
11957:{\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}
11232:is not a group, since it lacks a right identity.
9148:
8903:The third field contributing to group theory was
29746:
27341:
26733:Mathematicians: An Outer View of the Inner World
26639:
26314:
26260:
26221:
23409:. An important example of a gauge theory is the
23255:
23248:matrices, because it is given by the inequality
22974:in a category. Briefly, this is an object with
19099:, and the inverse of the element represented by
18977:
18528:, the four group elements can be represented by
17529:{\displaystyle \left(\mathbb {Q} ,\cdot \right)}
17433:{\displaystyle \left(\mathbb {Q} ,\cdot \right)}
17177:{\displaystyle \left(\mathbb {Z} ,\cdot \right)}
14022:and two left cosets are either equal or have an
8869:(1854) gives the first abstract definition of a
3702:The definition of a group does not require that
3479:{\displaystyle \mathbb {R} \smallsetminus \{0\}}
3131:. To avoid cumbersome notation, it is common to
2260:Mathematicians: An Outer View of the Inner World
225:. After contributions from other fields such as
28016:"An Introduction to Computational Group Theory"
27912:
26736:, Princeton, N.J.: Princeton University Press,
26669:
26344:
25778:
21385:Each solution can be obtained by replacing the
9002:(from the late 1930s) and later by the work of
180:is a Lie group consisting of the symmetries of
71:that associates an element of the set to every
28705:
28599:(in German), New York: Johnson Reprint Corp.,
27267:
27206:; Delgado Friedrichs, Olaf; Huson, Daniel H.;
26657:
25539:. A more involved example is the action of an
20843:General linear group and representation theory
19939:{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}
18071:{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}
15230:{\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}
12235:{\displaystyle \varphi \circ \psi =\iota _{H}}
12194:{\displaystyle \psi \circ \varphi =\iota _{G}}
11559:When studying sets, one uses concepts such as
9042:
7871:, then to compose the resulting symmetry with
7489:: Composition is a binary operation. That is,
3961:. Vertices are identified by color or number.
3116:of the group, and the operation is called the
1409:One of the more familiar groups is the set of
1404:
1399:
29238:
28740:
27967:Elementary Number Theory and its Applications
27880:Grundlehren der mathematischen Wissenschaften
27735:; Karrass, Abraham; Solitar, Donald (2004) ,
27364:, Oxford University Press, pp. 441â444,
27360:Ellis, Graham (2019), "6.4 Triangle groups",
27210:(2001), "On three-dimensional space groups",
22963:and is used in computing with groups and for
22540:. But there exist nonabelian groups of order
20946:{\displaystyle \mathrm {GL} (n,\mathbb {R} )}
19186:{\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}
18446:. The group operation, multiplication modulo
16833:A periodic wallpaper pattern gives rise to a
16083:Every group is isomorphic to a quotient of a
14553:(highlighted in green in the Cayley table of
12367:
12348:
12275:
12256:
1380:
1039:
199:The concept of a group arose in the study of
28023:Notices of the American Mathematical Society
27878:
27419:
26987:Notices of the American Mathematical Society
26377:
24656:
24638:
24089:
24083:
24041:
24035:
23993:
23987:
23188:is a group that also has the structure of a
22737:can sometimes be reduced to questions about
22684:
22678:
22365:Any finite abelian group is isomorphic to a
18880:{\displaystyle a\cdot b\equiv 1{\pmod {p}},}
17755:The hours on a clock form a group that uses
16928:can be smoothly contracted to a single point
16818:
16774:
16705:
16622:
16553:
16531:
16519:
16247:
16235:
15845:
15795:
14885:
14864:
14540:
14480:
13932:
13908:
13877:
13853:
13238:
13188:
11275:
11257:
11030:
11018:
8945:
3473:
3467:
1505:
1427:
212:
28617:The Theory of Unitary Group Representations
28075:
27658:Kurzweil, Hans; Stellmacher, Bernd (2004),
27268:Denecke, Klaus; Wismath, Shelly L. (2002),
26884:Introduction to the Theory of Finite Groups
25267:
25176:One usually avoids using fraction notation
25136:. This condition is subsumed by requiring "
23117:vary only a little. Such groups are called
18607:{\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}
17715:by other than zero is possible, such as in
17137:{\displaystyle \left(\mathbb {Z} ,+\right)}
9049:Elementary consequences of the group axioms
7431:{\displaystyle f_{\mathrm {h} }\circ r_{3}}
3099:
3034:
29245:
29231:
28747:
28733:
28655:The Theory of Groups and Quantum Mechanics
28097:Profinite Groups, Arithmetic, and Geometry
27883:, vol. 322, Berlin: Springer-Verlag,
27754:
27544:
27185:Chancey, C. C.; O'Brien, M. C. M. (2021),
26973:
26513:
26084:Coornaert, Delzant & Papadopoulos 1990
25519:
25164:
23413:, which describes three of the four known
22300:{\displaystyle \mathbb {F} _{p}^{\times }}
22067:may not exist, in which case the order of
20207:{\displaystyle \mathbb {F} _{p}^{\times }}
19311:{\displaystyle \mathbb {F} _{p}^{\times }}
19275:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
16477:{\displaystyle r^{4},f^{2},(r\cdot f)^{2}}
16049:
16032:
14127:{\displaystyle g_{1}^{-1}\cdot g_{2}\in H}
13938:{\displaystyle Hg=\{h\cdot g\mid h\in H\}}
13883:{\displaystyle gH=\{g\cdot h\mid h\in H\}}
12717:
12700:
10997:However, only assuming the existence of a
7382:red (upper left region). A left and right
2212:The integers, together with the operation
1387:
1373:
1046:
1032:
265:, completed in 2004. Since the mid-1980s,
28657:, translated by Robertson, H. P., Dover,
28200:
28152:
27575:
27456:
27223:
27088:
27041:
27031:
26898:
26886:, Oliver and Boyd, Edinburgh and London,
26878:
26609:
25999:
25904:
25892:
24631:
24618:
24587:
24557:
24527:
24496:
24466:
24227:{\displaystyle \operatorname {Hom} (x,x)}
24076:
24028:
23980:
23953:
22282:
21210:
21107:
21055:. It describes all possible rotations in
20936:
20498:
20189:
20098:. These numbers can be visualized as the
19923:
19910:
19293:
19268:
19255:
19208:
18055:
18042:
17723:
17626:
17584:
17551:
17511:
17415:
17384:
17293:
17159:
17119:
17090:
16849:
11827:It would be natural to require also that
11567:. When studying groups, one uses instead
10184:Equivalent definition with relaxed axioms
9242:
4268:leaving everything unchanged, denoted id;
3460:
3432:
3409:
3383:
3361:
3332:
3291:
3257:
3232:
3152:
1635:
1420:
731:
706:
669:
152:, groups arise naturally in the study of
27:Set with associative invertible operation
28472:
28373:
28273:
28123:An Introduction to Theoretical Chemistry
27869:
27714:Categories for the Working Mathematician
27707:
27480:
27295:, Oxford University Press, p. 265,
27049:
26918:
26799:
26769:
26621:
26549:
26413:
26104:
25975:
25832:
25688:
25596:
25437:
25246:
25242:
24431:arguments, for some nonnegative integer
22985:
22360:
21518:of the group. An important class is the
20860:
20569:
20122:-gon, as shown in blue in the image for
19337:
18335:{\displaystyle 9+4\equiv 1{\pmod {12}}.}
17750:
16904:
16828:
13490:consists of all products of elements of
38:
28671:
28549:
28440:MacTutor History of Mathematics Archive
28160:
28138:
28080:, Mathematical Association of America,
28049:Linear Representations of Finite Groups
27773:
27584:
27531:, Philadelphia: W.B. Saunders Company,
27498:
27362:An Invitation to Computational Homotopy
26461:
26362:
26071:
25880:
25808:
25796:
25772:
25748:
25712:
25700:
21880:that is, application of the operation "
21514:. The number of elements is called the
20824:, which characterizes functions having
16913:Elements of the fundamental group of a
13100:, equipped with the group operation on
9553:Therefore, it is customary to speak of
9519: is the identity element and
8072:can be checked using the Cayley table:
3486:and whose operation is multiplication.
3313:{\displaystyle (\mathbb {R} ,+,\cdot )}
3168:, which has the operations of addition
14:
29747:
28891:Classification of finite simple groups
28611:
28534:
28515:
28450:
28410:
28242:An introduction to homological algebra
28236:
28217:
28181:
28120:
28013:
27526:
27394:
27158:
27105:
26999:
26585:
26537:
26525:
26302:
26243:
26116:
25856:
25844:
25820:
25760:
25736:
25489:
25404:are other instances of this principle.
25318:
25280:
24234:is a group; conversely, given a group
23369:, are basic symmetries of the laws of
23121:and they are the group objects in the
22919:{\displaystyle G\times G\rightarrow G}
22855:, relate the monster group to certain
22809:Classification of finite simple groups
22803:Classification of finite simple groups
22224:is 4, as is the order of the subgroup
20798:ion, exhibits square-planar geometry
17639:{\displaystyle \mathbb {Q} ^{\times }}
17400:. There is still a minor obstacle for
11582:
9031:classification of finite simple groups
3445:{\displaystyle \mathbb {R} ^{\times }}
3346:to denote any of these three objects.
1849:gives the same final result as adding
604:Classification of finite simple groups
263:classification of finite simple groups
207:in the 1830s, who introduced the term
83:, and every element of the set has an
29226:
28728:
28706:
28634:
28385:
28297:
28222:, Berlin, New York: Springer-Verlag,
28094:
28053:, Berlin, New York: Springer-Verlag,
28041:
27991:
27964:
27938:
27842:
27792:
27429:Representation Theory: A First Course
27359:
27309:
27124:
26947:
26941:
26926:, Berlin, New York: Springer-Verlag,
26703:
26697:
26681:
26633:
26597:
26573:
26561:
26501:
26485:
26425:
26401:
26389:
26326:
26215:
26140:
25931:
25868:
25724:
25652:
24427:operation (i.e., an operation taking
22981:
19448:is a group all of whose elements are
18396:. Its elements can be represented by
17740:
16902:translate into properties of groups.
16175:in a vertical line (every element of
14891:{\displaystyle G/N=\{gN\mid g\in G\}}
12851:: it has a subset of the elements of
11891:{\displaystyle \varphi (1_{G})=1_{H}}
8320:, as it does not change any symmetry
7515:is a symmetry for any two symmetries
7378:whose Cayley table is highlighted in
3397:and whose operation is addition. The
3375:is the group whose underlying set is
1829:first, and then adding the result to
28649:
28544:(in French), Paris: Gauthier-Villars
28527:(Galois work was first published by
28525:(in French), Paris: Gauthier-Villars
28319:
27623:
27344:Stereochemistry of Organic Compounds
27290:
26852:
26819:
26749:
26729:
26489:
26449:
26437:
26266:
26254:
26227:
26200:
26188:
26176:
26164:
26152:
26128:
26047:
26035:
26023:
26011:
25987:
25960:
25948:
25919:
25676:
25664:
25640:
25628:
25608:
25535:for the impact of a group action on
25506:
25389:
25376:
25335:
25238:
25234:
25147:
22485:{\displaystyle \mathrm {Z} _{p^{2}}}
21490:fundamental theorem of Galois theory
17887:{\displaystyle a\equiv b{\pmod {n}}}
8817:(solutions). The elements of such a
7827:. The other way is to first compose
7386:of this subgroup are highlighted in
4748:) and then reflecting horizontally (
28591:
28341:
28283:, New York: John Wiley & Sons,
27999:, Wiley Classics, Wiley-Blackwell,
27845:The Geometry of Minkowski Spacetime
27687:Linear Algebra and Its Applications
27684:
26645:
26350:
26278:
25784:
25258:The word homomorphism derives from
24361:is defined only when the source of
22955:(which provides the inverse) and a
22818:list all groups of order up to 2000
21445:; analogous formulae are known for
20804:Finite symmetry groups such as the
18866:
18321:
17876:
16537:{\displaystyle \langle r,f\rangle }
16512:is the smallest normal subgroup of
16253:{\displaystyle \langle r,f\rangle }
16119:is generated by the right rotation
16065:. Surjective homomorphisms are the
15662:{\displaystyle U=f_{\mathrm {v} }R}
15572:The elements of the quotient group
15335:Cayley table of the quotient group
11565:quotient by an equivalence relation
11146:{\displaystyle e\cdot f=f\cdot f=f}
11100:{\displaystyle e\cdot e=f\cdot e=e}
9235:. It is thus customary to speak of
24:
28348:Quantum Field Theory in a Nutshell
27816:; Fogarty, J.; Kirwan, F. (1994),
27555:Proceedings of the Royal Society A
27529:Introduction to Topological Groups
27370:10.1093/oso/9780198832973.001.0001
27212:BeitrÀge zur Algebra und Geometrie
27073:"The groups of order at most 2000"
26860:(3rd ed.), Berlin, New York:
26339:Mumford, Fogarty & Kirwan 1994
26143:, p. 54, (Theorem 2.1).
26060:Magnus, Karrass & Solitar 2004
23424:
23403:Symmetries that vary with location
22576:
22518:
22503:
22465:
22405:
22396:is isomorphic to the cyclic group
22257:
22177:
21972:" represents multiplication, then
21679:
21622:
21588:
21534:
21196:
21193:
21030:
21027:
20922:
20919:
20541:
19389:is not, because the odd powers of
18009:
16782:
16652:
16269:
16184:
16159:
16099:
15937:
15904:
15888:{\displaystyle \mathrm {D} _{4}/R}
15867:
15802:
15799:
15752:
15737:
15716:
15698:
15650:
15602:{\displaystyle \mathrm {D} _{4}/R}
15581:
15365:{\displaystyle \mathrm {D} _{4}/R}
15344:
14788:
14764:
14746:
14728:
14710:
14673:
14646:
14619:
14564:
14534:
14519:
14504:
14489:
14468:
14320:
13669:
13654:
13623:
13620:
13594:
13344:{\displaystyle g^{-1}\cdot h\in H}
13259:necessary and sufficient condition
13195:
13192:
12785:. Injective homomorphisms are the
8751:
8724:
8709:
8665:
8637:
8602:
8489:
8456:
8423:
8390:
8359:
8356:
8304:
8301:
8250:
8235:
8200:
8182:
8107:
8092:
8040:
8022:
7991:
7976:
7741:
7634:
7600:
7585:
7455:
7409:
7265:
7262:
7234:
7231:
7122:
7092:
7062:
7032:
7002:
6944:
6941:
6860:
6830:
6800:
6770:
6740:
6654:
6651:
6598:
6568:
6538:
6508:
6478:
6364:
6361:
6336:
6306:
6276:
6246:
6216:
6184:
6154:
6124:
6094:
6010:
6007:
5924:
5894:
5864:
5834:
5778:
5775:
5664:
5634:
5604:
5574:
5546:
5543:
5404:
5374:
5344:
5314:
5202:
5199:
5176:
5173:
5146:
5116:
5086:
5056:
4944:
4941:
4889:
4855:
4827:
4796:
4763:
4574:
4543:
4481:
4448:
4411:
4378:
4240:
4201:
4162:
4124:
3982:
3979:
3939:
257:. A theory has been developed for
25:
29776:
28699:
28435:"The development of group theory"
27635:, Boston, MA: BirkhÀuser Boston,
27312:"On some old and new problems in
27243:Géométrie et théorie des groupes
26492:, p. 77 for similar results.
26191:, p. 22, §II.1 (example 11).
25392:, Theorem IV.1.9. The notions of
23192:; informally, this means that it
22894:equipped with a binary operation
22827:The classification of all finite
22307:of multiplication modulo a prime
22114:the order of any finite subgroup
17276:{\displaystyle b={\tfrac {1}{2}}}
12067:{\displaystyle \iota _{G}:G\to G}
11235:
9408: are inverses of each other)
8884:. After novel geometries such as
3924:has eight symmetries. These are:
3733:{\displaystyle a\cdot b=b\cdot a}
3489:More generally, one speaks of an
3139:For example, consider the set of
29727:
29726:
28644:, Mathematical Monographs, No. 1
28597:Gesammelte Abhandlungen. Band 1
27969:(4th ed.), Addison-Wesley,
27847:, New York: Dover Publications,
27110:, New York: Dover Publications,
26924:A Course in the Theory of Groups
26452:, p. 292, (Theorem VI.7.2).
25566:
25060:
22870:Groups with additional structure
22789:. A nontrivial group is called
22626:
22589:{\displaystyle \mathrm {D} _{4}}
22418:{\displaystyle \mathrm {Z} _{p}}
22266:{\displaystyle f_{\mathrm {v} }}
22190:{\displaystyle \mathrm {D} _{4}}
21692:{\displaystyle \mathrm {S} _{3}}
21635:{\displaystyle \mathrm {S} _{N}}
21601:{\displaystyle \mathrm {S} _{3}}
21556:, the groups of permutations of
21547:{\displaystyle \mathrm {S} _{N}}
21495:
21242:
21120:{\displaystyle \mathbb {R} ^{3}}
21046:{\displaystyle \mathrm {SO} (n)}
20746:
20739:
20732:
20725:
20514:{\displaystyle (\mathbb {Z} ,+)}
19327:
18370:multiplicative group of integers
18022:{\displaystyle \mathrm {Z} _{n}}
17307:has a (multiplicative) inverse.
16665:{\displaystyle \mathrm {D} _{4}}
16282:{\displaystyle \mathrm {D} _{4}}
16197:{\displaystyle \mathrm {D} _{4}}
16168:{\displaystyle f_{\mathrm {v} }}
16112:{\displaystyle \mathrm {D} _{4}}
16090:For example, the dihedral group
16072:
15950:{\displaystyle \mathrm {D} _{4}}
15917:{\displaystyle \mathrm {D} _{4}}
14819:is a normal subgroup of a group
14686:{\displaystyle \mathrm {D} _{4}}
14577:{\displaystyle \mathrm {D} _{4}}
14333:{\displaystyle \mathrm {D} _{4}}
13950:The left cosets of any subgroup
13603:{\displaystyle f_{\mathrm {v} }}
13030:{\displaystyle h_{1}\cdot h_{2}}
11512:associating to every element of
8764:{\displaystyle \mathrm {D} _{4}}
8615:{\displaystyle \mathrm {D} _{4}}
8498:{\displaystyle f_{\mathrm {c} }}
8465:{\displaystyle f_{\mathrm {d} }}
8432:{\displaystyle f_{\mathrm {v} }}
8399:{\displaystyle f_{\mathrm {h} }}
8346:: Each symmetry has an inverse:
7754:{\displaystyle \mathrm {D} _{4}}
7643:{\displaystyle f_{\mathrm {c} }}
7464:{\displaystyle f_{\mathrm {d} }}
7131:{\displaystyle f_{\mathrm {h} }}
7101:{\displaystyle f_{\mathrm {d} }}
7071:{\displaystyle f_{\mathrm {v} }}
7041:{\displaystyle f_{\mathrm {c} }}
7011:{\displaystyle f_{\mathrm {c} }}
6869:{\displaystyle f_{\mathrm {v} }}
6839:{\displaystyle f_{\mathrm {c} }}
6809:{\displaystyle f_{\mathrm {h} }}
6779:{\displaystyle f_{\mathrm {d} }}
6749:{\displaystyle f_{\mathrm {d} }}
6607:{\displaystyle f_{\mathrm {d} }}
6577:{\displaystyle f_{\mathrm {v} }}
6547:{\displaystyle f_{\mathrm {c} }}
6517:{\displaystyle f_{\mathrm {h} }}
6487:{\displaystyle f_{\mathrm {h} }}
6345:{\displaystyle f_{\mathrm {c} }}
6315:{\displaystyle f_{\mathrm {h} }}
6285:{\displaystyle f_{\mathrm {d} }}
6255:{\displaystyle f_{\mathrm {v} }}
6225:{\displaystyle f_{\mathrm {v} }}
6193:{\displaystyle f_{\mathrm {v} }}
6163:{\displaystyle f_{\mathrm {h} }}
6133:{\displaystyle f_{\mathrm {c} }}
6103:{\displaystyle f_{\mathrm {d} }}
5933:{\displaystyle f_{\mathrm {d} }}
5903:{\displaystyle f_{\mathrm {c} }}
5873:{\displaystyle f_{\mathrm {v} }}
5843:{\displaystyle f_{\mathrm {h} }}
5673:{\displaystyle f_{\mathrm {h} }}
5643:{\displaystyle f_{\mathrm {v} }}
5613:{\displaystyle f_{\mathrm {d} }}
5583:{\displaystyle f_{\mathrm {c} }}
5413:{\displaystyle f_{\mathrm {c} }}
5383:{\displaystyle f_{\mathrm {d} }}
5353:{\displaystyle f_{\mathrm {h} }}
5323:{\displaystyle f_{\mathrm {v} }}
5155:{\displaystyle f_{\mathrm {c} }}
5125:{\displaystyle f_{\mathrm {d} }}
5095:{\displaystyle f_{\mathrm {h} }}
5065:{\displaystyle f_{\mathrm {v} }}
4902:{\displaystyle \mathrm {D} _{4}}
4805:{\displaystyle f_{\mathrm {d} }}
4772:{\displaystyle f_{\mathrm {h} }}
4587:{\displaystyle \mathrm {D} _{4}}
4552:{\displaystyle f_{\mathrm {h} }}
4490:{\displaystyle f_{\mathrm {c} }}
4457:{\displaystyle f_{\mathrm {d} }}
4420:{\displaystyle f_{\mathrm {h} }}
4387:{\displaystyle f_{\mathrm {v} }}
4249:{\displaystyle f_{\mathrm {c} }}
4222:
4210:{\displaystyle f_{\mathrm {d} }}
4183:
4171:{\displaystyle f_{\mathrm {h} }}
4144:
4133:{\displaystyle f_{\mathrm {v} }}
4106:
4069:
4034:
3999:
3966:
3952:{\displaystyle \mathrm {D} _{4}}
3892:Second example: a symmetry group
3592:, and the inverse of an element
3517:, and the inverse of an element
3273:{\displaystyle (\mathbb {R} ,+)}
296:
217:) for the symmetry group of the
27323:Quasigroups and Related Systems
26507:
26479:
26474:Kurzweil & Stellmacher 2004
26233:
26107:, in particular §§I.12 and I.13
26089:
25561:Besche, Eick & O'Brien 2001
25550:
25525:
25512:
25495:
25474:
25443:
25422:
25407:
25382:
25354:
25341:
25324:
25299:
25286:
25273:
25252:
25227:
25170:
25153:
24168:A group can be thought of as a
23353:links continuous symmetries to
22835:of such groups, as well as 26 "
22094:states that for a finite group
22047:.) In infinite groups, such an
20681:describe crystal symmetries in
20606:on another mathematical object
20355:{\displaystyle 3^{2}=9\equiv 4}
18859:
18314:
17869:
17341:{\displaystyle {\frac {a}{b}}.}
16890:. By means of this connection,
14002:of all left cosets is equal to
12827:contained within a bigger one,
12513:{\displaystyle \varphi :G\to H}
12486:if there exists an isomorphism
12123:{\displaystyle \varphi :G\to H}
11686:{\displaystyle \varphi :G\to H}
10346:{\displaystyle f^{-1}\cdot f=e}
9239:identity element of the group.
9033:, with the final step taken by
8931:groups describing factorization
8917:(1798), and more explicitly by
1780:{\displaystyle (a+b)+c=a+(b+c)}
194:symmetry in molecular chemistry
28519:(1908), Tannery, Jules (ed.),
28327:, Princeton University Press,
28125:, Cambridge University Press,
28099:, Princeton University Press,
27799:, Princeton University Press,
27778:, Cambridge University Press,
27776:Theory of Finite Simple Groups
27189:, Princeton University Press,
27056:, Cambridge University Press,
26906:, New York: Barnes and Noble,
26488:, Proposition 6.4.3. See also
26315:Eliel, Wilen & Mander 1994
25140:" to be a binary operation on
25088:
24309:
24297:
24221:
24209:
24137:produces a group known as the
24098:
24072:
24050:
24024:
24002:
23976:
23279:{\displaystyle \det(A)\neq 0,}
23264:
23258:
23123:category of topological spaces
22948:{\displaystyle G\rightarrow G}
22939:
22910:
21746:is the least positive integer
21473:similar to the formula above.
21214:
21200:
21189:
21129:. A representation of a group
21040:
21034:
20940:
20926:
20812:, which is in turn applied in
20594:greatly simplify the study of
20508:
20494:
19946:introduced above, the element
19933:
19906:
19842:
19833:
19827:
19818:
19757:
19738:
18992:
18980:
18870:
18860:
18325:
18315:
18065:
18038:
17880:
17870:
16777:
16759:
16746:
16607:
16594:
16465:
16452:
16037:
16012:
15984:
15758:
15728:
15188:
15178:
15067:
15058:
14936:
14449:itself, or otherwise equal to
14362:of rotations, the left cosets
13755:by an arbitrary group element
13154:
12721:
12705:
12669:
12586:; then any statement true for
12504:
12362:
12356:
12270:
12264:
12146:
12114:
12058:
11942:
11935:
11926:
11910:
11872:
11859:
11747:
11741:
11732:
11726:
11717:
11705:
11677:
11648:
11636:
11616:
11604:
11472:
11219:
11207:
10905:
10883:
10862:
10840:
10730:
10713:
10692:
10670:
10655:
10638:
10617:
10598:
10576:
10573:
10558:
10541:
10520:
10498:
10492:
10464:
10447:
10444:
10423:
10401:
10066:
9795:
9433:
9421:
9384:
9372:
9353: is the identity element)
9149:Uniqueness of identity element
9128:
9116:
9098:
9086:
8219:
8191:
8113:
8083:
8059:
8031:
7997:
7967:
7944:
7932:
7914:
7902:
4686:after performing the symmetry
4256:(counter-diagonal reflection)
3307:
3287:
3267:
3253:
2616:
2604:
2586:
2574:
1774:
1762:
1744:
1732:
965:Infinite dimensional Lie group
13:
1:
28641:History of Modern Mathematics
28395:American Mathematical Society
27433:Graduate Texts in Mathematics
27090:10.1090/S1079-6762-01-00087-7
26829:Graduate Texts in Mathematics
26766:, an elementary introduction.
26691:
23361:, as well as translations in
23173:
22978:that mimic the group axioms.
21303:{\displaystyle ax^{2}+bx+c=0}
20713:spontaneous symmetry breaking
20429:{\displaystyle 3^{4}\equiv 1}
20392:{\displaystyle 3^{3}\equiv 2}
17850:to be equivalent, denoted by
17830:that differ by a multiple of
16811:, a graphical depiction of a
16486:. In fact, it turns out that
14068:{\displaystyle g_{1}H=g_{2}H}
13630:{\displaystyle \mathrm {id} }
12678:{\displaystyle \phi :G'\to G}
10023:{\displaystyle b\cdot a^{-1}}
9736:{\displaystyle a^{-1}\cdot b}
9629:, there is a unique solution
8972:modular representation theory
8925:made early attempts to prove
8861:{\displaystyle \theta ^{n}=1}
8366:{\displaystyle \mathrm {id} }
8311:{\displaystyle \mathrm {id} }
7272:{\displaystyle \mathrm {id} }
7241:{\displaystyle \mathrm {id} }
6951:{\displaystyle \mathrm {id} }
6661:{\displaystyle \mathrm {id} }
6371:{\displaystyle \mathrm {id} }
6017:{\displaystyle \mathrm {id} }
5785:{\displaystyle \mathrm {id} }
5553:{\displaystyle \mathrm {id} }
5209:{\displaystyle \mathrm {id} }
5183:{\displaystyle \mathrm {id} }
4951:{\displaystyle \mathrm {id} }
4101:(rotation by 270° clockwise)
3989:{\displaystyle \mathrm {id} }
3808:, and the group is called an
3564:. Similarly, one speaks of a
2239:
1789:. Expressed in words, adding
221:of an equation, now called a
108:generated by a single element
29531:Eigenvalues and eigenvectors
28420:, vol. II (1851â1860),
28076:Schwartzman, Steven (1994),
27106:Bishop, David H. L. (1993),
27033:10.3998/ark.5550190.p008.911
26904:Introduction to Group Theory
25667:, p. 3, I.§1 and p. 7, I.§2.
25583:
25309:of a group which equals the
24594:{\displaystyle \mathbb {C} }
24564:{\displaystyle \mathbb {R} }
24534:{\displaystyle \mathbb {Q} }
24503:{\displaystyle \mathbb {Z} }
24473:{\displaystyle \mathbb {N} }
23960:{\displaystyle \mathbb {N} }
23823:Commutative-and-associative
21578:symmetric group on 3 letters
20816:of transmitted data, and in
20602:objects. A group is said to
19362:is a primitive element, but
18616:, because the usual product
18100:as the identity element and
18000:, forms a group, denoted as
17730:{\displaystyle \mathbb {Q} }
17391:{\displaystyle \mathbb {Q} }
17350:Fractions of integers (with
17310:
17300:{\displaystyle \mathbb {Z} }
17097:{\displaystyle \mathbb {Z} }
16856:{\displaystyle \mathbb {Z} }
15993:{\displaystyle \phi :G\to H}
12792:
12155:{\displaystyle \psi :H\to G}
9482: is an inverse of
9226:{\displaystyle e=e\cdot f=f}
7390:green (in the last row) and
3416:{\displaystyle \mathbb {R} }
3390:{\displaystyle \mathbb {R} }
3368:{\displaystyle \mathbb {R} }
3339:{\displaystyle \mathbb {R} }
3324:. But it is common to write
3239:{\displaystyle \mathbb {R} }
3159:{\displaystyle \mathbb {R} }
1913:. This property is known as
1642:{\displaystyle \mathbb {Z} }
738:{\displaystyle \mathbb {Z} }
713:{\displaystyle \mathbb {Z} }
676:{\displaystyle \mathbb {Z} }
261:, which culminated with the
251:representations of the group
7:
29252:
29158:Infinite dimensional groups
28621:University of Chicago Press
28305:, Oxford: Clarendon Press,
28168:(4th ed.), CRC Press,
27660:The Theory of Finite Groups
27004:, Oxford University Press,
26670:Romanowska & Smith 2002
25076:List of group theory topics
25053:
19801:is called a generator or a
17077:
16882:founded what is now called
16069:in the category of groups.
12789:in the category of groups.
11281:{\displaystyle X=\{x,y,z\}}
11240:The following sections use
9745:. It follows that for each
9560:
8994:counterpart, the theory of
8914:Disquisitiones Arithmeticae
8911:'s number-theoretical work
8805:, gave a criterion for the
7477:blue (below table center).
4031:(rotation by 90° clockwise)
2796:Such an element is unique (
1405:First example: the integers
1400:Definition and illustration
463:List of group theory topics
10:
29781:
28754:
28422:Cambridge University Press
28377:
28095:Shatz, Stephen S. (1972),
27997:Fourier Analysis on Groups
27965:Rosen, Kenneth H. (2000),
27916:; Smith, J. D. H. (2002),
27843:Naber, Gregory L. (2003),
27818:Geometric Invariant Theory
27591:Princeton University Press
27510:Cambridge University Press
27310:Dudek, WiesĆaw A. (2001),
27108:Group Theory and Chemistry
26920:Robinson, Derek John Scott
26658:Denecke & Wismath 2002
26291:Chancey & O'Brien 2021
25294:SeifertâVan Kampen theorem
23177:
23010:
22806:
22456:is abelian, isomorphic to
21576:objects. For example, the
21499:
21246:
21079:in this group are used in
20846:
20822:differential Galois theory
20551:
20545:
19331:
18934:{\displaystyle a\cdot b-1}
18816:, there exists an integer
18126:as the inverse element of
17744:
17489:{\displaystyle x\cdot 0=1}
17239:{\displaystyle a\cdot b=1}
17049:
17046:benefit from the concept.
17032:computational group theory
16822:
16505:{\displaystyle \ker \phi }
16406:{\displaystyle \ker \phi }
16076:
15114:serves as the identity of
14792:
13707:
13067:{\displaystyle h_{1}^{-1}}
12796:
12074:that maps each element of
11622:{\displaystyle (G,\cdot )}
11586:
11239:
11225:{\displaystyle (G,\cdot )}
10250:{\displaystyle e\cdot f=f}
9985:{\displaystyle x\cdot a=b}
9696:{\displaystyle a\cdot x=b}
8797:, extending prior work of
8778:
8774:
8291:: The identity element is
2957:{\displaystyle b\cdot a=e}
2923:{\displaystyle a\cdot b=e}
2858:, there exists an element
2784:{\displaystyle a\cdot e=a}
2748:{\displaystyle e\cdot a=a}
255:computational group theory
29:
29722:
29691:
29675:
29604:
29511:
29450:
29371:
29278:
29260:
29197:
29157:
29033:
28881:
28815:
28762:
28280:Gravitation and Cosmology
27774:Michler, Gerhard (2006),
27585:Kuipers, Jack B. (1999),
27481:Gollmann, Dieter (2011),
27164:Simple Groups of Lie Type
27043:2027/spo.5550190.p008.911
26730:Cook, Mariana R. (2009),
25360:The same is true for any
24604:
24513:
24483:
24453:
23417:and classifies all known
22926:(the group operation), a
22833:several infinite families
22376:Any group of prime order
22165:give a partial converse.
21512:finite number of elements
20820:. Another application is
20745:
20738:
20731:
20724:
20592:Symmetries in mathematics
18998:{\displaystyle \gcd(a,p)}
18680:it yields a remainder of
17790:defines any two elements
17767:Modular arithmetic for a
16819:Examples and applications
15962:first isomorphism theorem
15169:in the quotient group is
13779:. In symbolic terms, the
13703:
11036:{\displaystyle G=\{e,f\}}
9960:, the unique solution to
8956:Ferdinand Georg Frobenius
8785:The modern concept of an
7251:
4178:(horizontal reflection)
3819:, the operation is often
2341:", that combines any two
1595:is also an integer; this
271:finitely generated groups
158:geometric transformations
43:The manipulations of the
29061:Special orthogonal group
28445:University of St Andrews
27793:Milne, James S. (1980),
27410:: 239â50, archived from
27050:Bersuker, Isaac (2006),
26378:Fulton & Harris 1991
26203:, pp. 26, 29, §I.5.
26074:, p. 30, Chapter I.
25081:
24922:N/A, 1, 2, respectively
23162:, which plays a role in
23077:must not vary wildly if
23040:{\displaystyle g\cdot h}
22814:Computer algebra systems
22717:itself, questions about
22429:). Any group of order
21706:The order of an element
21016:special orthogonal group
20893:-coordinate by factor 2.
20796:tetrachloroplatinate(II)
19990:. Any cyclic group with
19642:{\displaystyle a\cdot a}
19452:of a particular element
17539:However, the set of all
17104:under addition, denoted
14953:{\displaystyle G\to G/N}
13811:, containing an element
12577:{\displaystyle \varphi }
11840:{\displaystyle \varphi }
11481:{\displaystyle f:X\to Y}
10121:{\displaystyle x\cdot a}
9850:{\displaystyle a\cdot x}
7508:{\displaystyle a\circ b}
4659:{\displaystyle b\circ a}
4636:is written symbolically
4563:of degree four, denoted
3854:geometric transformation
3843:{\displaystyle f\circ g}
3690:{\displaystyle a\cdot b}
3112:. The set is called the
3104:Formally, a group is an
3100:Notation and terminology
3061:and is commonly denoted
2986:is the identity element.
2637:There exists an element
2449:{\displaystyle a\cdot b}
581:Elementary abelian group
458:Glossary of group theory
29605:Algebraic constructions
29308:Algebraic number theory
28613:Mackey, George Whitelaw
27948:Oxford University Press
27875:Algebraic Number Theory
27527:Husain, Taqdir (1966),
27291:Dove, Martin T (2003),
27130:Linear Algebraic Groups
26801:Herstein, Israel Nathan
26771:Herstein, Israel Nathan
26155:, p. 292, §VIII.1.
