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10783:{\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}}}
29:
20730:
3957:
29717:
16819:
16895:
9532:{\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}}}
287:
10971:
20737:
8273:
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21454:, 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
8064:
4585:. 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
20578:
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
13703:
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
8943:(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
13704:
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
10966:{\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}}}
21601:
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
8059:
23362:. 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
16914:
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)
15767:
14540:
21219:
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
17602:
17047:, 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
25591:
13242:, 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
11188:
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,
16789:
17637:. 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
14770:
8268:{\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}}}
10189:, 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.
19234:
23366:, 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
19581:
12724:
1505:
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3124:
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.
22987:
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
9128:
16050:
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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
21372:
11747:
222: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
24098:
24050:
24002:
21214:
24323:
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:
22525:
21216:
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.
28404:
20678:
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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14649:
13238:
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12283:
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19884:
10804:
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2225:, 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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13689:. 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.
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12229:
12188:
7952:
134:. 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.
20940:
20858:(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
19180:
8577:
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.
2239:, 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.
18874:
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11997:. 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
18329:
28466:
3307:
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17700:
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
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3439:
14885:
11885:
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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
22479:
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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
16531:
16247:
15656:
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11094:
4548:
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
23114:. 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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3801:. 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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234:. 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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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
3181:
1582:
27007:
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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In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein,
1404:
27300:
23924:
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.
16898:
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.
9053:
7752:, 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
21597:
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 (
28176:
19187:
22948:, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids
21822:
3905:. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called
17726:â fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.
7551:
4806:
21302:
20693:
and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft
18761:, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer
1375:
1034:
28496:
26975:
19675:
16538:
7887:
27388:
23152:
19797:
19794:
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
12677:
9142:
The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements
28368:
18358:
17023:, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as
15994:
25152:
database of mathematics publications lists 1,779 research papers on group theory and its generalizations written in 2020 alone. See
27702:
20474:
are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to
18828:
13633:
8675:
8616:
3635:. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition,
28428:
21003:. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the
16852:
of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains
11687:
28879:
28735:
24057:
24009:
23961:
23927:
For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a
22797:
17935:, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent
9019:
592:
251:
21165:
18283:
28677:
28651:
28452:
28391:
28345:
28321:
28299:
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28216:
28162:
28119:
28093:
28074:
28047:
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27918:
27877:
27841:
27814:
27793:
27772:
27735:
27711:
27687:
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27595:
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26998:
26953:
26920:
26858:
26825:
26771:
26730:
26709:
22486:
20655:
by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on.
20512:, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.
4492:
These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example,
21458:) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and
20674:. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of
22415:
22080:
21478:
17431:, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no
15774:
14835:
14598:
13167:
542:
26720:
24602:
21245:
were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the
12328:
12236:
22079:
More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups:
13247:
11553:
2559:
8778:
developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of
28335:
27543:
24276:
11892:
8523:
are their own inverse, because performing them twice brings the square back to its original orientation. The rotations
8054:{\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}
7641:). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table.
1368:
1027:
537:
17490:
17394:
17138:
3444:
29233:
28277:
27993:
27460:
27289:
27050:
23128:
17310:
126:
The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers,
23240:
19891:
18023:
15163:
12195:
12154:
3804:
Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are
27389:"Herstellung von Graphen mit vorgegebener abstrakter Gruppe [Construction of graphs with prescribed group]"
22076:
is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.
20903:
20704:
has found further application in elementary particle physics, where its occurrence is related to the appearance of
19137:
11230:
64:
of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is
28423:
27451:
21481:âa precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
18564:
17098:
15913:
is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the
7388:
27748:
List of papers reviewed on MathSciNet on "Group theory and its generalizations" (MSC code 20), published in 2020
22265:
20172:
19276:
19239:
16989:), because the loop cannot be smoothly deformed across the hole, so each class of loops is characterized by its
16411:
14072:
13884:
13829:
10344:), one can show that every left inverse is also a right inverse of the same element as follows. Indeed, one has
28666:
The
Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory
28539:
27409:
25282:
24188:
23111:
20784:
9026:
and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of
953:
28530:
Traité des substitutions et des équations algébriques [Study of
Substitutions and Algebraic Equations]
23147:; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the
19955:
is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are
8580:
In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in
1640:. The following properties of integer addition serve as a model for the group axioms in the definition below.
29748:
29671:
28383:
28171:
27421:
26817:
20701:
15762:{\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}
9042:
Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under
8960:
8881:
had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas,
8787:
7608:
that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (
3272:
1361:
1020:
22886:
17609:
8865:
Geometry was a second field in which groups were used systematically, especially symmetry groups as part of
3415:
29519:
27148:
26420:
25470:
22821:
17936:
17177:
form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example,
14535:{\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}
11841:
9018:, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the
3557:
whenever the group operation is notated as multiplication; in this case, the identity is typically denoted
28509:
22785:
22448:
17844:
28609:
28150:
25064:
22989:
16884:
16504:
16220:
15623:
11582:
Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A
11101:
11055:
8902:
7472:
Given this set of symmetries and the described operation, the group axioms can be understood as follows.
1230:
637:
451:
29195:
27576:
Quaternions and
Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality
17597:{\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}
15850:
15564:
15327:
13301:
10983:
The same result can be obtained by only assuming the existence of a right identity and a right inverse.
29692:
29450:
28728:
28410:
27579:
27498:
27413:
20822:
20810:
20592:
20166:
17240:
17020:
12027:
3694:
369:
243:
24409:
Finally, it is possible to generalize any of these concepts by replacing the binary operation with an
23156:
22559:
22388:
22236:
22160:
21662:
21605:
21571:
21517:
21090:
21011:
20479:
17992:
16635:
16252:
16167:
16138:
16082:
15920:
15887:
14656:
14547:
14303:
13573:
12989:
8734:
8585:
8468:
8435:
8402:
8369:
7724:
7647:: The associativity axiom deals with composing more than two symmetries: Starting with three elements
7613:
7434:
7101:
7071:
7041:
7011:
6981:
6839:
6809:
6779:
6749:
6719:
6577:
6547:
6517:
6487:
6457:
6315:
6285:
6255:
6225:
6195:
6163:
6133:
6103:
6073:
5903:
5873:
5843:
5813:
5643:
5613:
5583:
5553:
5383:
5353:
5323:
5293:
5125:
5095:
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5035:
4872:
4775:
4742:
4557:
4522:
4460:
4427:
4390:
4357:
4219:
4180:
4141:
4103:
3922:
3238:
27061:
22851:
The gap between the classification of simple groups and the classification of all groups lies in the
21566:
20681:
Group theory helps predict the changes in physical properties that occur when a material undergoes a
20315:
16863:
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by
16193:
is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism
15950:
14015:
13456:
12480:
12090:
11653:
10303:
8944:
3482:
whenever the group operation is notated as addition; in this case, the identity is typically denoted
1717:
27541:(1937), "Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy",
23139:. Other locally compact topological groups include the group of points of an algebraic group over a
22921:
21082:
are linear representations in which the group acts on a vector space, such as the three-dimensional
20430:
Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element
17135:, has been described above. The integers, with the operation of multiplication instead of addition,
3877:
is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.
29697:
29687:
29396:
28790:
28433:
21251:
21004:
20802:
20395:
20358:
16919:
and are the identity element of the fundamental group. A loop which wraps around the missing point
14021:
13604:
12640:
9986:
9699:
8915:
8827:
8340:
8285:
7246:
7215:
6925:
6635:
6345:
5991:
5759:
5527:
5183:
5157:
4925:
3963:
3898:
3842:
835:
569:
446:
334:
259:
146:
24571:
24541:
24511:
24480:
24450:
23937:
20821:
are well-behaved. Geometric properties that remain stable under group actions are investigated in
17707:
17368:
17277:
17074:
16833:
15960:
12122:
9187:
3393:
3367:
3345:
3316:
3216:
3136:
1619:
715:
690:
653:
29702:
29406:
29296:
29190:
27936:
25377:
For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See
23178:
22810:
22802:
20580:
19791:
19310:
18957:
18737:, and therefore that the modular product is nonzero. The identity element is represented by
18146:, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on
16628:
16067:
11236:
8769:
1321:
19:
This article is about basic notions of groups in mathematics. For a more advanced treatment, see
18953:
18901:
17456:
17206:
17011:
In addition to the above theoretical applications, many practical applications of groups exist.
16478:
16379:
13031:
11589:
11192:
10217:
9952:
9663:
2924:
2890:
2751:
2715:
29584:
29509:
29430:
29420:
29386:
29336:
29075:
27234:, Lecture Notes in Mathematics (in French), vol. 1441, Berlin, New York: Springer-Verlag,
26789:
26759:
25336:
The transition from the integers to the rationals by including fractions is generalized by the
25252:
23956:(including zero) under addition form a monoid, as do the nonzero integers under multiplication
23497:
23387:
23363:
22949:
20644:
18962:
17701:
17305:
16997:
13162:
In the example of symmetries of a square, the identity and the rotations constitute a subgroup
12623:
11498:
10997:
10177:
The group axioms for identity and inverses may be "weakened" to assert only the existence of a
9047:
3902:
3894:
3846:
3805:
2235:
The axioms for a group are short and natural ... Yet somehow hidden behind these axioms is the
985:
775:
255:
80:
27400:
27040:
26715:, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article.
23013:
20643:
and the composition of these operations follows the group law. For example, an element of the
19615:
14918:
12556:
11819:
11448:
10094:
9823:
7481:
4706:
lists the results of all such compositions possible. For example, rotating by 270° clockwise (
4632:
3816:
3663:
2422:
29743:
29579:
29573:
29346:
29226:
29061:
29013:
28721:
28471:
25529:
24877:
23441:
23426:
21760:
20894:
20841:
20818:
20588:
20520:
20278:
20141:. The group operation is multiplication of complex numbers. In the picture, multiplying with
20088:
20055:
18522:
16964:
11301:
9003:
8968:
8952:
8815:
8791:
3890:
2331:
1586:
859:
235:
37:
27725:
25600:, p. 1, §1.1: "The idea of a group is one which pervades the whole of mathematics both
23382:. The full symmetry group of Minkowski space, i.e., including translations, is known as the
23039:
21942:
21874:
19643:
16993:
around the missing point. The resulting group is isomorphic to the integers under addition.
15367:
13136:
11378:
11035:
10251:
10048:
9777:
9236:
The group axioms also imply that the inverse of each element is unique: Let a group element
9131:
4904:
3860:
3606:
3055:
2311:
29543:
29466:
29424:
29367:
29289:
29037:
29031:
28795:
28617:
28593:
28573:
28544:
28248:
28103:
28057:
28023:
27973:
27887:
27851:
27824:
27666:
27639:
27605:
27583:
27552:
27439:
27376:
27323:
27249:
27222:
27156:
27140:
27087:
26908:
26880:
26835:
26801:
26781:
26751:
25562:
25386:
24399:
23446:
23431:
23343:
23339:
22953:
22806:
22663:
22588:
22532:
22421:
22189:
21964:
21795:
21689:
21079:
20890:
20855:
20837:
20749:
20744:
20579:
theory (see below). Conceptually, group theory can be thought of as the study of symmetry.
19586:
19403:
19354:
17060:
17052:
16880:
16853:
16666:
16350:
16301:
16198:
16111:
15043:
14238:
13546:
13405:
12938:
12911:
11621:
10994:
inverse (or vice versa) is not sufficient to define a group. For example, consider the set
10182:
8897:
8553:
8526:
8499:
7337:
7306:
7275:
7187:
7159:
7131:
6951:
6897:
6869:
6689:
6661:
6607:
6427:
6399:
6371:
6045:
6017:
5963:
5935:
5785:
5731:
5703:
5675:
5499:
5471:
5443:
5415:
5265:
5237:
5209:
5007:
4979:
4951:
4711:
4495:
4323:
4294:
4265:
4066:
4031:
3996:
3809:
2093:
2059:
1968:
1934:
1308:
1300:
1272:
1267:
1258:
1215:
1157:
799:
787:
405:
339:
239:
65:
28067:
The Words of
Mathematics: An Etymological Dictionary of Mathematical Terms Used in English
27895:
25565:
for an example where symmetry greatly reduces the complexity analysis of physical systems.
21377:
15077:
218:
and geometry, the group notion was generalized and firmly established around 1870. Modern
8:
29600:
29524:
29470:
29446:
29390:
29331:
29316:
29306:
28770:
28419:
28128:
Solomon, Ronald (2018), "The classification of finite simple groups: A progress report",
27534:
27358:
25605:
25494:
action on the vector space of solutions of the differential equations is considered. See
25474:
25350:
25299:
24844:
24403:
23407:
22837:
22751:
22321:
21656:
20777:
20568:
20554:
20228:
20116:
19999:
elements is isomorphic to this group. A second example for cyclic groups is the group of
19438:
19021:
18492:
18410:
18213:
18092:
17964:
17910:
17670:
17640:
17180:
16888:
15233:
15108:
14890:
9134:
can be inserted anywhere within such a series of terms, parentheses are usually omitted.
8878:
8874:
8779:
3310:
3160:
3117:
1561:
1326:
1316:
1167:
1050:
374:
269:
189:
28528:
27727:
Combinatorial Group Theory: Presentations of Groups in Terms of
Generators and Relations
27587:
27556:
26967:
25533:
24813:
24782:
24328:
21469:
generalizes the above type of Galois groups by shifting to field theory and considering
18608:
18261:
16784:{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}
15138:
15018:
14995:
14972:
14354:
3638:
3531:
3191:
2175:
29559:
29476:
29380:
29311:
29279:
29258:
29253:
28970:
28786:
28780:
28669:
28624:
28561:
28488:
28267:
28233:, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press,
28195:
28036:
27618:
27208:
27192:
26963:
26945:
25422:
25418:
25337:
25320:
25056:
24379:
24355:
24252:
24228:
24166:
24135:
24127:
24107:
23857:
23386:. By the above, it plays a pivotal role in special relativity and, by implication, for
23379:
23318:
23298:
23278:
23220:
23200:
23089:
23069:
22866:
22729:
22709:
22689:
22643:
22623:
22368:
22299:
22216:
22131:
22106:
22086:
22059:
22039:
22017:
21993:
21918:
21896:
21738:
21718:
21698:
21636:
21548:
21417:
21397:
21246:
21141:
21121:
21047:
20974:
20950:
20863:
20689:
materials, where the change from a paraelectric to a ferroelectric state occurs at the
20663:
20624:
20600:
20542:
20457:
20435:
20256:
20206:
20144:
20094:
20033:
20004:
19982:
19960:
19938:
19773:
19444:
19381:
19334:
19113:
19091:
19069:
19047:
18999:
18935:
18879:
18808:
18786:
18764:
18742:
18718:
18696:
18674:
18652:
18630:
18468:
18440:
18388:
18366:
18338:
18239:
18191:
18169:
18149:
18120:
18072:
17942:
17888:
17822:
17802:
17782:
17762:
17745:
17735:
17434:
17342:
17048:
17005:
16942:
16922:
16872:
16813:
16328:
16281:
15914:
15601:
15539:
15518:
15497:
15474:
15453:
15432:
15409:
15388:
15317:
15289:
15265:
14952:
14813:
14791:
14765:{\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}
14578:
14421:
14401:
14379:
14334:
14272:
14216:
14196:
14176:
14154:
14130:
13994:
13988:
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13781:
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13502:
13482:
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13436:
13414:
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13343:
13279:
13255:
13114:
13092:
13070:
12967:
12889:
12867:
12845:
12821:
12799:
12755:
12731:
12627:
12600:
12578:
12534:
12514:
12454:
12434:
12402:
12380:
12308:
12288:
12066:
12007:
11978:
11956:
11794:
11772:
11752:
11577:
11561:
11526:
11504:
11480:
11426:
11404:
11356:
11332:
11308:
11283:
11171:
11149:
10980:) one may have, for example, that a left identity is not necessarily a right identity.
10281:
10195:
10150:
10124:
10074:
10026:
9930:
9908:
9883:
9857:
9803:
9757:
9737:
9641:
9621:
9599:
9577:
9557:
9279:
9259:
9239:
9165:
9145:
9027:
9023:
8803:
8312:
7865:
7841:
7819:
7797:
7775:
7755:
7700:
7676:
7652:
7529:
7507:
4680:
4658:
4608:
4588:
4260:
rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by
4254:
3854:
3774:
3752:
3732:
3584:
3562:
3509:
3487:
3033:
3005:
2983:
2958:
2870:
2850:
2828:
2806:
2691:
2669:
2649:
2629:
2535:
2511:
2487:
2463:
2398:
2376:
2356:
2336:
2287:
2261:
2206:
2153:
2127:
2039:
2017:
1914:
1883:
1861:
1841:
1821:
1801:
1781:
1693:
1671:
1649:
1593:
1536:
1514:
1132:
1123:
1081:
359:
331:
207:
182:
174:
104:
28505:
27859:
8929:
The convergence of these various sources into a uniform theory of groups started with
8783:
193:
29753:
29650:
29620:
29490:
29400:
29351:
29284:
28893:
28696:
28673:
28647:
28492:
28448:
28440:
28387:
28351:
28341:
28317:
28295:
28273:
28252:
28234:
28212:
28158:
28115:
28089:
28070:
28043:
28031:
27989:
27959:
27940:
27914:
27873:
27837:
27810:
27789:
27768:
27731:
27707:
27697:
27683:
27652:
27625:
27591:
27521:
27502:
27492:
27475:
27456:
27446:
27425:
27362:
27336:
27285:
27266:
27235:
27179:
27160:
27126:
27100:
27046:
27031:
26994:
26949:
26916:
26896:
26854:
26821:
26767:
26726:
26705:
25382:
25303:
25050:
24594:
23403:
23136:
23115:
23007:
23001:
22945:
22852:
22841:
21451:
21221:
21069:
20943:
20690:
20675:
16907:
16903:
16876:
16867:
groups to them and studying the properties of the corresponding groups. For example,
12423:
11278:
9007:
8907:
3850:
3121:
2256:
2244:
764:
607:
501:
53:
29132:
27783:
23383:
22186:
of symmetries of a square is a finite group of order 8. In this group, the order of
20817:
of a prescribed form, giving group-theoretic criteria for when solutions of certain
16868:
930:
166:
29720:
29605:
29274:
29219:
29200:
29137:
29112:
29104:
29096:
29088:
29080:
29067:
29049:
29043:
28553:
28517:
28480:
28462:
28185:
28137:
27910:
27891:
27560:
27354:
27196:
27073:
27026:
27016:
26888:
26868:
25601:
25521:
25295:
23453:
23436:
22845:
21504:
21447:
21439:
20845:
20825:
20761:. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.
20682:
20652:
17032:
17001:
16857:
12622:
The collection of all groups, together with the homomorphisms between them, form a
12619:, provided that any specific elements mentioned in the statement are also renamed.