26062:, pp. 56â67, §1.6.
24904:{\displaystyle a\neq 0}
24413:) and in the theory of
23375:Lorentz transformations
23190:differentiable manifold
23168:Ă©tale fundamental group
23146:and can be studied via
22961:existential quantifiers
22851:conjectures, proved by
21794:{\displaystyle a^{n}=e}
21642:for a suitable integer
21235:, especially (locally)
20312:{\displaystyle 3^{1}=3}
20089:{\displaystyle z^{n}=1}
19781:, etc. Such an element
19322:public-key cryptography
19119:is represented by
19075:is that represented by
18969:greatest common divisor
18561:{\displaystyle 1,2,3,4}
16991:{\displaystyle m\neq k}
16079:Presentation of a group
9557:inverse of an element.
9055:elementary group theory
8781:History of group theory
4217:(diagonal reflection)
2266:A group is a non-empty
1522:. For any two integers
92:mathematical structures
29348:Noncommutative algebra
29087:Exceptional Lie groups
28303:Codes and Cryptography
28218:Warner, Frank (1983),
27879:
27577:10.1098/rspa.1937.0142
27404:Compositio Mathematica
27053:The JahnâTeller Effect
27000:Awodey, Steve (2010),
25631:, p. 360, App. 2.
24919:0, 2, 1, respectively
24905:
24874:
24838:
24807:
24663:
24595:
24565:
24535:
24504:
24474:
24401:
24385:matches the target of
24377:
24353:
24322:
24274:
24250:
24228:
24188:
24157:
24129:
24105:
24057:
24009:
23961:
23430:Group-like structures
23399:quantum field theories
23340:
23320:
23300:
23280:
23242:
23222:
23164:infinite Galois theory
23111:
23091:
23071:
23070:{\displaystyle g^{-1}}
23041:
23008:
23007:(shown at the bottom).
22949:
22920:
22888:
22781:
22751:
22731:
22711:
22691:
22665:
22651:has a normal subgroup
22645:
22617:
22590:
22561:
22532:
22486:
22450:
22419:
22390:
22349:
22321:
22301:
22267:
22238:
22218:
22191:
22153:
22128:
22108:
22081:
22061:
22039:
22015:
21993:
21964:
21963:{\displaystyle \cdot }
21940:
21918:
21896:
21895:{\displaystyle \cdot }
21874:
21824:
21795:
21760:
21740:
21720:
21693:
21658:
21636:
21602:
21570:
21548:
21439:
21419:
21399:
21379:
21304:
21221:
21163:
21143:
21121:
21069:
21047:
20996:
20972:
20947:
20894:
20885:
20830:differential equations
20656:(2,3,7) triangle group
20646:
20622:
20583:
20515:
20479:
20457:
20430:
20393:
20356:
20313:
20278:
20256:
20228:
20208:
20176:rotation by 60°. From
20166:
20144:
20116:
20090:
20055:
20036:complex roots of unity
20026:
20004:
19982:
19960:
19940:
19891:
19795:
19773:
19675:
19674:{\displaystyle a^{-3}}
19643:
19615:
19588:
19466:
19441:
19432:
19403:
19383:
19356:
19312:
19276:
19241:
19187:
19135:
19113:
19091:
19069:
19049:
19021:
18999:
18967:and the fact that the
18963:can be found by using
18957:
18935:
18901:
18881:
18830:
18808:
18786:
18764:
18740:
18718:
18696:
18674:
18652:
18630:
18608:
18562:
18520:
18490:
18462:
18438:
18410:
18388:
18360:
18336:
18283:
18261:
18241:
18213:
18191:
18171:
18142:
18120:
18094:
18072:
18023:
17992:
17964:
17938:
17910:
17888:
17844:
17824:
17804:
17784:
17764:
17731:
17700:
17670:
17640:
17609:
17536:is still not a group.
17530:
17490:
17456:
17434:
17392:
17370:nonzero) are known as
17364:
17342:
17301:
17277:
17240:
17208:
17178:
17138:
17098:
17082:The group of integers
17030:knowledge obtained in
17009:geometric group theory
16992:
16964:
16944:
16910:
16892:topological properties
16857:
16838:
16796:
16696:yields an isomorphism
16688:
16666:
16629:
16538:
16506:
16478:
16407:
16379:
16350:
16330:
16303:
16283:
16254:
16220:
16198:
16169:
16140:
16113:
16057:
15994:
15951:
15918:
15889:
15852:
15774:
15663:
15623:
15603:
15561:
15540:
15519:
15496:
15475:
15454:
15431:
15410:
15389:
15388:{\displaystyle \cdot }
15366:
15311:
15287:
15263:
15231:
15163:
15138:
15108:
15077:
15043:
15020:
14997:
14974:
14954:
14920:
14892:
14835:
14813:
14777:
14687:
14656:
14600:
14578:
14547:
14443:
14423:
14401:
14379:
14356:
14334:
14294:
14272:
14238:
14218:
14198:
14176:
14152:
14128:
14069:
14016:
13990:
13964:
13939:
13884:
13827:
13803:
13771:
13749:
13729:
13692:
13631:
13604:
13575:
13546:
13524:
13504:
13484:
13458:
13436:
13407:
13385:
13365:
13345:
13301:
13277:
13261:for a nonempty subset
13245:
13164:
13163:{\displaystyle H\to G}
13136:
13114:
13092:
13068:
13031:
12989:
12967:
12940:
12911:
12889:
12867:
12843:
12821:
12777:
12753:
12731:
12679:
12622:
12600:
12578:
12556:
12536:
12514:
12476:
12456:
12424:
12402:
12382:
12330:
12310:
12290:
12236:
12195:
12156:
12124:
12088:
12068:
12029:
12000:
11978:
11958:
11892:
11841:
11816:
11794:
11774:
11754:
11687:
11655:
11623:
11548:
11526:
11502:
11482:
11448:
11426:
11406:
11405:{\displaystyle x\in X}
11378:
11354:
11330:
11305:
11282:
11226:
11193:
11171:
11147:
11101:
11057:
11056:{\displaystyle \cdot }
11037:
10978:
10795:
10347:
10303:
10283:
10282:{\displaystyle f^{-1}}
10251:
10217:
10172:
10146:
10128:is a bijection called
10122:
10096:
10076:
10075:{\displaystyle G\to G}
10048:
10024:
9986:
9952:
9930:
9905:
9879:
9851:
9825:
9805:
9804:{\displaystyle G\to G}
9779:
9759:
9737:
9697:
9663:
9643:
9621:
9599:
9579:
9544:
9301:
9281:
9261:
9243:Uniqueness of inverses
9227:
9187:
9167:
9135:
8998:, was first shaped by
8980:locally compact groups
8946:
8862:
8821:correspond to certain
8765:
8734:
8675:
8616:
8582:
8555:
8528:
8507:and the 180° rotation
8499:
8466:
8433:
8400:
8367:
8334:
8312:
8280:
8066:
7954:
7887:
7863:
7841:
7819:
7797:
7777:
7755:
7722:
7698:
7674:
7644:
7613:
7551:
7529:
7509:
7465:
7432:
7366:
7335:
7304:
7273:
7242:
7216:
7188:
7160:
7132:
7102:
7072:
7042:
7012:
6980:
6952:
6926:
6898:
6870:
6840:
6810:
6780:
6750:
6718:
6690:
6662:
6636:
6608:
6578:
6548:
6518:
6488:
6456:
6428:
6400:
6372:
6346:
6316:
6286:
6256:
6226:
6194:
6164:
6134:
6104:
6074:
6046:
6018:
5992:
5964:
5934:
5904:
5874:
5844:
5814:
5786:
5760:
5732:
5704:
5674:
5644:
5614:
5584:
5554:
5528:
5500:
5472:
5444:
5414:
5384:
5354:
5324:
5294:
5266:
5238:
5210:
5184:
5156:
5126:
5096:
5066:
5036:
5008:
4980:
4952:
4926:
4925:{\displaystyle \circ }
4903:
4868:
4806:
4773:
4740:
4702:
4680:
4660:
4630:
4610:
4588:
4553:
4524:
4491:
4458:
4429:), or through the two
4421:
4388:
4352:
4323:
4294:
4250:
4211:
4172:
4134:
4095:
4060:
4025:
3990:
3953:
3882:
3881:{\displaystyle \circ }
3844:
3796:
3774:
3754:
3734:
3691:
3663:
3638:
3637:{\displaystyle x^{-1}}
3606:
3584:
3556:
3531:
3509:
3480:
3446:
3417:
3391:
3369:
3340:
3314:
3274:
3240:
3216:
3188:
3160:
3087:
3086:{\displaystyle a^{-1}}
3055:
3027:
3005:
2980:
2958:
2924:
2892:
2872:
2850:
2828:
2785:
2749:
2713:
2691:
2671:
2651:
2623:
2557:
2533:
2509:
2485:
2450:
2420:
2404:to form an element of
2398:
2378:
2358:
2333:
2332:{\displaystyle \cdot }
2309:
2283:
2252:
2228:
2200:
2175:
2149:
2127:
2093:
2061:
2047:, there is an integer
2039:
2002:
1968:
1936:
1905:
1883:
1863:
1843:
1823:
1803:
1781:
1715:
1693:
1671:
1643:
1615:
1589:
1558:
1536:
1512:
997:Linear algebraic group
739:
714:
677:
267:geometric group theory
249:(that is, through the
213:
126:
52:
29585:Orthogonal complement
29358:Representation theory
29073:Special unitary group
28483:Mathematische Annalen
28374:Historical references
28121:Simons, Jack (2003),
28014:Seress, Ăkos (1997),
27168:John Wiley & Sons
26858:Undergraduate Algebra
26002:, p. 39, §II.12.
25895:, pp. 4â5, §1.2.
25541:absolute Galois group
25379:, p. 86, §III.1.
25098:under the operation "
24906:
24875:
24839:
24808:
24664:
24596:
24566:
24536:
24505:
24475:
24402:
24378:
24354:
24323:
24275:
24251:
24229:
24189:
24158:
24130:
24106:
24058:
24010:
23962:
23341:
23321:
23301:
23281:
23243:
23223:
23138:. These examples are
23112:
23092:
23072:
23042:
22989:
22965:computer-aided proofs
22950:
22921:
22889:
22797:JordanâHölder theorem
22782:
22752:
22732:
22712:
22692:
22690:{\displaystyle \{1\}}
22666:
22646:
22623:above is an example.
22618:
22616:{\displaystyle 2^{3}}
22591:
22567:; the dihedral group
22562:
22560:{\displaystyle p^{3}}
22533:
22487:
22451:
22449:{\displaystyle p^{2}}
22420:
22391:
22361:Finite abelian groups
22350:
22322:
22302:
22268:
22239:
22219:
22217:{\displaystyle r_{1}}
22192:
22154:
22129:
22109:
22082:
22062:
22040:
22016:
21994:
21992:{\displaystyle a^{n}}
21965:
21941:
21919:
21897:
21875:
21825:
21823:{\displaystyle a^{n}}
21796:
21761:
21741:
21721:
21694:
21659:
21637:
21603:
21571:
21549:
21457:exist in general for
21440:
21420:
21400:
21380:
21305:
21222:
21164:
21144:
21122:
21091:group representations
21087:Representation theory
21070:
21048:
20997:
20973:
20948:
20906:matrix multiplication
20886:
20864:
20853:Representation theory
20658:acts on a triangular
20647:
20623:
20573:
20516:
20480:
20458:
20431:
20394:
20357:
20314:
20284:is a generator since
20279:
20257:
20229:
20209:
20167:
20145:
20117:
20091:
20056:
20027:
20005:
19983:
19961:
19941:
19892:
19796:
19774:
19676:
19644:
19616:
19614:{\displaystyle a^{2}}
19589:
19467:
19433:
19431:{\displaystyle z^{2}}
19404:
19384:
19382:{\displaystyle z^{2}}
19357:
19341:
19313:
19277:
19242:
19188:
19136:
19114:
19092:
19070:
19050:
19022:
19000:
18958:
18936:
18902:
18882:
18831:
18809:
18787:
18765:
18741:
18719:
18697:
18675:
18653:
18631:
18609:
18563:
18521:
18491:
18463:
18439:
18411:
18389:
18361:
18344:For any prime number
18337:
18284:
18262:
18242:
18214:
18197:hours, it ends up on
18192:
18172:
18143:
18121:
18095:
18073:
18024:
17993:
17965:
17939:
17911:
17889:
17845:
17825:
17805:
17785:
17754:
17732:
17701:
17671:
17641:
17610:
17531:
17491:
17457:
17435:
17393:
17365:
17343:
17302:
17278:
17241:
17209:
17179:
17139:
17099:
16993:
16965:
16945:
16908:
16865:multiplicative groups
16858:
16832:
16797:
16689:
16687:{\displaystyle \phi }
16667:
16630:
16539:
16507:
16479:
16408:
16380:
16378:{\displaystyle f_{1}}
16351:
16331:
16329:{\displaystyle r_{1}}
16304:
16284:
16260:on two generators to
16255:
16221:
16219:{\displaystyle \phi }
16199:
16170:
16141:
16139:{\displaystyle r_{1}}
16114:
16058:
15995:
15952:
15919:
15890:
15853:
15775:
15664:
15624:
15604:
15562:
15541:
15520:
15497:
15476:
15455:
15432:
15411:
15390:
15367:
15312:
15288:
15264:
15232:
15164:
15146:, and the inverse of
15139:
15109:
15078:
15076:{\displaystyle (gh)N}
15044:
15021:
14998:
14975:
14960:sending each element
14955:
14921:
14893:
14836:
14814:
14778:
14688:
14657:
14601:
14579:
14548:
14444:
14424:
14402:
14380:
14357:
14335:
14295:
14273:
14271:{\displaystyle gH=Hg}
14239:
14219:
14199:
14184:. The left cosets of
14177:
14153:
14129:
14070:
14017:
13991:
13965:
13940:
13885:
13828:
13804:
13772:
13750:
13730:
13693:
13632:
13605:
13576:
13574:{\displaystyle r_{2}}
13547:
13525:
13505:
13485:
13459:
13437:
13408:
13386:
13366:
13346:
13302:
13278:
13246:
13165:
13137:
13115:
13093:
13076:, so the elements of
13069:
13032:
12990:
12968:
12966:{\displaystyle h_{2}}
12941:
12939:{\displaystyle h_{1}}
12912:
12895:must be contained in
12890:
12868:
12844:
12822:
12778:
12754:
12732:
12680:
12623:
12601:
12579:
12557:
12542:can be obtained from
12537:
12515:
12477:
12457:
12442:homomorphism. Groups
12425:
12403:
12383:
12331:
12311:
12291:
12244:, that is, such that
12237:
12196:
12157:
12125:
12089:
12069:
12030:
12013:identity homomorphism
12001:
11979:
11959:
11893:
11842:
11817:
11795:
11775:
11755:
11688:
11656:
11654:{\displaystyle (H,*)}
11624:
11549:
11527:
11503:
11483:
11449:
11427:
11407:
11379:
11355:
11331:
11306:
11283:
11227:
11194:
11172:
11148:
11102:
11058:
11038:
10979:
10796:
10348:
10304:
10284:
10259:) and a left inverse
10252:
10218:
10173:
10147:
10123:
10097:
10077:
10049:
10025:
9987:
9953:
9931:
9906:
9880:
9852:
9826:
9806:
9780:
9760:
9738:
9698:
9664:
9644:
9622:
9600:
9580:
9545:
9302:
9282:
9262:
9228:
9188:
9168:
9136:
9015:University of Chicago
8990:and many others. Its
8964:representation theory
8927:Fermat's Last Theorem
8896:founded the study of
8863:
8827:Augustin Louis Cauchy
8803:Joseph-Louis Lagrange
8766:
8735:
8676:
8617:
8583:
8581:{\displaystyle r_{1}}
8556:
8554:{\displaystyle r_{3}}
8529:
8527:{\displaystyle r_{2}}
8500:
8467:
8434:
8401:
8368:
8335:
8313:
8281:
8067:
7955:
7888:
7864:
7842:
7820:
7798:
7778:
7756:
7723:
7699:
7675:
7645:
7614:
7552:
7530:
7510:
7466:
7433:
7367:
7365:{\displaystyle r_{3}}
7336:
7334:{\displaystyle r_{2}}
7305:
7303:{\displaystyle r_{1}}
7274:
7243:
7217:
7215:{\displaystyle r_{2}}
7189:
7187:{\displaystyle r_{3}}
7161:
7159:{\displaystyle r_{1}}
7133:
7103:
7073:
7043:
7013:
6981:
6979:{\displaystyle r_{2}}
6953:
6927:
6925:{\displaystyle r_{1}}
6899:
6897:{\displaystyle r_{3}}
6871:
6841:
6811:
6781:
6751:
6719:
6717:{\displaystyle r_{1}}
6691:
6689:{\displaystyle r_{3}}
6663:
6637:
6635:{\displaystyle r_{2}}
6609:
6579:
6549:
6519:
6489:
6457:
6455:{\displaystyle r_{3}}
6429:
6427:{\displaystyle r_{1}}
6401:
6399:{\displaystyle r_{2}}
6373:
6347:
6317:
6287:
6257:
6227:
6195:
6165:
6135:
6105:
6075:
6073:{\displaystyle r_{2}}
6047:
6045:{\displaystyle r_{1}}
6019:
5993:
5991:{\displaystyle r_{3}}
5965:
5963:{\displaystyle r_{3}}
5935:
5905:
5875:
5845:
5815:
5813:{\displaystyle r_{1}}
5787:
5761:
5759:{\displaystyle r_{3}}
5733:
5731:{\displaystyle r_{2}}
5705:
5703:{\displaystyle r_{2}}
5675:
5645:
5615:
5585:
5555:
5529:
5527:{\displaystyle r_{3}}
5501:
5499:{\displaystyle r_{2}}
5473:
5471:{\displaystyle r_{1}}
5445:
5443:{\displaystyle r_{1}}
5415:
5385:
5355:
5325:
5295:
5293:{\displaystyle r_{3}}
5267:
5265:{\displaystyle r_{2}}
5239:
5237:{\displaystyle r_{1}}
5211:
5185:
5157:
5127:
5097:
5067:
5037:
5035:{\displaystyle r_{3}}
5009:
5007:{\displaystyle r_{2}}
4981:
4979:{\displaystyle r_{1}}
4953:
4927:
4904:
4869:
4807:
4774:
4741:
4739:{\displaystyle r_{3}}
4703:
4681:
4666:("apply the symmetry
4661:
4631:
4611:
4589:
4554:
4525:
4523:{\displaystyle r_{1}}
4492:
4459:
4422:
4389:
4353:
4351:{\displaystyle r_{3}}
4324:
4322:{\displaystyle r_{2}}
4295:
4293:{\displaystyle r_{1}}
4251:
4212:
4173:
4140:(vertical reflection)
4135:
4096:
4094:{\displaystyle r_{3}}
4061:
4059:{\displaystyle r_{2}}
4026:
4024:{\displaystyle r_{1}}
3996:(keeping it as it is)
3991:
3954:
3883:
3845:
3797:
3775:
3755:
3735:
3692:
3664:
3639:
3607:
3585:
3557:
3532:
3510:
3481:
3447:
3418:
3392:
3370:
3341:
3315:
3275:
3241:
3217:
3189:
3161:
3088:
3056:
3028:
3006:
2981:
2959:
2925:
2893:
2873:
2851:
2829:
2786:
2750:
2714:
2692:
2677:such that, for every
2672:
2652:
2624:
2558:
2534:
2510:
2486:
2451:
2421:
2399:
2379:
2359:
2334:
2310:
2284:
2244:
2229:
2201:
2176:
2150:
2128:
2126:{\displaystyle b+a=0}
2094:
2092:{\displaystyle a+b=0}
2062:
2040:
2003:
2001:{\displaystyle a+0=a}
1969:
1967:{\displaystyle 0+a=a}
1942:is any integer, then
1937:
1906:
1884:
1864:
1844:
1824:
1804:
1782:
1716:
1694:
1672:
1644:
1616:
1590:
1559:
1537:
1513:
740:
715:
678:
247:representation theory
127:
42:
29760:Algebraic structures
29683:Algebraic structures
29451:Algebraic structures
29436:Equivalence relation
29379:Algebraic expression
29170:Diffeomorphism group
29049:Special linear group
29043:General linear group
28556:Mathematics Magazine
28431:Robertson, Edmund F.
28393:, Providence, R.I.:
27208:Thurston, William P.
26755:Applied Group Theory
26179:, p. 26, §II.2.
26167:, p. 22, §II.1.
26038:, p. 45, §II.4.
26014:, p. 41, §II.4.
25990:, p. 19, §II.1.
25963:, p. 34, §II.3.
25922:, p. 17, §II.1.
25679:, p. 16, II.§1.
25574:Schwarzschild metric
25501:More precisely, the
25338:, p. 84, §II.1.
25102:", which means that
24889:
24856:
24825:
24794:
24614:
24583:
24553:
24523:
24492:
24462:
24411:fundamental groupoid
24391:
24367:
24340:
24330:. More generally, a
24288:
24264:
24240:
24200:
24178:
24147:
24119:
24069:
24021:
23973:
23949:
23419:elementary particles
23355:conserved quantities
23330:
23310:
23290:
23252:
23232:
23212:
23208:of the space of all
23101:
23081:
23051:
23025:
22933:
22898:
22878:
22763:
22741:
22721:
22701:
22675:
22655:
22635:
22600:
22571:
22544:
22498:
22460:
22433:
22400:
22380:
22333:
22311:
22277:
22248:
22228:
22201:
22172:
22143:
22118:
22098:
22071:
22051:
22029:
22005:
21976:
21954:
21930:
21908:
21886:
21834:
21807:
21772:
21750:
21730:
21710:
21701:equilateral triangle
21674:
21648:
21617:
21583:
21560:
21529:
21429:
21409:
21398:{\displaystyle \pm }
21389:
21314:
21263:
21177:
21153:
21133:
21102:
21059:
21023:
20986:
20962:
20915:
20910:general linear group
20875:
20849:General linear group
20761:icosahedral symmetry
20756:Buckminsterfullerene
20675:molecular symmetries
20636:
20612:
20491:
20469:
20465:, all the powers of
20447:
20407:
20370:
20327:
20290:
20268:
20240:
20218:
20214:is cyclic for prime
20184:
20156:
20128:
20106:
20067:
20045:
20016:
19994:
19972:
19950:
19903:
19809:
19785:
19687:
19655:
19627:
19598:
19476:
19456:
19415:
19393:
19366:
19346:
19288:
19251:
19199:
19149:
19125:
19103:
19081:
19059:
19033:
19011:
18974:
18947:
18913:
18891:
18840:
18820:
18798:
18776:
18754:
18730:
18708:
18686:
18664:
18642:
18620:
18576:
18534:
18504:
18480:
18452:
18422:
18400:
18378:
18368:, there is also the
18350:
18295:
18273:
18251:
18225:
18203:
18181:
18161:
18132:
18104:
18084:
18035:
18004:
17976:
17954:
17922:
17900:
17856:
17834:
17814:
17794:
17774:
17719:
17682:
17652:
17621:
17547:
17502:
17468:
17446:
17406:
17380:
17354:
17322:
17289:
17252:
17218:
17192:
17150:
17110:
17086:
16976:
16954:
16934:
16845:
16702:
16678:
16647:
16550:
16516:
16490:
16423:
16391:
16362:
16340:
16313:
16293:
16264:
16232:
16228:from the free group
16210:
16179:
16150:
16123:
16094:
16006:
15972:
15932:
15899:
15862:
15786:
15782:. Both the subgroup
15677:
15635:
15613:
15576:
15551:
15530:
15509:
15486:
15465:
15444:
15421:
15400:
15379:
15339:
15301:
15277:
15245:
15175:
15150:
15120:
15107:{\displaystyle eN=N}
15089:
15055:
15030:
15007:
14984:
14964:
14930:
14902:
14847:
14825:
14803:
14701:
14668:
14610:
14590:
14559:
14453:
14433:
14413:
14391:
14385:are either equal to
14366:
14346:
14315:
14284:
14250:
14228:
14208:
14188:
14166:
14142:
14084:
14033:
14006:
13980:
13954:
13896:
13841:
13817:
13793:
13761:
13739:
13719:
13645:
13616:
13585:
13558:
13536:
13514:
13494:
13474:
13448:
13426:
13415:. Knowing a group's
13397:
13375:
13355:
13313:
13291:
13267:
13179:
13148:
13126:
13104:
13082:
13043:
13001:
12979:
12950:
12923:
12901:
12879:
12857:
12833:
12811:
12767:
12743:
12689:
12652:
12612:
12590:
12568:
12546:
12526:
12492:
12466:
12446:
12414:
12392:
12340:
12320:
12300:
12248:
12207:
12166:
12134:
12102:
12096:inverse homomorphism
12078:
12039:
12035:is the homomorphism
12019:
11990:
11968:
11904:
11853:
11847:respect identities,
11831:
11806:
11784:
11764:
11699:
11665:
11633:
11601:
11538:
11516:
11492:
11460:
11438:
11416:
11390:
11368:
11344:
11320:
11295:
11248:
11242:mathematical symbols
11204:
11183:
11161:
11113:
11067:
11047:
11009:
10811:
10362:
10315:
10293:
10263:
10229:
10207:
10162:
10136:
10130:right multiplication
10106:
10086:
10060:
10038:
9998:
9964:
9942:
9920:
9895:
9869:
9835:
9815:
9789:
9769:
9749:
9711:
9675:
9653:
9633:
9611:
9589:
9569:
9314:
9291:
9271:
9251:
9199:
9177:
9157:
9065:
8909:Carl Friedrich Gauss
8839:
8791:polynomial equations
8746:
8687:
8628:
8597:
8565:
8538:
8511:
8480:
8447:
8414:
8381:
8352:
8324:
8297:
8076:
7964:
7899:
7877:
7853:
7831:
7809:
7787:
7767:
7736:
7712:
7688:
7664:
7625:
7563:
7541:
7519:
7493:
7473:, is highlighted in
7446:
7400:
7349:
7318:
7287:
7258:
7227:
7199:
7171:
7143:
7113:
7083:
7053:
7023:
6993:
6963:
6937:
6909:
6881:
6851:
6821:
6791:
6761:
6731:
6701:
6673:
6647:
6619:
6589:
6559:
6529:
6499:
6469:
6439:
6411:
6383:
6357:
6327:
6297:
6267:
6237:
6207:
6175:
6145:
6115:
6085:
6057:
6029:
6003:
5975:
5947:
5915:
5885:
5855:
5825:
5797:
5771:
5743:
5715:
5687:
5655:
5625:
5595:
5565:
5539:
5511:
5483:
5455:
5427:
5395:
5365:
5335:
5305:
5277:
5249:
5221:
5195:
5169:
5137:
5107:
5077:
5047:
5019:
4991:
4963:
4937:
4916:
4884:
4818:
4787:
4754:
4723:
4692:
4670:
4644:
4620:
4600:
4569:
4534:
4507:
4472:
4439:
4402:
4369:
4335:
4306:
4277:
4231:
4192:
4153:
4115:
4078:
4043:
4008:
3975:
3934:
3872:
3828:
3821:function composition
3786:
3764:
3744:
3706:
3675:
3650:
3618:
3596:
3574:
3566:multiplicative group
3543:
3521:
3499:
3456:
3427:
3405:
3399:multiplicative group
3379:
3357:
3328:
3284:
3250:
3228:
3203:
3172:
3148:
3129:mathematical objects
3067:
3045:
3017:
2995:
2970:
2936:
2902:
2882:
2862:
2840:
2818:
2800:). It is called the
2763:
2727:
2703:
2681:
2661:
2641:
2571:
2547:
2523:
2499:
2475:
2434:
2410:
2388:
2368:
2348:
2323:
2299:
2273:
2248:monster simple group
2218:
2187:
2181:and is denoted
2165:
2139:
2105:
2071:
2051:
2029:
1980:
1946:
1926:
1895:
1873:
1853:
1833:
1813:
1793:
1729:
1705:
1683:
1661:
1631:
1605:
1573:
1548:
1526:
1416:
1284:Group with operators
1227:Complemented lattice
1062:Algebraic structures
727:
702:
665:
201:polynomial equations
116:
29612:Composition algebra
29536:Inner product space
29514:multilinear algebra
29402:Polynomial function
29343:Multilinear algebra
29328:Homological algebra
29318:Commutative algebra
28995:Other finite groups
28782:Commutator subgroup
28636:Smith, David Eugene
28429:O'Connor, John J.;
27755:MathSciNet (2021),
27685:Lay, David (2003),
27599:1999qrsp.book.....K
27568:1937RSPSA.161..220J
27463:Classical Mechanics
27204:Conway, John Horton
26975:Aschbacher, Michael
26257:, pp. 197â202.
26026:, p. 12, §I.2.
25691:, p. 54, §2.6.
25655:, p. 40, §2.2.
25509:, pp. 105â113.
25311:singular cohomology
24873:{\displaystyle 1/a}
24447:
23431:
23119:topological groups,
22849:monstrous moonshine
22780:{\displaystyle G/N}
22348:{\displaystyle p-1}
22296:
22168:The dihedral group
21999:corresponds to the
21461:and higher. In the
20789:octahedral symmetry
20566:Symmetry in physics
20255:{\displaystyle p=5}
20203:
20143:{\displaystyle n=6}
19409:are not a power of
19307:
19048:{\displaystyle p=5}
18887:that is, such that
18704:. The primality of
18519:{\displaystyle p=5}
18498:. For example, for
18437:{\displaystyle p-1}
18240:{\displaystyle 9+4}
18119:{\displaystyle n-a}
17991:{\displaystyle n-1}
17937:{\displaystyle n-1}
17699:{\displaystyle b/a}
17669:{\displaystyle a/b}
17207:{\displaystyle a=2}
17019:and number theory.
16919:equivalence classes
16886:by introducing the
16637:. This is called a
16417:; examples include
16146:and the reflection
15372:
15262:{\displaystyle G/N}
15137:{\displaystyle G/N}
14919:{\displaystyle G/N}
14606:is normal, because
14104:
13170:is a homomorphism.
13063:
11583:Group homomorphisms
9863:left multiplication
8966:of finite groups),
8890:projective geometry
8624:, as, for example,
4909:
3962:
3896:Two figures in the
3866:automorphism groups
3187:{\displaystyle a+b}
1601:property says that
1588:{\displaystyle a+b}
1338:Composition algebra
1098:Quasigroup and loop
371:Group homomorphisms
281:Algebraic structure
29392:Quadratic equation
29323:Elementary algebra
29291:Algebraic geometry
29025:Rubik's Cube group
28982:Baby monster group
28792:Group homomorphism
28708:Weisstein, Eric W.
28681:Dover Publications
28496:10.1007/BF01443322
28452:Curtis, Charles W.
28238:Weibel, Charles A.
28141:Notices of the AMS
28043:Serre, Jean-Pierre
27709:Mac Lane, Saunders
27505:Algebraic Topology
27458:Goldstein, Herbert
26957:Dover Publications
26942:Special references
26698:General references
26293:, pp. 15, 16.
26240:Conway et al. 2001
26050:, p. 9, §I.2.
25951:, p. 7, §I.2.
25907:, p. 3, §I.1.
25531:See, for example,
25434:discrete logarithm
25432:protocol uses the
25349:field of fractions
25262:áœÎŒÏÏâthe same and
25233:See, for example,
25068:Mathematics portal
24901:
24870:
24837:{\displaystyle -a}
24834:
24806:{\displaystyle -a}
24803:
24659:
24591:
24561:
24531:
24500:
24470:
24445:
24397:
24373:
24352:{\displaystyle fg}
24349:
24318:
24270:
24246:
24224:
24184:
24153:
24139:Grothendieck group
24125:
24101:
24053:
24005:
23957:
23869:Commutative monoid
23429:
23415:fundamental forces
23391:special relativity
23336:
23316:
23296:
23276:
23238:
23218:
23194:looks locally like
23107:
23087:
23067:
23037:
23019:topological spaces
23009:
22982:Topological groups
22945:
22916:
22884:
22816:have been used to
22777:
22747:
22727:
22707:
22687:
22661:
22641:
22613:
22586:
22557:
22528:
22482:
22446:
22427:Lagrange's theorem
22425:(a consequence of
22415:
22386:
22345:
22317:
22297:
22280:
22263:
22234:
22214:
22187:
22149:
22124:
22104:
22092:Lagrange's Theorem
22077:
22057:
22035:
22011:
21989:
21960:
21936:
21914:
21892:
21870:
21866:
21854:
21820:
21791:
21756:
21736:
21716:
21689:
21654:
21632:
21598:
21566:
21544:
21506:A group is called
21435:
21415:
21395:
21375:
21300:
21258:quadratic equation
21233:topological groups
21217:
21159:
21139:
21117:
21065:
21043:
20992:
20968:
20943:
20895:
20881:
20687:quantum mechanical
20642:
20618:
20584:
20554:Molecular symmetry
20511:
20475:
20453:
20426:
20389:
20352:
20309:
20274:
20252:
20234:: for example, if
20224:
20204:
20187:
20162:
20140:
20112:
20086:
20051:
20022:
20000:
19978:
19956:
19936:
19887:
19791:
19769:
19671:
19639:
19611:
19584:
19462:
19442:
19428:
19399:
19379:
19352:
19308:
19291:
19272:
19237:
19183:
19131:
19109:
19087:
19065:
19045:
19017:
18995:
18953:
18931:
18897:
18877:
18826:
18804:
18782:
18760:
18736:
18714:
18692:
18670:
18660:: when divided by
18648:
18629:{\displaystyle 16}
18626:
18604:
18558:
18516:
18486:
18458:
18434:
18406:
18384:
18356:
18332:
18291:" or, in symbols,
18282:{\displaystyle 12}
18279:
18257:
18237:
18209:
18187:
18167:
18138:
18116:
18090:
18068:
18019:
17988:
17960:
17934:
17906:
17884:
17840:
17820:
17800:
17780:
17765:
17747:Modular arithmetic
17741:Modular arithmetic
17727:
17696:
17666:
17636:
17605:
17526:
17486:
17452:
17430:
17388:
17360:
17338:
17297:
17273:
17271:
17236:
17204:
17174:
17134:
17094:
17074:also form groups.
17017:algebraic geometry
16988:
16960:
16940:
16911:
16884:algebraic topology
16853:
16839:
16825:Examples of groups
16792:
16684:
16662:
16625:
16534:
16502:
16474:
16403:
16375:
16346:
16326:
16299:
16279:
16250:
16216:
16194:
16165:
16136:
16109:
16053:
15990:
15947:
15926:semidirect product
15914:
15885:
15848:
15770:
15659:
15619:
15599:
15557:
15536:
15515:
15492:
15471:
15450:
15427:
15406:
15385:
15362:
15334:
15329:universal property
15307:
15283:
15259:
15227:
15162:{\displaystyle gN}
15159:
15134:
15104:
15073:
15042:{\displaystyle hN}
15039:
15019:{\displaystyle gN}
15016:
14996:{\displaystyle gN}
14993:
14970:
14950:
14926:for which the map
14916:
14888:
14831:
14809:
14773:
14683:
14652:
14596:
14574:
14543:
14439:
14419:
14397:
14378:{\displaystyle gR}
14375:
14352:
14330:
14290:
14268:
14234:
14214:
14194:
14172:
14148:
14124:
14087:
14065:
14012:
13986:
13960:
13935:
13880:
13823:
13799:
13767:
13745:
13725:
13688:
13639:, and the element
13627:
13600:
13571:
13542:
13520:
13500:
13480:
13454:
13432:
13403:
13381:
13361:
13341:
13297:
13273:
13241:
13160:
13132:
13110:
13088:
13064:
13046:
13027:
12985:
12963:
12936:
12907:
12885:
12863:
12839:
12817:
12773:
12749:
12737:for some subgroup
12727:
12675:
12639:category of groups
12618:
12596:
12574:
12552:
12532:
12510:
12472:
12452:
12420:
12398:
12378:
12326:
12306:
12286:
12232:
12191:
12152:
12130:is a homomorphism
12120:
12098:of a homomorphism
12084:
12064:
12025:
11996:
11974:
11954:
11888:
11837:
11812:
11790:
11770:
11750:
11683:
11651:
11619:
11589:Group homomorphism
11544:
11522:
11498:
11478:
11444:
11422:
11402:
11374:
11350:
11326:
11301:
11278:
11222:
11189:
11167:
11143:
11097:
11053:
11043:with the operator
11033:
10974:
10972:
10791:
10789:
10343:
10299:
10279:
10247:
10213:
10168:
10142:
10118:
10092:
10072:
10044:
10020:
9982:
9948:
9926:
9901:
9875:
9847:
9821:
9801:
9775:
9755:
9733:
9693:
9659:
9639:
9617:
9595:
9575:
9540:
9538:
9307:as inverses. Then
9297:
9277:
9257:
9223:
9183:
9163:
9131:
8858:
8761:
8742:. In other words,
8730:
8671:
8612:
8578:
8551:
8524:
8495:
8462:
8429:
8396:
8375:, the reflections
8363:
8330:
8308:
8276:
8274:
8062:
7950:
7883:
7859:
7837:
7815:
7793:
7773:
7751:
7718:
7694:
7670:
7640:
7609:
7547:
7525:
7505:
7461:
7428:
7362:
7331:
7300:
7269:
7238:
7212:
7184:
7156:
7128:
7098:
7068:
7038:
7008:
6976:
6948:
6922:
6894:
6866:
6836:
6806:
6776:
6746:
6714:
6686:
6658:
6632:
6604:
6574:
6544:
6514:
6484:
6452:
6424:
6396:
6368:
6342:
6312:
6282:
6252:
6222:
6190:
6160:
6130:
6100:
6070:
6042:
6014:
5988:
5960:
5930:
5900:
5870:
5840:
5810:
5782:
5756:
5728:
5700:
5670:
5640:
5610:
5580:
5550:
5524:
5496:
5468:
5440:
5410:
5380:
5350:
5320:
5290:
5262:
5234:
5206:
5180:
5152:
5122:
5092:
5062:
5032:
5004:
4976:
4948:
4922:
4899:
4876:
4864:
4802:
4769:
4736:
4698:
4676:
4656:
4638:from right to left
4626:
4606:
4584:
4549:
4520:
4487:
4454:
4417:
4384:
4348:
4319:
4290:
4266:identity operation
4246:
4207:
4168:
4130:
4091:
4066:(rotation by 180°)
4056:
4021:
3986:
3949:
3927:
3878:
3862:permutation groups
3840:
3792:
3770:
3750:
3730:
3687:
3662:{\displaystyle ab}
3659:
3634:
3602:
3580:
3555:{\displaystyle -x}
3552:
3527:
3505:
3476:
3442:
3413:
3387:
3365:
3336:
3310:
3270:
3236:
3215:{\displaystyle ab}
3212:
3184:
3156:
3083:
3051:
3023:
3001:
2976:
2954:
2920:
2888:
2868:
2846:
2824:
2781:
2745:
2709:
2687:
2667:
2647:
2619:
2553:
2529:
2505:
2481:
2446:
2416:
2394:
2374:
2354:
2329:
2305:
2279:
2224:
2199:{\displaystyle -a}
2196:
2171:
2145:
2123:
2089:
2057:
2035:
2023:For every integer
1998:
1964:
1932:
1901:
1879:
1859:
1839:
1819:
1799:
1777:
1711:
1689:
1667:
1639:
1611:
1585:
1554:
1532:
1508:
847:Special orthogonal
735:
710:
673:
554:Lagrange's theorem
186:special relativity
122:
100:addition operation
53:
49:Rubik's Cube group
29740:
29739:
29662:Symmetric algebra
29632:Geometric algebra
29412:Linear inequality
29363:Universal algebra
29296:Algebraic variety
29220:
29219:
28905:Alternating group
28690:978-0-486-45868-7
28664:978-0-486-60269-1
28474:von Dyck, Walther
28465:978-0-8218-2677-5
28404:978-0-8218-0288-5
28358:978-0-691-14034-6
28334:978-0-691-02374-8
28312:978-0-19-853287-3
28251:978-0-521-55987-4
28229:978-0-387-90894-6
28175:978-1-4822-4582-0
28132:978-0-521-53047-7
28106:978-0-691-08017-8
28087:978-0-88385-511-9
28060:978-0-387-90190-9
27976:978-0-201-87073-2
27957:978-0-19-280723-6
27931:978-981-02-4942-7
27914:Romanowska, A. B.
27890:978-3-540-65399-8
27854:978-0-486-43235-9
27827:978-3-540-56963-3
27806:978-0-691-08238-7
27785:978-0-521-86625-5
27748:978-0-486-43830-6
27724:978-0-387-98403-2
27700:978-0-201-70970-4
27669:978-0-387-40510-0
27642:978-0-8176-3688-3
27608:978-0-691-05872-6
27538:978-0-89874-193-3
27519:978-0-521-79540-1
27492:978-0-470-74115-3
27483:Computer Security
27442:978-0-387-97495-8
27379:978-0-19-883298-0
27353:978-0-471-01670-0
27283:978-1-58488-254-1
27252:978-3-540-52977-4
27196:978-0-691-22534-0
27177:978-0-471-50683-6
27143:978-0-387-97370-8
27117:978-0-486-67355-4
27011:978-0-19-958736-0
26966:978-0-486-62342-9
26933:978-0-387-94461-6
26900:Ledermann, Walter
26880:Ledermann, Walter
26871:978-0-387-22025-3
26838:978-0-387-95385-4
26805:Topics in Algebra
26784:978-0-13-374562-7
26743:978-0-691-13951-7
26722:978-0-13-468960-9
25599:, p. 26, §2.
25428:For example, the
25315:classifying space
25112:is an element of
25051:
25050:
24606:Integers modulo 3
24400:{\displaystyle g}
24376:{\displaystyle f}
24273:{\displaystyle x}
24249:{\displaystyle G}
24187:{\displaystyle x}
24156:{\displaystyle M}
24128:{\displaystyle M}
24063:produces a group
23933:
23932:
23351:Noether's theorem
23339:{\displaystyle n}
23319:{\displaystyle n}
23299:{\displaystyle A}
23241:{\displaystyle n}
23221:{\displaystyle n}
23148:harmonic analysis
23127:topological field
23110:{\displaystyle h}
23090:{\displaystyle g}
23013:Topological group
22957:nullary operation
22887:{\displaystyle G}
22864:extension problem
22857:modular functions
22853:Richard Borcherds
22750:{\displaystyle N}
22730:{\displaystyle G}
22710:{\displaystyle G}
22664:{\displaystyle N}
22644:{\displaystyle G}
22389:{\displaystyle p}
22320:{\displaystyle p}
22237:{\displaystyle R}
22152:{\displaystyle G}
22127:{\displaystyle H}
22107:{\displaystyle G}
22080:{\displaystyle a}
22060:{\displaystyle n}
22038:{\displaystyle a}
22014:{\displaystyle n}
21939:{\displaystyle a}
21917:{\displaystyle n}
21863:
21839:
21837:
21759:{\displaystyle n}
21739:{\displaystyle G}
21719:{\displaystyle a}
21657:{\displaystyle N}
21569:{\displaystyle N}
21463:quadratic formula
21451:quartic equations
21438:{\displaystyle -}
21418:{\displaystyle +}
21370:
21359:
21162:{\displaystyle n}
21142:{\displaystyle G}
21081:computer graphics
21077:Rotation matrices
21068:{\displaystyle n}
20995:{\displaystyle n}
20971:{\displaystyle n}
20884:{\displaystyle x}
20802:
20801:
20702:Curie temperature
20645:{\displaystyle X}
20621:{\displaystyle X}
20478:{\displaystyle a}
20456:{\displaystyle a}
20277:{\displaystyle 3}
20227:{\displaystyle p}
20174:counter-clockwise
20172:corresponds to a
20165:{\displaystyle z}
20115:{\displaystyle n}
20054:{\displaystyle z}
20025:{\displaystyle n}
20003:{\displaystyle n}
19981:{\displaystyle 1}
19959:{\displaystyle 1}
19803:primitive element
19794:{\displaystyle a}
19465:{\displaystyle a}
19402:{\displaystyle z}
19355:{\displaystyle z}
19320:, are crucial to
19134:{\displaystyle 2}
19112:{\displaystyle 3}
19090:{\displaystyle 4}
19068:{\displaystyle 4}
19020:{\displaystyle 1}
18965:BĂ©zout's identity
18956:{\displaystyle b}
18900:{\displaystyle p}
18829:{\displaystyle b}
18807:{\displaystyle p}
18792:not divisible by
18785:{\displaystyle a}
18763:{\displaystyle 1}
18739:{\displaystyle p}
18717:{\displaystyle p}
18695:{\displaystyle 1}
18673:{\displaystyle 5}
18651:{\displaystyle 1}
18636:is equivalent to
18570:. In this group,
18489:{\displaystyle p}
18461:{\displaystyle p}
18409:{\displaystyle 1}
18387:{\displaystyle p}
18359:{\displaystyle p}
18260:{\displaystyle 1}
18212:{\displaystyle 1}
18190:{\displaystyle 4}
18170:{\displaystyle 9}
18141:{\displaystyle a}
18093:{\displaystyle 0}
17963:{\displaystyle 0}
17909:{\displaystyle 0}
17843:{\displaystyle n}
17823:{\displaystyle b}
17803:{\displaystyle a}
17783:{\displaystyle n}
17543:rational numbers
17455:{\displaystyle x}
17363:{\displaystyle b}
17333:
17270:
17013:hyperbolic groups
16963:{\displaystyle m}
16943:{\displaystyle k}
16923:smoothly deformed
16915:topological space
16888:fundamental group
16349:{\displaystyle f}
16302:{\displaystyle r}
16046:
15964:implies that any
15895:are abelian, but
15858:and the quotient
15622:{\displaystyle R}
15570:
15569:
15560:{\displaystyle R}
15539:{\displaystyle U}
15518:{\displaystyle U}
15495:{\displaystyle U}
15474:{\displaystyle R}
15453:{\displaystyle R}
15430:{\displaystyle U}
15409:{\displaystyle R}
15310:{\displaystyle N}
15286:{\displaystyle G}
14973:{\displaystyle g}
14834:{\displaystyle G}
14812:{\displaystyle N}
14599:{\displaystyle R}
14442:{\displaystyle R}
14429:is an element of
14422:{\displaystyle g}
14400:{\displaystyle R}
14355:{\displaystyle R}
14293:{\displaystyle H}
14237:{\displaystyle G}
14217:{\displaystyle g}
14197:{\displaystyle H}
14175:{\displaystyle H}
14151:{\displaystyle H}
14029:. The first case
14015:{\displaystyle G}
13989:{\displaystyle G}
13963:{\displaystyle H}
13826:{\displaystyle g}
13802:{\displaystyle H}
13770:{\displaystyle g}
13748:{\displaystyle H}
13728:{\displaystyle H}
13545:{\displaystyle S}
13523:{\displaystyle G}
13503:{\displaystyle S}
13483:{\displaystyle S}
13457:{\displaystyle G}
13435:{\displaystyle S}
13422:Given any subset
13406:{\displaystyle H}
13384:{\displaystyle h}
13364:{\displaystyle g}
13351:for all elements
13300:{\displaystyle G}
13276:{\displaystyle H}
13135:{\displaystyle H}
13113:{\displaystyle G}
13091:{\displaystyle H}
12988:{\displaystyle H}
12910:{\displaystyle H}
12888:{\displaystyle G}
12866:{\displaystyle G}
12842:{\displaystyle G}
12820:{\displaystyle H}
12776:{\displaystyle G}
12752:{\displaystyle H}
12714:
12621:{\displaystyle H}
12599:{\displaystyle G}
12555:{\displaystyle G}
12535:{\displaystyle H}
12475:{\displaystyle H}
12455:{\displaystyle G}
12423:{\displaystyle H}
12401:{\displaystyle h}
12329:{\displaystyle G}
12309:{\displaystyle g}
12087:{\displaystyle G}
12028:{\displaystyle G}
11999:{\displaystyle G}
11977:{\displaystyle a}
11815:{\displaystyle G}
11793:{\displaystyle b}
11773:{\displaystyle a}
11760:for all elements
11547:{\displaystyle Y}
11525:{\displaystyle X}
11501:{\displaystyle f}
11447:{\displaystyle X}
11432:is an element of
11425:{\displaystyle x}
11377:{\displaystyle z}
11353:{\displaystyle y}
11329:{\displaystyle x}
11304:{\displaystyle X}
11192:{\displaystyle e}
11170:{\displaystyle e}
10968:
10947:
10920:
10871:
10785:
10764:
10701:
10626:
10529:
10432:
10302:{\displaystyle f}
10289:for each element
10216:{\displaystyle e}
10171:{\displaystyle a}
10154:right translation
10145:{\displaystyle a}
10095:{\displaystyle x}
10047:{\displaystyle a}
9951:{\displaystyle b}
9929:{\displaystyle a}
9916:Similarly, given
9904:{\displaystyle a}
9878:{\displaystyle a}
9824:{\displaystyle x}
9778:{\displaystyle G}
9758:{\displaystyle a}
9662:{\displaystyle G}
9642:{\displaystyle x}
9620:{\displaystyle G}
9598:{\displaystyle b}
9578:{\displaystyle a}
9534:
9520:
9512:
9491:
9483:
9475:
9448:
9409:
9401:
9393:
9354:
9346:
9300:{\displaystyle c}
9280:{\displaystyle b}
9260:{\displaystyle a}
9186:{\displaystyle f}
9166:{\displaystyle e}
9019:Daniel Gorenstein
8919:Leopold Kronecker
8333:{\displaystyle a}
7886:{\displaystyle a}
7862:{\displaystyle c}
7840:{\displaystyle b}
7818:{\displaystyle c}
7796:{\displaystyle b}
7776:{\displaystyle a}
7721:{\displaystyle c}
7697:{\displaystyle b}
7673:{\displaystyle a}
7550:{\displaystyle b}
7528:{\displaystyle a}
7481:
7480:
4701:{\displaystyle a}
4679:{\displaystyle b}
4629:{\displaystyle b}
4609:{\displaystyle a}
4261:
4260:
3795:{\displaystyle G}
3773:{\displaystyle b}
3753:{\displaystyle a}
3740:for all elements
3605:{\displaystyle x}
3583:{\displaystyle 1}
3530:{\displaystyle x}
3508:{\displaystyle 0}
3054:{\displaystyle a}
3026:{\displaystyle b}
3004:{\displaystyle a}
2979:{\displaystyle e}
2891:{\displaystyle G}
2871:{\displaystyle b}
2849:{\displaystyle G}
2827:{\displaystyle a}
2712:{\displaystyle G}
2690:{\displaystyle a}
2670:{\displaystyle G}
2650:{\displaystyle e}
2556:{\displaystyle G}
2532:{\displaystyle c}
2508:{\displaystyle b}
2484:{\displaystyle a}
2462:, are satisfied:
2419:{\displaystyle G}
2397:{\displaystyle G}
2377:{\displaystyle b}
2357:{\displaystyle a}
2308:{\displaystyle G}
2282:{\displaystyle G}
2256:Richard Borcherds
2227:{\displaystyle +}
2174:{\displaystyle a}
2148:{\displaystyle b}
2060:{\displaystyle b}
2038:{\displaystyle a}
1935:{\displaystyle a}
1904:{\displaystyle c}
1882:{\displaystyle b}
1862:{\displaystyle a}
1842:{\displaystyle c}
1822:{\displaystyle b}
1802:{\displaystyle a}
1714:{\displaystyle c}
1692:{\displaystyle b}
1670:{\displaystyle a}
1655:For all integers
1614:{\displaystyle +}
1557:{\displaystyle b}
1535:{\displaystyle a}
1397:
1396:
1056:
1055:
631:
630:
513:Alternating group
470:
469:
125:{\displaystyle 1}
18:MathematicalGrouP
16:(Redirected from
29772:
29730:
29729:
29617:Exterior algebra
29286:Abstract algebra
29247:
29240:
29233:
29224:
29223:
29212:Abstract algebra
29149:Quaternion group
29079:Symplectic group
29055:Orthogonal group
28749:
28742:
28735:
28726:
28725:
28721:
28720:
28693:
28667:
28645:
28631:
28607:
28587:
28545:
28529:Joseph Liouville
28526:
28517:Galois, Ăvariste
28511:
28506:, archived from
28468:
28447:
28424:
28407:
28369:
28337:
28315:
28293:
28275:Weinberg, Steven
28270:
28232:
28213:
28204:
28178:
28157:
28156:
28154:10.1090/noti1689
28135:
28117:
28090:
28071:
28052:
28037:
28020:
28009:
27987:
27960:
27934:
27922:World Scientific
27909:
27882:
27871:Neukirch, JĂŒrgen
27865:
27838:
27809:
27796:Ătale Cohomology
27788:
27770:
27769:
27767:
27762:
27751:
27727:
27703:
27680:
27653:
27634:
27619:
27580:
27579:
27562:(905): 220â235,
27541:
27522:
27495:
27476:
27453:
27415:
27390:
27356:
27337:
27320:
27315:
27305:
27286:
27263:
27236:
27227:
27199:
27180:
27160:Carter, Roger W.
27154:
27120:
27101:
27092:
27066:
27046:
27045:
27035:
27014:
26995:
26983:
26969:
26936:
26914:
26894:
26874:
26849:
26815:
26795:
26775:Abstract Algebra
26765:
26746:
26725:
26685:
26679:
26673:
26667:
26661:
26655:
26649:
26643:
26637:
26631:
26625:
26619:
26613:
26607:
26601:
26595:
26589:
26583:
26577:
26571:
26565:
26559:
26553:
26547:
26541:
26535:
26529:
26523:
26517:
26511:
26505:
26499:
26493:
26483:
26477:
26471:
26465:
26459:
26453:
26447:
26441:
26435:
26429:
26423:
26417:
26411:
26405:
26399:
26393:
26387:
26381:
26375:
26366:
26360:
26354:
26348:
26342:
26336:
26330:
26324:
26318:
26312:
26306:
26300:
26294:
26288:
26282:
26276:
26270:
26264:
26258:
26252:
26246:
26237:
26231:
26225:
26219:
26213:
26204:
26198:
26192:
26186:
26180:
26174:
26168:
26162:
26156:
26150:
26144:
26138:
26132:
26126:
26120:
26114:
26108:
26093:
26087:
26081:
26075:
26069:
26063:
26057:
26051:
26045:
26039:
26033:
26027:
26021:
26015:
26009:
26003:
25997:
25991:
25985:
25979:
25973:
25964:
25958:
25952:
25946:
25935:
25929:
25923:
25917:
25908:
25902:
25896:
25890:
25884:
25878:
25872:
25866:
25860:
25854:
25848:
25842:
25836:
25830:
25824:
25818:
25812:
25806:
25800:
25794:
25788:
25782:
25776:
25770:
25764:
25758:
25752:
25746:
25740:
25734:
25728:
25722:
25716:
25710:
25704:
25698:
25692:
25686:
25680:
25674:
25668:
25662:
25656:
25650:
25644:
25638:
25632:
25626:
25620:
25606:
25600:
25594:
25577:
25570:
25564:
25554:
25548:
25545:Ă©tale cohomology
25529:
25523:
25516:
25510:
25499:
25493:
25486:Frucht's theorem
25478:
25472:
25470:
25464:
25458:
25447:
25441:
25426:
25420:
25411:
25405:
25386:
25380:
25374:
25368:
25358:
25352:
25345:
25339:
25328:
25322:
25307:group cohomology
25303:
25297:
25290:
25284:
25277:
25271:
25268:Schwartzman 1994
25266:âstructure. See
25256:
25250:
25231:
25225:
25223:
25213:
25203:
25197:
25196:
25194:
25193:
25188:
25185:
25174:
25168:
25157:
25151:
25146:. See Lang
25145:
25139:
25135:
25129:
25123:
25117:
25111:
25101:
25092:
25070:
25065:
25064:
24912:
24910:
24908:
24907:
24902:
24881:
24879:
24877:
24876:
24871:
24866:
24845:
24843:
24841:
24840:
24835:
24814:
24812:
24810:
24809:
24804:
24670:
24668:
24666:
24665:
24660:
24634:
24626:
24621:
24602:
24600:
24598:
24597:
24592:
24590:
24572:
24570:
24568:
24567:
24562:
24560:
24542:
24540:
24538:
24537:
24532:
24530:
24515:Rational numbers
24511:
24509:
24507:
24506:
24501:
24499:
24481:
24479:
24477:
24476:
24471:
24469:
24448:
24444:
24439:
24434:
24430:
24424:
24408:
24406:
24404:
24403:
24398:
24384:
24382:
24380:
24379:
24374:
24360:
24358:
24356:
24355:
24350:
24329:
24327:
24325:
24324:
24319:
24281:
24279:
24277:
24276:
24271:
24257:
24255:
24253:
24252:
24247:
24233:
24231:
24230:
24225:
24195:
24193:
24191:
24190:
24185:
24172:with one object
24164:
24162:
24160:
24159:
24154:
24136:
24134:
24132:
24131:
24126:
24112:
24110:
24108:
24107:
24102:
24079:
24062:
24060:
24059:
24054:
24031:
24016:
24014:
24012:
24011:
24006:
23983:
23966:
23964:
23963:
23958:
23956:
23432:
23428:
23345:
23343:
23342:
23337:
23325:
23323:
23322:
23317:
23305:
23303:
23302:
23297:
23285:
23283:
23282:
23277:
23247:
23245:
23244:
23239:
23227:
23225:
23224:
23219:
23135:
23116:
23114:
23113:
23108:
23096:
23094:
23093:
23088:
23076:
23074:
23073:
23068:
23066:
23065:
23046:
23044:
23043:
23038:
22954:
22952:
22951:
22946:
22925:
22923:
22922:
22917:
22893:
22891:
22890:
22885:
22788:
22786:
22784:
22783:
22778:
22773:
22756:
22754:
22753:
22748:
22736:
22734:
22733:
22728:
22716:
22714:
22713:
22708:
22696:
22694:
22693:
22688:
22670:
22668:
22667:
22662:
22650:
22648:
22647:
22642:
22622:
22620:
22619:
22614:
22612:
22611:
22595:
22593:
22592:
22587:
22585:
22584:
22579:
22566:
22564:
22563:
22558:
22556:
22555:
22539:
22537:
22535:
22534:
22529:
22527:
22526:
22521:
22512:
22511:
22506:
22491:
22489:
22488:
22483:
22481:
22480:
22479:
22478:
22468:
22455:
22453:
22452:
22447:
22445:
22444:
22424:
22422:
22421:
22416:
22414:
22413:
22408:
22395:
22393:
22392:
22387:
22356:
22354:
22352:
22351:
22346:
22326:
22324:
22323:
22318:
22306:
22304:
22303:
22298:
22295:
22290:
22285:
22272:
22270:
22269:
22264:
22262:
22261:
22260:
22243:
22241:
22240:
22235:
22223:
22221:
22220:
22215:
22213:
22212:
22196:
22194:
22193:
22188:
22186:
22185:
22180:
22160:
22158:
22156:
22155:
22150:
22133:
22131:
22130:
22125:
22113:
22111:
22110:
22105:
22086:
22084:
22083:
22078:
22066:
22064:
22063:
22058:
22046:
22044:
22042:
22041:
22036:
22022:
22020:
22018:
22017:
22012:
21998:
21996:
21995:
21990:
21988:
21987:
21971:
21969:
21967:
21966:
21961:
21947:
21945:
21943:
21942:
21937:
21923:
21921:
21920:
21915:
21903:
21901:
21899:
21898:
21893:
21879:
21877:
21876:
21871:
21865:
21864:
21861:
21855:
21850:
21829:
21827:
21826:
21821:
21819:
21818:
21802:
21800:
21798:
21797:
21792:
21784:
21783:
21765:
21763:
21762:
21757:
21745:
21743:
21742:
21737:
21725:
21723:
21722:
21717:
21698:
21696:
21695:
21690:
21688:
21687:
21682:
21668:Cayley's theorem
21665:
21663:
21661:
21660:
21655:
21641:
21639:
21638:
21633:
21631:
21630:
21625:
21607:
21605:
21604:
21599:
21597:
21596:
21591:
21575:
21573:
21572:
21567:
21555:
21553:
21551:
21550:
21545:
21543:
21542:
21537:
21521:symmetric groups
21482:field extensions
21444:
21442:
21441:
21436:
21424:
21422:
21421:
21416:
21404:
21402:
21401:
21396:
21384:
21382:
21381:
21376:
21371:
21369:
21361:
21360:
21346:
21345:
21336:
21324:
21309:
21307:
21306:
21301:
21278:
21277:
21226:
21224:
21223:
21218:
21213:
21199:
21168:
21166:
21165:
21160:
21148:
21146:
21145:
21140:
21128:
21126:
21124:
21123:
21118:
21116:
21115:
21110:
21074:
21072:
21071:
21066:
21054:
21052:
21050:
21049:
21044:
21033:
21003:
21001:
20999:
20998:
20993:
20979:
20977:
20975:
20974:
20969:
20953:consists of all
20952:
20950:
20949:
20944:
20939:
20925:
20892:
20890:
20888:
20887:
20882:
20857:Character theory
20837:invariant theory
20814:error correction
20750:
20743:
20736:
20729:
20722:
20721:
20717:Goldstone bosons
20694:phase transition
20664:hyperbolic plane
20653:
20651:
20649:
20648:
20643:
20629:
20627:
20625:
20624:
20619:
20522:
20520:
20518:
20517:
20512:
20501:
20484:
20482:
20481:
20476:
20464:
20462:
20460:
20459:
20454:
20437:
20435:
20433:
20432:
20427:
20419:
20418:
20400:
20398:
20396:
20395:
20390:
20382:
20381:
20363:
20361:
20359:
20358:
20353:
20339:
20338:
20320:
20318:
20316:
20315:
20310:
20302:
20301:
20283:
20281:
20280:
20275:
20263:
20261:
20259:
20258:
20253:
20233:
20231:
20230:
20225:
20213:
20211:
20210:
20205:
20202:
20197:
20192:
20171:
20169:
20168:
20163:
20151:
20149:
20147:
20146:
20141:
20121:
20119:
20118:
20113:
20097:
20095:
20093:
20092:
20087:
20079:
20078:
20060:
20058:
20057:
20052:
20033:
20031:
20029:
20028:
20023:
20009:
20007:
20006:
20001:
19989:
19987:
19985:
19984:
19979:
19965:
19963:
19962:
19957:
19945:
19943:
19942:
19937:
19926:
19918:
19913:
19896:
19894:
19893:
19888:
19800:
19798:
19797:
19792:
19780:
19778:
19776:
19775:
19770:
19768:
19767:
19734:
19733:
19718:
19717:
19702:
19701:
19680:
19678:
19677:
19672:
19670:
19669:
19650:
19648:
19646:
19645:
19640:
19620:
19618:
19617:
19612:
19610:
19609:
19593:
19591:
19590:
19585:
19574:
19573:
19561:
19560:
19542:
19541:
19529:
19528:
19513:
19512:
19497:
19496:
19471:
19469:
19468:
19463:
19439:
19437:
19435:
19434:
19429:
19427:
19426:
19408:
19406:
19405:
19400:
19388:
19386:
19385:
19380:
19378:
19377:
19361:
19359:
19358:
19353:
19319:
19317:
19315:
19314:
19309:
19306:
19301:
19296:
19281:
19279:
19278:
19273:
19271:
19263:
19258:
19246:
19244:
19243:
19238:
19236:
19232:
19225:
19211:
19194:
19192:
19190:
19189:
19184:
19182:
19181:
19142:
19140:
19138:
19137:
19132:
19118:
19116:
19115:
19110:
19098:
19096:
19094:
19093:
19088:
19074:
19072:
19071:
19066:
19054:
19052:
19051:
19046:
19028:
19026:
19024:
19023:
19018:
19004:
19002:
19001:
18996:
18962:
18960:
18959:
18954:
18942:
18940:
18938:
18937:
18932:
18906:
18904:
18903:
18898:
18886:
18884:
18883:
18878:
18873:
18835:
18833:
18832:
18827:
18815:
18813:
18811:
18810:
18805:
18791:
18789:
18788:
18783:
18771:
18769:
18767:
18766:
18761:
18747:
18745:
18743:
18742:
18737:
18723:
18721:
18720:
18715:
18703:
18701:
18699:
18698:
18693:
18679:
18677:
18676:
18671:
18659:
18657:
18655:
18654:
18649:
18635:
18633:
18632:
18627:
18615:
18613:
18611:
18610:
18605:
18603:
18602:
18569:
18567:
18565:
18564:
18559:
18527:
18525:
18523:
18522:
18517:
18497:
18495:
18493:
18492:
18487:
18469:
18467:
18465:
18464:
18459:
18445:
18443:
18441:
18440:
18435:
18415:
18413:
18412:
18407:
18395:
18393:
18391:
18390:
18385:
18367:
18365:
18363:
18362:
18357:
18341:
18339:
18338:
18333:
18328:
18290:
18288:
18286:
18285:
18280:
18266:
18264:
18263:
18258:
18247:is congruent to
18246:
18244:
18243:
18238:
18220:
18218:
18216:
18215:
18210:
18196:
18194:
18193:
18188:
18177:and is advanced
18176:
18174:
18173:
18168:
18149:
18147:
18145:
18144:
18139:
18125:
18123:
18122:
18117:
18099:
18097:
18096:
18091:
18079:
18077:
18075:
18074:
18069:
18058:
18050:
18045:
18028:
18026:
18025:
18020:
18018:
18017:
18012:
17999:
17997:
17995:
17994:
17989:
17969:
17967:
17966:
17961:
17945:
17943:
17941:
17940:
17935:
17915:
17913:
17912:
17907:
17895:
17893:
17891:
17890:
17885:
17883:
17849:
17847:
17846:
17841:
17829:
17827:
17826:
17821:
17809:
17807:
17806:
17801:
17789:
17787:
17786:
17781:
17762:
17759: 12. Here,
17736:
17734:
17733:
17728:
17726:
17707:
17705:
17703:
17702:
17697:
17692:
17675:
17673:
17672:
17667:
17662:
17647:
17645:
17643:
17642:
17637:
17635:
17634:
17629:
17614:
17612:
17611:
17606:
17604:
17600:
17587:
17568:
17554:
17535:
17533:
17532:
17527:
17525:
17521:
17514:
17497:
17495:
17493:
17492:
17487:
17461:
17459:
17458:
17453:
17441:
17439:
17437:
17436:
17431:
17429:
17425:
17418:
17399:
17397:
17395:
17394:
17389:
17387:
17372:rational numbers
17369:
17367:
17366:
17361:
17347:
17345:
17344:
17339:
17334:
17326:
17306:
17304:
17303:
17298:
17296:
17284:
17282:
17280:
17279:
17274:
17272:
17263:
17246:in this case is
17245:
17243:
17242:
17237:
17213:
17211:
17210:
17205:
17183:
17181:
17180:
17175:
17173:
17169:
17162:
17145:
17143:
17141:
17140:
17135:
17133:
17129:
17122:
17103:
17101:
17100:
17095:
17093:
17044:computer science
16999:
16997:
16995:
16994:
16989:
16969:
16967:
16966:
16961:
16949:
16947:
16946:
16941:
16869:abstract algebra
16862:
16860:
16859:
16854:
16852:
16803:
16801:
16799:
16798:
16793:
16791:
16790:
16785:
16767:
16766:
16742:
16741:
16729:
16728:
16695:
16693:
16691:
16690:
16685:
16671:
16669:
16668:
16663:
16661:
16660:
16655:
16636:
16634:
16632:
16631:
16626:
16615:
16614:
16590:
16589:
16577:
16576:
16543:
16541:
16540:
16535:
16511:
16509:
16508:
16503:
16485:
16483:
16481:
16480:
16475:
16473:
16472:
16448:
16447:
16435:
16434:
16412:
16410:
16409:
16404:
16386:
16384:
16382:
16381:
16376:
16374:
16373:
16355:
16353:
16352:
16347:
16335:
16333:
16332:
16327:
16325:
16324:
16308:
16306:
16305:
16300:
16288:
16286:
16285:
16280:
16278:
16277:
16272:
16259:
16257:
16256:
16251:
16227:
16225:
16223:
16222:
16217:
16203:
16201:
16200:
16195:
16193:
16192:
16187:
16174:
16172:
16171:
16166:
16164:
16163:
16162:
16145:
16143:
16142:
16137:
16135:
16134:
16118:
16116:
16115:
16110:
16108:
16107:
16102:
16087:, in many ways.