9011:
8988:
8948:
8940:
8870:
8795:
3886:
2279:
2005:
1611:
1152:
915:
907:
899:
891:
883:
871:
811:
751:
741:
583:
525:
400:
162:
69:
57:
27078:
22959:
This way of defining groups lends itself to generalizations such as the notion of
1177:
29640:
29630:
29549:
29376:
29301:
29153:
28987:
28856:
28765:
28613:
28589:
28569:
28263:
28244:
28099:
28053:
28019:
27969:
27902:
27883:
27847:
27820:
27662:
27635:
27601:
27435:
27372:
27319:
27245:
27218:
27136:
27122:
27121:, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York:
27083:
27021:
26876:
26850:
26831:
26797:
26777:
26747:
25507:
This was crucial to the classification of finite simple groups, for example. See
24503:
23587:
23564:
23367:
22916:
22825:
21509:
21474:
21470:
21083:
21065:
20705:
20671:
17360:
17044:
16823:
14291:
8984:
2146:
1244:
1238:
1225:
1205:
1196:
1162:
1099:
999:
992:
978:
935:
823:
746:
576:
490:
430:
310:
131:
127:
73:
33:
28004:
27933:
Symmetry and the
Monster: The Story of One of the Greatest Quests of Mathematics
23394:
are central to the modern description of physical interactions with the help of
22262:
etc. is 2. Both orders divide 8, as predicted by
Lagrange's theorem. The groups
20697:
mode, a vibrational lattice mode that goes to zero frequency at the transition.
29655:
29644:
29563:
29414:
29326:
29158:
29000:
28953:
28946:
28939:
28932:
28904:
28871:
28841:
28836:
28826:
28775:
28524:
28226:
27721:
27679:
26739:
25390:
24563:
24443:
24158:
23932:
23722:
23699:
23399:
23391:
23182:
23148:
22829:
22355:
22151:
21455:
21435:
20814:
20559:
20536:
20028:
19229:{\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}
16990:
16916:
16801:
14783:
14065:
12864:, with the same operation. Concretely, this means that the identity element of
11565:
8956:
8930:
8799:
8775:
4549:
3184:
1286:
1006:
942:
632:
612:
549:
514:
435:
425:
410:
395:
349:
326:
227:
158:
150:
96:
24424:). With the proper generalization of the group axioms, this gives a notion of
20515:
The study of finitely generated abelian groups is quite mature, including the
17015:
relies on the combination of the abstract group theory approach together with
8976:
29737:
29569:
29534:
29168:
29127:
29055:
28976:
28961:
28909:
28861:
28816:
28601:
28400:
28355:
28287:
27802:
27538:
27488:
27384:
26701:
26693:
25525:
25404:
25248:
25029:
25009:
24999:
23901:
23119:
22984:
22833:
21466:
21225:
20798:
20794:
20686:
20685:, for example, from a cubic to a tetrahedral crystalline form. An example is
20024:
18143:
17304:
The desire for the existence of multiplicative inverses suggests considering
17274:, which is a rational number, but not an integer. Hence not every element of
14193:
may or may not be the same as its right cosets. If they are (that is, if all
13724:
determines left and right cosets, which can be thought of as translations of
13243:
10178:
8893:
8819:
3798:
2236:
1904:
1172:
1137:
1094:
925:
847:
681:
554:
420:
215:
153:
of the object, and the transformations of a given type form a general group.
28256:
20519:; and reflecting this state of affairs, many group-related notions, such as
14651:
and similarly for the other elements of the group. (In fact, in the case of
4519:
sends a point to its rotation 90° clockwise around the square's center, and
29553:
29486:
29460:
29341:
29122:
28922:
28884:
28831:
28821:
28811:
28752:
28700:
28661:
28639:
28447:, History of Mathematics, Providence, R.I.: American Mathematical Society,
28375:
28309:
27981:
27565:
27114:
26089:
23677:
23654:
23475:
23395:
23132:
22960:
22780:
21490:
21237:
20999:
20886:
20648:
20564:
19322:
17056:
17012:
16797:
15989:
factors canonically as a quotient homomorphism followed by an isomorphism:
12775:
10192:
In particular, assuming associativity and the existence of a left identity
8996:
8992:
8972:
8923:
8911:
8859:
8807:
4866:
4703:
3098:
3094:
1346:
1277:
1111:
780:
479:
468:
415:
390:
385:
344:
315:
278:
247:
231:
219:
211:
92:
61:
20:
27455:(2nd ed.), Reading, MA: Addison-Wesley Publishing, pp. 588â596,
26900:
26123:
24185:
in which every morphism is an isomorphism: given such a category, the set
23193:
A standard example is the general linear group introduced above: it is an
18713:
ensures that the usual product of two representatives is not divisible by
16830:
Examples and applications of groups abound. A starting point is the group
15660:. The group operation on the quotient is shown in the table. For example,
12427:
is a homomorphism that has an inverse homomorphism; equivalently, it is a
29624:
29610:
29528:
29514:
29480:
28851:
28846:
28445:
Pioneers of
Representation Theory: Frobenius, Burnside, Schur, and Brauer
28331:
27613:
26085:
26042:
25024:
25014:
24533:
24425:
23371:
23194:
23140:
22980:
20667:
20659:
20550:
20546:
16055:
9015:
8964:
8919:
8866:
8811:
3794:
3129:
1336:
1331:
1220:
1210:
1184:
178:
45:
28628:
27213:
26066:
25647:
25548:
isomorphism, there are about 49 billion groups of order up to 2000. See
22820:
groups was a major achievement in contemporary group theory. There are
20850:
20563:
The (2,3,7) triangle group, a hyperbolic reflection group, acts on this
29634:
29614:
29539:
29163:
28581:
28565:
28484:
28199:
27928:
26937:
26842:
26809:
25149:
23813:
23790:
23632:
23609:
23187:
23144:
22975:
21500:
20524:
19576:{\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}
16818:
16661:
by generators and relations, because the first isomorphism theorem for
16073:
15954:
8882:
3793:. If this additional condition holds, then the operation is said to be
1086:
947:
675:
142:
27066:
Electronic
Research Announcements of the American Mathematical Society
13543:. In the example of symmetries of a square, the subgroup generated by
11568:. These are the analogues that take the group structure into account.
9006:'s 1960â61 Group Theory Year brought together group theorists such as
2009:
of addition because adding it to any integer returns the same integer.
1500:{\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}
29023:
28705:
28142:
27424:, Readings in Mathematics, vol. 129, New York: Springer-Verlag,
27262:
27191:
26820:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
26228:
25491:
23767:
23744:
23375:
23359:
23168:
22993:
21598:
21159:
20806:
20722:
20162:
19761:{\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}
18460:
17028:
17016:
14012:
12634:
12428:
10977:
9847:
9050:
applications of the associativity axiom show that the unambiguity of
8886:
4803:). Using the above symbols, highlighted in blue in the Cayley table:
3841:; then the identity may be denoted id. In the more specific cases of
1556:
1341:
1147:
1104:
1072:
768:
170:
154:
28557:
28511:
Manuscrits de Ăvariste Galois [Ăvariste Galois' Manuscripts]
28190:
26273:
26171:
25268:
However, a group is not determined by its lattice of subgroups. See
16894:
16617:{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }
29410:
29372:
29321:
28760:
27474:(2nd ed.), West Sussex, England: John Wiley & Sons, Ltd.,
26456:
26432:
25402:
The stated property is a possible definition of prime numbers. See
24320:
23542:
23519:
23347:
22964:
21459:
20584:
19184:. Hence all group axioms are fulfilled. This example is similar to
16911:
12787:
11557:
9123:{\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}
7364:
4419:
3906:
1508:
1142:
305:
223:
138:
88:
27809:, vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag,
27259:
Universal Algebra and Applications in Theoretical Computer Science
22233:
that this element generates. The order of the reflection elements
20729:
9130:
generalizes to more than three factors. Because this implies that
7383:
yellow (last column), respectively. The result of the composition
29242:
24473:
22813:
all finite groups is a problem considered too hard to be solved.
22123:
20831:
20754:
18210:, as shown in the illustration. This is expressed by saying that
17024:
16864:
12719:{\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}
8980:
1399:
647:
561:
262:
as geometric objects, has become an active area in group theory.
84:
28:
28340:(second ed.), Princeton, N.J.: Princeton University Press,
26321:
17363:. The set of all such irreducible fractions is commonly denoted
12674:
factors canonically as an isomorphism followed by an inclusion,
25034:
25019:
25004:
24994:
24989:
23928:
23835:
20764:
20715:
20694:
19327:
16045:{\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}
14887:
denotes its set of cosets. Then there is a unique group law on
11549:
8896:. Certain abelian group structures had been used implicitly in
8824:
On the theory of groups, as depending on the symbolic equation
8782:
of degree higher than 4. The 19th-century French mathematician
4212:
4173:
4059:
4024:
3910:
3097:
of a set and a binary operation on this set that satisfies the
1076:
286:
26968:"The status of the classification of the finite simple groups"
26429:, Chapter VI (see in particular p. 273 for concrete examples).
26297:
26054:
25438:
The additive notation for elements of a cyclic group would be
22992:, such as the red arc in the figure, looks like a part of the
21862:{\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}
20619:
if every group element can be associated to some operation on
17939:. Modular addition, defined in this way for the integers from
17740:
17604:
does form an abelian group under multiplication, also denoted
16910:
of loops, where loops are considered equivalent if one can be
16856:. These groups are predecessors of important constructions in
9037:
4134:
4096:
3989:
3956:
123:(these properties characterize the integers in a unique way).
27176:
The JahnâTeller Effect in C60 and Other Icosahedral Complexes
25982:
25545:
24410:
23351:
20993:
matrices with real entries. Its subgroups are referred to as
20527:, describe the extent to which a given group is not abelian.
19163:
18584:
15316:. The quotient group can alternatively be characterized by a
13698:
7601:{\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}
7372:
4856:{\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}
4770:) is the same as performing a reflection along the diagonal (
4699:"). This is the usual notation for composition of functions.