16064:
16062:
16060:
16059:
16054:
16048:
16047:
16045:
16040:
16035:
16022:
15999:
15997:
15996:
15991:
15956:
15954:
15953:
15948:
15946:
15945:
15940:
15923:
15921:
15920:
15915:
15913:
15912:
15907:
15894:
15892:
15891:
15886:
15881:
15876:
15875:
15870:
15857:
15855:
15854:
15849:
15844:
15843:
15831:
15830:
15818:
15817:
15805:
15781:
15779:
15777:
15776:
15771:
15757:
15756:
15755:
15742:
15741:
15740:
15721:
15720:
15719:
15703:
15702:
15701:
15670:
15668:
15666:
15665:
15660:
15655:
15654:
15653:
15628:
15626:
15625:
15620:
15608:
15606:
15605:
15600:
15595:
15590:
15589:
15584:
15566:
15564:
15563:
15558:
15545:
15543:
15542:
15537:
15524:
15522:
15521:
15516:
15501:
15499:
15498:
15493:
15480:
15478:
15477:
15472:
15459:
15457:
15456:
15451:
15436:
15434:
15433:
15428:
15415:
15413:
15412:
15407:
15394:
15392:
15391:
15386:
15373:
15371:
15369:
15368:
15363:
15358:
15353:
15352:
15347:
15333:
15318:
15316:
15314:
15313:
15308:
15294:
15292:
15290:
15289:
15284:
15270:
15268:
15266:
15265:
15260:
15255:
15238:
15236:
15234:
15233:
15228:
15223:
15219:
15218:
15199:
15198:
15168:
15166:
15165:
15160:
15145:
15143:
15141:
15140:
15135:
15130:
15113:
15111:
15110:
15105:
15084:
15082:
15080:
15079:
15074:
15048:
15046:
15045:
15040:
15025:
15023:
15022:
15017:
15002:
15000:
14999:
14994:
14979:
14977:
14976:
14971:
14959:
14957:
14956:
14951:
14946:
14925:
14923:
14922:
14917:
14912:
14897:
14895:
14894:
14889:
14857:
14842:
14840:
14838:
14837:
14832:
14818:
14816:
14815:
14810:
14784:
14782:
14780:
14779:
14774:
14769:
14768:
14767:
14751:
14750:
14749:
14733:
14732:
14731:
14715:
14714:
14713:
14694:
14692:
14690:
14689:
14684:
14682:
14681:
14676:
14661:
14659:
14658:
14653:
14651:
14650:
14649:
14624:
14623:
14622:
14605:
14603:
14602:
14597:
14586:). The subgroup
14585:
14583:
14581:
14580:
14575:
14573:
14572:
14567:
14552:
14550:
14549:
14544:
14539:
14538:
14537:
14524:
14523:
14522:
14509:
14508:
14507:
14494:
14493:
14492:
14473:
14472:
14471:
14448:
14446:
14445:
14440:
14428:
14426:
14425:
14420:
14408:
14406:
14404:
14403:
14398:
14384:
14382:
14381:
14376:
14361:
14359:
14358:
14353:
14341:
14339:
14337:
14336:
14331:
14329:
14328:
14323:
14300:is said to be a
14299:
14297:
14296:
14291:
14279:
14277:
14275:
14274:
14269:
14243:
14241:
14240:
14235:
14223:
14221:
14220:
14215:
14203:
14201:
14200:
14195:
14183:
14181:
14179:
14178:
14173:
14159:
14157:
14155:
14154:
14149:
14135:
14133:
14131:
14130:
14125:
14117:
14116:
14103:
14095:
14074:
14072:
14071:
14066:
14061:
14060:
14045:
14044:
14021:
14019:
14018:
14013:
13997:
13995:
13993:
13992:
13987:
13969:
13967:
13966:
13961:
13946:
13944:
13942:
13941:
13936:
13889:
13887:
13886:
13881:
13834:
13832:
13830:
13829:
13824:
13810:
13808:
13806:
13805:
13800:
13778:
13776:
13774:
13773:
13768:
13754:
13752:
13751:
13746:
13734:
13732:
13731:
13726:
13699:
13697:
13695:
13694:
13689:
13687:
13686:
13674:
13673:
13672:
13659:
13658:
13657:
13638:
13636:
13634:
13633:
13628:
13626:
13609:
13607:
13606:
13601:
13599:
13598:
13597:
13580:
13578:
13577:
13572:
13570:
13569:
13553:
13551:
13549:
13548:
13543:
13529:
13527:
13526:
13521:
13509:
13507:
13506:
13501:
13489:
13487:
13486:
13481:
13465:
13463:
13461:
13460:
13455:
13441:
13439:
13438:
13433:
13414:
13412:
13410:
13409:
13404:
13390:
13388:
13387:
13382:
13370:
13368:
13367:
13362:
13350:
13348:
13347:
13342:
13328:
13327:
13308:
13306:
13304:
13303:
13298:
13284:
13282:
13280:
13279:
13274:
13252:
13250:
13248:
13247:
13242:
13237:
13236:
13224:
13223:
13211:
13210:
13198:
13169:
13167:
13166:
13161:
13143:
13141:
13139:
13138:
13133:
13119:
13117:
13116:
13111:
13099:
13097:
13095:
13094:
13089:
13075:
13073:
13071:
13070:
13065:
13062:
13054:
13036:
13034:
13033:
13028:
13026:
13025:
13013:
13012:
12996:
12994:
12992:
12991:
12986:
12972:
12970:
12969:
12964:
12962:
12961:
12945:
12943:
12942:
12937:
12935:
12934:
12918:
12916:
12914:
12913:
12908:
12894:
12892:
12891:
12886:
12874:
12872:
12870:
12869:
12864:
12850:
12848:
12846:
12845:
12840:
12826:
12824:
12823:
12818:
12784:
12782:
12780:
12779:
12774:
12760:
12758:
12756:
12755:
12750:
12736:
12734:
12733:
12728:
12716:
12715:
12713:
12708:
12703:
12699:
12684:
12682:
12681:
12676:
12668:
12629:
12627:
12625:
12624:
12619:
12605:
12603:
12602:
12597:
12585:
12583:
12581:
12580:
12575:
12561:
12559:
12558:
12553:
12541:
12539:
12538:
12533:
12522:. In this case,
12521:
12519:
12517:
12516:
12511:
12481:
12479:
12478:
12473:
12461:
12459:
12458:
12453:
12431:
12429:
12427:
12426:
12421:
12407:
12405:
12404:
12399:
12387:
12385:
12384:
12379:
12371:
12370:
12352:
12351:
12335:
12333:
12332:
12327:
12315:
12313:
12312:
12307:
12295:
12293:
12292:
12287:
12279:
12278:
12260:
12259:
12243:
12241:
12239:
12238:
12233:
12231:
12230:
12200:
12198:
12197:
12192:
12190:
12189:
12161:
12159:
12158:
12153:
12129:
12127:
12126:
12121:
12093:
12091:
12090:
12085:
12073:
12071:
12070:
12065:
12051:
12050:
12034:
12032:
12031:
12026:
12007:
12005:
12003:
12002:
11997:
11983:
11981:
11980:
11975:
11963:
11961:
11960:
11955:
11953:
11952:
11925:
11924:
11900:, and inverses,
11899:
11897:
11895:
11894:
11889:
11887:
11886:
11871:
11870:
11846:
11844:
11843:
11838:
11823:
11821:
11819:
11818:
11813:
11799:
11797:
11796:
11791:
11779:
11777:
11776:
11771:
11759:
11757:
11756:
11751:
11692:
11690:
11689:
11684:
11660:
11658:
11657:
11652:
11628:
11626:
11625:
11620:
11563:, function, and
11555:
11553:
11551:
11550:
11545:
11531:
11529:
11528:
11523:
11507:
11505:
11504:
11499:
11487:
11485:
11484:
11479:
11455:
11453:
11451:
11450:
11445:
11431:
11429:
11428:
11423:
11411:
11409:
11408:
11403:
11385:
11383:
11381:
11380:
11375:
11361:
11359:
11357:
11356:
11351:
11337:
11335:
11333:
11332:
11327:
11310:
11308:
11307:
11302:
11287:
11285:
11284:
11279:
11231:
11229:
11228:
11223:
11198:
11196:
11195:
11190:
11178:
11176:
11174:
11173:
11168:
11154:
11152:
11150:
11149:
11144:
11106:
11104:
11103:
11098:
11062:
11060:
11059:
11054:
11042:
11040:
11039:
11034:
10983:
10981:
10980:
10975:
10973:
10969:
10966:
10963:
10952:
10948:
10945:
10942:
10925:
10921:
10918:
10915:
10904:
10903:
10876:
10872:
10869:
10866:
10855:
10854:
10800:
10798:
10797:
10792:
10790:
10786:
10783:
10780:
10769:
10765:
10762:
10759:
10757:
10756:
10741:
10740:
10728:
10727:
10706:
10702:
10699:
10696:
10691:
10690:
10666:
10665:
10653:
10652:
10631:
10627:
10624:
10621:
10616:
10615:
10591:
10590:
10569:
10568:
10556:
10555:
10534:
10530:
10527:
10524:
10519:
10518:
10491:
10490:
10475:
10474:
10462:
10461:
10437:
10433:
10430:
10427:
10422:
10421:
10387:
10386:
10354:
10352:
10350:
10349:
10344:
10330:
10329:
10308:
10306:
10305:
10300:
10288:
10286:
10285:
10280:
10278:
10277:
10258:
10256:
10254:
10253:
10248:
10222:
10220:
10219:
10214:
10198:one-sided axioms
10179:
10177:
10175:
10174:
10169:
10151:
10149:
10148:
10143:
10127:
10125:
10124:
10119:
10101:
10099:
10098:
10093:
10081:
10079:
10078:
10073:
10055:
10053:
10051:
10050:
10045:
10031:
10029:
10027:
10026:
10021:
10019:
10018:
9991:
9989:
9988:
9983:
9959:
9957:
9955:
9954:
9949:
9935:
9933:
9932:
9927:
9912:
9910:
9908:
9907:
9902:
9887:left translation
9884:
9882:
9881:
9876:
9856:
9854:
9853:
9848:
9830:
9828:
9827:
9822:
9810:
9808:
9807:
9802:
9784:
9782:
9781:
9776:
9764:
9762:
9761:
9756:
9744:
9742:
9740:
9739:
9734:
9726:
9725:
9704:
9702:
9700:
9699:
9694:
9669:to the equation
9668:
9666:
9665:
9660:
9648:
9646:
9645:
9640:
9628:
9626:
9624:
9623:
9618:
9604:
9602:
9601:
9596:
9584:
9582:
9581:
9576:
9549:
9547:
9546:
9541:
9539:
9535:
9532:
9521:
9518:
9513:
9510:
9507:
9496:
9492:
9489:
9484:
9481:
9476:
9473:
9470:
9453:
9449:
9446:
9443:
9414:
9410:
9407:
9402:
9399:
9394:
9391:
9388:
9359:
9355:
9352:
9347:
9344:
9341:
9306:
9304:
9303:
9298:
9286:
9284:
9283:
9278:
9266:
9264:
9263:
9258:
9234:
9232:
9230:
9229:
9224:
9192:
9190:
9189:
9184:
9172:
9170:
9169:
9164:
9140:
9138:
9137:
9132:
9023:John G. Thompson
9000:Claude Chevalley
8996:algebraic groups
8960:William Burnside
8952:Walther von Dyck
8949:
8882:Erlangen program
8867:
8865:
8864:
8859:
8851:
8850:
8771:is not abelian.
8770:
8768:
8767:
8762:
8760:
8759:
8754:
8741:
8739:
8737:
8736:
8731:
8729:
8728:
8727:
8714:
8713:
8712:
8699:
8698:
8680:
8678:
8677:
8672:
8670:
8669:
8668:
8655:
8654:
8642:
8641:
8640:
8623:
8621:
8619:
8618:
8613:
8611:
8610:
8605:
8587:
8585:
8584:
8579:
8577:
8576:
8560:
8558:
8557:
8552:
8550:
8549:
8533:
8531:
8530:
8525:
8523:
8522:
8506:
8504:
8502:
8501:
8496:
8494:
8493:
8492:
8473:
8471:
8469:
8468:
8463:
8461:
8460:
8459:
8440:
8438:
8436:
8435:
8430:
8428:
8427:
8426:
8407:
8405:
8403:
8402:
8397:
8395:
8394:
8393:
8374:
8372:
8370:
8369:
8364:
8362:
8339:
8337:
8336:
8331:
8319:
8317:
8315:
8314:
8309:
8307:
8289:Identity element
8285:
8283:
8282:
8277:
8275:
8268:
8267:
8255:
8254:
8253:
8240:
8239:
8238:
8218:
8217:
8205:
8204:
8203:
8187:
8186:
8185:
8171:
8170:
8158:
8157:
8145:
8144:
8128:
8127:
8112:
8111:
8110:
8097:
8096:
8095:
8071:
8069:
8068:
8063:
8058:
8057:
8045:
8044:
8043:
8027:
8026:
8025:
8012:
8011:
7996:
7995:
7994:
7981:
7980:
7979:
7959:
7957:
7956:
7951:
7894:
7892:
7890:
7889:
7884:
7870:
7868:
7866:
7865:
7860:
7846:
7844:
7843:
7838:
7826:
7824:
7822:
7821:
7816:
7802:
7800:
7799:
7794:
7782:
7780:
7779:
7774:
7762:
7760:
7758:
7757:
7752:
7750:
7749:
7744:
7729:
7727:
7725:
7724:
7719:
7705:
7703:
7701:
7700:
7695:
7681:
7679:
7677:
7676:
7671:
7651:
7649:
7647:
7646:
7641:
7639:
7638:
7637:
7618:
7616:
7615:
7610:
7605:
7604:
7603:
7590:
7589:
7588:
7575:
7574:
7558:
7556:
7554:
7553:
7548:
7534:
7532:
7531:
7526:
7514:
7512:
7511:
7506:
7487:Binary operation
7476:
7472:
7470:
7468:
7467:
7462:
7460:
7459:
7458:
7439:
7437:
7435:
7434:
7429:
7427:
7426:
7414:
7413:
7412:
7393:
7389:
7381:
7373:
7371:
7369:
7368:
7363:
7361:
7360:
7342:
7340:
7338:
7337:
7332:
7330:
7329:
7311:
7309:
7307:
7306:
7301:
7299:
7298:
7280:
7278:
7276:
7275:
7270:
7268:
7247:
7245:
7244:
7239:
7237:
7221:
7219:
7218:
7213:
7211:
7210:
7193:
7191:
7190:
7185:
7183:
7182:
7165:
7163:
7162:
7157:
7155:
7154:
7137:
7135:
7134:
7129:
7127:
7126:
7125:
7107:
7105:
7104:
7099:
7097:
7096:
7095:
7077:
7075:
7074:
7069:
7067:
7066:
7065:
7047:
7045:
7044:
7039:
7037:
7036:
7035:
7017:
7015:
7014:
7009:
7007:
7006:
7005:
6985:
6983:
6982:
6977:
6975:
6974:
6957:
6955:
6954:
6949:
6947:
6931:
6929:
6928:
6923:
6921:
6920:
6903:
6901:
6900:
6895:
6893:
6892:
6875:
6873:
6872:
6867:
6865:
6864:
6863:
6845:
6843:
6842:
6837:
6835:
6834:
6833:
6815:
6813:
6812:
6807:
6805:
6804:
6803:
6785:
6783:
6782:
6777:
6775:
6774:
6773:
6755:
6753:
6752:
6747:
6745:
6744:
6743:
6723:
6721:
6720:
6715:
6713:
6712:
6695:
6693:
6692:
6687:
6685:
6684:
6667:
6665:
6664:
6659:
6657:
6641:
6639:
6638:
6633:
6631:
6630:
6613:
6611:
6610:
6605:
6603:
6602:
6601:
6583:
6581:
6580:
6575:
6573:
6572:
6571:
6553:
6551:
6550:
6545:
6543:
6542:
6541:
6523:
6521:
6520:
6515:
6513:
6512:
6511:
6493:
6491:
6490:
6485:
6483:
6482:
6481:
6461:
6459:
6458:
6453:
6451:
6450:
6433:
6431:
6430:
6425:
6423:
6422:
6405:
6403:
6402:
6397:
6395:
6394:
6377:
6375:
6374:
6369:
6367:
6351:
6349:
6348:
6343:
6341:
6340:
6339:
6321:
6319:
6318:
6313:
6311:
6310:
6309:
6291:
6289:
6288:
6283:
6281:
6280:
6279:
6261:
6259:
6258:
6253:
6251:
6250:
6249:
6231:
6229:
6228:
6223:
6221:
6220:
6219:
6199:
6197:
6196:
6191:
6189:
6188:
6187:
6169:
6167:
6166:
6161:
6159:
6158:
6157:
6139:
6137:
6136:
6131:
6129:
6128:
6127:
6109:
6107:
6106:
6101:
6099:
6098:
6097:
6079:
6077:
6076:
6071:
6069:
6068:
6051:
6049:
6048:
6043:
6041:
6040:
6023:
6021:
6020:
6015:
6013:
5997:
5995:
5994:
5989:
5987:
5986:
5969:
5967:
5966:
5961:
5959:
5958:
5939:
5937:
5936:
5931:
5929:
5928:
5927:
5909:
5907:
5906:
5901:
5899:
5898:
5897:
5879:
5877:
5876:
5871:
5869:
5868:
5867:
5849:
5847:
5846:
5841:
5839:
5838:
5837:
5819:
5817:
5816:
5811:
5809:
5808:
5791:
5789:
5788:
5783:
5781:
5765:
5763:
5762:
5757:
5755:
5754:
5737:
5735:
5734:
5729:
5727:
5726:
5709:
5707:
5706:
5701:
5699:
5698:
5679:
5677:
5676:
5671:
5669:
5668:
5667:
5649:
5647:
5646:
5641:
5639:
5638:
5637:
5619:
5617:
5616:
5611:
5609:
5608:
5607:
5589:
5587:
5586:
5581:
5579:
5578:
5577:
5559:
5557:
5556:
5551:
5549:
5533:
5531:
5530:
5525:
5523:
5522:
5505:
5503:
5502:
5497:
5495:
5494:
5477:
5475:
5474:
5469:
5467:
5466:
5449:
5447:
5446:
5441:
5439:
5438:
5419:
5417:
5416:
5411:
5409:
5408:
5407:
5389:
5387:
5386:
5381:
5379:
5378:
5377:
5359:
5357:
5356:
5351:
5349:
5348:
5347:
5329:
5327:
5326:
5321:
5319:
5318:
5317:
5299:
5297:
5296:
5291:
5289:
5288:
5271:
5269:
5268:
5263:
5261:
5260:
5243:
5241:
5240:
5235:
5233:
5232:
5215:
5213:
5212:
5207:
5205:
5189:
5187:
5186:
5181:
5179:
5161:
5159:
5158:
5153:
5151:
5150:
5149:
5131:
5129:
5128:
5123:
5121:
5120:
5119:
5101:
5099:
5098:
5093:
5091:
5090:
5089:
5071:
5069:
5068:
5063:
5061:
5060:
5059:
5041:
5039:
5038:
5033:
5031:
5030:
5013:
5011:
5010:
5005:
5003:
5002:
4985:
4983:
4982:
4977:
4975:
4974:
4957:
4955:
4954:
4949:
4947:
4931:
4929:
4928:
4923:
4910:
4908:
4906:
4905:
4900:
4898:
4897:
4892:
4875:
4873:
4871:
4870:
4865:
4860:
4859:
4858:
4845:
4844:
4832:
4831:
4830:
4813:
4811:
4809:
4808:
4803:
4801:
4800:
4799:
4780:
4778:
4776:
4775:
4770:
4768:
4767:
4766:
4747:
4745:
4743:
4742:
4737:
4735:
4734:
4709:
4707:
4705:
4704:
4699:
4685:
4683:
4682:
4677:
4665:
4663:
4662:
4657:
4635:
4633:
4632:
4627:
4615:
4613:
4612:
4607:
4595:
4593:
4591:
4590:
4585:
4583:
4582:
4577:
4558:
4556:
4555:
4550:
4548:
4547:
4546:
4529:
4527:
4526:
4521:
4519:
4518:
4498:
4496:
4494:
4493:
4488:
4486:
4485:
4484:
4465:
4463:
4461:
4460:
4455:
4453:
4452:
4451:
4428:
4426:
4424:
4423:
4418:
4416:
4415:
4414:
4395:
4393:
4391:
4390:
4385:
4383:
4382:
4381:
4359:
4357:
4355:
4354:
4349:
4347:
4346:
4328:
4326:
4325:
4320:
4318:
4317:
4301:
4299:
4297:
4296:
4291:
4289:
4288:
4255:
4253:
4252:
4247:
4245:
4244:
4243:
4226:
4216:
4214:
4213:
4208:
4206:
4205:
4204:
4187:
4177:
4175:
4174:
4169:
4167:
4166:
4165:
4148:
4139:
4137:
4136:
4131:
4129:
4128:
4127:
4110:
4100:
4098:
4097:
4092:
4090:
4089:
4073:
4065:
4063:
4062:
4057:
4055:
4054:
4038:
4030:
4028:
4027:
4022:
4020:
4019:
4003:
3995:
3993:
3992:
3987:
3985:
3970:
3963:
3960:
3958:
3956:
3955:
3950:
3948:
3947:
3942:
3926:
3887:
3885:
3884:
3879:
3851:
3849:
3847:
3846:
3841:
3803:
3801:
3799:
3798:
3793:
3779:
3777:
3776:
3771:
3759:
3757:
3756:
3751:
3739:
3737:
3736:
3731:
3698:
3696:
3694:
3693:
3688:
3668:
3666:
3665:
3660:
3645:
3643:
3641:
3640:
3635:
3633:
3632:
3611:
3609:
3608:
3603:
3591:
3589:
3587:
3586:
3581:
3563:
3561:
3559:
3558:
3553:
3536:
3534:
3533:
3528:
3516:
3514:
3512:
3511:
3506:
3485:
3483:
3482:
3477:
3463:
3451:
3449:
3448:
3443:
3441:
3440:
3435:
3422:
3420:
3419:
3414:
3412:
3396:
3394:
3393:
3388:
3386:
3374:
3372:
3371:
3366:
3364:
3345:
3343:
3342:
3337:
3335:
3319:
3317:
3316:
3311:
3294:
3280:is a group, and
3279:
3277:
3276:
3271:
3260:
3245:
3243:
3242:
3237:
3235:
3223:
3221:
3219:
3218:
3213:
3193:
3191:
3190:
3185:
3167:
3165:
3163:
3162:
3157:
3155:
3094:
3092:
3090:
3089:
3084:
3082:
3081:
3060:
3058:
3057:
3052:
3037:); it is called
3032:
3030:
3029:
3024:
3012:
3010:
3008:
3007:
3002:
2985:
2983:
2982:
2977:
2965:
2963:
2961:
2960:
2955:
2929:
2927:
2926:
2921:
2897:
2895:
2894:
2889:
2877:
2875:
2874:
2869:
2857:
2855:
2853:
2852:
2847:
2833:
2831:
2830:
2825:
2802:identity element
2792:
2790:
2788:
2787:
2782:
2756:
2754:
2752:
2751:
2746:
2720:
2718:
2716:
2715:
2710:
2696:
2694:
2693:
2688:
2676:
2674:
2673:
2668:
2656:
2654:
2653:
2648:
2634:Identity element
2630:
2628:
2626:
2625:
2620:
2564:
2562:
2560:
2559:
2554:
2540:
2538:
2536:
2535:
2530:
2516:
2514:
2512:
2511:
2506:
2492:
2490:
2488:
2487:
2482:
2457:
2455:
2453:
2452:
2447:
2427:
2425:
2423:
2422:
2417:
2403:
2401:
2400:
2395:
2383:
2381:
2380:
2375:
2363:
2361:
2360:
2355:
2340:
2338:
2336:
2335:
2330:
2317:, here denoted "
2316:
2314:
2312:
2311:
2306:
2291:binary operation
2289:together with a
2288:
2286:
2285:
2280:
2262:
2235:
2233:
2231:
2230:
2225:
2207:
2205:
2203:
2202:
2197:
2180:
2178:
2177:
2172:
2154:
2152:
2151:
2146:
2134:
2132:
2130:
2129:
2124:
2098:
2096:
2095:
2090:
2066:
2064:
2063:
2058:
2046:
2044:
2042:
2041:
2036:
2017:identity element
2009:
2007:
2005:
2004:
1999:
1973:
1971:
1970:
1965:
1941:
1939:
1938:
1933:
1912:
1910:
1908:
1907:
1902:
1888:
1886:
1885:
1880:
1868:
1866:
1865:
1860:
1848:
1846:
1845:
1840:
1828:
1826:
1825:
1820:
1808:
1806:
1805:
1800:
1788:
1786:
1784:
1783:
1778:
1722:
1720:
1718:
1717:
1712:
1698:
1696:
1695:
1690:
1678:
1676:
1674:
1673:
1668:
1650:
1648:
1646:
1645:
1640:
1638:
1623:binary operation
1620:
1618:
1617:
1612:
1594:
1592:
1591:
1586:
1565:
1563:
1561:
1560:
1555:
1541:
1539:
1538:
1533:
1517:
1515:
1514:
1509:
1423:
1389:
1382:
1375:
1164:Commutative ring
1093:Rack and quandle
1058:
1057:
1048:
1041:
1034:
990:Algebraic groups
763:Hyperbolic group
753:Arithmetic group
744:
742:
741:
736:
734:
719:
717:
716:
711:
709:
682:
680:
679:
674:
672:
595:Schur multiplier
549:Cauchy's theorem
537:Quaternion group
485:
484:
311:
310:
300:
287:
276:
275:
269:, which studies
216:
203:, starting with
174:particle physics
143:polynomial roots
139:geometric shapes
133:
131:
129:
128:
123:
106:group, which is
81:identity element
21:
29780:
29779:
29775:
29774:
29773:
29771:
29770:
29769:
29745:
29744:
29741:
29736:
29718:
29687:
29671:
29652:Quotient object
29642:Polynomial ring
29600:
29561:Linear subspace
29513:
29507:
29446:
29388:Linear equation
29367:
29313:Category theory
29274:
29256:
29251:
29221:
29216:
29193:
29165:Conformal group
29153:
29127:
29119:
29111:
29103:
29095:
29029:
29021:
29008:
28999:Symmetric group
28978:
28968:
28961:
28954:
28947:
28939:
28930:
28926:
28916:Sporadic groups
28910:
28901:
28883:Discrete groups
28877:
28868:Wallpaper group
28848:Solvable groups
28816:Types of groups
28811:
28777:Normal subgroup
28758:
28753:
28702:
28697:
28691:
28665:
28569:10.2307/2690312
28551:Kleiner, Israel
28536:Jordan, Camille
28466:
28405:
28382:
28376:
28359:
28335:
28313:
28291:
28252:
28230:
28202:10.2307/1990375
28176:
28133:
28107:
28088:
28061:
28018:
28007:
27977:
27958:
27932:
27891:
27855:
27828:
27807:
27786:
27765:
27763:
27749:
27733:Magnus, Wilhelm
27725:
27701:
27670:
27643:
27609:
27539:
27520:
27493:
27474:
27443:
27421:Fulton, William
27380:
27354:
27318:
27313:
27303:
27284:
27253:
27225:math.MG/9911185
27197:
27178:
27144:
27134:Springer-Verlag
27118:
27064:
27012:
27002:Category Theory
26981:
26967:
26944:
26934:
26872:
26862:Springer-Verlag
26839:
26785:
26744:
26723:
26700:
26694:
26689:
26688:
26680:
26676:
26668:
26664:
26656:
26652:
26644:
26640:
26632:
26628:
26620:
26616:
26608:
26604:
26596:
26592:
26584:
26580:
26572:
26568:
26560:
26556:
26548:
26544:
26536:
26532:
26524:
26520:
26514:Aschbacher 2004
26512:
26508:
26500:
26496:
26484:
26480:
26472:
26468:
26460:
26456:
26448:
26444:
26436:
26432:
26424:
26420:
26416:, p. viii.
26412:
26408:
26400:
26396:
26388:
26384:
26376:
26369:
26361:
26357:
26349:
26345:
26337:
26333:
26325:
26321:
26313:
26309:
26301:
26297:
26289:
26285:
26277:
26273:
26265:
26261:
26253:
26249:
26238:
26234:
26226:
26222:
26214:
26207:
26199:
26195:
26187:
26183:
26175:
26171:
26163:
26159:
26151:
26147:
26139:
26135:
26127:
26123:
26115:
26111:
26094:
26090:
26082:
26078:
26070:
26066:
26058:
26054:
26046:
26042:
26034:
26030:
26022:
26018:
26010:
26006:
25998:
25994:
25986:
25982:
25974:
25967:
25959:
25955:
25947:
25938:
25930:
25926:
25918:
25911:
25903:
25899:
25891:
25887:
25879:
25875:
25867:
25863:
25855:
25851:
25843:
25839:
25831:
25827:
25819:
25815:
25807:
25803:
25795:
25791:
25783:
25779:
25771:
25767:
25759:
25755:
25747:
25743:
25735:
25731:
25723:
25719:
25711:
25707:
25699:
25695:
25687:
25683:
25675:
25671:
25663:
25659:
25651:
25647:
25639:
25635:
25627:
25623:
25607:
25603:
25595:
25591:
25586:
25581:
25580:
25571:
25567:
25555:
25551:
25530:
25526:
25520:Aschbacher 2004
25517:
25513:
25500:
25496:
25479:
25475:
25466:
25460:
25450:
25448:
25444:
25427:
25423:
25412:
25408:
25402:simple algebras
25387:
25383:
25370:
25364:
25359:
25355:
25346:
25342:
25329:
25325:
25304:
25300:
25296:for an example.
25291:
25287:
25278:
25274:
25257:
25253:
25232:
25228:
25215:
25205:
25199:
25189:
25186:
25181:
25180:
25178:
25177:
25175:
25171:
25165:MathSciNet 2021
25158:
25154:
25141:
25137:
25131:
25125:
25119:
25113:
25103:
25099:
25093:
25089:
25084:
25066:
25059:
25056:
24890:
24887:
24886:
24884:
24882:
24862:
24857:
24854:
24853:
24851:
24826:
24823:
24822:
24820:
24795:
24792:
24791:
24789:
24630:
24622:
24617:
24615:
24612:
24611:
24609:
24608:
24586:
24584:
24581:
24580:
24578:
24575:Complex numbers
24573:
24556:
24554:
24551:
24550:
24548:
24543:
24526:
24524:
24521:
24520:
24518:
24495:
24493:
24490:
24489:
24487:
24465:
24463:
24460:
24459:
24457:
24455:Natural numbers
24437:
24432:
24428:
24422:
24392:
24389:
24388:
24386:
24368:
24365:
24364:
24362:
24341:
24338:
24337:
24335:
24289:
24286:
24285:
24283:
24265:
24262:
24261:
24259:
24241:
24238:
24237:
24235:
24201:
24198:
24197:
24179:
24176:
24175:
24173:
24148:
24145:
24144:
24142:
24120:
24117:
24116:
24114:
24075:
24070:
24067:
24066:
24064:
24027:
24022:
24019:
24018:
23979:
23974:
23971:
23970:
23968:
23952:
23950:
23947:
23946:
23944:natural numbers
23427:
23425:Generalizations
23385:âas a model of
23379:Minkowski space
23331:
23328:
23327:
23311:
23308:
23307:
23291:
23288:
23287:
23253:
23250:
23249:
23233:
23230:
23229:
23213:
23210:
23209:
23182:
23176:
23142:, so they have
23140:locally compact
23131:
23102:
23099:
23098:
23082:
23079:
23078:
23058:
23054:
23052:
23049:
23048:
23026:
23023:
23022:
23015:
22984:
22934:
22931:
22930:
22928:unary operation
22899:
22896:
22895:
22879:
22876:
22875:
22872:
22837:sporadic groups
22811:
22805:
22769:
22764:
22761:
22760:
22758:
22742:
22739:
22738:
22722:
22719:
22718:
22702:
22699:
22698:
22676:
22673:
22672:
22656:
22653:
22652:
22636:
22633:
22632:
22629:
22607:
22603:
22601:
22598:
22597:
22580:
22575:
22574:
22572:
22569:
22568:
22551:
22547:
22545:
22542:
22541:
22522:
22517:
22516:
22507:
22502:
22501:
22499:
22496:
22495:
22493:
22474:
22470:
22469:
22464:
22463:
22461:
22458:
22457:
22440:
22436:
22434:
22431:
22430:
22409:
22404:
22403:
22401:
22398:
22397:
22381:
22378:
22377:
22363:
22334:
22331:
22330:
22328:
22312:
22309:
22308:
22291:
22286:
22281:
22278:
22275:
22274:
22256:
22255:
22251:
22249:
22246:
22245:
22229:
22226:
22225:
22208:
22204:
22202:
22199:
22198:
22181:
22176:
22175:
22173:
22170:
22169:
22144:
22141:
22140:
22138:
22119:
22116:
22115:
22099:
22096:
22095:
22072:
22069:
22068:
22052:
22049:
22048:
22030:
22027:
22026:
22024:
22006:
22003:
22002:
22000:
21983:
21979:
21977:
21974:
21973:
21955:
21952:
21951:
21949:
21931:
21928:
21927:
21925:
21909:
21906:
21905:
21887:
21884:
21883:
21881:
21860:
21856:
21840:
21838:
21835:
21832:
21831:
21814:
21810:
21808:
21805:
21804:
21779:
21775:
21773:
21770:
21769:
21767:
21751:
21748:
21747:
21731:
21728:
21727:
21711:
21708:
21707:
21683:
21678:
21677:
21675:
21672:
21671:
21666:, according to
21649:
21646:
21645:
21643:
21626:
21621:
21620:
21618:
21615:
21614:
21592:
21587:
21586:
21584:
21581:
21580:
21561:
21558:
21557:
21538:
21533:
21532:
21530:
21527:
21526:
21524:
21504:
21498:
21486:splitting field
21430:
21427:
21426:
21410:
21407:
21406:
21390:
21387:
21386:
21362:
21341:
21337:
21335:
21325:
21323:
21315:
21312:
21311:
21273:
21269:
21264:
21261:
21260:
21251:
21245:
21209:
21192:
21178:
21175:
21174:
21154:
21151:
21150:
21134:
21131:
21130:
21111:
21106:
21105:
21103:
21100:
21099:
21097:
21095:Euclidean space
21060:
21057:
21056:
21026:
21024:
21021:
21020:
21018:
20987:
20984:
20983:
20981:
20963:
20960:
20959:
20957:
20935:
20918:
20916:
20913:
20912:
20876:
20873:
20872:
20870:
20859:
20847:Main articles:
20845:
20826:antiderivatives
20787:
20785:
20781:
20771:
20759:
20683:crystallography
20637:
20634:
20633:
20631:
20613:
20610:
20609:
20607:
20587:Symmetry groups
20568:
20550:
20544:
20542:Symmetry groups
20497:
20492:
20489:
20488:
20486:
20470:
20467:
20466:
20448:
20445:
20444:
20442:
20414:
20410:
20408:
20405:
20404:
20402:
20377:
20373:
20371:
20368:
20367:
20365:
20334:
20330:
20328:
20325:
20324:
20322:
20297:
20293:
20291:
20288:
20287:
20285:
20269:
20266:
20265:
20241:
20238:
20237:
20235:
20219:
20216:
20215:
20198:
20193:
20188:
20185:
20182:
20181:
20157:
20154:
20153:
20129:
20126:
20125:
20123:
20107:
20104:
20103:
20074:
20070:
20068:
20065:
20064:
20062:
20046:
20043:
20042:
20040:complex numbers
20017:
20014:
20013:
20011:
19995:
19992:
19991:
19973:
19970:
19969:
19967:
19951:
19948:
19947:
19922:
19914:
19909:
19904:
19901:
19900:
19810:
19807:
19806:
19786:
19783:
19782:
19760:
19756:
19726:
19722:
19710:
19706:
19694:
19690:
19688:
19685:
19684:
19682:
19662:
19658:
19656:
19653:
19652:
19628:
19625:
19624:
19622:
19605:
19601:
19599:
19596:
19595:
19569:
19565:
19556:
19552:
19537:
19533:
19521:
19517:
19505:
19501:
19489:
19485:
19477:
19474:
19473:
19457:
19454:
19453:
19422:
19418:
19416:
19413:
19412:
19410:
19394:
19391:
19390:
19373:
19369:
19367:
19364:
19363:
19347:
19344:
19343:
19336:
19330:
19302:
19297:
19292:
19289:
19286:
19285:
19283:
19267:
19259:
19254:
19252:
19249:
19248:
19215:
19207:
19206:
19202:
19200:
19197:
19196:
19177:
19173:
19150:
19147:
19146:
19144:
19126:
19123:
19122:
19120:
19104:
19101:
19100:
19082:
19079:
19078:
19076:
19060:
19057:
19056:
19034:
19031:
19030:
19012:
19009:
19008:
19006:
18975:
18972:
18971:
18948:
18945:
18944:
18914:
18911:
18910:
18908:
18907:evenly divides
18892:
18889:
18888:
18858:
18841:
18838:
18837:
18821:
18818:
18817:
18799:
18796:
18795:
18793:
18777:
18774:
18773:
18755:
18752:
18751:
18749:
18731:
18728:
18727:
18725:
18709:
18706:
18705:
18687:
18684:
18683:
18681:
18665:
18662:
18661:
18643:
18640:
18639:
18637:
18621:
18618:
18617:
18598:
18594:
18577:
18574:
18573:
18571:
18535:
18532:
18531:
18529:
18505:
18502:
18501:
18499:
18481:
18478:
18477:
18475:
18474:of division by
18453:
18450:
18449:
18447:
18423:
18420:
18419:
18417:
18401:
18398:
18397:
18379:
18376:
18375:
18373:
18351:
18348:
18347:
18345:
18313:
18296:
18293:
18292:
18274:
18271:
18270:
18268:
18252:
18249:
18248:
18226:
18223:
18222:
18204:
18201:
18200:
18198:
18182:
18179:
18178:
18162:
18159:
18158:
18133:
18130:
18129:
18127:
18105:
18102:
18101:
18085:
18082:
18081:
18054:
18046:
18041:
18036:
18033:
18032:
18030:
18013:
18008:
18007:
18005:
18002:
18001:
17977:
17974:
17973:
17971:
17955:
17952:
17951:
17923:
17920:
17919:
17917:
17901:
17898:
17897:
17868:
17857:
17854:
17853:
17851:
17835:
17832:
17831:
17815:
17812:
17811:
17795:
17792:
17791:
17775:
17772:
17771:
17760:
17757:addition modulo
17749:
17743:
17722:
17720:
17717:
17716:
17688:
17683:
17680:
17679:
17677:
17658:
17653:
17650:
17649:
17630:
17625:
17624:
17622:
17619:
17618:
17616:
17583:
17576:
17572:
17558:
17550:
17548:
17545:
17544:
17510:
17509:
17505:
17503:
17500:
17499:
17469:
17466:
17465:
17463:
17447:
17444:
17443:
17414:
17413:
17409:
17407:
17404:
17403:
17401:
17383:
17381:
17378:
17377:
17375:
17355:
17352:
17351:
17325:
17323:
17320:
17319:
17313:
17292:
17290:
17287:
17286:
17261:
17253:
17250:
17249:
17247:
17219:
17216:
17215:
17193:
17190:
17189:
17158:
17157:
17153:
17151:
17148:
17147:
17118:
17117:
17113:
17111:
17108:
17107:
17105:
17089:
17087:
17084:
17083:
17080:
17052:
16977:
16974:
16973:
16971:
16955:
16952:
16951:
16935:
16932:
16931:
16848:
16846:
16843:
16842:
16835:wallpaper group
16827:
16821:
16786:
16781:
16780:
16762:
16758:
16737:
16733:
16724:
16720:
16703:
16700:
16699:
16697:
16679:
16676:
16675:
16673:
16656:
16651:
16650:
16648:
16645:
16644:
16610:
16606:
16585:
16581:
16572:
16568:
16551:
16548:
16547:
16545:
16517:
16514:
16513:
16491:
16488:
16487:
16468:
16464:
16443:
16439:
16430:
16426:
16424:
16421:
16420:
16418:
16392:
16389:
16388:
16369:
16365:
16363:
16360:
16359:
16357:
16341:
16338:
16337:
16320:
16316:
16314:
16311:
16310:
16294:
16291:
16290:
16273:
16268:
16267:
16265:
16262:
16261:
16233:
16230:
16229:
16211:
16208:
16207:
16205:
16188:
16183:
16182:
16180:
16177:
16176:
16158:
16157:
16153:
16151:
16148:
16147:
16130:
16126:
16124:
16121:
16120:
16103:
16098:
16097:
16095:
16092:
16091:
16081:
16075:
16041:
16036:
16034:
16033:
16018:
16007:
16004:
16003:
16001:
15973:
15970:
15969:
15957:is an example.