27229:
26796:(2nd ed.), Lexington, Mass.: Xerox College Publishing,
26766:(3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc.,
26652:
26072:
25875:
25083:
Some authors include an additional axiom referred to as the
23338:
Lie groups are of fundamental importance in modern physics:
19461:. In multiplicative notation, the elements of the group are
13680:{\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}
10172:
8722:{\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}
8663:{\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}
7884:. These two ways must give always the same result, that is,
3893:
if one can be changed into the other using a combination of
27060:
Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001),
27006:
26640:
26362:
26360:
26111:
25469:
More rigorously, every group is the symmetry group of some
23355:
21659:. Parallel to the group of symmetries of the square above,
17697:, therefore the axiom of the inverse element is satisfied.
7792:
into a single symmetry, then to compose that symmetry with
4352:
reflections about the horizontal and vertical middle line (
2000:
83:
are groups endowed with other properties. For example, the
29211:
26183:
26135:
25904:
25902:
25887:
25319:
Elements which do have multiplicative inverses are called
24102:, and likewise adjoining inverses to any (abelian) monoid
22828:" that do not belong to any of the families. The largest
21367:{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
20736:
19271:
that have a multiplicative inverse. These groups, denoted
17885:. Every integer is equivalent to one of the integers from
12551:
simply by renaming its elements according to the function
11742:{\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)}
8329:
when composed with it either on the left or on the right.
28713:
28542:(1986), "The evolution of group theory: A brief survey",
28467:"Gruppentheoretische Studien (Group-theoretical studies)"
25791:
25755:
25671:
25193:
is abelian, because of the ambiguity of whether it means
24093:{\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )}
24045:{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}
23997:{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}
22791:
21688:
can also be interpreted as the group of symmetries of an
19236:
above: it consists of exactly those elements in the ring
18142:
A familiar example is addition of hours on the face of a
14992:
is a homomorphism. Explicitly, the product of two cosets
14331:, the group of symmetries of a square, with its subgroup
28380:
Essays in the History of Lie Groups and Algebraic Groups
28174:(1951), "On the lattice of subgroups of finite groups",
26444:
26396:
26357:
26285:
25933:
25931:
25929:
25611:
22360:
fundamental theorem of finitely generated abelian groups
21209:{\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )}
20517:
fundamental theorem of finitely generated abelian groups
18459:, replaces the usual product by its representative, the
17051:
and fields. Further abstract algebraic concepts such as
17000:
employs geometric concepts, for example in the study of
13133:, indeed form a group. In this case, the inclusion map
8967:'s papers. The theory of Lie groups, and more generally
3441:
whose underlying set is the set of nonzero real numbers
157:
appear in symmetry groups in geometry, and also in the
26237:
26159:
26147:
26018:
25994:
25970:
25960:
25958:
25943:
25899:
25579:
23118:, such as the field of complex numbers or the field of
22520:{\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}
22358:
of finite cyclic groups; this statement is part of the
16796:
A presentation of a group can be used to construct the
10793:
Similarly, the left identity is also a right identity:
2447:, such that the following three requirements, known as
149:: The symmetries of an object form a group, called the
28209:
Foundations of Differentiable Manifolds and Lie Groups
27620:
Galois' Dream: Group Theory and Differential Equations
26592:
26508:
26006:
25779:
25731:
25659:
25635:
17251:
14149:. Similar considerations apply to the right cosets of
14125:, i.e., when the two elements differ by an element of
3116:
A group and its underlying set are thus two different
27230:
Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990),
27059:
26604:
26030:
25926:
25549:
24880:
24847:
24816:
24785:
24605:
24574:
24544:
24514:
24483:
24453:
24382:
24358:
24331:
24279:
24255:
24231:
24191:
24169:
24138:
24110:
24060:
24012:
23964:
23940:
23321:
23301:
23281:
23243:
23223:
23203:
23155:. A generalization used in algebraic geometry is the
23092:
23072:
23042:
23016:
22924:
22889:
22869:
22754:
22732:
22712:
22692:
22666:
22646:
22626:
22591:
22562:
22535:
22489:
22451:
22424:
22391:
22371:
22324:
22302:
22268:
22239:
22219:
22192:
22163:
22134:
22109:
22089:
22062:
22042:
22020:
21996:
21967:
21945:
21921:
21899:
21877:
21825:
21798:
21763:
21741:
21721:
21701:
21665:
21639:
21608:
21574:
21551:
21520:
21420:
21400:
21380:
21305:
21254:
21168:
21144:
21124:
21093:
21050:
21014:
20977:
20953:
20906:
20866:
20627:
20603:
20482:
20460:
20438:
20398:
20361:
20318:
20281:
20259:
20231:
20209:
20175:
20147:
20119:
20097:
20058:
20036:
20007:
19985:
19963:
19941:
19894:
19800:
19776:
19678:
19646:
19618:
19589:
19467:
19447:
19406:
19384:
19357:
19337:
19279:
19242:
19190:
19140:
19116:
19094:
19072:
19050:
19024:
19002:
18965:
18938:
18904:
18882:
18831:
18811:
18789:
18767:
18745:
18721:
18699:
18677:
18655:
18633:
18611:
18567:
18525:
18495:
18471:
18443:
18413:
18391:
18369:
18341:
18286:
18264:
18242:
18216:
18194:
18172:
18152:
18123:
18095:
18075:
18026:
17995:
17967:
17945:
17913:
17891:
17847:
17825:
17805:
17785:
17765:
17710:
17673:
17643:
17612:
17538:
17493:
17459:
17437:
17397:
17371:
17345:
17313:
17280:
17243:
17209:
17203:
is an integer, but the only solution to the equation
17183:
17141:
17101:
17077:
17004:. Further branches crucially applying groups include
16967:
16945:
16925:
16836:
16693:
16669:
16638:
16541:
16507:
16481:
16414:
16382:
16353:
16331:
16304:
16284:
16255:
16223:
16201:
16170:
16141:
16114:
16085:
15997:
15963:
15923:
15890:
15853:
15840:{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}
15777:
15668:
15626:
15604:
15567:
15542:
15521:
15500:
15477:
15456:
15435:
15412:
15391:
15370:
15330:
15292:
15268:
15236:
15166:
15141:
15111:
15080:
15046:
15021:
14998:
14975:
14955:
14921:
14893:
14838:
14816:
14794:
14692:
14684:, the cosets generated by reflections are all equal:
14659:
14644:{\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}
14601:
14581:
14550:
14444:
14424:
14404:
14382:
14357:
14337:
14306:
14275:
14241:
14219:
14199:
14179:
14157:
14133:
14075:
14024:
13997:
13971:
13945:
13887:
13832:
13808:
13784:
13752:
13730:
13710:
13636:
13607:
13599:
consists of these two elements, the identity element
13576:
13549:
13527:
13505:
13485:
13465:
13439:
13417:
13388:
13366:
13346:
13304:
13282:
13258:
13233:{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}
13170:
13139:
13117:
13095:
13073:
13034:
12992:
12970:
12941:
12914:
12892:
12870:
12848:
12824:
12802:
12758:
12734:
12680:
12643:
12603:
12581:
12559:
12537:
12517:
12483:
12457:
12437:
12405:
12383:
12331:
12311:
12291:
12239:
12198:
12157:
12125:
12093:
12069:
12030:
12010:
11981:
11959:
11895:
11844:
11822:
11797:
11775:
11755:
11690:
11656:
11624:
11592:
11529:
11507:
11483:
11451:
11429:
11407:
11381:
11359:
11335:
11311:
11286:
11239:
11195:
11174:
11152:
11104:
11058:
11038:
11000:
10802:
10353:
10306:
10284:
10254:
10220:
10198:
10153:
10127:
10097:
10077:
10051:
10029:
9989:
9955:
9933:
9911:
9886:
9860:
9826:
9806:
9780:
9760:
9740:
9702:
9666:
9644:
9624:
9602:
9580:
9560:
9305:
9282:
9262:
9242:
9190:
9182:
of a group are equal, because the group axioms imply
9168:
9148:
9056:
8936:
Traité des substitutions et des équations algébriques
8830:
8737:
8678:
8619:
8588:
8556:
8529:
8502:
8471:
8438:
8405:
8372:
8343:
8315:
8288:
8067:
7955:
7890:
7868:
7844:
7822:
7800:
7778:
7758:
7727:
7703:
7679:
7655:
7616:
7554:
7532:
7510:
7484:
7437:
7391:
7340:
7309:
7278:
7249:
7218:
7190:
7162:
7134:
7104:
7074:
7044:
7014:
6984:
6954:
6928:
6900:
6872:
6842:
6812:
6782:
6752:
6722:
6692:
6664:
6638:
6610:
6580:
6550:
6520:
6490:
6460:
6430:
6402:
6374:
6348:
6318:
6288:
6258:
6228:
6198:
6166:
6136:
6106:
6076:
6048:
6020:
5994:
5966:
5938:
5906:
5876:
5846:
5816:
5788:
5762:
5734:
5706:
5678:
5646:
5616:
5586:
5556:
5530:
5502:
5474:
5446:
5418:
5386:
5356:
5326:
5296:
5268:
5240:
5212:
5186:
5160:
5128:
5098:
5068:
5038:
5010:
4982:
4954:
4928:
4907:
4875:
4809:
4778:
4745:
4714:
4683:
4661:
4635:
4611:
4591:
4560:
4525:
4498:
4463:
4430:
4393:
4360:
4326:
4297:
4268:
4222:
4183:
4144:
4106:
4069:
4034:
3999:
3966:
3925:
3863:
3819:
3777:
3755:
3735:
3697:
3666:
3641:
3609:
3587:
3565:
3534:
3512:
3490:
3447:
3418:
3396:
3370:
3348:
3319:
3275:
3241:
3219:
3194:
3163:
3139:
3058:
3036:
3008:
2986:
2961:
2927:
2893:
2873:
2853:
2831:
2809:
2754:
2718:
2694:
2672:
2652:
2632:
2562:
2538:
2514:
2490:
2466:
2425:
2401:
2379:
2359:
2339:
2314:
2290:
2264:
2209:
2178:
2156:
2130:
2096:
2062:
2042:
2020:
1971:
1937:
1917:
1886:
1864:
1844:
1824:
1804:
1784:
1720:
1696:
1674:
1652:
1622:
1596:
1564:
1539:
1517:
1407:
718:
693:
656:
107:
27801:
27720:
27331:
Eliel, Ernest; Wilen, Samuel; Mander, Lewis (1994),
26746:, American Elsevier Publishing Co., Inc., New York,
26532:
26327:
26200:
26198:
26048:
25955:
25815:
25046:
24651:{\displaystyle \mathbb {Z} /n\mathbb {Z} =\{0,1,2\}}
19331:
The 6th complex roots of unity form a cyclic group.
13408:
is important in understanding the group as a whole.
12370:{\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}
12278:{\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}
11144:. This structure does have a left identity (namely,
8798:
of a particular polynomial equation in terms of the
7942:{\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}
28369:
Historically important publications in group theory
27706:(2nd ed.), Berlin, New York: Springer-Verlag,
27651:, Universitext, Berlin, New York: Springer-Verlag,
27646:
27282:
Structure and Dynamics: An Atomic View of Materials
26568:
26462:
26345:
25914:
25863:
25695:
25683:
24006:. Adjoining inverses of all elements of the monoid
22858:
19879:{\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}
13499:and their inverses. It is the smallest subgroup of
3880:
2786:
2611:{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}
28406:The Collected Mathematical Papers of Arthur Cayley
28035:
27745:
27617:
27173:
26664:
26616:
26580:
26520:
26279:
26261:
26099:
25839:
25827:
25803:
25743:
25719:
24892:
24861:
24825:
24794:
24650:
24582:
24552:
24522:
24491:
24461:
24388:
24364:
24340:
24309:
24261:
24237:
24215:
24175:
24144:
24116:
24092:
24044:
23996:
23948:
23370:. The latter servesâin the absence of significant
23327:
23307:
23287:
23267:
23229:
23209:
23098:
23078:
23058:
23028:
22936:
22907:
22875:
22768:
22738:
22718:
22698:
22678:
22652:
22632:
22604:
22577:
22548:
22519:
22473:
22437:
22406:
22377:
22336:
22308:
22288:
22254:
22225:
22205:
22178:
22140:
22115:
22095:
22068:
22048:
22026:
22002:
21980:
21951:
21927:
21905:
21883:
21861:
21811:
21782:
21747:
21727:
21707:
21680:
21645:
21623:
21589:
21557:
21535:
21426:
21406:
21386:
21366:
21291:
21208:
21150:
21130:
21108:
21056:
21034:
20983:
20959:
20934:
20872:
20633:
20609:
20502:
20466:
20444:
20417:
20380:
20343:
20300:
20265:
20243:
20215:
20195:
20153:
20131:
20103:
20077:
20042:
20013:
19991:
19969:
19947:
19927:
19878:
19782:
19760:
19662:
19630:
19602:
19575:
19453:
19419:
19390:
19370:
19343:
19299:
19263:
19228:
19174:
19122:
19100:
19078:
19056:
19036:
19008:
18986:
18944:
18922:
18888:
18868:
18817:
18795:
18773:
18751:
18727:
18705:
18683:
18661:
18639:
18617:
18595:
18549:
18507:
18477:
18449:
18425:
18397:
18375:
18347:
18323:
18270:
18248:
18228:
18200:
18178:
18158:
18129:
18107:
18081:
18059:
18010:
17979:
17951:
17925:
17897:
17875:
17831:
17811:
17791:
17771:
17718:
17687:
17657:
17627:
17596:
17517:
17477:
17443:
17421:
17379:
17351:
17329:
17288:
17264:
17227:
17195:
17165:
17125:
17085:
17043:Many number systems, such as the integers and the
16979:
16951:
16931:
16844:
16783:
16675:
16653:
16616:
16525:
16493:
16465:
16394:
16366:
16337:
16317:
16290:
16270:
16241:
16207:
16185:
16156:
16127:
16100:
16044:
15981:
15938:
15905:
15876:
15839:
15761:
15650:
15610:
15590:
15548:
15527:
15506:
15483:
15462:
15441:
15418:
15397:
15376:
15353:
15298:
15274:
15250:
15218:
15150:
15125:
15095:
15064:
15030:
15007:
14984:
14961:
14941:
14907:
14879:
14822:
14800:
14764:
14674:
14643:
14587:
14565:
14534:
14430:
14410:
14388:
14366:
14343:
14321:
14281:
14259:
14225:
14205:
14185:
14163:
14139:
14115:
14056:
14003:
13977:
13951:
13926:
13871:
13814:
13790:
13758:
13736:
13716:
13679:
13618:
13591:
13562:
13533:
13511:
13491:
13471:
13445:
13423:
13394:
13372:
13352:
13332:
13288:
13264:
13232:
13151:
13123:
13101:
13079:
13055:
13018:
12976:
12954:
12927:
12898:
12876:
12854:
12830:
12808:
12764:
12740:
12718:
12666:
12609:
12587:
12565:
12543:
12523:
12501:
12463:
12443:
12411:
12389:
12369:
12317:
12297:
12277:
12223:
12182:
12143:
12111:
12075:
12055:
12016:
11987:
11965:
11945:
11879:
11828:
11803:
11781:
11761:
11741:
11674:
11642:
11610:
11535:
11513:
11489:
11469:
11435:
11413:
11393:
11365:
11341:
11317:
11292:
11269:
11213:
11180:
11168:), and each element has a right inverse (which is
11158:
11134:
11088:
11044:
11024:
10965:
10782:
10334:
10290:
10270:
10238:
10204:
10159:
10133:
10109:
10083:
10063:
10035:
10011:
9973:
9939:
9917:
9892:
9866:
9838:
9812:
9792:
9766:
9746:
9724:
9684:
9650:
9630:
9608:
9586:
9566:
9531:
9288:
9268:
9248:
9214:
9174:
9154:
9122:
8849:
8752:
8721:
8662:
8603:
8569:
8542:
8515:
8486:
8453:
8420:
8387:
8354:
8321:
8299:
8267:
8053:
7941:
7874:
7850:
7828:
7806:
7784:
7764:
7742:
7709:
7685:
7661:
7631:
7600:
7538:
7516:
7496:
7452:
7419:
7353:
7322:
7291:
7260:
7229:
7203:
7175:
7147:
7119:
7089:
7059:
7029:
6999:
6967:
6939:
6913:
6885:
6857:
6827:
6797:
6767:
6737:
6705:
6677:
6649:
6623:
6595:
6565:
6535:
6505:
6475:
6443:
6415:
6387:
6359:
6333:
6303:
6273:
6243:
6213:
6181:
6151:
6121:
6091:
6061:
6033:
6005:
5979:
5951:
5921:
5891:
5861:
5831:
5801:
5773:
5747:
5719:
5691:
5661:
5631:
5601:
5571:
5541:
5515:
5487:
5459:
5431:
5401:
5371:
5341:
5311:
5281:
5253:
5225:
5197:
5171:
5143:
5113:
5083:
5053:
5023:
4995:
4967:
4939:
4913:
4890:
4855:
4793:
4760:
4727:
4689:
4667:
4647:
4617:
4597:
4575:
4540:
4511:
4478:
4445:
4408:
4375:
4339:
4310:
4281:
4237:
4198:
4159:
4121:
4082:
4047:
4012:
3977:
3940:
3917:The elements of the symmetry group of the square,
3869:
3831:
3783:
3761:
3741:
3721:
3678:
3650:
3625:
3593:
3571:
3543:
3518:
3496:
3467:
3433:
3404:
3378:
3356:
3327:
3301:
3261:
3227:
3203:
3175:
3147:
3074:
3042:
3014:
2992:
2967:
2945:
2911:
2879:
2859:
2837:
2815:
2772:
2736:
2700:
2678:
2658:
2638:
2610:
2544:
2520:
2496:
2472:
2437:
2407:
2385:
2365:
2345:
2320:
2296:
2270:
2215:
2187:
2162:
2136:
2114:
2080:
2048:
2026:
1989:
1955:
1923:
1892:
1870:
1850:
1830:
1810:
1790:
1768:
1702:
1680:
1658:
1630:
1602:
1576:
1545:
1523:
1499:
726:
701:
664:
113:
28417:
28177:Transactions of the American Mathematical Society
26556:
26544:
26484:
26408:
26384:
26372:
26309:
26195:
25851:
25707:
25623:
24398:. Groupoids arise in topology (for instance, the
24310:{\displaystyle \operatorname {Hom} (x,x)\simeq G}
24247:, one can build a small category with one object
21477:of a polynomial. This theory establishesâvia the
21162:real vector space is simply a group homomorphism
19044:above, the inverse of the element represented by
16939:times cannot be deformed into a loop which wraps
13298:to be a subgroup: it is sufficient to check that
11946:{\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}
11221:is not a group, since it lacks a right identity.