15941:
15936:
15935:
15933:
15930:
15929:
15908:
15903:
15902:
15900:
15897:
15896:
15877:
15871:
15866:
15865:
15863:
15860:
15859:
15839:
15835:
15826:
15822:
15813:
15809:
15798:
15787:
15784:
15783:
15751:
15750:
15746:
15736:
15735:
15731:
15715:
15714:
15710:
15697:
15696:
15692:
15678:
15675:
15674:
15672:
15649:
15648:
15644:
15636:
15633:
15632:
15630:
15614:
15611:
15610:
15591:
15585:
15580:
15579:
15577:
15574:
15573:
15552:
15549:
15548:
15531:
15528:
15527:
15510:
15507:
15506:
15487:
15484:
15483:
15466:
15463:
15462:
15445:
15442:
15441:
15422:
15419:
15418:
15401:
15398:
15397:
15380:
15377:
15376:
15354:
15348:
15343:
15342:
15340:
15337:
15336:
15319:", is called a
15302:
15299:
15298:
15296:
15278:
15275:
15274:
15272:
15251:
15246:
15243:
15242:
15240:
15211:
15207:
15203:
15191:
15187:
15176:
15173:
15172:
15170:
15151:
15148:
15147:
15126:
15121:
15118:
15117:
15115:
15090:
15087:
15086:
15056:
15053:
15052:
15050:
15031:
15028:
15027:
15008:
15005:
15004:
14985:
14982:
14981:
14965:
14962:
14961:
14942:
14931:
14928:
14927:
14908:
14903:
14900:
14899:
14853:
14848:
14845:
14844:
14826:
14823:
14822:
14820:
14804:
14801:
14800:
14797:
14791:
14789:Quotient groups
14763:
14762:
14758:
14745:
14744:
14740:
14727:
14726:
14722:
14709:
14708:
14704:
14702:
14699:
14698:
14696:
14677:
14672:
14671:
14669:
14666:
14665:
14663:
14645:
14644:
14640:
14618:
14617:
14613:
14611:
14608:
14607:
14591:
14588:
14587:
14568:
14563:
14562:
14560:
14557:
14556:
14554:
14533:
14532:
14528:
14518:
14517:
14513:
14503:
14502:
14498:
14488:
14487:
14483:
14467:
14466:
14462:
14454:
14451:
14450:
14434:
14431:
14430:
14414:
14411:
14410:
14392:
14389:
14388:
14386:
14367:
14364:
14363:
14347:
14344:
14343:
14324:
14319:
14318:
14316:
14313:
14312:
14310:
14303:normal subgroup
14285:
14282:
14281:
14251:
14248:
14247:
14245:
14229:
14226:
14225:
14209:
14206:
14205:
14189:
14186:
14185:
14167:
14164:
14163:
14161:
14143:
14140:
14139:
14137:
14112:
14108:
14096:
14091:
14085:
14082:
14081:
14079:
14056:
14052:
14040:
14036:
14034:
14031:
14030:
14007:
14004:
14003:
13998:; that is, the
13981:
13978:
13977:
13975:
13955:
13952:
13951:
13948:
13947:, respectively.
13897:
13894:
13893:
13891:
13842:
13839:
13838:
13818:
13815:
13814:
13812:
13794:
13791:
13790:
13788:
13762:
13759:
13758:
13756:
13740:
13737:
13736:
13720:
13717:
13716:
13712:
13706:
13682:
13678:
13668:
13667:
13663:
13653:
13652:
13648:
13646:
13643:
13642:
13640:
13619:
13617:
13614:
13613:
13611:
13593:
13592:
13588:
13586:
13583:
13582:
13565:
13561:
13559:
13556:
13555:
13537:
13534:
13533:
13531:
13515:
13512:
13511:
13495:
13492:
13491:
13475:
13472:
13471:
13466:, the subgroup
13449:
13446:
13445:
13443:
13427:
13424:
13423:
13398:
13395:
13394:
13392:
13376:
13373:
13372:
13356:
13353:
13352:
13320:
13316:
13314:
13311:
13310:
13292:
13289:
13288:
13286:
13268:
13265:
13264:
13262:
13232:
13228:
13219:
13215:
13206:
13202:
13191:
13180:
13177:
13176:
13174:
13149:
13146:
13145:
13127:
13124:
13123:
13121:
13105:
13102:
13101:
13083:
13080:
13079:
13077:
13055:
13050:
13044:
13041:
13040:
13038:
13021:
13017:
13008:
13004:
13002:
12999:
12998:
12980:
12977:
12976:
12974:
12957:
12953:
12951:
12948:
12947:
12930:
12926:
12924:
12921:
12920:
12919:, and whenever
12902:
12899:
12898:
12896:
12880:
12877:
12876:
12858:
12855:
12854:
12852:
12834:
12831:
12830:
12828:
12812:
12809:
12808:
12801:
12795:
12768:
12765:
12764:
12762:
12744:
12741:
12740:
12738:
12709:
12704:
12702:
12701:
12692:
12690:
12687:
12686:
12661:
12653:
12650:
12649:
12613:
12610:
12609:
12607:
12591:
12588:
12587:
12569:
12566:
12565:
12563:
12547:
12544:
12543:
12527:
12524:
12523:
12493:
12490:
12489:
12487:
12467:
12464:
12463:
12447:
12444:
12443:
12415:
12412:
12411:
12409:
12393:
12390:
12389:
12366:
12365:
12347:
12346:
12341:
12338:
12337:
12321:
12318:
12317:
12301:
12298:
12297:
12274:
12273:
12255:
12254:
12249:
12246:
12245:
12226:
12222:
12208:
12205:
12204:
12202:
12185:
12181:
12167:
12164:
12163:
12135:
12132:
12131:
12103:
12100:
12099:
12079:
12076:
12075:
12046:
12042:
12040:
12037:
12036:
12020:
12017:
12016:
11991:
11988:
11987:
11985:
11969:
11966:
11965:
11945:
11941:
11917:
11913:
11905:
11902:
11901:
11882:
11878:
11866:
11862:
11854:
11851:
11850:
11848:
11832:
11829:
11828:
11825:
11807:
11804:
11803:
11801:
11785:
11782:
11781:
11765:
11762:
11761:
11700:
11697:
11696:
11666:
11663:
11662:
11634:
11631:
11630:
11602:
11599:
11598:
11591:
11585:
11577:quotient groups
11557:
11539:
11536:
11535:
11533:
11517:
11514:
11513:
11493:
11490:
11489:
11461:
11458:
11457:
11456:. The notation
11439:
11436:
11435:
11433:
11417:
11414:
11413:
11391:
11388:
11387:
11369:
11366:
11365:
11363:
11345:
11342:
11341:
11339:
11321:
11318:
11317:
11315:
11296:
11293:
11292:
11249:
11246:
11245:
11238:
11205:
11202:
11201:
11184:
11181:
11180:
11162:
11159:
11158:
11156:
11114:
11111:
11110:
11108:
11068:
11065:
11064:
11048:
11045:
11044:
11010:
11007:
11006:
11001:identity and a
10971:
10970:
10967:(left identity)
10965:
10962:
10950:
10949:
10946:(right inverse)
10944:
10941:
10923:
10922:
10919:(associativity)
10917:
10914:
10896:
10892:
10874:
10873:
10868:
10865:
10847:
10843:
10827:
10814:
10812:
10809:
10808:
10788:
10787:
10782:
10779:
10767:
10766:
10763:(left identity)
10761:
10758:
10749:
10745:
10733:
10729:
10720:
10716:
10704:
10703:
10698:
10695:
10683:
10679:
10658:
10654:
10645:
10641:
10629:
10628:
10625:(associativity)
10623:
10620:
10608:
10604:
10583:
10579:
10561:
10557:
10548:
10544:
10532:
10531:
10526:
10523:
10511:
10507:
10483:
10479:
10467:
10463:
10454:
10450:
10435:
10434:
10431:(left identity)
10429:
10426:
10414:
10410:
10388:
10379:
10375:
10365:
10363:
10360:
10359:
10322:
10318:
10316:
10313:
10312:
10310:
10294:
10291:
10290:
10270:
10266:
10264:
10261:
10260:
10230:
10227:
10226:
10224:
10208:
10205:
10204:
10186:
10163:
10160:
10159:
10157:
10137:
10134:
10133:
10107:
10104:
10103:
10087:
10084:
10083:
10082:that maps each
10061:
10058:
10057:
10056:, the function
10039:
10036:
10035:
10033:
10011:
10007:
9999:
9996:
9995:
9993:
9965:
9962:
9961:
9943:
9940:
9939:
9937:
9921:
9918:
9917:
9896:
9893:
9892:
9890:
9870:
9867:
9866:
9861:; it is called
9836:
9833:
9832:
9816:
9813:
9812:
9811:that maps each
9790:
9787:
9786:
9785:, the function
9770:
9767:
9766:
9750:
9747:
9746:
9718:
9714:
9712:
9709:
9708:
9706:
9676:
9673:
9672:
9670:
9654:
9651:
9650:
9634:
9631:
9630:
9612:
9609:
9608:
9606:
9590:
9587:
9586:
9570:
9567:
9566:
9565:Given elements
9563:
9537:
9536:
9531:
9517:
9509:
9506:
9494:
9493:
9488:
9480:
9472:
9469:
9451:
9450:
9447:(associativity)
9445:
9442:
9412:
9411:
9406:
9400: and
9398:
9390:
9387:
9357:
9356:
9351:
9343:
9340:
9324:
9317:
9315:
9312:
9311:
9292:
9289:
9288:
9272:
9269:
9268:
9252:
9249:
9248:
9245:
9200:
9197:
9196:
9194:
9178:
9175:
9174:
9158:
9155:
9154:
9151:
9066:
9063:
9062:
9057:. For example,
9051:
8982:was studied by
8962:(who worked on
8846:
8842:
8840:
8837:
8836:
8795:Ăvariste Galois
8783:
8777:
8755:
8750:
8749:
8747:
8744:
8743:
8723:
8722:
8718:
8708:
8707:
8703:
8694:
8690:
8688:
8685:
8684:
8682:
8664:
8663:
8659:
8650:
8646:
8636:
8635:
8631:
8629:
8626:
8625:
8606:
8601:
8600:
8598:
8595:
8594:
8592:
8572:
8568:
8566:
8563:
8562:
8545:
8541:
8539:
8536:
8535:
8518:
8514:
8512:
8509:
8508:
8488:
8487:
8483:
8481:
8478:
8477:
8475:
8455:
8454:
8450:
8448:
8445:
8444:
8442:
8422:
8421:
8417:
8415:
8412:
8411:
8409:
8389:
8388:
8384:
8382:
8379:
8378:
8376:
8355:
8353:
8350:
8349:
8347:
8344:Inverse element
8325:
8322:
8321:
8300:
8298:
8295:
8294:
8292:
8273:
8272:
8263:
8259:
8249:
8248:
8244:
8234:
8233:
8229:
8222:
8213:
8209:
8199:
8198:
8194:
8181:
8180:
8176:
8173:
8172:
8166:
8162:
8153:
8149:
8140:
8136:
8129:
8123:
8119:
8106:
8105:
8101:
8091:
8090:
8086:
8079:
8077:
8074:
8073:
8053:
8049:
8039:
8038:
8034:
8021:
8020:
8016:
8007:
8003:
7990:
7989:
7985:
7975:
7974:
7970:
7965:
7962:
7961:
7900:
7897:
7896:
7878:
7875:
7874:
7872:
7854:
7851:
7850:
7848:
7832:
7829:
7828:
7810:
7807:
7806:
7804:
7788:
7785:
7784:
7768:
7765:
7764:
7745:
7740:
7739:
7737:
7734:
7733:
7731:
7713:
7710:
7709:
7707:
7689:
7686:
7685:
7683:
7665:
7662:
7661:
7659:
7633:
7632:
7628:
7626:
7623:
7622:
7620:
7599:
7598:
7594:
7584:
7583:
7579:
7570:
7566:
7564:
7561:
7560:
7559:. For example,
7542:
7539:
7538:
7536:
7520:
7517:
7516:
7494:
7491:
7490:
7474:
7454:
7453:
7449:
7447:
7444:
7443:
7441:
7440:, the symmetry
7422:
7418:
7408:
7407:
7403:
7401:
7398:
7397:
7395:
7391:
7387:
7379:
7356:
7352:
7350:
7347:
7346:
7344:
7325:
7321:
7319:
7316:
7315:
7313:
7294:
7290:
7288:
7285:
7284:
7282:
7261:
7259:
7256:
7255:
7253:
7230:
7228:
7225:
7224:
7206:
7202:
7200:
7197:
7196:
7178:
7174:
7172:
7169:
7168:
7150:
7146:
7144:
7141:
7140:
7121:
7120:
7116:
7114:
7111:
7110:
7091:
7090:
7086:
7084:
7081:
7080:
7061:
7060:
7056:
7054:
7051:
7050:
7031:
7030:
7026:
7024:
7021:
7020:
7001:
7000:
6996:
6994:
6991:
6990:
6970:
6966:
6964:
6961:
6960:
6940:
6938:
6935:
6934:
6916:
6912:
6910:
6907:
6906:
6888:
6884:
6882:
6879:
6878:
6859:
6858:
6854:
6852:
6849:
6848:
6829:
6828:
6824:
6822:
6819:
6818:
6799:
6798:
6794:
6792:
6789:
6788:
6769:
6768:
6764:
6762:
6759:
6758:
6739:
6738:
6734:
6732:
6729:
6728:
6708:
6704:
6702:
6699:
6698:
6680:
6676:
6674:
6671:
6670:
6650:
6648:
6645:
6644:
6626:
6622:
6620:
6617:
6616:
6597:
6596:
6592:
6590:
6587:
6586:
6567:
6566:
6562:
6560:
6557:
6556:
6537:
6536:
6532:
6530:
6527:
6526:
6507:
6506:
6502:
6500:
6497:
6496:
6477:
6476:
6472:
6470:
6467:
6466:
6446:
6442:
6440:
6437:
6436:
6418:
6414:
6412:
6409:
6408:
6390:
6386:
6384:
6381:
6380:
6360:
6358:
6355:
6354:
6335:
6334:
6330:
6328:
6325:
6324:
6305:
6304:
6300:
6298:
6295:
6294:
6275:
6274:
6270:
6268:
6265:
6264:
6245:
6244:
6240:
6238:
6235:
6234:
6215:
6214:
6210:
6208:
6205:
6204:
6183:
6182:
6178:
6176:
6173:
6172:
6153:
6152:
6148:
6146:
6143:
6142:
6123:
6122:
6118:
6116:
6113:
6112:
6093:
6092:
6088:
6086:
6083:
6082:
6064:
6060:
6058:
6055:
6054:
6036:
6032:
6030:
6027:
6026:
6006:
6004:
6001:
6000:
5982:
5978:
5976:
5973:
5972:
5954:
5950:
5948:
5945:
5944:
5923:
5922:
5918:
5916:
5913:
5912:
5893:
5892:
5888:
5886:
5883:
5882:
5863:
5862:
5858:
5856:
5853:
5852:
5833:
5832:
5828:
5826:
5823:
5822:
5804:
5800:
5798:
5795:
5794:
5774:
5772:
5769:
5768:
5750:
5746:
5744:
5741:
5740:
5722:
5718:
5716:
5713:
5712:
5694:
5690:
5688:
5685:
5684:
5663:
5662:
5658:
5656:
5653:
5652:
5633:
5632:
5628:
5626:
5623:
5622:
5603:
5602:
5598:
5596:
5593:
5592:
5573:
5572:
5568:
5566:
5563:
5562:
5542:
5540:
5537:
5536:
5518:
5514:
5512:
5509:
5508:
5490:
5486:
5484:
5481:
5480:
5462:
5458:
5456:
5453:
5452:
5434:
5430:
5428:
5425:
5424:
5403:
5402:
5398:
5396:
5393:
5392:
5373:
5372:
5368:
5366:
5363:
5362:
5343:
5342:
5338:
5336:
5333:
5332:
5313:
5312:
5308:
5306:
5303:
5302:
5284:
5280:
5278:
5275:
5274:
5256:
5252:
5250:
5247:
5246:
5228:
5224:
5222:
5219:
5218:
5198:
5196:
5193:
5192:
5172:
5170:
5167:
5166:
5145:
5144:
5140:
5138:
5135:
5134:
5115:
5114:
5110:
5108:
5105:
5104:
5085:
5084:
5080:
5078:
5075:
5074:
5055:
5054:
5050:
5048:
5045:
5044:
5026:
5022:
5020:
5017:
5016:
4998:
4994:
4992:
4989:
4988:
4970:
4966:
4964:
4961:
4960:
4940:
4938:
4935:
4934:
4917:
4914:
4913:
4893:
4888:
4887:
4885:
4882:
4881:
4854:
4853:
4849:
4840:
4836:
4826:
4825:
4821:
4819:
4816:
4815:
4795:
4794:
4790:
4788:
4785:
4784:
4782:
4762:
4761:
4757:
4755:
4752:
4751:
4749:
4730:
4726:
4724:
4721:
4720:
4718:
4693:
4690:
4689:
4687:
4671:
4668:
4667:
4645:
4642:
4641:
4621:
4618:
4617:
4601:
4598:
4597:
4578:
4573:
4572:
4570:
4567:
4566:
4564:
4542:
4541:
4537:
4535:
4532:
4531:
4514:
4510:
4508:
4505:
4504:
4480:
4479:
4475:
4473:
4470:
4469:
4467:
4447:
4446:
4442:
4440:
4437:
4436:
4434:
4410:
4409:
4405:
4403:
4400:
4399:
4397:
4377:
4376:
4372:
4370:
4367:
4366:
4364:
4360:, respectively;
4342:
4338:
4336:
4333:
4332:
4330:
4313:
4309:
4307:
4304:
4303:
4284:
4280:
4278:
4275:
4274:
4272:
4239:
4238:
4234:
4232:
4229:
4228:
4227:
4200:
4199:
4195:
4193:
4190:
4189:
4188:
4161:
4160:
4156:
4154:
4151:
4150:
4149:
4123:
4122:
4118:
4116:
4113:
4112:
4111:
4085:
4081:
4079:
4076:
4075:
4074:
4050:
4046:
4044:
4041:
4040:
4039:
4015:
4011:
4009:
4006:
4005:
4004:
3978:
3976:
3973:
3972:
3971:
3943:
3938:
3937:
3935:
3932:
3931:
3929:
3894:
3873:
3870:
3869:
3829:
3826:
3825:
3823:
3787:
3784:
3783:
3781:
3765:
3762:
3761:
3745:
3742:
3741:
3707:
3704:
3703:
3676:
3673:
3672:
3670:
3651:
3648:
3647:
3625:
3621:
3619:
3616:
3615:
3613:
3597:
3594:
3593:
3575:
3572:
3571:
3569:
3544:
3541:
3540:
3538:
3522:
3519:
3518:
3500:
3497:
3496:
3494:
3459:
3457:
3454:
3453:
3436:
3431:
3430:
3428:
3425:
3424:
3408:
3406:
3403:
3402:
3382:
3380:
3377:
3376:
3360:
3358:
3355:
3354:
3331:
3329:
3326:
3325:
3290:
3285:
3282:
3281:
3256:
3251:
3248:
3247:
3231:
3229:
3226:
3225:
3204:
3201:
3200:
3198:
3173:
3170:
3169:
3151:
3149:
3146:
3145:
3143:
3118:group operation
3102:
3074:
3070:
3068:
3065:
3064:
3062:
3046:
3043:
3042:
3018:
3015:
3014:
2996:
2993:
2992:
2990:
2971:
2968:
2967:
2937:
2934:
2933:
2931:
2903:
2900:
2899:
2883:
2880:
2879:
2863:
2860:
2859:
2841:
2838:
2837:
2835:
2819:
2816:
2815:
2811:Inverse element
2808:) of the group.
2806:neutral element
2764:
2761:
2760:
2758:
2728:
2725:
2724:
2722:
2704:
2701:
2700:
2698:
2682:
2679:
2678:
2662:
2659:
2658:
2642:
2639:
2638:
2572:
2569:
2568:
2566:
2548:
2545:
2544:
2542:
2524:
2521:
2520:
2518:
2500:
2497:
2496:
2494:
2476:
2473:
2472:
2470:
2435:
2432:
2431:
2429:
2411:
2408:
2407:
2405:
2389:
2386:
2385:
2369:
2366:
2365:
2349:
2346:
2345:
2324:
2321:
2320:
2318:
2300:
2297:
2296:
2294:
2274:
2271:
2270:
2264:
2254:
2242:
2219:
2216:
2215:
2213:
2188:
2185:
2184:
2182:
2166:
2163:
2162:
2161:of the integer
2158:inverse element
2140:
2137:
2136:
2106:
2103:
2102:
2100:
2072:
2069:
2068:
2052:
2049:
2048:
2030:
2027:
2026:
2024:
1981:
1978:
1977:
1975:
1947:
1944:
1943:
1927:
1924:
1923:
1896:
1893:
1892:
1890:
1874:
1871:
1870:
1854:
1851:
1850:
1834:
1831:
1830:
1814:
1811:
1810:
1794:
1791:
1790:
1730:
1727:
1726:
1724:
1706:
1703:
1702:
1700:
1684:
1681:
1680:
1662:
1659:
1658:
1656:
1634:
1632:
1629:
1628:
1626:
1606:
1603:
1602:
1574:
1571:
1570:
1549:
1546:
1545:
1543:
1527:
1524:
1523:
1419:
1417:
1414:
1413:
1407:
1402:
1393:
1364:
1363:
1362:
1333:Non-associative
1315:
1304:
1303:
1293:
1273:
1262:
1261:
1250:Map of lattices
1246:
1242:Boolean algebra
1237:Heyting algebra
1211:
1200:
1199:
1193:
1174:Integral domain
1138:
1127:
1126:
1120:
1074:
1052:
1023:
1022:
1011:Abelian variety
1004:Reductive group
992:
982:
981:
980:
979:
930:
922:
914:
906:
898:
871:Special unitary
782:
768:
767:
749:
748:
730:
728:
725:
724:
705:
703:
700:
699:
668:
666:
663:
662:
654:
653:
644:Discrete groups
633:
632:
588:Frobenius group
533:
520:
509:
502:Symmetric group
498:
482:
472:
471:
322:Normal subgroup
308:
288:
279:
239:quotient groups
205:Ăvariste Galois
117:
114:
113:
111:
85:inverse element
35:
28:
23:
22:
15:
12:
11:
5:
29778:
29768:
29767:
29762:
29757:
29738:
29737:
29735:
29734:
29723:
29720:
29719:
29717:
29716:
29711:
29706:
29704:Linear algebra
29701:
29695:
29693:
29689:
29688:
29686:
29685:
29679:
29677:
29673:
29672:
29670:
29669:
29667:Tensor algebra
29664:
29659:
29656:Quotient group
29649:
29639:
29629:
29619:
29614:
29608:
29606:
29602:
29601:
29599:
29598:
29593:
29588:
29578:
29575:Euclidean norm
29568:
29558:
29548:
29543:
29533:
29528:
29523:
29517:
29515:
29509:
29508:
29506:
29505:
29495:
29485:
29475:
29465:
29454:
29452:
29448:
29447:
29445:
29444:
29439:
29429:
29426:Multiplication
29415:
29405:
29395:
29381:
29375:
29373:
29372:Basic concepts
29369:
29368:
29366:
29365:
29360:
29355:
29350:
29345:
29340:
29338:Linear algebra
29335:
29330:
29325:
29320:
29315:
29310:
29305:
29304:
29303:
29298:
29288:
29282:
29280:
29276:
29275:
29273:
29272:
29267:
29261:
29258:
29257:
29250:
29249:
29242:
29235:
29227:
29218:
29217:
29215:
29214:
29209:
29204:
29198:
29195:
29194:
29192:
29191:
29188:
29185:
29182:
29177:
29172:
29167:
29161:
29159:
29155:
29154:
29152:
29151:
29146:
29144:Poincaré group
29141:
29136:
29130:
29129:
29125:
29121:
29117:
29113:
29109:
29105:
29101:
29097:
29093:
29089:
29083:
29082:
29076:
29070:
29064:
29058:
29052:
29046:
29039:
29037:
29031:
29030:
29028:
29027:
29022:
29017:
29012:Dihedral group
29009:
29004:
28996:
28992:
28991:
28985:
28979:
28976:
28970:
28966:
28959:
28952:
28945:
28940:
28937:
28931:
28928:
28924:
28918:
28912:
28911:
28908:
28902:
28899:
28893:
28887:
28885:
28879:
28878:
28876:
28875:
28870:
28865:
28860:
28855:
28853:Symmetry group
28850:
28845:
28840:
28838:Infinite group
28835:
28830:
28828:Abelian groups
28825:
28819:
28817:
28813:
28812:
28810:
28809:
28804:
28802:direct product
28794:
28789:
28787:Quotient group
28784:
28779:
28774:
28768:
28766:
28760:
28759:
28752:
28751:
28744:
28737:
28729:
28723:
28722:
28701:
28700:External links
28698:
28696:
28695:
28689:
28669:
28663:
28647:
28632:
28609:
28589:
28563:(4): 195â215,
28547:
28532:
28513:
28470:
28464:
28448:
28426:
28412:Cayley, Arthur
28408:
28403:
28375:
28372:
28371:
28370:
28357:
28339:
28333:
28317:
28311:
28299:Welsh, Dominic
28295:
28289:
28271:
28250:
28234:
28228:
28215:
28195:(2): 345â371,
28183:Suzuki, Michio
28179:
28174:
28158:
28136:
28131:
28118:
28105:
28092:
28086:
28073:
28059:
28039:
28029:(6): 671â679,
28011:
28005:
27989:
27975:
27962:
27956:
27936:
27930:
27910:
27889:
27867:
27853:
27840:
27826:
27814:Mumford, David
27810:
27805:
27790:
27784:
27771:
27752:
27747:
27729:
27723:
27705:
27699:
27691:Addison-Wesley
27682:
27668:
27655:
27641:
27621:
27607:
27582:
27542:
27537:
27524:
27518:
27500:Hatcher, Allen
27496:
27491:
27478:
27472:
27454:
27441:
27417:
27392:
27378:
27357:
27352:
27339:
27307:
27301:
27288:
27282:
27265:
27251:
27238:
27218:(2): 475â507,
27200:
27195:
27182:
27176:
27156:
27142:
27122:
27116:
27103:
27068:
27062:
27047:
27015:
27010:
26997:
26971:
26965:
26943:
26940:
26939:
26938:
26932:
26916:
26896:
26876:
26870:
26850:
26837:
26817:
26797:
26783:
26767:
26747:
26742:
26727:
26721:
26705:Artin, Michael
26699:
26696:
26695:
26693:
26690:
26687:
26686:
26674:
26662:
26650:
26638:
26626:
26614:
26610:Goldstein 1980
26602:
26590:
26578:
26566:
26554:
26542:
26530:
26518:
26516:, p. 737.
26506:
26494:
26478:
26466:
26454:
26442:
26430:
26418:
26406:
26394:
26382:
26367:
26355:
26343:
26331:
26319:
26307:
26295:
26283:
26281:, p. 228.
26271:
26259:
26247:
26232:
26220:
26205:
26193:
26181:
26169:
26157:
26145:
26133:
26131:, Chapter VII.
26121:
26109:
26088:
26076:
26064:
26052:
26040:
26028:
26016:
26004:
26000:Ledermann 1973
25992:
25980:
25965:
25953:
25936:
25924:
25909:
25905:Ledermann 1973
25897:
25893:Ledermann 1953
25885:
25873:
25861:
25849:
25837:
25825:
25813:
25801:
25799:, p. 204.
25789:
25777:
25765:
25753:
25751:, p. 202.
25741:
25729:
25717:
25705:
25693:
25681:
25669:
25657:
25645:
25633:
25621:
25601:
25588:
25587:
25585:
25582:
25579:
25578:
25565:
25549:
25537:simple modules
25524:
25511:
25494:
25473:
25442:
25430:DiffieâHellman
25421:
25406:
25381:
25353:
25340:
25323:
25305:An example is
25298:
25285:
25272:
25270:, p. 108.
25251:
25226:
25169:
25152:
25086:
25085:
25083:
25080:
25079:
25078:
25072:
25071:
25055:
25052:
25049:
25048:
25043:
25038:
25033:
25028:
25023:
25018:
25013:
25008:
25003:
24998:
24994:
24993:
24990:
24987:
24984:
24981:
24978:
24975:
24972:
24969:
24966:
24963:
24959:
24958:
24955:
24952:
24949:
24946:
24943:
24940:
24937:
24934:
24931:
24928:
24924:
24923:
24920:
24917:
24914:
24900:
24897:
24894:
24869:
24865:
24861:
24849:
24846:
24833:
24830:
24818:
24815:
24802:
24799:
24787:
24784:
24781:
24777:
24776:
24773:
24770:
24767:
24764:
24761:
24758:
24755:
24752:
24749:
24746:
24742:
24741:
24738:
24735:
24732:
24729:
24726:
24723:
24720:
24717:
24714:
24711:
24707:
24706:
24703:
24700:
24697:
24694:
24691:
24688:
24685:
24682:
24679:
24676:
24672:
24671:
24658:
24655:
24652:
24649:
24646:
24643:
24640:
24637:
24633:
24629:
24625:
24620:
24603:
24589:
24559:
24529:
24512:
24498:
24482:
24468:
24452:
24396:
24372:
24348:
24345:
24317:
24314:
24311:
24308:
24305:
24302:
24299:
24296:
24293:
24269:
24245:
24223:
24220:
24217:
24214:
24211:
24208:
24205:
24183:
24170:small category
24152:
24124:
24100:
24097:
24094:
24091:
24088:
24085:
24082:
24078:
24074:
24052:
24049:
24046:
24043:
24040:
24037:
24034:
24030:
24026:
24004:
24001:
23998:
23995:
23992:
23989:
23986:
23982:
23978:
23955:
23931:
23930:
23927:
23924:
23921:
23918:
23915:
23909:
23908:
23905:
23902:
23899:
23896:
23893:
23887:
23886:
23883:
23880:
23877:
23874:
23871:
23865:
23864:
23861:
23858:
23855:
23852:
23849:
23843:
23842:
23839:
23836:
23833:
23830:
23827:
23820:
23819:
23816:
23813:
23810:
23807:
23804:
23797:
23796:
23793:
23790:
23787:
23784:
23781:
23774:
23773:
23770:
23767:
23764:
23761:
23758:
23752:
23751:
23748:
23745:
23742:
23739:
23736:
23729:
23728:
23725:
23722:
23719:
23716:
23713:
23707:
23706:
23703:
23700:
23697:
23694:
23691:
23684:
23683:
23680:
23677:
23674:
23671:
23668:
23662:
23661:
23658:
23655:
23652:
23649:
23646:
23639:
23638:
23635:
23632:
23629:
23626:
23623:
23617:
23616:
23613:
23610:
23607:
23604:
23601:
23594:
23593:
23590:
23587:
23584:
23581:
23578:
23572:
23571:
23568:
23565:
23562:
23559:
23556:
23549:
23548:
23545:
23542:
23539:
23536:
23533:
23527:
23526:
23523:
23520:
23517:
23514:
23511:
23509:Small category
23505:
23504:
23501:
23498:
23495:
23492:
23489:
23483:
23482:
23479:
23476:
23473:
23470:
23467:
23461:
23460:
23455:
23450:
23445:
23440:
23435:
23426:
23423:
23411:Standard Model
23395:Poincaré group
23335:
23315:
23295:
23275:
23272:
23269:
23266:
23263:
23260:
23257:
23237:
23217:
23178:Main article:
23175:
23172:
23160:Krull topology
23106:
23086:
23064:
23061:
23057:
23036:
23033:
23030:
23011:Main article:
22983:
22980:
22944:
22941:
22938:
22915:
22912:
22909:
22906:
22903:
22883:
22871:
22868:
22843:is called the
22841:sporadic group
22807:Main article:
22804:
22801:
22776:
22772:
22768:
22746:
22726:
22706:
22686:
22683:
22680:
22660:
22640:
22628:
22625:
22610:
22606:
22583:
22578:
22554:
22550:
22525:
22520:
22515:
22510:
22505:
22477:
22473:
22467:
22443:
22439:
22412:
22407:
22385:
22362:
22359:
22344:
22341:
22338:
22316:
22294:
22289:
22284:
22259:
22254:
22233:
22211:
22207:
22184:
22179:
22163:Sylow theorems
22148:
22123:
22103:
22076:
22056:
22034:
22010:
21986:
21982:
21959:
21935:
21913:
21891:
21869:
21859:
21853:
21849:
21846:
21843:
21817:
21813:
21790:
21787:
21782:
21778:
21755:
21735:
21715:
21686:
21681:
21653:
21629:
21624:
21595:
21590:
21565:
21541:
21536:
21500:Main article:
21497:
21494:
21484:formed as the
21434:
21414:
21394:
21374:
21368:
21365:
21358:
21355:
21352:
21349:
21344:
21340:
21334:
21331:
21328:
21322:
21319:
21299:
21296:
21293:
21290:
21287:
21284:
21281:
21276:
21272:
21268:
21247:Main article:
21244:
21241:
21237:compact groups
21216:
21212:
21208:
21205:
21202:
21198:
21195:
21191:
21188:
21185:
21182:
21158:
21138:
21114:
21109:
21064:
21042:
21039:
21036:
21032:
21029:
20991:
20967:
20942:
20938:
20934:
20931:
20928:
20924:
20921:
20904:together with
20880:
20844:
20841:
20806:Mathieu groups
20800:
20799:
20792:
20783:
20779:
20773:
20769:
20763:
20752:
20751:
20744:
20737:
20730:
20669:In chemistry,
20641:
20617:
20548:Symmetry group
20546:Main article:
20543:
20540:
20510:
20507:
20504:
20500:
20496:
20474:
20452:
20425:
20422:
20417:
20413:
20388:
20385:
20380:
20376:
20351:
20348:
20345:
20342:
20337:
20333:
20308:
20305:
20300:
20296:
20273:
20251:
20248:
20245:
20223:
20201:
20196:
20191:
20161:
20139:
20136:
20133:
20111:
20085:
20082:
20077:
20073:
20050:
20021:
19999:
19977:
19955:
19935:
19932:
19929:
19925:
19921:
19917:
19912:
19908:
19899:In the groups
19886:
19883:
19880:
19877:
19874:
19871:
19868:
19865:
19862:
19859:
19856:
19853:
19850:
19847:
19844:
19841:
19838:
19835:
19832:
19829:
19826:
19823:
19820:
19817:
19814:
19790:
19766:
19763:
19759:
19755:
19752:
19749:
19746:
19743:
19740:
19737:
19732:
19729:
19725:
19721:
19716:
19713:
19709:
19705:
19700:
19697:
19693:
19668:
19665:
19661:
19638:
19635:
19632:
19608:
19604:
19583:
19580:
19577:
19572:
19568:
19564:
19559:
19555:
19551:
19548:
19545:
19540:
19536:
19532:
19527:
19524:
19520:
19516:
19511:
19508:
19504:
19500:
19495:
19492:
19488:
19484:
19481:
19461:
19425:
19421:
19398:
19376:
19372:
19351:
19332:Main article:
19329:
19326:
19305:
19300:
19295:
19270:
19266:
19262:
19257:
19235:
19231:
19228:
19224:
19221:
19218:
19214:
19210:
19205:
19180:
19176:
19172:
19169:
19166:
19163:
19160:
19157:
19154:
19130:
19108:
19086:
19064:
19044:
19041:
19038:
19029:. In the case
19016:
18994:
18991:
18988:
18985:
18982:
18979:
18952:
18943:. The inverse
18930:
18927:
18924:
18921:
18918:
18896:
18876:
18872:
18869:
18865:
18862:
18857:
18854:
18851:
18848:
18845:
18825:
18803:
18781:
18759:
18735:
18713:
18691:
18669:
18647:
18625:
18601:
18597:
18593:
18590:
18587:
18584:
18581:
18557:
18554:
18551:
18548:
18545:
18542:
18539:
18515:
18512:
18509:
18485:
18457:
18433:
18430:
18427:
18405:
18383:
18355:
18331:
18327:
18324:
18320:
18317:
18312:
18309:
18306:
18303:
18300:
18278:
18256:
18236:
18233:
18230:
18208:
18186:
18166:
18137:
18115:
18112:
18109:
18089:
18067:
18064:
18061:
18057:
18053:
18049:
18044:
18040:
18016:
18011:
17987:
17984:
17981:
17959:
17948:representative
17933:
17930:
17927:
17905:
17882:
17879:
17875:
17872:
17867:
17864:
17861:
17839:
17819:
17799:
17779:
17745:Main article:
17742:
17739:
17725:
17695:
17691:
17687:
17665:
17661:
17657:
17633:
17628:
17603:
17599:
17596:
17593:
17590:
17586:
17582:
17579:
17575:
17571:
17567:
17564:
17561:
17557:
17553:
17524:
17520:
17517:
17513:
17508:
17485:
17482:
17479:
17476:
17473:
17451:
17428:
17424:
17421:
17417:
17412:
17386:
17359:
17337:
17332:
17329:
17312:
17309:
17295:
17269:
17266:
17260:
17257:
17235:
17232:
17229:
17226:
17223:
17203:
17200:
17197:
17172:
17168:
17165:
17161:
17156:
17132:
17128:
17125:
17121:
17116:
17092:
17079:
17076:
17051:
17048:
17002:winding number
16987:
16984:
16981:
16959:
16939:
16880:Henri Poincaré
16851:
16823:Main article:
16820:
16817:
16813:discrete group
16789:
16784:
16779:
16776:
16773:
16770:
16765:
16761:
16757:
16754:
16751:
16748:
16745:
16740:
16736:
16732:
16727:
16723:
16719:
16716:
16713:
16710:
16707:
16683:
16659:
16654:
16624:
16621:
16618:
16613:
16609:
16605:
16602:
16599:
16596:
16593:
16588:
16584:
16580:
16575:
16571:
16567:
16564:
16561:
16558:
16555:
16533:
16530:
16527:
16524:
16521:
16501:
16498:
16495:
16471:
16467:
16463:
16460:
16457:
16454:
16451:
16446:
16442:
16438:
16433:
16429:
16402:
16399:
16396:
16387:. Elements in
16372:
16368:
16345:
16323:
16319:
16298:
16276:
16271:
16249:
16246:
16243:
16240:
16237:
16215:
16191:
16186:
16161:
16156:
16133:
16129:
16106:
16101:
16077:Main article:
16074:
16071:
16052:
16044:
16039:
16031:
16028:
16025:
16021:
16017:
16014:
16011:
15989:
15986:
15983:
15980:
15977:
15944:
15939:
15928:construction;
15911:
15906:
15884:
15880:
15874:
15869:
15847:
15842:
15838:
15834:
15829:
15825:
15821:
15816:
15812:
15808:
15804:
15801:
15797:
15794:
15791:
15769:
15766:
15763:
15760:
15754:
15749:
15745:
15739:
15734:
15730:
15727:
15724:
15718:
15713:
15709:
15706:
15700:
15695:
15691:
15688:
15685:
15682:
15658:
15652:
15647:
15643:
15640:
15618:
15598:
15594:
15588:
15583:
15568:
15567:
15556:
15546:
15535:
15525:
15514:
15503:
15502:
15491:
15481:
15470:
15460:
15449:
15438:
15437:
15426:
15416:
15405:
15395:
15384:
15361:
15357:
15351:
15346:
15321:quotient group
15306:
15282:
15258:
15254:
15250:
15226:
15222:
15217:
15214:
15210:
15206:
15202:
15197:
15194:
15190:
15186:
15183:
15180:
15158:
15155:
15133:
15129:
15125:
15103:
15100:
15097:
15094:
15072:
15069:
15066:
15063:
15060:
15038:
15035:
15015:
15012:
14992:
14989:
14969:
14949:
14945:
14941:
14938:
14935:
14915:
14911:
14907:
14887:
14884:
14881:
14878:
14875:
14872:
14869:
14866:
14863:
14860:
14856:
14852:
14830:
14808:
14795:Quotient group
14793:Main article:
14790:
14787:
14772:
14766:
14761:
14757:
14754:
14748:
14743:
14739:
14736:
14730:
14725:
14721:
14718:
14712:
14707:
14680:
14675:
14648:
14643:
14639:
14636:
14633:
14630:
14627:
14621:
14616:
14595:
14571:
14566:
14542:
14536:
14531:
14527:
14521:
14516:
14512:
14506:
14501:
14497:
14491:
14486:
14482:
14479:
14476:
14470:
14465:
14461:
14458:
14438:
14418:
14396:
14374:
14371:
14351:
14327:
14322:
14289:
14267:
14264:
14261:
14258:
14255:
14233:
14213:
14193:
14171:
14147:
14123:
14120:
14115:
14111:
14107:
14102:
14099:
14094:
14090:
14077:precisely when
14064:
14059:
14055:
14051:
14048:
14043:
14039:
14011:
13985:
13959:
13934:
13931:
13928:
13925:
13922:
13919:
13916:
13913:
13910:
13907:
13904:
13901:
13879:
13876:
13873:
13870:
13867:
13864:
13861:
13858:
13855:
13852:
13849:
13846:
13837:
13822:
13798:
13766:
13744:
13724:
13708:Main article:
13705:
13702:
13685:
13681:
13677:
13671:
13666:
13662:
13656:
13651:
13625:
13622:
13596:
13591:
13568:
13564:
13541:
13519:
13499:
13479:
13453:
13431:
13402:
13380:
13360:
13340:
13337:
13334:
13331:
13326:
13323:
13319:
13296:
13272:
13240:
13235:
13231:
13227:
13222:
13218:
13214:
13209:
13205:
13201:
13197:
13194:
13190:
13187:
13184:
13159:
13156:
13153:
13131:
13120:restricted to
13109:
13087:
13061:
13058:
13053:
13049:
13024:
13020:
13016:
13011:
13007:
12997:, then so are
12984:
12960:
12956:
12933:
12929:
12906:
12884:
12862:
12838:
12816:
12803:Informally, a
12797:Main article:
12794:
12791:
12772:
12748:
12726:
12723:
12720:
12712:
12707:
12698:
12695:
12674:
12671:
12667:
12664:
12660:
12657:
12617:
12595:
12573:
12551:
12531:
12509:
12506:
12503:
12500:
12497:
12471:
12451:
12419:
12397:
12377:
12374:
12369:
12364:
12361:
12358:
12355:
12350:
12345:
12336:and such that
12325:
12305:
12285:
12282:
12277:
12272:
12269:
12266:
12263:
12258:
12253:
12229:
12225:
12221:
12218:
12215:
12212:
12188:
12184:
12180:
12177:
12174:
12171:
12151:
12148:
12145:
12142:
12139:
12119:
12116:
12113:
12110:
12107:
12094:to itself. An
12083:
12063:
12060:
12057:
12054:
12049:
12045:
12024:
11995:
11973:
11951:
11948:
11944:
11940:
11937:
11934:
11931:
11928:
11923:
11920:
11916:
11912:
11909:
11885:
11881:
11877:
11874:
11869:
11865:
11861:
11858:
11836:
11811:
11789:
11769:
11749:
11746:
11743:
11740:
11737:
11734:
11731:
11728:
11725:
11722:
11719:
11716:
11713:
11710:
11707:
11704:
11695:
11682:
11679:
11676:
11673:
11670:
11661:is a function
11650:
11647:
11644:
11641:
11638:
11618:
11615:
11612:
11609:
11606:
11587:Main article:
11584:
11581:
11543:
11532:an element of
11521:
11497:
11477:
11474:
11471:
11468:
11465:
11443:
11421:
11412:to state that
11401:
11398:
11395:
11373:
11349:
11325:
11300:
11277:
11274:
11271:
11268:
11265:
11262:
11259:
11256:
11253:
11237:
11236:Basic concepts
11234:
11221:
11218:
11215:
11212:
11209:
11188:
11166:
11142:
11139:
11136:
11133:
11130:
11127:
11124:
11121:
11118:
11096:
11093:
11090:
11087:
11084:
11081:
11078:
11075:
11072:
11052:
11032:
11029:
11026:
11023:
11020:
11017:
11014:
10985:
10984:
10964:
10961:
10958:
10955:
10953:
10951:
10943:
10940:
10937:
10934:
10931:
10928:
10926:
10924:
10916:
10913:
10910:
10907:
10902:
10899:
10895:
10891:
10888:
10885:
10882:
10879:
10877:
10875:
10870:(left inverse)
10867:
10864:
10861:
10858:
10853:
10850:
10846:
10842:
10839:
10836:
10833:
10830:
10828:
10826:
10823:
10820:
10817:
10816:
10802:
10801:
10784:(left inverse)
10781:
10778:
10775:
10772:
10770:
10768:
10760:
10755:
10752:
10748:
10744:
10739:
10736:
10732:
10726:
10723:
10719:
10715:
10712:
10709:
10707:
10705:
10700:(left inverse)
10697:
10694:
10689:
10686:
10682:
10678:
10675:
10672:
10669:
10664:
10661:
10657:
10651:
10648:
10644:
10640:
10637:
10634:
10632:
10630:
10622:
10619:
10614:
10611:
10607:
10603:
10600:
10597:
10594:
10589:
10586:
10582:
10578:
10575:
10572:
10567:
10564:
10560:
10554:
10551:
10547:
10543:
10540:
10537:
10535:
10533:
10528:(left inverse)
10525:
10522:
10517:
10514:
10510:
10506:
10503:
10500:
10497:
10494:
10489:
10486:
10482:
10478:
10473:
10470:
10466:
10460:
10457:
10453:
10449:
10446:
10443:
10440:
10438:
10436:
10428:
10425:
10420:
10417:
10413:
10409:
10406:
10403:
10400:
10397:
10394:
10391:
10389:
10385:
10382:
10378:
10374:
10371:
10368:
10367:
10342:
10339:
10336:
10333:
10328:
10325:
10321:
10298:
10276:
10273:
10269:
10246:
10243:
10240:
10237:
10234:
10212:
10185:
10182:
10167:
10141:
10117:
10114:
10111:
10091:
10071:
10068:
10065:
10043:
10017:
10014:
10010:
10006:
10003:
9981:
9978:
9975:
9972:
9969:
9947:
9925:
9900:
9874:
9846:
9843:
9840:
9820:
9800:
9797:
9794:
9774:
9754:
9732:
9729:
9724:
9721:
9717:
9692:
9689:
9686:
9683:
9680:
9658:
9638:
9616:
9594:
9574:
9562:
9559:
9551:
9550:
9530:
9527:
9524:
9516:
9508:
9505:
9502:
9499:
9497:
9495:
9487:
9479:
9471:
9468:
9465:
9462:
9459:
9456:
9454:
9452:
9444:
9441:
9438:
9435:
9432:
9429:
9426:
9423:
9420:
9417:
9415:
9413:
9405:
9397:
9389:
9386:
9383:
9380:
9377:
9374:
9371:
9368:
9365:
9362:
9360:
9358:
9350:
9342:
9339:
9336:
9333:
9330:
9327:
9325:
9323:
9320:
9319:
9296:
9276:
9256:
9244:
9241:
9222:
9219:
9216:
9213:
9210:
9207:
9204:
9182:
9162:
9150:
9147:
9130:
9127:
9124:
9121:
9118:
9115:
9112:
9109:
9106:
9103:
9100:
9097:
9094:
9091:
9088:
9085:
9082:
9079:
9076:
9073:
9070:
9050:
9047:
9043:examples below
8968:Richard Brauer
8942:Camille Jordan
8929:by developing
8857:
8854:
8849:
8845:
8811:symmetry group
8787:abstract group
8779:Main article:
8776:
8773:
8758:
8753:
8726:
8721:
8717:
8711:
8706:
8702:
8697:
8693:
8667:
8662:
8658:
8653:
8649:
8645:
8639:
8634:
8609:
8604:
8575:
8571:
8548:
8544:
8521:
8517:
8491:
8486:
8458:
8453:
8425:
8420:
8392:
8387:
8361:
8358:
8329:
8306:
8303:
8271:
8266:
8262:
8258:
8252:
8247:
8243:
8237:
8232:
8228:
8225:
8223:
8221:
8216:
8212:
8208:
8202:
8197:
8193:
8190:
8184:
8179:
8175:
8174:
8169:
8165:
8161:
8156:
8152:
8148:
8143:
8139:
8135:
8132:
8130:
8126:
8122:
8118:
8115:
8109:
8104:
8100:
8094:
8089:
8085:
8082:
8081:
8061:
8056:
8052:
8048:
8042:
8037:
8033:
8030:
8024:
8019:
8015:
8010:
8006:
8002:
7999:
7993:
7988:
7984:
7978:
7973:
7969:
7949:
7946:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7922:
7919:
7916:
7913:
7910:
7907:
7904:
7882:
7858:
7836:
7814:
7792:
7772:
7748:
7743:
7717:
7693:
7669:
7636:
7631:
7608:
7602:
7597:
7593:
7587:
7582:
7578:
7573:
7569:
7546:
7524:
7504:
7501:
7498:
7479:
7478:
7457:
7452:
7425:
7421:
7417:
7411:
7406:
7359:
7355:
7328:
7324:
7297:
7293:
7267:
7264:
7249:
7248:
7236:
7233:
7222:
7209:
7205:
7194:
7181:
7177:
7166:
7153:
7149:
7138:
7124:
7119:
7108:
7094:
7089:
7078:
7064:
7059:
7048:
7034:
7029:
7018:
7004:
6999:
6987:
6986:
6973:
6969:
6958:
6946:
6943:
6932:
6919:
6915:
6904:
6891:
6887:
6876:
6862:
6857:
6846:
6832:
6827:
6816:
6802:
6797:
6786:
6772:
6767:
6756:
6742:
6737:
6725:
6724:
6711:
6707:
6696:
6683:
6679:
6668:
6656:
6653:
6642:
6629:
6625:
6614:
6600:
6595:
6584:
6570:
6565:
6554:
6540:
6535:
6524:
6510:
6505:
6494:
6480:
6475:
6463:
6462:
6449:
6445:
6434:
6421:
6417:
6406:
6393:
6389:
6378:
6366:
6363:
6352:
6338:
6333:
6322:
6308:
6303:
6292:
6278:
6273:
6262:
6248:
6243:
6232:
6218:
6213:
6201:
6200:
6186:
6181:
6170:
6156:
6151:
6140:
6126:
6121:
6110:
6096:
6091:
6080:
6067:
6063:
6052:
6039:
6035:
6024:
6012:
6009:
5998:
5985:
5981:
5970:
5957:
5953:
5941:
5940:
5926:
5921:
5910:
5896:
5891:
5880:
5866:
5861:
5850:
5836:
5831:
5820:
5807:
5803:
5792:
5780:
5777:
5766:
5753:
5749:
5738:
5725:
5721:
5710:
5697:
5693:
5681:
5680:
5666:
5661:
5650:
5636:
5631:
5620:
5606:
5601:
5590:
5576:
5571:
5560:
5548:
5545:
5534:
5521:
5517:
5506:
5493:
5489:
5478:
5465:
5461:
5450:
5437:
5433:
5421:
5420:
5406:
5401:
5390:
5376:
5371:
5360:
5346:
5341:
5330:
5316:
5311:
5300:
5287:
5283:
5272:
5259:
5255:
5244:
5231:
5227:
5216:
5204:
5201:
5190:
5178:
5175:
5163:
5162:
5148:
5143:
5132:
5118:
5113:
5102:
5088:
5083:
5072:
5058:
5053:
5042:
5029:
5025:
5014:
5001:
4997:
4986:
4973:
4969:
4958:
4946:
4943:
4932:
4921:
4896:
4891:
4863:
4857:
4852:
4848:
4843:
4839:
4835:
4829:
4824:
4798:
4793:
4765:
4760:
4733:
4729:
4697:
4675:
4655:
4652:
4649:
4625:
4605:
4581:
4576:
4561:dihedral group
4545:
4540:
4517:
4513:
4501:
4500:
4483:
4478:
4450:
4445:
4413:
4408:
4380:
4375:
4361:
4345:
4341:
4316:
4312:
4287:
4283:
4269:
4259:
4258:
4242:
4237:
4219:
4203:
4198:
4180:
4164:
4159:
4141:
4126:
4121:
4103:
4102:
4088:
4084:
4067:
4053:
4049:
4032:
4018:
4014:
3997:
3984:
3981:
3946:
3941:
3893:
3890:
3877:
3839:
3836:
3833:
3791:
3769:
3749:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3686:
3683:
3680:
3658:
3655:
3631:
3628:
3624:
3601:
3579:
3551:
3548:
3526:
3504:
3491:additive group
3475:
3472:
3469:
3466:
3462:
3439:
3434:
3411:
3385:
3363:
3351:additive group
3334:
3309:
3306:
3303:
3300:
3297:
3293:
3289:
3269:
3266:
3263:
3259:
3255:
3234:
3211:
3208:
3196:multiplication
3183:
3180:
3177:
3154:
3133:abuse notation
3114:underlying set
3101:
3098:
3097:
3096:
3080:
3077:
3073:
3050:
3022:
3013:, the element
3000:
2987:
2975:
2953:
2950:
2947:
2944:
2941:
2919:
2916:
2913:
2910:
2907:
2887:
2867:
2845:
2823:
2812:
2809:
2804:(or sometimes
2794:
2780:
2777:
2774:
2771:
2768:
2744:
2741:
2738:
2735:
2732:
2708:
2686:
2666:
2646:
2635:
2632:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2552:
2528:
2504:
2480:
2467:
2445:
2442:
2439:
2415:
2393:
2373:
2353:
2328:
2304:
2278:
2243:
2241:
2238:
2223:
2210:
2209:
2195:
2192:
2170:
2155:is called the
2144:
2135:. The integer
2122:
2119:
2116:
2113:
2110:
2088:
2085:
2082:
2079:
2076:
2056:
2034:
2021:
2014:is called the
1997:
1994:
1991:
1988:
1985:
1963:
1960:
1957:
1954:
1951:
1931:
1920:
1900:
1878:
1869:to the sum of
1858:
1838:
1818:
1798:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1710:
1688:
1666:
1637:
1610:
1584:
1581:
1578:
1553:
1531:
1518:together with
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1422:
1406:
1403:
1401:
1398:
1395:
1394:
1392:
1391:
1384:
1377:
1369:
1366:
1365:
1361:
1360:
1355:
1350:
1345:
1340:
1335:
1330:
1324:
1323:
1322:
1316:
1310:
1309:
1306:
1305:
1302:
1301:
1298:Linear algebra
1292:
1291:
1286:
1281:
1275:
1274:
1268:
1267:
1264:
1263:
1260:
1259:
1256:Lattice theory
1252:
1245:
1244:
1239:
1234:
1229:
1224:
1219:
1213:
1212:
1206:
1205:
1202:
1201:
1192:
1191:
1186:
1181:
1176:
1171:
1166:
1161:
1156:
1151:
1146:
1140:
1139:
1133:
1132:
1129:
1128:
1119:
1118:
1113:
1108:
1102:
1101:
1100:
1095:
1090:
1081:
1075:
1069:
1068:
1065:
1064:
1054:
1053:
1051:
1050:
1043:
1036:
1028:
1025:
1024:
1021:
1020:
1018:Elliptic curve
1014:
1013:
1007:
1006:
1000:
999:
993:
988:
987:
984:
983:
978:
977:
974:
971:
967:
963:
962:
961:
956:
954:Diffeomorphism
950:
949:
944:
939:
933:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
891:
890:
879:
878:
867:
866:
855:
854:
843:
842:
831:
830:
819:
818:
811:Special linear
807:
806:
799:General linear
795:
794:
789:
783:
774:
773:
770:
769:
766:
765:
760:
755:
747:
746:
733:
721:
708:
695:
693:Modular groups
691:
690:
689:
684:
671:
655:
652:
651:
646:
640:
639:
638:
635:
634:
629:
628:
627:
626:
621:
616:
613:
607:
606:
600:
599:
598:
597:
591:
590:
584:
583:
578:
569:
568:
566:Hall's theorem
563:
561:Sylow theorems
557:
556:
551:
543:
542:
541:
540:
534:
529:
526:Dihedral group
522:
521:
516:
510:
505:
499:
494:
483:
478:
477:
474:
473:
468:
467:
466:
465:
460:
452:
451:
450:
449:
444:
439:
434:
429:
424:
419:
417:multiplicative
414:
409:
404:
399:
391:
390:
389:
388:
383:
375:
374:
366:
365:
364:
363:
361:Wreath product
358:
353:
348:
346:direct product
340:
338:Quotient group
332:
331:
330:
329:
324:
319:
309:
306:
305:
302:
301:
293:
292:
178:Poincaré group
170:Standard Model
162:symmetry group
121:
26:
9:
6:
4:
3:
2:
29777:
29766:
29763:
29761:
29758:
29756:
29753:
29752:
29750:
29743:
29733:
29725:
29724:
29721:
29715:
29712:
29710:
29707:
29705:
29702:
29700:
29697:
29696:
29694:
29690:
29684:
29681:
29680:
29678:
29674:
29668:
29665:
29663:
29660:
29657:
29653:
29650:
29647:
29643:
29640:
29637:
29633:
29630:
29627:
29623:
29620:
29618:
29615:
29613:
29610:
29609:
29607:
29603:
29597:
29594:
29592:
29589:
29586:
29582:
29581:Orthogonality
29579:
29576:
29572:
29569:
29566:
29562:
29559:
29556:
29552:
29549:
29547:
29546:Hilbert space
29544:
29541:
29537:
29534:
29532:
29529:
29527:
29524:
29522:
29519:
29518:
29516:
29510:
29503:
29499:
29496:
29493:
29489:
29486:
29483:
29479:
29476:
29473:
29469:
29466:
29463:
29459:
29456:
29455:
29453:
29449:
29443:
29440:
29437:
29433:
29430:
29427:
29423:
29419:
29416:
29413:
29409:
29406:
29403:
29399:
29396:
29393:
29389:
29385:
29382:
29380:
29377:
29376:
29374:
29370:
29364:
29361:
29359:
29356:
29354:
29351:
29349:
29346:
29344:
29341:
29339:
29336:
29334:
29331:
29329:
29326:
29324:
29321:
29319:
29316:
29314:
29311:
29309:
29306:
29302:
29299:
29297:
29294:
29293:
29292:
29289:
29287:
29284:
29283:
29281:
29277:
29271:
29268:
29266:
29263:
29262:
29259:
29255:
29248:
29243:
29241:
29236:
29234:
29229:
29228:
29225:
29213:
29210:
29208:
29205:
29203:
29200:
29199:
29196:
29189:
29186:
29183:
29181:
29180:Quantum group
29178:
29176:
29173:
29171:
29168:
29166:
29163:
29162:
29160:
29156:
29150:
29147:
29145:
29142:
29140:
29139:Lorentz group
29137:
29135:
29132:
29131:
29128:
29122:
29120:
29114:
29112:
29106:
29104:
29098:
29096:
29090:
29088:
29085:
29084:
29080:
29077:
29074:
29071:
29068:
29067:Unitary group
29065:
29062:
29059:
29056:
29053:
29050:
29047:
29044:
29041:
29040:
29038:
29036:
29032:
29026:
29023:
29020:
29016:
29013:
29010:
29007:
29003:
29000:
28997:
28994:
28993:
28989:
28988:Monster group
28986:
28983:
28980:
28974:
28973:Fischer group
28971:
28969:
28962:
28955:
28948:
28942:Janko groups
28941:
28935:
28932:
28922:
28921:Mathieu group
28919:
28917:
28914:
28913:
28906:
28903:
28897:
28894:
28892:
28889:
28888:
28886:
28884:
28880:
28874:
28873:Trivial group
28871:
28869:
28866:
28864:
28861:
28859:
28856:
28854:
28851:
28849:
28846:
28844:
28843:Simple groups
28841:
28839:
28836:
28834:
28833:Cyclic groups
28831:
28829:
28826:
28824:
28823:Finite groups
28821:
28820:
28818:
28814:
28808:
28805:
28803:
28799:
28795:
28793:
28790:
28788:
28785:
28783:
28780:
28778:
28775:
28773:
28770:
28769:
28767:
28765:
28764:Basic notions
28761:
28757:
28750:
28745:
28743:
28738:
28736:
28731:
28730:
28727:
28719:
28718:
28713:
28709:
28704:
28703:
28692:
28686:
28682:
28678:
28674:
28673:Wussing, Hans
28670:
28666:
28660:
28656:
28652:
28651:Weyl, Hermann
28648:
28643:
28642:
28637:
28633:
28630:
28626:
28622:
28618:
28614:
28610:
28606:
28602:
28598:
28594:
28590:
28586:
28582:
28578:
28574:
28570:
28566:
28562:
28558:
28557:
28552:
28548:
28543:
28542:
28537:
28533:
28530:
28524:
28523:
28518:
28514:
28510:on 2014-02-22
28509:
28505:
28501:
28497:
28493:
28489:
28486:(in German),
28485:
28484:
28479:
28475:
28471:
28467:
28461:
28457:
28453:
28449:
28446:
28442:
28441:
28436:
28432:
28427:
28423:
28419:
28418:
28413:
28409:
28406:
28400:
28396:
28392:
28388:
28387:Borel, Armand
28384:
28383:
28381:
28368:
28364:
28360:
28354:
28350:
28349:
28344:
28340:
28336:
28330:
28326:
28322:
28321:Weyl, Hermann
28318:
28314:
28308:
28304:
28300:
28296:
28292:
28290:0-471-92567-5
28286:
28282:
28281:
28276:
28272:
28269:
28265:
28261:
28257:
28253:
28247:
28243:
28239:
28235:
28231:
28225:
28221:
28216:
28212:
28208:
28203:
28198:
28194:
28190:
28189:
28184:
28180:
28177:
28171:
28167:
28166:Galois Theory
28163:
28159:
28155:
28150:
28146:
28142:
28137:
28134:
28128:
28124:
28119:
28116:
28112:
28108:
28102:
28098:
28093:
28089:
28083:
28079:
28074:
28070:
28066:
28062:
28056:
28051:
28050:
28044:
28040:
28036:
28032:
28028:
28024:
28017:
28012:
28008:
28006:0-471-52364-X
28002:
27998:
27994:
27993:Rudin, Walter
27990:
27986:
27982:
27978:
27972:
27968:
27963:
27959:
27953:
27949:
27945:
27941:
27937:
27933:
27927:
27923:
27919:
27915:
27911:
27908:
27904:
27900:
27896:
27892:
27886:
27881:
27876:
27872:
27868:
27864:
27860:
27856:
27850:
27846:
27841:
27837:
27833:
27829:
27823:
27819:
27815:
27811:
27808:
27802:
27798:
27797:
27791:
27787:
27781:
27777:
27772:
27761:
27760:
27753:
27750:
27744:
27740:
27739:
27734:
27730:
27726:
27720:
27716:
27715:
27710:
27706:
27702:
27696:
27692:
27688:
27683:
27679:
27675:
27671:
27665:
27661:
27656:
27652:
27648:
27644:
27638:
27633:
27632:
27626:
27622:
27618:
27614:
27610:
27604:
27600:
27596:
27592:
27588:
27583:
27578:
27573:
27569:
27565:
27561:
27557:
27556:
27551:
27547:
27543:
27540:
27534:
27530:
27525:
27521:
27515:
27511:
27507:
27506:
27501:
27497:
27494:
27488:
27484:
27479:
27475:
27473:0-201-02918-9
27469:
27465:
27464:
27459:
27455:
27452:
27448:
27444:
27438:
27434:
27430:
27426:
27422:
27418:
27414:on 2008-12-01
27413:
27409:
27406:(in German),
27405:
27401:
27397:
27393:
27389:
27385:
27381:
27375:
27371:
27367:
27363:
27358:
27355:
27349:
27345:
27340:
27336:
27332:
27328:
27324:
27317:
27308:
27304:
27302:0-19-850678-3
27298:
27294:
27289:
27285:
27279:
27275:
27271:
27266:
27262:
27258:
27254:
27248:
27244:
27239:
27235:
27231:
27226:
27221:
27217:
27213:
27209:
27205:
27201:
27198:
27192:
27188:
27183:
27179:
27173:
27169:
27165:
27161:
27157:
27153:
27149:
27145:
27139:
27135:
27131:
27127:
27126:Borel, Armand
27123:
27119:
27113:
27109:
27104:
27100:
27096:
27091:
27086:
27082:
27078:
27074:
27069:
27065:
27063:0-521-82212-2
27059:
27055:
27054:
27048:
27044:
27039:
27034:
27029:
27025:
27021:
27016:
27013:
27007:
27003:
26998:
26993:
26989:
26988:
26980:
26976:
26972:
26968:
26962:
26958:
26954:
26953:Galois Theory
26950:
26946:
26945:
26935:
26929:
26925:
26921:
26917:
26913:
26909:
26905:
26901:
26897:
26893:
26889:
26885:
26881:
26877:
26873:
26867:
26863:
26859:
26855:
26851:
26848:
26844:
26840:
26834:
26830:
26826:
26822:
26818:
26814:
26810:
26806:
26802:
26798:
26794:
26790:
26786:
26780:
26776:
26772:
26768:
26764:
26760:
26756:
26752:
26748:
26745:
26739:
26735:
26734:
26728:
26724:
26718:
26714:
26713:Prentice Hall
26710:
26706:
26702:
26701:
26683:
26678:
26671:
26666:
26659:
26654:
26647:
26642:
26635:
26630:
26623:
26622:Weinberg 1972
26618:
26611:
26606:
26599:
26594:
26587:
26582:
26575:
26570:
26563:
26558:
26551:
26550:Neukirch 1999
26546:
26539:
26534:
26527:
26522:
26515:
26510:
26503:
26498:
26491:
26487:
26482:
26475:
26470:
26463:
26458:
26451:
26446:
26439:
26434:
26427:
26422:
26415:
26414:Robinson 1996
26410:
26403:
26398:
26391:
26386:
26379:
26374:
26372:
26364:
26359:
26352:
26347:
26340:
26335:
26328:
26323:
26317:, p. 82.
26316:
26311:
26304:
26299:
26292:
26287:
26280:
26275:
26268:
26263:
26256:
26251:
26245:
26241:
26236:
26229:
26224:
26217:
26212:
26210:
26202:
26197:
26190:
26185:
26178:
26173:
26166:
26161:
26154:
26149:
26142:
26137:
26130:
26125:
26118:
26113:
26106:
26105:Neukirch 1999
26102:
26101:Picard groups
26098:
26095:For example,
26092:
26085:
26080:
26073:
26068:
26061:
26056:
26049:
26044:
26037:
26032:
26025:
26020:
26013:
26008:
26001:
25996:
25989:
25984:
25977:
25976:Mac Lane 1998
25972:
25970:
25962:
25957:
25950:
25945:
25943:
25941:
25934:, p. 40.
25933:
25928:
25921:
25916:
25914:
25906:
25901:
25894:
25889:
25882:
25877:
25870:
25865:
25858:
25853:
25846:
25841:
25834:
25833:von Dyck 1882
25829:
25822:
25817:
25810:
25805:
25798:
25793:
25786:
25781:
25774:
25769:
25762:
25757:
25750:
25745:
25738:
25733:
25726:
25721:
25714:
25709:
25702:
25697:
25690:
25689:Herstein 1975
25685:
25678:
25673:
25666:
25661:
25654:
25649:
25643:, p. 24.