9137:
8892:The third field contributing to group theory was
29735:
27330:
26722:Mathematicians: An Outer View of the Inner World
26628:
26303:
26249:
26210:
23398:. An important example of a gauge theory is the
23244:
23237:matrices, because it is given by the inequality
22963:in a category. Briefly, this is an object with
19088:, and the inverse of the element represented by
18966:
18517:, the four group elements can be represented by
17518:{\displaystyle \left(\mathbb {Q} ,\cdot \right)}
17422:{\displaystyle \left(\mathbb {Q} ,\cdot \right)}
17166:{\displaystyle \left(\mathbb {Z} ,\cdot \right)}
14011:and two left cosets are either equal or have an
8858:(1854) gives the first abstract definition of a
3691:The definition of a group does not require that
3468:{\displaystyle \mathbb {R} \smallsetminus \{0\}}
3120:. To avoid cumbersome notation, it is common to
2249:Mathematicians: An Outer View of the Inner World
214:. After contributions from other fields such as
28005:"An Introduction to Computational Group Theory"
27901:
26725:, Princeton, N.J.: Princeton University Press,
26658:
26333:
25767:
21374:Each solution can be obtained by replacing the
8991:(from the late 1930s) and later by the work of
169:is a Lie group consisting of the symmetries of
60:that associates an element of the set to every
28694:
28588:(in German), New York: Johnson Reprint Corp.,
27256:
27195:; Delgado Friedrichs, Olaf; Huson, Daniel H.;
26646:
25528:. A more involved example is the action of an
20832:General linear group and representation theory
19928:{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}
18060:{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}
15219:{\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}
12224:{\displaystyle \varphi \circ \psi =\iota _{H}}
12183:{\displaystyle \psi \circ \varphi =\iota _{G}}
11548:When studying sets, one uses concepts such as
9031:
7860:, then to compose the resulting symmetry with
7478:: Composition is a binary operation. That is,
3950:. Vertices are identified by color or number.
3105:of the group, and the operation is called the
1398:One of the more familiar groups is the set of
1393:
1388:
29227:
28729:
27956:Elementary Number Theory and its Applications
27869:Grundlehren der mathematischen Wissenschaften
27724:; Karrass, Abraham; Solitar, Donald (2004) ,
27353:, Oxford University Press, pp. 441â444,
27349:Ellis, Graham (2019), "6.4 Triangle groups",
27199:(2001), "On three-dimensional space groups",
22952:and is used in computing with groups and for
22529:. But there exist nonabelian groups of order
20935:{\displaystyle \mathrm {GL} (n,\mathbb {R} )}
19175:{\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}
18435:. The group operation, multiplication modulo
16822:A periodic wallpaper pattern gives rise to a
16072:Every group is isomorphic to a quotient of a
14542:(highlighted in green in the Cayley table of
12356:
12337:
12264:
12245:
1369:
1028:
188:The concept of a group arose in the study of
28012:Notices of the American Mathematical Society
27867:
27408:
26976:Notices of the American Mathematical Society
26366:
24645:
24627:
24078:
24072:
24030:
24024:
23982:
23976:
23177:is a group that also has the structure of a
22726:can sometimes be reduced to questions about
22673:
22667:
22354:Any finite abelian group is isomorphic to a
18869:{\displaystyle a\cdot b\equiv 1{\pmod {p}},}
17744:The hours on a clock form a group that uses
16917:can be smoothly contracted to a single point
16807:
16763:
16694:
16611:
16542:
16520:
16508:
16236:
16224:
15834:
15784:
14874:
14853:
14529:
14469:
13921:
13897:
13866:
13842:
13227:
13177:
11264:
11246:
11019:
11007:
8934:
3462:
3456:
1494:
1416:
201:
28606:The Theory of Unitary Group Representations
28064:
27647:Kurzweil, Hans; Stellmacher, Bernd (2004),
27257:Denecke, Klaus; Wismath, Shelly L. (2002),
26873:Introduction to the Theory of Finite Groups
25256:
25165:One usually avoids using fraction notation
25125:. This condition is subsumed by requiring "
23106:vary only a little. Such groups are called
18596:{\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}
17704:by other than zero is possible, such as in
17126:{\displaystyle \left(\mathbb {Z} ,+\right)}
9038:Elementary consequences of the group axioms
7420:{\displaystyle f_{\mathrm {h} }\circ r_{3}}
3088:
3023:
29234:
29220:
28736:
28722:
28644:The Theory of Groups and Quantum Mechanics
28086:Profinite Groups, Arithmetic, and Geometry
27872:, vol. 322, Berlin: Springer-Verlag,
27743:
27533:
27174:Chancey, C. C.; O'Brien, M. C. M. (2021),
26962:
26502:
26073:Coornaert, Delzant & Papadopoulos 1990
25508:
25153:
23402:, which describes three of the four known
22289:{\displaystyle \mathbb {F} _{p}^{\times }}
22056:may not exist, in which case the order of
20196:{\displaystyle \mathbb {F} _{p}^{\times }}
19300:{\displaystyle \mathbb {F} _{p}^{\times }}
19264:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
16466:{\displaystyle r^{4},f^{2},(r\cdot f)^{2}}
16038:
16021:
14116:{\displaystyle g_{1}^{-1}\cdot g_{2}\in H}
13927:{\displaystyle Hg=\{h\cdot g\mid h\in H\}}
13872:{\displaystyle gH=\{g\cdot h\mid h\in H\}}
12706:
12689:
10986:However, only assuming the existence of a
7371:red (upper left region). A left and right
2201:The integers, together with the operation
1376:
1362:
1035:
1021:
254:, completed in 2004. Since the mid-1980s,
28646:, translated by Robertson, H. P., Dover,
28189:
28141:
27564:
27445:
27212:
27077:
27030:
27020:
26887:
26875:, Oliver and Boyd, Edinburgh and London,
26867:
26598:
25988:
25893:
25881:
24620:
24607:
24576:
24546:
24516:
24485:
24455:
24216:{\displaystyle \operatorname {Hom} (x,x)}
24065:
24017:
23969:
23942:
22271:
21199:
21096:
21044:. It describes all possible rotations in
20925:
20487:
20178:
20087:. These numbers can be visualized as the
19912:
19899:
19282:
19257:
19244:
19197:
18044:
18031:
17712:
17615:
17573:
17540:
17500:
17404:
17373:
17282:
17148:
17108:
17079:
16838:
11816:It would be natural to require also that
11556:. When studying groups, one uses instead
10173:Equivalent definition with relaxed axioms
9231:
4257:leaving everything unchanged, denoted id;
3449:
3421:
3398:
3372:
3350:
3321:
3280:
3246:
3221:
3141:
1624:
1409:
720:
695:
658:
141:, groups arise naturally in the study of
16:Set with associative invertible operation
28461:
28362:
28262:
28112:An Introduction to Theoretical Chemistry
27858:
27703:Categories for the Working Mathematician
27696:
27469:
27284:, Oxford University Press, p. 265,
27038:
26907:
26788:
26758:
26610:
26538:
26402:
26093:
25964:
25821:
25677:
25585:
25426:
25235:
25231:
24420:arguments, for some nonnegative integer
22974:
22349:
21507:of the group. An important class is the
20849:
20558:
20111:-gon, as shown in blue in the image for
19326:
18324:{\displaystyle 9+4\equiv 1{\pmod {12}}.}
17739:
16893:
16817:
13479:consists of all products of elements of
27:
28660:
28538:
28429:MacTutor History of Mathematics Archive
28149:
28127:
28069:, Mathematical Association of America,
28038:Linear Representations of Finite Groups
27762:
27573:
27520:, Philadelphia: W.B. Saunders Company,
27487:
27351:An Invitation to Computational Homotopy
26450:
26351:
26060:
25869:
25797:
25785:
25761:
25737:
25701:
25689:
21869:that is, application of the operation "
21503:. The number of elements is called the
20813:, which characterizes functions having
16902:Elements of the fundamental group of a
13089:, equipped with the group operation on
9542:Therefore, it is customary to speak of
9508: is the identity element and
8061:can be checked using the Cayley table:
3475:and whose operation is multiplication.
3302:{\displaystyle (\mathbb {R} ,+,\cdot )}
3157:, which has the operations of addition
29736:
28880:Classification of finite simple groups
28600:
28523:
28504:
28439:
28399:
28231:An introduction to homological algebra
28225:
28206:
28170:
28109:
28002:
27515:
27383:
27147:
27094:
26988:
26574:
26526:
26514:
26291:
26232:
26105:
25845:
25833:
25809:
25749:
25725:
25478:
25393:are other instances of this principle.
25307:
25269:
24223:is a group; conversely, given a group
23358:, are basic symmetries of the laws of
23110:and they are the group objects in the
22908:{\displaystyle G\times G\rightarrow G}
22844:, relate the monster group to certain
22798:Classification of finite simple groups
22792:Classification of finite simple groups
22213:is 4, as is the order of the subgroup
20787:ion, exhibits square-planar geometry
17628:{\displaystyle \mathbb {Q} ^{\times }}
17389:. There is still a minor obstacle for
11571:
9020:classification of finite simple groups
3434:{\displaystyle \mathbb {R} ^{\times }}
3335:to denote any of these three objects.
1838:gives the same final result as adding
593:Classification of finite simple groups
252:classification of finite simple groups
196:in the 1830s, who introduced the term
72:, and every element of the set has an
29215:
28717:
28695:
28623:
28374:
28286:
28211:, Berlin, New York: Springer-Verlag,
28083:
28042:, Berlin, New York: Springer-Verlag,
28030:
27980:
27953:
27927:
27831:
27781:
27418:Representation Theory: A First Course
27348:
27298:
27113:
26936:
26930:
26915:, Berlin, New York: Springer-Verlag,
26692:
26686:
26670:
26622:
26586:
26562:
26550:
26490:
26474:
26414:
26390:
26378:
26315:
26204:
26129:
25920:
25857:
25713:
25641:
24416:operation (i.e., an operation taking
22970:
19437:is a group all of whose elements are
18385:. Its elements can be represented by
17729:
16891:translate into properties of groups.