25642:
25637:
25630:
25625:
25618:
25614:
25610:
25605:
25598:
25597:Herstein 1975
25593:
25589:
25575:
25569:
25562:
25558:
25553:
25546:
25542:
25538:
25534:
25533:Schur's Lemma
25528:
25521:
25515:
25508:
25504:
25498:
25491:
25487:
25483:
25477:
25469:
25463:
25457:
25453:
25446:
25439:
25438:Gollmann 2011
25435:
25431:
25425:
25418:
25417:
25416:Prime element
25410:
25403:
25399:
25395:
25391:
25385:
25378:
25373:
25367:
25363:
25357:
25350:
25344:
25337:
25333:
25327:
25320:
25316:
25312:
25308:
25302:
25295:
25289:
25282:
25276:
25269:
25265:
25261:
25255:
25248:
25247:Herstein 1975
25244:
25243:Herstein 1996
25240:
25236:
25230:
25222:
25218:
25212:
25208:
25202:
25192:
25184:
25173:
25166:
25162:
25156:
25149:
25144:
25134:
25128:
25122:
25116:
25110:
25106:
25097:
25091:
25087:
25077:
25074:
25073:
25069:
25063:
25058:
25047:
25044:
25042:
25041:abelian group
25039:
25037:
25034:
25032:
25029:
25027:
25024:
25022:
25021:abelian group
25019:
25017:
25014:
25012:
25011:abelian group
25009:
25007:
25004:
25002:
24999:
24996:
24995:
24991:
24988:
24985:
24982:
24979:
24976:
24973:
24970:
24967:
24964:
24961:
24960:
24956:
24953:
24950:
24947:
24944:
24941:
24938:
24935:
24932:
24929:
24926:
24925:
24921:
24918:
24915:
24898:
24895:
24892:
24867:
24863:
24859:
24850:
24847:
24831:
24828:
24819:
24816:
24800:
24797:
24788:
24785:
24782:
24779:
24778:
24774:
24771:
24768:
24765:
24762:
24759:
24756:
24753:
24750:
24747:
24744:
24743:
24739:
24736:
24733:
24730:
24727:
24724:
24721:
24718:
24715:
24712:
24709:
24708:
24704:
24701:
24698:
24695:
24692:
24689:
24686:
24683:
24680:
24677:
24674:
24673:
24653:
24650:
24647:
24644:
24641:
24635:
24627:
24623:
24607:
24576:
24546:
24516:
24486:
24456:
24450:
24449:
24443:
24441:
24426:
24418:
24416:
24412:
24394:
24370:
24346:
24343:
24333:
24315:
24312:
24306:
24303:
24300:
24294:
24291:
24267:
24243:
24218:
24215:
24212:
24206:
24203:
24181:
24171:
24166:
24150:
24140:
24122:
24095:
24092:
24086:
24080:
24047:
24044:
24038:
24032:
23999:
23996:
23990:
23984:
23945:
23941:
23936:
23928:
23925:
23922:
23919:
23916:
23914:
23913:Abelian group
23911:
23910:
23906:
23903:
23900:
23897:
23894:
23892:
23889:
23888:
23884:
23881:
23878:
23875:
23872:
23870:
23867:
23866:
23862:
23859:
23856:
23853:
23850:
23848:
23845:
23844:
23840:
23837:
23834:
23831:
23828:
23826:
23822:
23821:
23817:
23814:
23811:
23808:
23805:
23803:
23799:
23798:
23794:
23791:
23788:
23785:
23782:
23780:
23776:
23775:
23771:
23768:
23765:
23762:
23759:
23757:
23754:
23753:
23749:
23746:
23743:
23740:
23737:
23735:
23731:
23730:
23726:
23723:
23720:
23717:
23714:
23712:
23709:
23708:
23704:
23701:
23698:
23695:
23692:
23690:
23686:
23685:
23681:
23678:
23675:
23672:
23669:
23667:
23664:
23663:
23659:
23656:
23653:
23650:
23647:
23645:
23641:
23640:
23636:
23633:
23630:
23627:
23624:
23622:
23619:
23618:
23614:
23611:
23608:
23605:
23602:
23600:
23596:
23595:
23591:
23588:
23585:
23582:
23579:
23577:
23574:
23573:
23569:
23566:
23563:
23560:
23557:
23555:
23551:
23550:
23546:
23543:
23540:
23537:
23534:
23532:
23529:
23528:
23524:
23521:
23518:
23515:
23512:
23510:
23507:
23506:
23502:
23499:
23496:
23493:
23490:
23488:
23485:
23484:
23480:
23477:
23474:
23471:
23468:
23466:
23465:Partial magma
23463:
23462:
23459:
23456:
23454:
23451:
23449:
23446:
23444:
23441:
23439:
23436:
23434:
23433:
23422:
23420:
23416:
23412:
23408:
23404:
23400:
23396:
23392:
23388:
23384:
23380:
23376:
23372:
23368:
23364:
23360:
23356:
23352:
23347:
23333:
23313:
23293:
23273:
23270:
23267:
23261:
23235:
23215:
23207:
23202:
23200:
23195:
23191:
23187:
23181:
23171:
23169:
23165:
23161:
23157:
23153:
23149:
23145:
23144:Haar measures
23141:
23137:
23136:-adic numbers
23134:
23128:
23124:
23120:
23104:
23084:
23062:
23059:
23055:
23034:
23031:
23028:
23020:
23014:
23006:
23002:
22997:
22996:complex plane
22993:
22988:
22979:
22977:
22973:
22968:
22966:
22962:
22958:
22942:
22936:
22929:
22913:
22907:
22904:
22901:
22881:
22867:
22865:
22860:
22858:
22854:
22850:
22846:
22845:monster group
22842:
22838:
22834:
22830:
22825:
22823:
22819:
22815:
22810:
22800:
22798:
22794:
22793:
22774:
22770:
22766:
22744:
22724:
22704:
22681:
22658:
22638:
22631:When a group
22627:Simple groups
22624:
22608:
22604:
22581:
22552:
22548:
22523:
22513:
22508:
22475:
22471:
22441:
22437:
22428:
22410:
22383:
22374:
22372:
22368:
22358:
22342:
22339:
22336:
22314:
22292:
22287:
22252:
22231:
22209:
22205:
22182:
22166:
22164:
22146:
22137:the order of
22136:
22121:
22101:
22093:
22088:
22074:
22054:
22032:
22008:
21984:
21980:
21957:
21933:
21911:
21889:
21867:
21862: factors
21857:
21851:
21847:
21844:
21841:
21815:
21811:
21788:
21785:
21780:
21776:
21753:
21733:
21713:
21704:
21702:
21684:
21669:
21651:
21627:
21611:
21593:
21579:
21563:
21539:
21523:
21522:
21517:
21513:
21509:
21503:
21496:Finite groups
21493:
21491:
21487:
21483:
21479:
21478:Galois theory
21474:
21472:
21468:
21464:
21460:
21456:
21452:
21448:
21432:
21412:
21392:
21372:
21366:
21363:
21356:
21353:
21350:
21347:
21342:
21338:
21332:
21329:
21326:
21320:
21317:
21310:are given by
21297:
21294:
21291:
21288:
21285:
21282:
21279:
21274:
21270:
21266:
21259:
21255:
21254:Galois groups
21250:
21243:Galois groups
21240:
21238:
21234:
21228:
21206:
21203:
21186:
21183:
21180:
21172:
21156:
21136:
21112:
21096:
21092:
21088:
21084:
21082:
21078:
21062:
21037:
21017:
21013:
21012:
21011:linear groups
21007:
21006:matrix groups
20989:
20965:
20956:
20932:
20929:
20911:
20907:
20903:
20899:
20898:Matrix groups
20878:
20868:
20863:
20858:
20854:
20850:
20840:
20838:
20835:
20831:
20827:
20823:
20819:
20815:
20811:
20810:coding theory
20807:
20797:
20793:
20790:
20777:
20774:
20767:
20764:
20762:
20757:
20754:
20753:
20749:
20742:
20735:
20728:
20723:
20720:
20718:
20714:
20709:
20707:
20703:
20699:
20698:ferroelectric
20695:
20690:
20688:
20684:
20680:
20676:
20672:
20667:
20665:
20661:
20657:
20639:
20615:
20605:
20601:
20597:
20593:
20588:
20581:
20577:
20572:
20567:
20563:
20559:
20555:
20549:
20539:
20537:
20533:
20529:
20524:
20505:
20502:
20472:
20450:
20439:
20423:
20420:
20415:
20411:
20386:
20383:
20378:
20374:
20349:
20346:
20343:
20340:
20335:
20331:
20306:
20303:
20298:
20294:
20271:
20249:
20246:
20243:
20221:
20199:
20194:
20179:
20175:
20159:
20137:
20134:
20131:
20109:
20102:on a regular
20101:
20083:
20080:
20075:
20071:
20048:
20041:
20037:
20019:
19997:
19975:
19953:
19930:
19927:
19919:
19915:
19897:
19884:
19881:
19878:
19875:
19872:
19869:
19866:
19863:
19860:
19857:
19854:
19851:
19848:
19845:
19839:
19836:
19830:
19824:
19821:
19815:
19812:
19804:
19788:
19764:
19761:
19753:
19750:
19747:
19744:
19741:
19735:
19730:
19727:
19723:
19719:
19714:
19711:
19707:
19703:
19698:
19695:
19691:
19666:
19663:
19659:
19636:
19633:
19630:
19606:
19602:
19581:
19578:
19575:
19570:
19566:
19562:
19557:
19553:
19549:
19546:
19543:
19538:
19534:
19530:
19525:
19522:
19518:
19514:
19509:
19506:
19502:
19498:
19493:
19490:
19486:
19482:
19479:
19459:
19451:
19447:
19423:
19419:
19396:
19374:
19370:
19349:
19340:
19335:
19328:Cyclic groups
19325:
19323:
19303:
19298:
19264:
19260:
19233:
19229:
19226:
19222:
19219:
19216:
19212:
19203:
19178:
19170:
19167:
19164:
19161:
19158:
19155:
19152:
19128:
19106:
19084:
19062:
19042:
19039:
19036:
19014:
18989:
18986:
18983:
18970:
18966:
18950:
18928:
18925:
18922:
18919:
18916:
18894:
18874:
18867:
18863:
18855:
18852:
18849:
18846:
18843:
18823:
18801:
18779:
18757:
18733:
18711:
18689:
18667:
18645:
18623:
18599:
18591:
18588:
18585:
18582:
18579:
18555:
18552:
18549:
18546:
18543:
18540:
18537:
18513:
18510:
18507:
18483:
18473:
18455:
18431:
18428:
18425:
18403:
18381:
18371:
18353:
18342:
18329:
18322:
18318:
18310:
18307:
18304:
18301:
18298:
18276:
18254:
18234:
18231:
18228:
18206:
18184:
18164:
18156:
18151:
18135:
18113:
18110:
18107:
18087:
18062:
18059:
18051:
18047:
18014:
17985:
17982:
17979:
17957:
17949:
17931:
17928:
17925:
17903:
17877:
17873:
17865:
17862:
17859:
17837:
17817:
17797:
17777:
17770:
17758:
17753:
17748:
17738:
17714:
17709:
17693:
17689:
17685:
17663:
17659:
17655:
17631:
17601:
17597:
17594:
17591:
17588:
17580:
17577:
17573:
17569:
17565:
17562:
17559:
17555:
17542:
17537:
17522:
17518:
17515:
17506:
17483:
17480:
17477:
17474:
17471:
17449:
17426:
17422:
17419:
17410:
17373:
17357:
17348:
17335:
17330:
17327:
17318:
17308:
17267:
17264:
17258:
17255:
17233:
17230:
17227:
17224:
17221:
17201:
17198:
17195:
17187:
17170:
17166:
17163:
17154:
17130:
17126:
17123:
17114:
17075:
17073:
17069:
17068:vector spaces
17065:
17061:
17057:
17047:
17045:
17041:
17037:
17033:
17029:
17028:algorithmical
17025:
17020:
17018:
17014:
17010:
17005:
17003:
16985:
16982:
16979:
16957:
16937:
16929:
16924:
16920:
16916:
16907:
16903:
16901:
16897:
16893:
16889:
16885:
16881:
16877:
16872:
16870:
16866:
16836:
16831:
16826:
16816:
16814:
16810:
16805:
16787:
16771:
16768:
16763:
16755:
16752:
16749:
16743:
16738:
16734:
16730:
16725:
16721:
16717:
16714:
16711:
16708:
16681:
16657:
16642:
16641:
16619:
16616:
16611:
16603:
16600:
16597:
16591:
16586:
16582:
16578:
16573:
16569:
16565:
16562:
16559:
16556:
16528:
16525:
16522:
16499:
16496:
16493:
16469:
16461:
16458:
16455:
16449:
16444:
16440:
16436:
16431:
16427:
16416:
16400:
16397:
16394:
16370:
16366:
16343:
16321:
16317:
16296:
16274:
16244:
16241:
16238:
16213:
16189:
16154:
16131:
16127:
16104:
16088:
16086:
16080:
16073:Presentations
16070:
16068:
16050:
16042:
16029:
16026:
16023:
16019:
16015:
16009:
15987:
15981:
15978:
15975:
15968:homomorphism
15967:
15963:
15958:
15942:
15927:
15909:
15882:
15878:
15872:
15840:
15836:
15832:
15827:
15823:
15819:
15814:
15810:
15806:
15792:
15789:
15767:
15764:
15761:
15747:
15743:
15732:
15725:
15722:
15711:
15707:
15704:
15693:
15689:
15686:
15683:
15680:
15656:
15645:
15641:
15638:
15616:
15596:
15592:
15586:
15554:
15547:
15533:
15526:
15512:
15505:
15504:
15489:
15482:
15468:
15461:
15447:
15440:
15439:
15424:
15417:
15403:
15396:
15382:
15375:
15374:
15359:
15355:
15349:
15332:
15330:
15326:
15322:
15304:
15280:
15256:
15252:
15248:
15224:
15220:
15215:
15212:
15208:
15204:
15200:
15195:
15192:
15184:
15181:
15156:
15153:
15131:
15127:
15123:
15101:
15098:
15095:
15092:
15070:
15064:
15061:
15036:
15033:
15013:
15010:
14990:
14987:
14967:
14947:
14943:
14939:
14933:
14913:
14909:
14905:
14882:
14879:
14876:
14873:
14870:
14867:
14861:
14858:
14854:
14850:
14828:
14806:
14799:Suppose that
14796:
14786:
14770:
14759:
14755:
14752:
14741:
14737:
14734:
14723:
14719:
14716:
14705:
14678:
14641:
14637:
14634:
14631:
14628:
14625:
14614:
14593:
14569:
14529:
14525:
14514:
14510:
14499:
14495:
14484:
14477:
14474:
14463:
14459:
14456:
14436:
14416:
14394:
14372:
14369:
14349:
14325:
14307:
14305:
14304:
14287:
14265:
14262:
14259:
14256:
14253:
14231:
14211:
14191:
14169:
14145:
14121:
14118:
14113:
14109:
14105:
14100:
14097:
14092:
14088:
14078:
14062:
14057:
14053:
14049:
14046:
14041:
14037:
14028:
14025:
14009:
14001:
13983:
13973:
13957:
13929:
13926:
13923:
13920:
13917:
13914:
13911:
13905:
13902:
13899:
13874:
13871:
13868:
13865:
13862:
13859:
13856:
13850:
13847:
13844:
13836:
13820:
13796:
13786:
13782:
13764:
13742:
13722:
13711:
13701:
13683:
13679:
13675:
13664:
13660:
13649:
13589:
13566:
13562:
13539:
13517:
13497:
13477:
13469:
13451:
13429:
13420:
13418:
13400:
13378:
13358:
13338:
13335:
13332:
13329:
13324:
13321:
13317:
13294:
13270:
13260:
13256:
13255:subgroup test
13233:
13229:
13225:
13220:
13216:
13212:
13207:
13203:
13199:
13185:
13182:
13171:
13157:
13151:
13129:
13107:
13085:
13059:
13056:
13051:
13047:
13022:
13018:
13014:
13009:
13005:
12982:
12958:
12954:
12931:
12927:
12904:
12882:
12860:
12836:
12814:
12806:
12800:
12790:
12788:
12787:monomorphisms
12770:
12746:
12724:
12718:
12710:
12696:
12693:
12672:
12665:
12662:
12658:
12655:
12648:homomorphism
12647:
12642:
12640:
12636:
12631:
12615:
12593:
12571:
12549:
12529:
12507:
12501:
12498:
12495:
12485:
12469:
12449:
12441:
12437:
12436:
12417:
12395:
12375:
12372:
12359:
12353:
12343:
12323:
12303:
12283:
12280:
12267:
12261:
12251:
12227:
12223:
12219:
12216:
12213:
12210:
12186:
12182:
12178:
12175:
12172:
12169:
12149:
12143:
12140:
12137:
12117:
12111:
12108:
12105:
12097:
12081:
12061:
12055:
12052:
12047:
12043:
12022:
12014:
12009:
11993:
11971:
11949:
11946:
11938:
11932:
11929:
11921:
11918:
11914:
11907:
11883:
11879:
11875:
11867:
11863:
11856:
11834:
11809:
11787:
11767:
11744:
11738:
11735:
11729:
11723:
11720:
11714:
11711:
11708:
11702:
11694:
11680:
11674:
11671:
11668:
11645:
11642:
11639:
11613:
11610:
11607:
11597:from a group
11596:
11590:
11580:
11578:
11574:
11573:homomorphisms
11570:
11566:
11562:
11541:
11519:
11511:
11495:
11475:
11469:
11466:
11463:
11441:
11419:
11399:
11396:
11393:
11371:
11347:
11323:
11314:
11298:
11291:
11272:
11269:
11266:
11263:
11260:
11254:
11251:
11243:
11233:
11216:
11213:
11210:
11186:
11164:
11140:
11137:
11134:
11131:
11128:
11125:
11122:
11119:
11116:
11094:
11091:
11088:
11085:
11082:
11079:
11076:
11073:
11070:
11050:
11027:
11024:
11021:
11015:
11012:
11004:
11000:
10995:
10992:
10990:
10959:
10956:
10954:
10938:
10935:
10932:
10929:
10927:
10911:
10908:
10900:
10897:
10893:
10889:
10886:
10880:
10878:
10859:
10856:
10851:
10848:
10844:
10837:
10834:
10831:
10829:
10824:
10821:
10818:
10807:
10806:
10805:
10776:
10773:
10771:
10753:
10750:
10746:
10742:
10737:
10734:
10724:
10721:
10717:
10710:
10708:
10687:
10684:
10680:
10676:
10673:
10667:
10662:
10659:
10649:
10646:
10642:
10635:
10633:
10612:
10609:
10605:
10601:
10595:
10592:
10587:
10584:
10580:
10570:
10565:
10562:
10552:
10549:
10545:
10538:
10536:
10515:
10512:
10508:
10504:
10501:
10495:
10487:
10484:
10480:
10476:
10471:
10468:
10458:
10455:
10451:
10441:
10439:
10418:
10415:
10411:
10407:
10404:
10398:
10395:
10392:
10390:
10383:
10380:
10376:
10372:
10369:
10358:
10357:
10356:
10340:
10337:
10334:
10331:
10326:
10323:
10319:
10296:
10274:
10271:
10267:
10244:
10241:
10238:
10235:
10232:
10210:
10201:
10199:
10196:. From these
10195:
10194:left inverses
10191:
10190:left identity
10181:
10165:
10155:
10139:
10131:
10115:
10112:
10109:
10089:
10069:
10063:
10041:
10015:
10012:
10008:
10004:
10001:
9979:
9976:
9973:
9970:
9967:
9945:
9923:
9914:
9898:
9888:
9872:
9864:
9860:
9844:
9841:
9838:
9818:
9798:
9792:
9772:
9752:
9730:
9727:
9722:
9719:
9715:
9690:
9687:
9684:
9681:
9678:
9656:
9636:
9614:
9592:
9572:
9558:
9556:
9528:
9525:
9522:
9514:
9503:
9500:
9498:
9485:
9477:
9466:
9463:
9460:
9457:
9455:
9439:
9436:
9430:
9427:
9424:
9418:
9416:
9403:
9395:
9381:
9378:
9375:
9369:
9366:
9363:
9361:
9348:
9337:
9334:
9331:
9328:
9326:
9321:
9310:
9309:
9308:
9294:
9274:
9254:
9240:
9238:
9220:
9217:
9214:
9211:
9208:
9205:
9202:
9180:
9160:
9146:
9144:
9125:
9122:
9119:
9113:
9110:
9107:
9104:
9101:
9095:
9092:
9089:
9083:
9080:
9077:
9074:
9071:
9068:
9060:
9056:
9046:
9044:
9040:
9036:
9032:
9028:
9024:
9020:
9016:
9011:
9009:
9005:
9001:
8997:
8993:
8989:
8985:
8981:
8977:
8973:
8969:
8965:
8961:
8957:
8953:
8948:
8943:
8938:
8936:
8935:prime numbers
8932:
8928:
8924:
8920:
8916:
8915:
8910:
8906:
8905:number theory
8901:
8899:
8895:
8891:
8887:
8883:
8879:
8874:
8872:
8868:
8855:
8852:
8847:
8843:
8832:
8831:Arthur Cayley
8828:
8824:
8820:
8816:
8812:
8808:
8804:
8800:
8799:Paolo Ruffini
8796:
8792:
8788:
8782:
8772:
8756:
8719:
8715:
8704:
8700:
8695:
8691:
8660:
8656:
8651:
8647:
8643:
8632:
8607:
8589:
8573:
8569:
8546:
8542:
8519:
8515:
8484:
8451:
8418:
8385:
8345:
8341:
8327:
8290:
8286:
8269:
8264:
8260:
8256:
8245:
8241:
8230:
8226:
8224:
8214:
8210:
8206:
8195:
8188:
8177:
8167:
8163:
8159:
8154:
8150:
8146:
8141:
8137:
8133:
8131:
8124:
8120:
8116:
8102:
8098:
8087:
8054:
8050:
8046:
8035:
8028:
8017:
8013:
8008:
8004:
8000:
7986:
7982:
7971:
7960:For example,
7947:
7941:
7938:
7935:
7929:
7926:
7923:
7920:
7917:
7911:
7908:
7905:
7880:
7856:
7834:
7812:
7790:
7770:
7746:
7715:
7691:
7667:
7657:
7656:Associativity
7653:
7629:
7606:
7595:
7591:
7580:
7576:
7571:
7567:
7544:
7522:
7502:
7499:
7496:
7488:
7484:
7450:
7423:
7419:
7415:
7404:
7385:
7377:
7357:
7353:
7326:
7322:
7295:
7291:
7252:The elements
7250:
7223:
7207:
7203:
7195:
7179:
7175:
7167:
7151:
7147:
7139:
7117:
7109:
7087:
7079:
7057:
7049:
7027:
7019:
6997:
6989:
6988:
6971:
6967:
6959:
6933:
6917:
6913:
6905:
6889:
6885:
6877:
6855:
6847:
6825:
6817:
6795:
6787:
6765:
6757:
6735:
6727:
6726:
6709:
6705:
6697:
6681:
6677:
6669:
6643:
6627:
6623:
6615:
6593:
6585:
6563:
6555:
6533:
6525:
6503:
6495:
6473:
6465:
6464:
6447:
6443:
6435:
6419:
6415:
6407:
6391:
6387:
6379:
6353:
6331:
6323:
6301:
6293:
6271:
6263:
6241:
6233:
6211:
6203:
6202:
6179:
6171:
6149:
6141:
6119:
6111:
6089:
6081:
6065:
6061:
6053:
6037:
6033:
6025:
5999:
5983:
5979:
5971:
5955:
5951:
5943:
5942:
5919:
5911:
5889:
5881:
5859:
5851:
5829:
5821:
5805:
5801:
5793:
5767:
5751:
5747:
5739:
5723:
5719:
5711:
5695:
5691:
5683:
5682:
5659:
5651:
5629:
5621:
5599:
5591:
5569:
5561:
5535:
5519:
5515:
5507:
5491:
5487:
5479:
5463:
5459:
5451:
5435:
5431:
5423:
5422:
5399:
5391:
5369:
5361:
5339:
5331:
5309:
5301:
5285:
5281:
5273:
5257:
5253:
5245:
5229:
5225:
5217:
5191:
5165:
5164:
5141:
5133:
5111:
5103:
5081:
5073:
5051:
5043:
5027:
5023:
5015:
4999:
4995:
4987:
4971:
4967:
4959:
4933:
4919:
4912:
4911:
4894:
4879:
4874:
4861:
4850:
4846:
4841:
4837:
4833:
4822:
4791:
4758:
4731:
4727:
4716:
4711:
4695:
4673:
4653:
4650:
4647:
4639:
4623:
4603:
4579:
4562:
4538:
4515:
4511:
4476:
4443:
4432:
4406:
4373:
4362:
4343:
4339:
4314:
4310:
4285:
4281:
4270:
4267:
4263:
4262:
4257:
4235:
4225:
4220:
4218:
4196:
4186:
4181:
4179:
4157:
4147:
4142:
4119:
4109:
4105:
4104:
4086:
4082:
4072:
4068:
4051:
4047:
4037:
4033:
4016:
4012:
4002:
3998:
3969:
3965:
3964:
3944:
3925:
3923:
3919:
3915:
3911:
3907:
3903:
3899:
3889:
3875:
3868:, the symbol
3867:
3863:
3859:
3855:
3837:
3834:
3831:
3822:
3818:
3813:
3811:
3810:abelian group
3807:
3789:
3767:
3747:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3700:
3684:
3681:
3678:
3656:
3653:
3629:
3626:
3622:
3599:
3577:
3567:
3549:
3546:
3524:
3502:
3492:
3487:
3470:
3464:
3437:
3423:is the group
3401:of the field
3400:
3353:of the field
3352:
3347:
3323:
3304:
3301:
3298:
3295:
3264:
3261:
3209:
3206:
3197:
3181:
3178:
3175:
3142:
3137:
3134:
3130:
3125:
3123:
3119:
3115:
3111:
3107:
3078:
3075:
3071:
3048:
3040:
3036:
3020:
2998:
2988:
2973:
2951:
2948:
2945:
2942:
2939:
2917:
2914:
2911:
2908:
2905:
2885:
2865:
2843:
2821:
2813:
2810:
2807:
2803:
2799:
2795:
2778:
2775:
2772:
2769:
2766:
2742:
2739:
2736:
2733:
2730:
2706:
2684:
2664:
2644:
2636:
2633:
2613:
2610:
2607:
2601:
2598:
2595:
2592:
2589:
2583:
2580:
2577:
2550:
2526:
2502:
2478:
2468:
2466:Associativity
2465:
2464:
2463:
2461:
2443:
2440:
2437:
2413:
2391:
2371:
2351:
2344:
2326:
2302:
2292:
2276:
2269:
2263:
2261:
2257:
2251:
2249:
2237:
2221:
2193:
2190:
2168:
2160:
2159:
2142:
2120:
2117:
2114:
2111:
2108:
2086:
2083:
2080:
2077:
2074:
2054:
2032:
2022:
2019:
2018:
2013:
1995:
1992:
1989:
1986:
1983:
1961:
1958:
1955:
1952:
1949:
1929:
1921:
1918:
1917:
1916:associativity
1898:
1876:
1856:
1836:
1816:
1796:
1771:
1768:
1765:
1759:
1756:
1753:
1750:
1747:
1741:
1738:
1735:
1708:
1686:
1664:
1654:
1653:
1652:
1624:
1608:
1600:
1599:
1582:
1579:
1576:
1569:
1551:
1529:
1521:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1424:
1412:
1390:
1385:
1383:
1378:
1376:
1371:
1370:
1368:
1367:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1325:
1321:
1318:
1317:
1313:
1308:
1307:
1300:
1299:
1295:
1294:
1290:
1287:
1285:
1282:
1280:
1277:
1276:
1271:
1266:
1265:
1258:
1257:
1253:
1251:
1248:
1247:
1243:
1240:
1238:
1235:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1214:
1209:
1204:
1203:
1198:
1197:
1190:
1187:
1185:
1184:Division ring
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1141:
1136:
1131:
1130:
1125:
1124:
1117:
1114:
1112:
1109:
1107:
1106:Abelian group
1104:
1103:
1099:
1096:
1094:
1091:
1089:
1085:
1082:
1080:
1077:
1076:
1072:
1067:
1066:
1063:
1060:
1059:
1049:
1044:
1042:
1037:
1035:
1030:
1029:
1027:
1026:
1019:
1016:
1015:
1012:
1009:
1008:
1005:
1002:
1001:
998:
995:
994:
991:
986:
985:
975:
972:
969:
968:
966:
960:
957:
955:
952:
951:
948:
945:
943:
940:
938:
935:
934:
931:
925:
923:
917:
915:
909:
907:
901:
899:
893:
892:
888:
884:
881:
880:
876:
872:
869:
868:
864:
860:
857:
856:
852:
848:
845:
844:
840:
836:
833:
832:
828:
824:
821:
820:
816:
812:
809:
808:
804:
800:
797:
796:
793:
790:
788:
785:
784:
781:
777:
772:
771:
764:
761:
759:
756:
754:
751:
750:
722:
697:
696:
694:
688:
685:
660:
657:
656:
650:
647:
645:
642:
641:
637:
636:
625:
622:
620:
617:
614:
611:
610:
609:
608:
605:
602:
601:
596:
593:
592:
589:
586:
585:
582:
579:
577:
575:
571:
570:
567:
564:
562:
559:
558:
555:
552:
550:
547:
546:
545:
544:
538:
535:
532:
527:
524:
523:
519:
514:
511:
508:
503:
500:
497:
492:
489:
488:
487:
486:
481:
480:Finite groups
476:
475:
464:
461:
459:
456:
455:
454:
453:
448:
445:
443:
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
394:
393:
392:
387:
384:
382:
379:
378:
377:
376:
373:
372:
368:
367:
362:
359:
357:
354:
352:
349:
347:
344:
341:
339:
336:
335:
334:
333:
328:
325:
323:
320:
318:
315:
314:
313:
312:
307:Basic notions
304:
303:
299:
295:
294:
291:
286:
282:
278:
277:
274:
272:
268:
264:
260:
259:finite groups
256:
252:
248:
244:
243:simple groups
240:
236:
232:
228:
227:number theory
224:
220:
215:
210:
206:
202:
197:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
146:
144:
140:
135:
119:
109:
105:
101:
97:
93:
88:
86:
82:
78:
74:
70:
66:
62:
58:
50:
46:
41:
37:
33:
19:
29755:Group theory
29742:
29709:Order theory
29699:Field theory
29565:Affine space
29498:Vector space
29467:
29353:Order theory
29207:Applications
29134:Circle group
29018:
29014:
29005:
29001:
28934:Conway group
28896:Cyclic group
28755:
28715:
28679:, New York:
28676:
28654:
28640:
28616:
28596:
28560:
28554:
28540:
28521:
28508:the original
28487:
28481:
28455:
28438:
28416:
28390:
28347:
28324:
28302:
28279:
28241:
28219:
28192:
28186:
28165:
28162:Stewart, Ian
28144:
28140:
28122:
28096:
28077:
28048:
28026:
28022:
27996:
27966:
27943:
27917:
27874:
27844:
27817:
27795:
27775:
27764:, retrieved
27758:
27737:
27712:
27686:
27659:
27630:
27625:Kuga, Michio
27586:
27559:
27553:
27528:
27504:
27482:
27461:
27428:
27412:the original
27407:
27403:
27361:
27343:
27326:
27322:
27316:-ary groups"
27292:
27269:
27242:
27215:
27211:
27186:
27166:, New York:
27163:
27129:
27107:
27080:
27076:
27052:
27026:(2): 64â75,
27023:
27019:
27001:
26994:(7): 736â740
26991:
26985:
26955:, New York:
26952:
26923:
26903:
26883:
26857:
26824:
26804:
26774:
26754:
26732:
26708:
26677:
26665:
26653:
26641:
26629:
26617:
26605:
26593:
26581:
26569:
26557:
26545:
26533:
26521:
26509:
26497:
26481:
26469:
26462:Stewart 2015
26457:
26445:
26433:
26421:
26409:
26397:
26385:
26363:Kuipers 1999
26358:
26346:
26334:
26322:
26310:
26298:
26286:
26274:
26262:
26250:
26235:
26223:
26196:
26184:
26172:
26160:
26148:
26136:
26124:
26112:
26097:class groups
26091:
26079:
26072:Hatcher 2002
26067:
26055:
26043:
26031:
26019:
26007:
25995:
25983:
25956:
25927:
25900:
25888:
25881:Solomon 2018
25876:
25864:
25852:
25840:
25828:
25816:
25809:Wussing 2007
25804:
25797:Kleiner 1986
25792:
25780:
25773:Wussing 2007
25768:
25756:
25749:Kleiner 1986
25744:
25732:
25720:
25713:Kleiner 1986
25708:
25701:Wussing 2007
25696:
25684:
25672:
25660:
25648:
25636:
25624:
25604:
25592:
25568:
25552:
25527:
25514:
25497:
25476:
25467:
25461:
25455:
25451:
25445:
25424:
25414:
25409:
25384:
25371:
25365:
25356:
25343:
25326:
25301:
25288:
25275:
25254:
25229:
25220:
25216:
25210:
25206:
25200:
25190:
25182:
25172:
25155:
25142:
25132:
25126:
25120:
25114:
25108:
25104:
25095:
25090:
24962:Commutative
24927:Associative
24545:Real numbers
24419:
24167:
23937:
23934:
23890:
23800:Associative
23777:Commutative
23732:Commutative
23689:unital magma
23687:Commutative
23666:Unital magma
23642:Commutative
23597:Commutative
23552:Commutative
23487:Semigroupoid
23453:Cancellation
23407:gauge theory
23348:
23203:
23185:
23183:
23132:
23118:
23016:
22972:group object
22969:
22873:
22861:
22828:
22826:
22812:
22790:
22630:
22375:
22364:
22167:
22089:
22023:th power of
21705:
21519:
21510:if it has a
21507:
21505:
21502:Finite group
21475:
21454:
21253:
21252:
21249:Galois group
21229:
21086:
21085:
21075:dimensions.
21009:
21005:
20909:
20896:
20808:are used in
20803:
20710:
20691:
20679:space groups
20671:point groups
20668:
20586:
20585:
20525:
20440:
20180:, the group
20178:field theory
19898:
19446:cyclic group
19445:
19443:
19334:Cyclic group
19005:equals
18343:
18152:
17768:
17766:
17710:
17540:
17538:
17349:
17314:
17185:
17081:
17053:
17024:Cryptography
17021:
17006:
16970:times (with
16912:
16873:
16840:
16809:Cayley graph
16806:
16640:presentation
16638:
16414:
16089:
16082:
16067:epimorphisms
15959:
15571:
15325:factor group
15324:
15320:
15239:. The group
15085:, the coset
14798:
14308:
14301:
14027:intersection
13949:
13784:
13780:
13713:
13421:
13172:
12973:are both in
12804:
12802:
12643:
12632:
12606:is true for
12483:
12433:
12095:
12012:
12010:
11826:
11595:homomorphism
11594:
11592:
11558:
11288:to denote a
11002:
10998:
10996:
10993:
10986:
10803:
10202:
10197:
10187:
10153:
10129:
9915:
9886:
9862:
9564:
9554:
9552:
9246:
9236:
9152:
9054:
9052:
9045:illustrate.
9012:
9008:Jacques Tits
9004:Armand Borel
8984:Hermann Weyl
8939:
8923:Ernst Kummer
8912:
8902:
8875:
8871:finite group
8834:
8823:permutations
8819:Galois group
8784:
8590:
8343:
8342:
8288:
8287:
7655:
7654:
7486:
7485:
7482:
4878:Cayley table
4715:Cayley table
4712:
4637:
4502:
4221:
4182:
4143:
3914:translations
3895:
3814:
3701:
3565:
3490:
3488:
3398:
3350:
3348:
3224:. Formally,
3141:real numbers
3138:
3126:
3121:
3117:
3113:
3110:group axioms
3106:ordered pair
3103:
3038:
2805:
2801:
2460:group axioms
2459:
2265:
2259:
2253:
2245:
2211:
2156:
2015:
1914:
1596:
1408:
1358:Hopf algebra
1296:
1289:Vector space
1254:
1194:
1123:Group theory
1121:
1086: /
1078:
1070:
886:
874:
862:
850:
838:
826:
814:
802:
573:
530:
517:
506:
495:
491:Cyclic group
369:
356:Free product
327:Group action
290:Group theory
285:Group theory
284:
231:group theory
223:Galois group
208:
198:
190:Point groups
147:
136:
89:
79:, it has an
60:
54:
45:Rubik's Cube
36:
32:Group theory
29714:Ring theory
29676:Topic lists
29636:Multivector
29622:Free object
29540:Dot product
29526:Determinant
29512:Linear and
28863:Point group
28858:Space group
28593:Lie, Sophus
28490:(1): 1â44,
27940:Ronan, Mark
27741:, Courier,
27425:Harris, Joe
26949:Artin, Emil
26854:Lang, Serge
26821:Lang, Serge
26751:Hall, G. G.
26586:Warner 1983
26538:Husain 1966
26526:Awodey 2010
26303:Simons 2003
26244:Bishop 1993
26242:. See also
26117:Seress 1997
25857:Mackey 1976
25845:Curtis 2003
25821:Jordan 1870
25761:Cayley 1889
25737:Galois 1908
25490:Frucht 1939
25369:instead of
25319:Weibel 1994
25281:Suzuki 1951
25036:quasi-group
25026:quasi-group
23458:Commutative
23443:Associative
23383:gravitation
23306:denotes an
23206:open subset
23152:local field
23001:small piece
22992:unit circle
22822:classifying
22671:other than
22327:have order
21830:represents
21726:in a group
21467:solvability
21171:dimensional
20900:consist of
20834:(geometric)
20596:geometrical
20562:Point group
20558:Space group
20061:satisfying
20038:, given by
19681:stands for
18836:such that
16876:associating
16413:are called
15271:, read as "
13530:containing
13442:of a group
13285:of a group
13257:provides a
12807:is a group
12482:are called
12435:isomorphism
12015:of a group
11629:to a group
11311:containing
11063:satisfying
10032:. For each
9605:of a group
9143:parentheses
9027:Walter Feit
8988:Ălie Cartan
8976:Issai Schur
8921:. In 1847,
8878:Felix Klein
8807:solvability
3910:reflections
3806:commutative
3669:instead of
3612:is denoted
3537:is denoted
3039:the inverse
3033:is unique (
1343:Lie algebra
1328:Associative
1232:Total order
1222:Semilattice
1196:Ring theory
776:Topological
615:alternating
77:associative
57:mathematics
29749:Categories
29692:Glossaries
29646:Polynomial
29626:Free group
29551:Linear map
29408:Inequality
29175:Loop group
29035:Lie groups
28807:direct sum
28378:See also:
27907:0956.11021
27550:Teller, E.