16164:in a vertical line (every element of
14880:{\displaystyle G/N=\{gN\mid g\in G\}}
12840:: it has a subset of the elements of
11880:{\displaystyle \varphi (1_{G})=1_{H}}
8309:, as it does not change any symmetry
7504:is a symmetry for any two symmetries
7367:whose Cayley table is highlighted in
3386:and whose operation is addition. The
3364:is the group whose underlying set is
1818:first, and then adding the result to
28638:
28533:(in French), Paris: Gauthier-Villars
28516:(Galois work was first published by
28514:(in French), Paris: Gauthier-Villars
28308:
27612:
27333:Stereochemistry of Organic Compounds
27279:
26841:
26808:
26738:
26718:
26478:
26438:
26426:
26255:
26243:
26216:
26189:
26177:
26165:
26153:
26141:
26117:
26036:
26024:
26012:
26000:
25976:
25949:
25937:
25908:
25665:
25653:
25629:
25617:
25597:
25524:for the impact of a group action on
25495:
25378:
25365:
25324:
25227:
25223:
25136:
22474:{\displaystyle \mathrm {Z} _{p^{2}}}
21479:fundamental theorem of Galois theory
17876:{\displaystyle a\equiv b{\pmod {n}}}
8806:(solutions). The elements of such a
7816:. The other way is to first compose
7375:of this subgroup are highlighted in
4737:) and then reflecting horizontally (
28580:
28330:
28272:, New York: John Wiley & Sons,
27988:, Wiley Classics, Wiley-Blackwell,
27834:The Geometry of Minkowski Spacetime
27676:Linear Algebra and Its Applications
27673:
26634:
26339:
26267:
25773:
25247:The word homomorphism derives from
24350:is defined only when the source of
22944:(which provides the inverse) and a
22807:list all groups of order up to 2000
21434:; analogous formulae are known for
20793:Finite symmetry groups such as the
18855:
18310:
17865:
16526:{\displaystyle \langle r,f\rangle }
16501:is the smallest normal subgroup of
16242:{\displaystyle \langle r,f\rangle }
16108:is generated by the right rotation
16054:. Surjective homomorphisms are the
15651:{\displaystyle U=f_{\mathrm {v} }R}
15561:The elements of the quotient group
15324:Cayley table of the quotient group
11554:quotient by an equivalence relation
11135:{\displaystyle e\cdot f=f\cdot f=f}
11089:{\displaystyle e\cdot e=f\cdot e=e}
9224:. It is thus customary to speak of
13:
28337:Quantum Field Theory in a Nutshell
27805:; Fogarty, J.; Kirwan, F. (1994),
27544:Proceedings of the Royal Society A
27518:Introduction to Topological Groups
27359:10.1093/oso/9780198832973.001.0001
27201:BeitrÀge zur Algebra und Geometrie
27062:"The groups of order at most 2000"
26849:(3rd ed.), Berlin, New York:
26328:Mumford, Fogarty & Kirwan 1994
26132:, p. 54, (Theorem 2.1).
26049:Magnus, Karrass & Solitar 2004
23413:
23392:Symmetries that vary with location
22565:
22507:
22492:
22454:
22394:
22385:is isomorphic to the cyclic group
22246:
22166:
21961:" represents multiplication, then
21668:
21611:
21577:
21523:
21185:
21182:
21019:
21016:
20911:
20908:
20530:
19378:is not, because the odd powers of
17998:
16771:
16641:
16258:
16173:
16148:
16088:
15926:
15893:
15877:{\displaystyle \mathrm {D} _{4}/R}
15856:
15791:
15788:
15741:
15726:
15705:
15687:
15639:
15591:{\displaystyle \mathrm {D} _{4}/R}
15570:
15354:{\displaystyle \mathrm {D} _{4}/R}
15333:
14777:
14753:
14735:
14717:
14699:
14662:
14635:
14608:
14553:
14523:
14508:
14493:
14478:
14457:
14309:
13658:
13643:
13612:
13609:
13583:
13333:{\displaystyle g^{-1}\cdot h\in H}
13248:necessary and sufficient condition
13184:
13181:
12774:. Injective homomorphisms are the
8740:
8713:
8698:
8654:
8626:
8591:
8478:
8445:
8412:
8379:
8348:
8345:
8293:
8290:
8239:
8224:
8189:
8171:
8096:
8081:
8029:
8011:
7980:
7965:
7730:
7623:
7589:
7574:
7444:
7398:
7254:
7251:
7223:
7220:
7111:
7081:
7051:
7021:
6991:
6933:
6930:
6849:
6819:
6789:
6759:
6729:
6643:
6640:
6587:
6557:
6527:
6497:
6467:
6353:
6350:
6325:
6295:
6265:
6235:
6205:
6173:
6143:
6113:
6083:
5999:
5996:
5913:
5883:
5853:
5823:
5767:
5764:
5653:
5623:
5593:
5563:
5535:
5532:
5393:
5363:
5333:
5303:
5191:
5188:
5165:
5162:
5135:
5105:
5075:
5045:
4933:
4930:
4878:
4844:
4816:
4785:
4752:
4563:
4532:
4470:
4437:
4400:
4367:
4229:
4190:
4151:
4113:
3971:
3968:
3928:
246:. A theory has been developed for
14:
29765:
28688:
28424:"The development of group theory"
27624:, Boston, MA: BirkhÀuser Boston,
27301:"On some old and new problems in
27232:Géométrie et théorie des groupes
26481:, p. 77 for similar results.
26180:, p. 22, §II.1 (example 11).
25381:, Theorem IV.1.9. The notions of
23181:; informally, this means that it
22883:equipped with a binary operation
22816:The classification of all finite
22296:of multiplication modulo a prime
22103:the order of any finite subgroup
17265:{\displaystyle b={\tfrac {1}{2}}}
12056:{\displaystyle \iota _{G}:G\to G}
11224:
9397: are inverses of each other)
8873:. After novel geometries such as
3913:has eight symmetries. These are:
3722:{\displaystyle a\cdot b=b\cdot a}
3478:More generally, one speaks of an
3128:For example, consider the set of
29716:
29715:
28633:, Mathematical Monographs, No. 1
28586:Gesammelte Abhandlungen. Band 1
27958:(4th ed.), Addison-Wesley,
27836:, New York: Dover Publications,
27099:, New York: Dover Publications,
26913:A Course in the Theory of Groups
26441:, p. 292, (Theorem VI.7.2).
25555:
25049:
22859:Groups with additional structure
22778:. A nontrivial group is called
22615:
22578:{\displaystyle \mathrm {D} _{4}}
22407:{\displaystyle \mathrm {Z} _{p}}
22255:{\displaystyle f_{\mathrm {v} }}
22179:{\displaystyle \mathrm {D} _{4}}
21681:{\displaystyle \mathrm {S} _{3}}
21624:{\displaystyle \mathrm {S} _{N}}
21590:{\displaystyle \mathrm {S} _{3}}
21545:, the groups of permutations of
21536:{\displaystyle \mathrm {S} _{N}}
21484:
21231:
21109:{\displaystyle \mathbb {R} ^{3}}
21035:{\displaystyle \mathrm {SO} (n)}
20735:
20728:
20721:
20714:
20503:{\displaystyle (\mathbb {Z} ,+)}
19316:
18359:multiplicative group of integers
18011:{\displaystyle \mathrm {Z} _{n}}
17296:has a (multiplicative) inverse.