27396:Frucht, R.
27272:, London:
26692:References
26682:Dudek 2001
26634:Naber 2003
26598:Borel 1991
26574:Milne 1980
26562:Shatz 1972
26502:Ronan 2007
26486:Artin 2018
26426:Artin 1998
26402:Rudin 1990
26390:Serre 1977
26327:Welsh 1989
26216:Ellis 2019
26141:Rosen 2000
25932:Artin 2018
25869:Borel 2001
25725:Smith 1906
25653:Artin 2018
25440:, §15.3.2.
25161:MathSciNet
25118:for every
24997:Structure
24675:Operation
24440:-ary group
23825:quasigroup
23802:quasigroup
23644:quasigroup
23621:Quasigroup
23174:Lie groups
23156:adele ring
21924:copies of
21766:such that
20955:invertible
20818:CD players
20600:analytical
20580:hyperbolic
20552:See also:
20536:commutator
17462:such that
16900:continuity
16085:free group
15966:surjective
13787:cosets of
12484:isomorphic
12162:such that
11693:such that
10309:(that is,
10223:(that is,
9267:have both
9035:Aschbacher
8898:Lie groups
8894:Sophus Lie
8886:hyperbolic
3918:symmetries
3246:is a set,
2898:such that
2721:, one has
2565:, one has
2428:, denoted
2240:Definition
2067:such that
1723:, one has
883:Symplectic
823:Orthogonal
780:Lie groups
687:Free group
412:continuous
351:Direct sum
166:Lie groups
154:symmetries
29418:Operation
28717:MathWorld
28653:(1950) ,
28531:in 1843).
28504:179178038
28367:768477138
27346:, Wiley,
27329:: 15â36,
27274:CRC Press
26490:Lang 2002
26450:Lang 2002
26438:Lang 2002
26305:, §4.2.1.
26267:Dove 2003
26255:Weyl 1950
26228:Weyl 1952
26201:Lang 2002
26189:Lang 2005
26177:Lang 2005
26165:Lang 2005
26153:Lang 2005
26129:Lang 2005
26048:Lang 2002
26036:Lang 2005
26024:Lang 2002
26012:Lang 2005
25988:Lang 2005
25961:Lang 2005
25949:Lang 2002
25920:Lang 2005
25811:, §I.3.4.
25775:, §III.2.
25677:Lang 2005
25665:Lang 2002
25641:Cook 2009
25629:Lang 2005
25609:Hall 1967
25584:Citations
25507:Kuga 1993
25503:monodromy
25390:Lang 2002
25377:Lang 2005
25336:Lang 2002
25239:Lang 2005
25235:Lang 2002
24896:≠
24829:−
24798:−
24745:Identity
24446:Examples
24313:≃
24295:
24282:in which
24207:
24096:⋅
24081:∖
24048:⋅
24033:∖
24000:⋅
23985:∖
23929:Required
23907:Unneeded
23885:Required
23863:Unneeded
23841:Required
23818:Unneeded
23795:Required
23779:semigroup
23772:Unneeded
23756:Semigroup
23750:Required
23727:Unneeded
23705:Required
23682:Unneeded
23660:Required
23637:Unneeded
23615:Required
23592:Unneeded
23570:Required
23547:Unneeded
23525:Unneeded
23503:Unneeded
23481:Unneeded
23387:spacetime
23371:mechanics
23268:≠
23186:Lie group
23180:Lie group
23060:−
23032:⋅
23005:real line
22976:morphisms
22940:→
22911:→
22905:×
22596:of order
22514:×
22340:−
22293:×
21958:⋅
21890:⋅
21852:⏟
21845:⋯
21610:factorial
21453:, but do
21433:−
21393:±
21348:−
21333:±
21327:−
21190:→
21181:ρ
20673:describe
20421:≡
20384:≡
20347:≡
20200:×
19882:…
19849:−
19837:−
19822:−
19813:…
19762:−
19751:⋅
19745:⋅
19728:−
19720:⋅
19712:−
19704:⋅
19696:−
19664:−
19634:⋅
19579:…
19523:−
19507:−
19491:−
19480:…
19304:×
19230:⋅
19213:∖
19168:≡
19156:⋅
18926:−
18920:⋅
18853:≡
18847:⋅
18589:≡
18583:⋅
18472:remainder
18429:−
18308:≡
18111:−
17983:−
17929:−
17863:≡
17761:9 + 4 ⥠1
17632:×
17595:≠
17589:∣
17581:∈
17556:∖
17519:⋅
17475:⋅
17423:⋅
17317:fractions
17311:Rationals
17225:⋅
17167:⋅
17056:rationals
17040:chemistry
16983:≠
16896:proximity
16778:→
16775:⟩
16753:⋅
16718:∣
16706:⟨
16682:ϕ
16623:⟩
16601:⋅
16566:∣
16554:⟨
16532:⟩
16520:⟨
16500:ϕ
16497:
16459:⋅
16415:relations
16401:ϕ
16398:
16248:⟩
16236:⟨
16214:ϕ
16043:∼
16038:→
16030:ϕ
16027:
16013:→
15985:→
15976:ϕ
15744:⋅
15708:⋅
15684:⋅
15383:⋅
15213:−
15193:−
14937:→
14880:∈
14874:∣
14119:∈
14106:⋅
14098:−
13972:partition
13927:∈
13921:∣
13915:⋅
13872:∈
13866:∣
13860:⋅
13676:⋅
13468:generated
13417:subgroups
13336:∈
13330:⋅
13322:−
13155:→
13057:−
13015:⋅
12793:Subgroups
12722:↪
12711:∼
12706:→
12670:→
12656:ϕ
12646:injective
12572:φ
12505:→
12496:φ
12440:bijective
12354:ψ
12344:φ
12262:φ
12252:ψ
12224:ι
12217:ψ
12214:∘
12211:φ
12183:ι
12176:φ
12173:∘
12170:ψ
12147:→
12138:ψ
12115:→
12106:φ
12059:→
12044:ι
11947:−
11933:φ
11919:−
11908:φ
11857:φ
11835:φ
11739:φ
11736:∗
11724:φ
11712:⋅
11703:φ
11678:→
11669:φ
11646:∗
11614:⋅
11569:subgroups
11473:→
11397:∈
11217:⋅
11132:⋅
11120:⋅
11086:⋅
11074:⋅
11051:⋅
10989:semigroup
10936:⋅
10909:⋅
10898:−
10890:⋅
10857:⋅
10849:−
10838:⋅
10822:⋅
10751:−
10743:⋅
10735:−
10722:−
10685:−
10677:⋅
10668:⋅
10660:−
10647:−
10610:−
10602:⋅
10593:⋅
10585:−
10571:⋅
10563:−
10550:−
10513:−
10505:⋅
10496:⋅
10485:−
10477:⋅
10469:−
10456:−
10416:−
10408:⋅
10399:⋅
10381:−
10373:⋅
10332:⋅
10324:−
10272:−
10236:⋅
10113:⋅
10067:→
10013:−
10005:⋅
9971:⋅
9859:bijection
9842:⋅
9796:→
9728:⋅
9720:−
9705:, namely
9682:⋅
9464:⋅
9437:⋅
9428:⋅
9379:⋅
9370:⋅
9335:⋅
9212:⋅
9123:⋅
9114:⋅
9102:⋅
9093:⋅
9078:⋅
9072:⋅
8992:algebraic
8900:in 1884.
8844:θ
8701:∘
8644:∘
8242:∘
8207:∘
8189:∘
8147:∘
8117:∘
8099:∘
8047:∘
8029:∘
8001:∘
7983:∘
7939:∘
7930:∘
7918:∘
7909:∘
7577:∘
7500:∘
7416:∘
4920:∘
4834:∘
4651:∘
4616:and then
4431:diagonals
3906:rotations
3902:congruent
3876:∘
3835:∘
3817:functions
3725:⋅
3713:⋅
3682:⋅
3627:−
3547:−
3465:∖
3438:×
3305:⋅
3122:group law
3076:−
3035:see below
2989:For each
2943:⋅
2909:⋅
2814:For each
2798:see below
2770:⋅
2734:⋅
2611:⋅
2602:⋅
2590:⋅
2581:⋅
2441:⋅
2327:⋅
2191:−
1889:and
1699:and
1503:…
1464:−
1455:−
1446:−
1437:−
1431:…
1353:Bialgebra
1159:Near-ring
1116:Lie group
1084:Semigroup
947:Conformal
835:Euclidean
442:nilpotent
253:) and of
235:subgroups
211:(French:
192:describe
182:spacetime
98:with the
69:operation
47:form the
29765:Symmetry
29732:Category
29442:Variable
29432:Relation
29422:Addition
29398:Function
29384:Equation
29333:K-theory
28772:Subgroup
28675:(2007),
28638:(1906),
28615:(1976),
28595:(1973),
28538:(1870),
28476:(1882),
28454:(2003),
28414:(1889),
28389:(2001),
28345:(2010),
28325:Symmetry
28323:(1952),
28301:(1989),
28277:(1972),
28268:36131259
28240:(1994),
28164:(2015),
28147:(6): 1,
28045:(1977),
27995:(1990),
27942:(2007),
27873:(1999),
27711:(1998),
27627:(1993),
27546:Jahn, H.
27502:(2002),
27460:(1980),
27427:(1991),
27398:(1939),
27162:(1989),
27128:(1991),
26977:(2004),
26951:(1998),
26922:(1996),
26902:(1973),
26882:(1953),
26856:(2005),
26823:(2002),
26803:(1975),
26773:(1996),
26753:(1967),
26707:(2018),
26646:Zee 2010
26464:, §12.1.
26351:Lay 2003
26279:Zee 2010
25785:Lie 1973
25459:, where
25292:See the
25054:See also
24780:Inverse
24485:Integers
24332:groupoid
23926:Required
23923:Required
23920:Required
23917:Required
23904:Required
23901:Required
23898:Required
23895:Required
23882:Unneeded
23879:Required
23876:Required
23873:Required
23860:Unneeded
23857:Required
23854:Required
23851:Required
23838:Required
23835:Unneeded
23832:Required
23829:Required
23815:Required
23812:Unneeded
23809:Required
23806:Required
23792:Unneeded
23789:Unneeded
23786:Required
23783:Required
23769:Unneeded
23766:Unneeded
23763:Required
23760:Required
23747:Required
23744:Required
23741:Unneeded
23738:Required
23724:Required
23721:Required
23718:Unneeded
23715:Required
23702:Unneeded
23699:Required
23696:Unneeded
23693:Required
23679:Unneeded
23676:Required
23673:Unneeded
23670:Required
23657:Required
23654:Unneeded
23651:Unneeded
23648:Required
23634:Required
23631:Unneeded
23628:Unneeded
23625:Required
23612:Unneeded
23609:Unneeded
23606:Unneeded
23603:Required
23589:Unneeded
23586:Unneeded
23583:Unneeded
23580:Required
23567:Required
23564:Required
23561:Required
23558:Unneeded
23554:Groupoid
23544:Required
23541:Required
23538:Required
23535:Unneeded
23531:Groupoid
23522:Unneeded
23519:Required
23516:Required
23513:Unneeded
23500:Unneeded
23497:Unneeded
23494:Required
23491:Unneeded
23478:Unneeded
23475:Unneeded
23472:Unneeded
23469:Unneeded
23448:Identity
23359:Rotation
23346:matrix.
21803:, where
21459:degree 5
21405:sign by
20902:matrices
20786:features
20758:displays
20677:, while
20100:vertices
18267:"modulo
17713:division
17078:Integers
17072:algebras
16894:such as
16289:sending
14280:), then
14244:satisfy
14075:happens
12805:subgroup
12799:Subgroup
12697:′
12666:′
12635:category
12388:for all
12296:for all
11964:for all
11510:function
11313:elements
11244:such as
9561:Division
9059:repeated
8950:(1870).
8880:'s 1872
7376:subgroup
3860:groups,
3858:symmetry
3856:groups,
2966:, where
2469:For all
2343:elements
1520:addition
1411:integers
1189:Lie ring
1154:Semiring
942:Poincaré
787:Solenoid
659:Integers
649:Lattices
624:sporadic
619:Lie type
447:solvable
437:dihedral
422:additive
407:infinite
317:Subgroup
150:geometry
104:infinite
102:form an
96:integers
67:with an
29644: (
29634: (
29500: (
29490: (
29480: (
29470: (
29460: (
29270:History
29265:Outline
29254:Algebra
29202:History
28712:"Group"
28629:0396826
28605:0392459
28585:0863090
28577:2690312
28343:Zee, A.
28260:1269324
28211:1990375
28115:0347778
28069:0450380
28035:1452069
27985:1739433
27899:1697859
27863:2044239
27836:1304906
27678:2014408
27651:1199112
27617:1670862
27595:Bibcode
27564:Bibcode
27451:1153249
27388:3971587
27335:1876783
27261:1075994
27234:1865535
27152:1102012
27099:1826989
27083:: 1â4,
27020:Arkivoc
26892:0054593
26847:1878556
26825:Algebra
26813:0356988
26793:1375019
26763:0219593
26709:Algebra
26528:, §4.1.
26476:, p. 3.
25617:applied
25394:torsion
25321:, §8.2.
25313:of its
25198:unless
25195:
25179:
25096:closure
24911:
24885:
24880:
24852:
24844:
24821:
24813:
24790:
24710:Closed
24669:
24610:
24601:
24579:
24571:
24549:
24541:
24519:
24510:
24488:
24480:
24458:
24407:
24387:
24383:
24363:
24359:
24336:
24328:
24284:
24280:
24260:
24256:
24236:
24194:
24174:
24163:
24143:
24135:
24115:
24111:
24065:
24015:
23969:
23438:Closure
22994:in the
22847:. The
22787:
22759:
22538:
22494:
22367:product
22355:
22329:
22159:
22139:
22135:divides
22045:
22025:
22021:
22001:
21970:
21950:
21948:. (If "
21946:
21926:
21902:
21882:
21801:
21768:
21664:
21644:
21554:
21525:
21476:Modern
21127:
21098:
21053:
21019:
21002:
20982:
20978:
20958:
20891:
20871:
20867:vectors
20766:Ammonia
20662:of the
20652:
20632:
20628:
20608:
20578:of the
20521:
20487:
20463:
20443:
20436:
20403:
20399:
20366:
20362:
20323:
20319:
20286:
20262:
20236:
20150:
20124:
20096:
20063:
20032:
20012:
19988:
19968:
19779:
19683:
19649:
19623:
19438:
19411:
19318:
19284:
19193:
19145:
19141:
19121:
19097:
19077:
19027:
19007:
18941:
18909:
18814:
18794:
18770:
18750:
18746:
18726:
18702:
18682:
18658:
18638:
18614:
18572:
18568:
18530:
18526:
18500:
18496:
18476:
18468:
18448:
18444:
18418:
18394:
18374:
18372:modulo
18366:
18346:
18289:
18269:
18219:
18199:
18148:
18128:
18080:, with
18078:
18031:
17998:
17972:
17944:
17918:
17894:
17852:
17769:modulus
17706:
17678:
17646:
17617:
17541:nonzero
17496:
17464:
17440:
17402:
17398:
17376:
17283:
17248:
17144:
17106:
17064:modules
17050:Numbers
17036:physics
16998:
16972:
16802:
16698:
16694:
16674:
16635:
16546:
16484:
16419:
16385:
16358:
16226:
16206:
16063:
16002:
15780:
15673:
15669:
15631:
15317:
15297:
15295:modulo
15293:
15273:
15269:
15241:
15237:
15171:
15144:
15116:
15083:
15051:
14841:
14821:
14783:
14697:
14693:
14664:
14584:
14555:
14407:
14387:
14340:
14311:
14278:
14246:
14182:
14162:
14158:
14138:
14134:
14080:
13996:
13976:
13970:form a
13945:
13892:
13833:
13813:
13809:
13789:
13777:
13757:
13698:
13641:
13637:
13612:
13552:
13532:
13464:
13444:
13413:
13393:
13307:
13287:
13283:
13263:
13251:
13175:
13142:
13122:
13098:
13078:
13074:
13039:
12995:
12975:
12917:
12897:
12873:
12853:
12849:
12829:
12783:
12763:
12759:
12739:
12628:
12608:
12584:
12564:
12520:
12488:
12430:
12410:
12242:
12203:
12006:
11986:
11898:
11849:
11822:
11802:
11554:
11534:
11454:
11434:
11384:
11364:
11360:
11340:
11336:
11316:
11177:
11157:
11153:
11109:
10353:
10311:
10257:
10225:
10178:
10158:
10054:
10034:
10030:
9994:
9958:
9938:
9911:
9891:
9743:
9707:
9703:
9671:
9627:
9607:
9233:
9195:
8813:of its
8775:History
8740:
8683:
8622:
8593:
8505:
8476:
8472:
8443:
8439:
8410:
8406:
8377:
8373:
8348:
8318:
8293:
7893:
7873:
7869:
7849:
7825:
7805:
7761:
7732:
7728:
7708:
7704:
7684:
7680:
7660:
7650:
7621:
7557:
7537:
7471:
7442:
7438:
7396:
7374:form a
7372:
7345:
7341:
7314:
7310:
7283:
7279:
7254:
4812:
4783:
4779:
4750:
4746:
4719:
4708:
4688:
4594:
4565:
4497:
4468:
4464:
4435:
4427:
4398:
4394:
4365:
4358:
4331:
4300:
4273:
3959:
3930:
3850:
3824:
3802:
3782:
3697:
3671:
3644:
3614:
3590:
3570:
3562:
3539:
3515:
3495:
3222:
3199:
3166:
3144:
3120:or the
3093:
3063:
3011:
2991:
2964:
2932:
2856:
2836:
2791:
2759:
2755:
2723:
2719:
2699:
2629:
2567:
2563:
2543:
2539:
2519:
2515:
2495:
2491:
2471:
2456:
2430:
2426:
2406:
2339:
2319:
2315:
2295:
2234:
2214:
2206:
2183:
2133:
2101:
2045:
2025:
2008:
1976:
1911:
1891:
1787:
1725:
1721:
1701:
1677:
1657:
1649:
1627:
1598:closure
1564:
1544:
1320:Algebra
1312:Algebra
1217:Lattice
1208:Lattice
937:Lorentz
859:Unitary
758:Lattice
698:PSL(2,
432:abelian
343:(Semi-)
132:
112:
110:called
29658:, ...)
29628:, ...)
29555:Matrix
29502:Vector
29492:theory
29482:theory
29478:Module
29472:theory
29462:theory
29301:Scheme
28977:22..24
28929:22..24
28925:11..12
28756:Groups
28687:
28661:
28627:
28603:
28583:
28575:
28502:
28462:
28401:
28365:
28355:
28331:
28309:
28287:
28266:
28258:
28248:
28226:
28209:
28172:
28129:
28113:
28103:
28084:
28067:
28057:
28033:
28003:
27983:
27973:
27954:
27928:
27905:
27897:
27887:
27861:
27851:
27834:
27824:
27803:
27782:
27766:14 May
27745:
27721:
27697:
27676:
27666:
27649:
27639:
27615:
27605:
27535:
27516:
27489:
27470:
27449:
27439:
27386:
27376:
27350:
27333:
27299:
27280:
27259:
27249:
27232:
27193:
27174:
27150:
27140:
27114:
27097:
27060:
27008:
26963:
26930:
26912:795613
26910:
26890:
26868:
26845:
26835:
26811:
26791:
26781:
26761:
26740:
26719:
26103:; see
25484:; see
25465:is in
25436:. See
25398:module
25375:. See
25334:, see
25317:, see
25046:monoid
25031:monoid
25016:monoid
25006:monoid
25001:monoid
24577:
24547:
24517:
24415:stacks
23942:. The
23940:monoid
23847:Monoid
23286:where
23199:smooth
22829:simple
22820:. But
22792:simple
22161:. The
21508:finite
21149:on an
20908:. The
20855:, and
20776:Cubane
20706:phonon
20660:tiling
20576:tiling
20564:, and
20532:center
20401:, and
19621:means
19594:where
19450:powers
14843:, and
13835:, are
13704:Cosets
12637:, the
11575:, and
11561:subset
11488:means
11362:, and
7475:
7392:
7388:
7380:
7343:, and
3922:square
3912:, and
3864:, and
2258:,
1566:, the
1348:Graded
1279:Module
1270:Module
1169:Domain
1088:Monoid
792:Circle
723:SL(2,
612:cyclic
576:-group
427:cyclic
402:finite
397:simple
381:kernel
214:groupe
176:. The
29596:Trace
29521:Basis
29468:Group
29458:Field
29279:Areas
29190:Sp(â)
29187:SU(â)
29081:Sp(n)
29075:SU(n)
29063:SO(n)
29051:SL(n)
29045:GL(n)
28798:Semi-
28573:JSTOR
28500:S2CID
28207:JSTOR
28019:(PDF)
27918:Modes
27319:(PDF)
27220:arXiv
26982:(PDF)
25557:Up to
25482:graph
25396:of a
25362:field
25332:units
25264:ÎŒÎżÏÏÎź
25260:Greek
25082:Notes
23891:Group
23599:magma
23576:Magma
23363:space
23017:Some
21904:" to
21516:order
21471:roots
21447:cubic
20711:Such
20582:plane
19143:, as
18155:clock
17060:rings
14409:, if
14024:empty
14000:union
13785:right
13710:Coset
12432:. An
11508:is a
11386:, or
11003:right
9857:is a
9039:proof
8933:into
8815:roots
7384:coset
3898:plane
3322:field
3320:is a
1621:is a
1314:-like
1272:-like
1210:-like
1179:Field
1137:-like
1111:Magma
1079:Group
1073:-like
1071:Group
976:Sp(â)
973:SU(â)
386:image
219:roots
209:group
90:Many
63:is a
61:group
29591:Rank
29571:Norm
29488:Ring
29184:O(â)
29069:U(n)
29057:O(n)
28938:1..3
28685:ISBN
28659:ISBN
28460:ISBN
28399:ISBN
28363:OCLC
28353:ISBN
28329:ISBN
28307:ISBN
28285:ISBN
28264:OCLC
28246:ISBN
28224:ISBN
28170:ISBN
28127:ISBN
28101:ISBN
28082:ISBN
28055:ISBN
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27780:ISBN
27768:2021
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27695:ISBN
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27024:2015
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26961:ISBN
26928:ISBN
26908:OCLC
26866:ISBN
26833:ISBN
26779:ISBN
26738:ISBN
26717:ISBN
26099:and
25615:and
25613:pure
25572:See
25400:and
25245:and
25159:The
25148:2002
25124:and
24992:Yes
24989:Yes
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24713:Yes
24451:Set
24425:-ary
23734:loop
23711:Loop
23367:time
23365:and
23326:-by-
23228:-by-
23097:and
23047:and
22990:The
22757:and
22697:and
21449:and
20980:-by-
20865:Two
20794:The
20768:, NH
20534:and
17810:and
17070:and
17042:and
16917:are
16898:and
16336:and
15960:The
15629:and
15609:are
15026:and
13890:and
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13581:and
13371:and
13037:and
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12011:The
11780:and
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10192:and
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9025:and
9013:The
9006:and
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7706:and
7535:and
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4264:the
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3349:The
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2012:Zero
1974:and
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970:O(â)
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241:and
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28565:doi
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15982:G
15979::
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15938:D
15910:4
15905:D
15883:R
15879:/
15873:4
15868:D
15846:}
15841:3
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15833:,
15828:2
15824:r
15820:,
15815:1
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15803:d
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15796:{
15793:=
15790:R
15768:R
15765:=
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15759:)
15753:v
15748:f
15738:v
15733:f
15729:(
15726:=
15723:R
15717:v
15712:f
15705:R
15699:v
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15690:=
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15657:R
15651:v
15646:f
15642:=
15639:U
15617:R
15597:R
15593:/
15587:4
15582:D
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15513:U
15490:U
15469:R
15448:R
15425:U
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15360:R
15356:/
15350:4
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15305:N
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15253:/
15249:G
15225:N
15221:)
15216:1
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15205:(
15201:=
15196:1
15189:)
15185:N
15182:g
15179:(
15157:N
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15128:/
15124:G
15102:N
15099:=
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15093:e
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15068:)
15065:h
15062:g
15059:(
15037:N
15034:h
15014:N
15011:g
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14988:g
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14948:N
14944:/
14940:G
14934:G
14914:N
14910:/
14906:G
14886:}
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14868:g
14865:{
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14635:=
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14565:D
14541:}
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14260:=
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14114:2
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14050:=
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13984:G
13958:H
13933:}
13930:H
13924:h
13918:g
13912:h
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13906:=
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13875:H
13869:h
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13854:{
13851:=
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13845:g
13821:g
13797:H
13765:g
13743:H
13723:H
13684:2
13680:r
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13661:=
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13624:d
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13567:2
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13189:{
13186:=
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12837:G
12815:H
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12747:H
12725:G
12719:H
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12673:G
12663:G
12659::
12616:H
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12550:G
12530:H
12508:H
12502:G
12499::
12470:H
12450:G
12418:H
12396:h
12376:h
12373:=
12368:)
12363:)
12360:h
12357:(
12349:(
12324:G
12304:g
12284:g
12281:=
12276:)
12271:)
12268:g
12265:(
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12220:=
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12179:=
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12144:H
12141::
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12109::
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12062:G
12056:G
12053::
12048:G
12023:G
11994:G
11972:a
11950:1
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11939:a
11936:(
11930:=
11927:)
11922:1
11915:a
11911:(
11884:H
11880:1
11876:=
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11868:G
11864:1
11860:(
11824:.
11810:G
11788:b
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11748:)
11745:b
11742:(
11733:)
11730:a
11727:(
11721:=
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11715:b
11709:a
11706:(
11681:H
11675:G
11672::
11649:)
11643:,
11640:H
11637:(
11617:)
11611:,
11608:G
11605:(
11556:.
11542:Y
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11299:X
11276:}
11273:z
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11264:,
11261:x
11258:{
11255:=
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11220:)
11214:,
11211:G
11208:(
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11165:e
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11138:=
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11126:=
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11117:e
11095:e
11092:=
11089:e
11083:f
11080:=
11077:e
11071:e
11031:}
11028:f
11025:,
11022:e
11019:{
11016:=
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10960:f
10957:=
10939:f
10933:e
10930:=
10912:f
10906:)
10901:1
10894:f
10887:f
10884:(
10881:=
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10860:f
10852:1
10845:f
10841:(
10835:f
10832:=
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10819:f
10777:e
10774:=
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10747:f
10738:1
10731:)
10725:1
10718:f
10714:(
10711:=
10693:)
10688:1
10681:f
10674:e
10671:(
10663:1
10656:)
10650:1
10643:f
10639:(
10636:=
10618:)
10613:1
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10599:)
10596:f
10588:1
10581:f
10577:(
10574:(
10566:1
10559:)
10553:1
10546:f
10542:(
10539:=
10521:)
10516:1
10509:f
10502:f
10499:(
10493:)
10488:1
10481:f
10472:1
10465:)
10459:1
10452:f
10448:(
10445:(
10442:=
10424:)
10419:1
10412:f
10405:f
10402:(
10396:e
10393:=
10384:1
10377:f
10370:f
10341:e
10338:=
10335:f
10327:1
10320:f
10297:f
10275:1
10268:f
10245:f
10242:=
10239:f
10233:e
10211:e
10166:a
10140:a
10116:a
10110:x
10090:x
10070:G
10064:G
10042:a
10016:1
10009:a
10002:b
9980:b
9977:=
9974:a
9968:x
9946:b
9924:a
9899:a
9873:a
9845:x
9839:a
9819:x
9799:G
9793:G
9773:G
9753:a
9731:b
9723:1
9716:a
9691:b
9688:=
9685:x
9679:a
9657:G
9637:x
9615:G
9593:b
9573:a
9533:)
9529:c
9526:=
9523:b
9515:e
9511:(
9504:c
9501:=
9490:)
9486:a
9478:b
9474:(
9467:c
9461:e
9458:=
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9434:)
9431:a
9425:b
9422:(
9419:=
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9396:c
9392:(
9385:)
9382:c
9376:a
9373:(
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9364:=
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9345:(
9338:e
9332:b
9329:=
9322:b
9295:c
9275:b
9255:a
9221:f
9218:=
9215:f
9209:e
9206:=
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9181:f
9161:e
9129:)
9126:c
9120:b
9117:(
9111:a
9108:=
9105:c
9099:)
9096:b
9090:a
9087:(
9084:=
9081:c
9075:b
9069:a
8856:1
8853:=
8848:n
8757:4
8752:D
8725:d
8720:f
8716:=
8710:h
8705:f
8696:1
8692:r
8666:c
8661:f
8657:=
8652:1
8648:r
8638:h
8633:f
8608:4
8603:D
8574:1
8570:r
8547:3
8543:r
8520:2
8516:r
8490:c
8485:f
8457:d
8452:f
8424:v
8419:f
8391:h
8386:f
8360:d
8357:i
8328:a
8305:d
8302:i
8270:.
8265:1
8261:r
8257:=
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8246:f
8236:d
8231:f
8227:=
8220:)
8215:2
8211:r
8201:v
8196:f
8192:(
8183:d
8178:f
8168:1
8164:r
8160:=
8155:2
8151:r
8142:3
8138:r
8134:=
8125:2
8121:r
8114:)
8108:v
8103:f
8093:d
8088:f
8084:(
8060:)
8055:2
8051:r
8041:v
8036:f
8032:(
8023:d
8018:f
8014:=
8009:2
8005:r
7998:)
7992:v
7987:f
7977:d
7972:f
7968:(
7948:,
7945:)
7942:c
7936:b
7933:(
7927:a
7924:=
7921:c
7915:)
7912:b
7906:a
7903:(
7881:a
7857:c
7835:b
7813:c
7791:b
7771:a
7747:4
7742:D
7716:c
7692:b
7668:a
7635:c
7630:f
7607:,
7601:c
7596:f
7592:=
7586:h
7581:f
7572:3
7568:r
7545:b
7523:a
7503:b
7497:a
7456:d
7451:f
7424:3
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7410:h
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7358:3
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7296:1
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7266:d
7263:i
7235:d
7232:i
7208:2
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7180:3
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7152:1
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7093:d
7088:f
7063:v
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7033:c
7028:f
7003:c
6998:f
6972:2
6968:r
6945:d
6942:i
6918:1
6914:r
6890:3
6886:r
6861:v
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6831:c
6826:f
6801:h
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6771:d
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6741:d
6736:f
6710:1
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6682:3
6678:r
6655:d
6652:i
6628:2
6624:r
6599:d
6594:f
6569:v
6564:f
6539:c
6534:f
6509:h
6504:f
6479:h
6474:f
6448:3
6444:r
6420:1
6416:r
6392:2
6388:r
6365:d
6362:i
6337:c
6332:f
6307:h
6302:f
6277:d
6272:f
6247:v
6242:f
6217:v
6212:f
6185:v
6180:f
6155:h
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6125:c
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6095:d
6090:f
6066:2
6062:r
6038:1
6034:r
6011:d
6008:i
5984:3
5980:r
5956:3
5952:r
5925:d
5920:f
5895:c
5890:f
5865:v
5860:f
5835:h
5830:f
5806:1
5802:r
5779:d
5776:i
5752:3
5748:r
5724:2
5720:r
5696:2
5692:r
5665:h
5660:f
5635:v
5630:f
5605:d
5600:f
5575:c
5570:f
5547:d
5544:i
5520:3
5516:r
5492:2
5488:r
5464:1
5460:r
5436:1
5432:r
5405:c
5400:f
5375:d
5370:f
5345:h
5340:f
5315:v
5310:f
5286:3
5282:r
5258:2
5254:r
5230:1
5226:r
5203:d
5200:i
5177:d
5174:i
5147:c
5142:f
5117:d
5112:f
5087:h
5082:f
5057:v
5052:f
5028:3
5024:r
5000:2
4996:r
4972:1
4968:r
4945:d
4942:i
4895:4
4890:D
4862:.
4856:d
4851:f
4847:=
4842:3
4838:r
4828:h
4823:f
4797:d
4792:f
4764:h
4759:f
4732:3
4728:r
4696:a
4674:b
4654:a
4648:b
4624:b
4604:a
4580:4
4575:D
4544:h
4539:f
4516:1
4512:r
4482:c
4477:f
4449:d
4444:f
4433:(
4412:h
4407:f
4379:v
4374:f
4344:3
4340:r
4315:2
4311:r
4286:1
4282:r
4241:c
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4202:d
4197:f
4163:h
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4125:v
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4087:3
4083:r
4052:2
4048:r
4017:1
4013:r
3983:d
3980:i
3945:4
3940:D
3838:g
3832:f
3790:G
3768:b
3748:a
3728:a
3722:b
3719:=
3716:b
3710:a
3685:b
3679:a
3657:b
3654:a
3630:1
3623:x
3600:x
3578:1
3550:x
3525:x
3503:0
3474:}
3471:0
3468:{
3461:R
3433:R
3410:R
3384:R
3362:R
3333:R
3308:)
3302:,
3299:+
3296:,
3292:R
3288:(
3268:)
3265:+
3262:,
3258:R
3254:(
3233:R
3210:b
3207:a
3182:b
3179:+
3176:a
3153:R
3095:.
3079:1
3072:a
3049:a
3021:b
2999:a
2974:e
2952:e
2949:=
2946:a
2940:b
2918:e
2915:=
2912:b
2906:a
2886:G
2866:b
2844:G
2822:a
2793:.
2779:a
2776:=
2773:e
2767:a
2743:a
2740:=
2737:a
2731:e
2707:G
2685:a
2665:G
2645:e
2631:.
2617:)
2614:c
2608:b
2605:(
2599:a
2596:=
2593:c
2587:)
2584:b
2578:a
2575:(
2551:G
2527:c
2503:b
2479:a
2444:b
2438:a
2414:G
2392:G
2372:b
2352:a
2303:G
2277:G
2222:+
2208:.
2194:a
2169:a
2143:b
2121:0
2118:=
2115:a
2112:+
2109:b
2087:0
2084:=
2081:b
2078:+
2075:a
2055:b
2033:a
1996:a
1993:=
1990:0
1987:+
1984:a
1962:a
1959:=
1956:a
1953:+
1950:0
1930:a
1919:.
1899:c
1877:b
1857:a
1837:c
1817:b
1797:a
1775:)
1772:c
1769:+
1766:b
1763:(
1760:+
1757:a
1754:=
1751:c
1748:+
1745:)
1742:b
1739:+
1736:a
1733:(
1709:c
1687:b
1665:a
1636:Z
1609:+
1583:b
1580:+
1577:a
1552:b
1530:a
1506:}
1500:,
1497:4
1494:,
1491:3
1488:,
1485:2
1482:,
1479:1
1476:,
1473:0
1470:,
1467:1
1461:,
1458:2
1452:,
1449:3
1443:,
1440:4
1434:,
1428:{
1425:=
1421:Z
1388:e
1381:t
1374:v
1047:e
1040:t
1033:v
929:8
927:E
921:7
919:E
913:6
911:E
905:4
903:F
897:2
895:G
889:)
887:n
877:)
875:n
865:)
863:n
853:)
851:n
841:)
839:n
829:)
827:n
817:)
815:n
805:)
803:n
745:)
732:Z
720:)
707:Z
683:)
670:Z
661:(
574:p
539:Q
531:n
528:D
518:n
515:A
507:n
504:S
496:n
493:Z
120:1
51:.
34:.
20:)
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