16654:{\displaystyle \mathrm {D} _{4}}
16271:{\displaystyle \mathrm {D} _{4}}
16186:{\displaystyle \mathrm {D} _{4}}
16157:{\displaystyle f_{\mathrm {v} }}
16101:{\displaystyle \mathrm {D} _{4}}
16079:For example, the dihedral group
16061:
15939:{\displaystyle \mathrm {D} _{4}}
15906:{\displaystyle \mathrm {D} _{4}}
14808:is a normal subgroup of a group
14675:{\displaystyle \mathrm {D} _{4}}
14566:{\displaystyle \mathrm {D} _{4}}
14322:{\displaystyle \mathrm {D} _{4}}
13939:The left cosets of any subgroup
13592:{\displaystyle f_{\mathrm {v} }}
13019:{\displaystyle h_{1}\cdot h_{2}}
11501:associating to every element of
8753:{\displaystyle \mathrm {D} _{4}}
8604:{\displaystyle \mathrm {D} _{4}}
8487:{\displaystyle f_{\mathrm {c} }}
8454:{\displaystyle f_{\mathrm {d} }}
8421:{\displaystyle f_{\mathrm {v} }}
8388:{\displaystyle f_{\mathrm {h} }}
8335:: Each symmetry has an inverse:
7743:{\displaystyle \mathrm {D} _{4}}
7632:{\displaystyle f_{\mathrm {c} }}
7453:{\displaystyle f_{\mathrm {d} }}
7120:{\displaystyle f_{\mathrm {h} }}
7090:{\displaystyle f_{\mathrm {d} }}
7060:{\displaystyle f_{\mathrm {v} }}
7030:{\displaystyle f_{\mathrm {c} }}
7000:{\displaystyle f_{\mathrm {c} }}
6858:{\displaystyle f_{\mathrm {v} }}
6828:{\displaystyle f_{\mathrm {c} }}
6798:{\displaystyle f_{\mathrm {h} }}
6768:{\displaystyle f_{\mathrm {d} }}
6738:{\displaystyle f_{\mathrm {d} }}
6596:{\displaystyle f_{\mathrm {d} }}
6566:{\displaystyle f_{\mathrm {v} }}
6536:{\displaystyle f_{\mathrm {c} }}
6506:{\displaystyle f_{\mathrm {h} }}
6476:{\displaystyle f_{\mathrm {h} }}
6334:{\displaystyle f_{\mathrm {c} }}
6304:{\displaystyle f_{\mathrm {h} }}
6274:{\displaystyle f_{\mathrm {d} }}
6244:{\displaystyle f_{\mathrm {v} }}
6214:{\displaystyle f_{\mathrm {v} }}
6182:{\displaystyle f_{\mathrm {v} }}
6152:{\displaystyle f_{\mathrm {h} }}
6122:{\displaystyle f_{\mathrm {c} }}
6092:{\displaystyle f_{\mathrm {d} }}
5922:{\displaystyle f_{\mathrm {d} }}
5892:{\displaystyle f_{\mathrm {c} }}
5862:{\displaystyle f_{\mathrm {v} }}
5832:{\displaystyle f_{\mathrm {h} }}
5662:{\displaystyle f_{\mathrm {h} }}
5632:{\displaystyle f_{\mathrm {v} }}
5602:{\displaystyle f_{\mathrm {d} }}
5572:{\displaystyle f_{\mathrm {c} }}
5402:{\displaystyle f_{\mathrm {c} }}
5372:{\displaystyle f_{\mathrm {d} }}
5342:{\displaystyle f_{\mathrm {h} }}
5312:{\displaystyle f_{\mathrm {v} }}
5144:{\displaystyle f_{\mathrm {c} }}
5114:{\displaystyle f_{\mathrm {d} }}
5084:{\displaystyle f_{\mathrm {h} }}
5054:{\displaystyle f_{\mathrm {v} }}
4891:{\displaystyle \mathrm {D} _{4}}
4794:{\displaystyle f_{\mathrm {d} }}
4761:{\displaystyle f_{\mathrm {h} }}
4576:{\displaystyle \mathrm {D} _{4}}
4541:{\displaystyle f_{\mathrm {h} }}
4479:{\displaystyle f_{\mathrm {c} }}
4446:{\displaystyle f_{\mathrm {d} }}
4409:{\displaystyle f_{\mathrm {h} }}
4376:{\displaystyle f_{\mathrm {v} }}
4238:{\displaystyle f_{\mathrm {c} }}
4211:
4199:{\displaystyle f_{\mathrm {d} }}
4172:
4160:{\displaystyle f_{\mathrm {h} }}
4133:
4122:{\displaystyle f_{\mathrm {v} }}
4095:
4058:
4023:
3988:
3955:
3941:{\displaystyle \mathrm {D} _{4}}
3881:Second example: a symmetry group
3581:, and the inverse of an element
3506:, and the inverse of an element
3262:{\displaystyle (\mathbb {R} ,+)}
285:
206:) for the symmetry group of the
27312:Quasigroups and Related Systems
26496:
26468:
26463:Kurzweil & Stellmacher 2004
26222:
26096:, in particular §§I.12 and I.13
26078:
25550:Besche, Eick & O'Brien 2001
25539:
25514:
25501:
25484:
25463:
25432:
25411:
25396:
25371:
25343:
25330:
25313:
25288:
25275:
25262:
25241:
25216:
25159:
25142:
24157:A group can be thought of as a
23342:links continuous symmetries to
22824:of such groups, as well as 26 "
22083:states that for a finite group
22036:.) In infinite groups, such an
20670:describe crystal symmetries in
20595:on another mathematical object
20344:{\displaystyle 3^{2}=9\equiv 4}
18848:
18303:
17858:
17330:{\displaystyle {\frac {a}{b}}.}
16879:. By means of this connection,
13991:of all left cosets is equal to
12816:contained within a bigger one,
12502:{\displaystyle \varphi :G\to H}
12475:if there exists an isomorphism
12112:{\displaystyle \varphi :G\to H}
11675:{\displaystyle \varphi :G\to H}
10335:{\displaystyle f^{-1}\cdot f=e}
9228:identity element of the group.
9022:, with the final step taken by
8920:groups describing factorization
8906:(1798), and more explicitly by
1769:{\displaystyle (a+b)+c=a+(b+c)}
183:symmetry in molecular chemistry
28508:(1908), Tannery, Jules (ed.),
28316:, Princeton University Press,
28114:, Cambridge University Press,
28088:, Princeton University Press,
27788:, Princeton University Press,
27767:, Cambridge University Press,
27765:Theory of Finite Simple Groups
27178:, Princeton University Press,
27045:, Cambridge University Press,
26895:, New York: Barnes and Noble,
26477:, Proposition 6.4.3. See also
26304:Eliel, Wilen & Mander 1994
25129:" to be a binary operation on
25077:
24298:
24286:
24210:
24198:
24126:produces a group known as the
24087:
24061:
24039:
24013:
23991:
23965:
23268:{\displaystyle \det(A)\neq 0,}
23253:
23247:
23112:category of topological spaces
22937:{\displaystyle G\rightarrow G}
22928:
22899:
21735:is the least positive integer
21462:similar to the formula above.
21203:
21189:
21178:
21118:. A representation of a group
21029:
21023:
20929:
20915:
20801:, which is in turn applied in
20583:greatly simplify the study of
20497:
20483:
19935:introduced above, the element
19922:
19895:
19831:
19822:
19816:
19807:
19746:
19727:
18981:
18969:
18859:
18849:
18314:
18304:
18054:
18027:
17869:
17859:
16766:
16748:
16735:
16596:
16583:
16454:
16441:
16026:
16001:
15973:
15747:
15717:
15177:
15167:
15056:
15047:
14925:
14438:itself, or otherwise equal to
14351:of rotations, the left cosets
13744:by an arbitrary group element
13143:
12710:
12694:
12658:
12575:; then any statement true for
12493:
12351:
12345:
12259:
12253:
12135:
12103:
12047:
11931:
11924:
11915:
11899:
11861:
11848:
11736:
11730:
11721:
11715:
11706:
11694:
11666:
11637:
11625:
11605:
11593:
11461:
11208:
11196:
10894:
10872:
10851:
10829:
10719:
10702:
10681:
10659:
10644:
10627:
10606:
10587:
10565:
10562:
10547:
10530:
10509:
10487:
10481:
10453:
10436:
10433:
10412:
10390:
10055:
9784:
9422:
9410:
9373:
9361:
9342: is the identity element)
9138:Uniqueness of identity element
9117:
9105:
9087:
9075:
8208:
8180:
8102:
8072:
8048:
8020:
7986:
7956:
7933:
7921:
7903:
7891:
4675:after performing the symmetry
4245:(counter-diagonal reflection)
3296:
3276:
3256:
3242:
2605:
2593:
2575:
2563:
1763:
1751:
1733:
1721:
954:Infinite dimensional Lie group
1:
28630:History of Modern Mathematics
28384:American Mathematical Society
27422:Graduate Texts in Mathematics
27079:10.1090/S1079-6762-01-00087-7
26818:Graduate Texts in Mathematics
26755:, an elementary introduction.
26680:
23350:, as well as translations in
23162:
22967:that mimic the group axioms.
21292:{\displaystyle ax^{2}+bx+c=0}
20702:spontaneous symmetry breaking
20418:{\displaystyle 3^{4}\equiv 1}
20381:{\displaystyle 3^{3}\equiv 2}
17839:to be equivalent, denoted by
17819:that differ by a multiple of
16800:, a graphical depiction of a
16475:. In fact, it turns out that
14057:{\displaystyle g_{1}H=g_{2}H}
13619:{\displaystyle \mathrm {id} }
12667:{\displaystyle \phi :G'\to G}
10012:{\displaystyle b\cdot a^{-1}}
9725:{\displaystyle a^{-1}\cdot b}
9618:, there is a unique solution
8961:modular representation theory
8914:made early attempts to prove
8850:{\displaystyle \theta ^{n}=1}
8355:{\displaystyle \mathrm {id} }
8300:{\displaystyle \mathrm {id} }
7261:{\displaystyle \mathrm {id} }
7230:{\displaystyle \mathrm {id} }
6940:{\displaystyle \mathrm {id} }
6650:{\displaystyle \mathrm {id} }
6360:{\displaystyle \mathrm {id} }
6006:{\displaystyle \mathrm {id} }
5774:{\displaystyle \mathrm {id} }
5542:{\displaystyle \mathrm {id} }
5198:{\displaystyle \mathrm {id} }
5172:{\displaystyle \mathrm {id} }
4940:{\displaystyle \mathrm {id} }
4090:(rotation by 270° clockwise)
3978:{\displaystyle \mathrm {id} }
3797:, and the group is called an
3553:. Similarly, one speaks of a
2228:
1778:. Expressed in words, adding
210:of an equation, now called a
97:generated by a single element
29520:Eigenvalues and eigenvectors
28409:, vol. II (1851â1860),
28065:Schwartzman, Steven (1994),
27095:Bishop, David H. L. (1993),
27022:10.3998/ark.5550190.p008.911
26893:Introduction to Group Theory
25656:, p. 3, I.§1 and p. 7, I.§2.
25572:
25298:of a group which equals the
24583:{\displaystyle \mathbb {C} }
24553:{\displaystyle \mathbb {R} }
24523:{\displaystyle \mathbb {Q} }
24492:{\displaystyle \mathbb {Z} }
24462:{\displaystyle \mathbb {N} }
23949:{\displaystyle \mathbb {N} }
23812:Commutative-and-associative
21567:symmetric group on 3 letters
20805:of transmitted data, and in
20591:objects. A group is said to
19351:is a primitive element, but
18605:, because the usual product
18089:as the identity element and
17989:, forms a group, denoted as
17719:{\displaystyle \mathbb {Q} }
17380:{\displaystyle \mathbb {Q} }
17339:Fractions of integers (with
17299:
17289:{\displaystyle \mathbb {Z} }
17086:{\displaystyle \mathbb {Z} }
16845:{\displaystyle \mathbb {Z} }
15982:{\displaystyle \phi :G\to H}
12781:
12144:{\displaystyle \psi :H\to G}
9471: is an inverse of
9215:{\displaystyle e=e\cdot f=f}
7379:green (in the last row) and
3405:{\displaystyle \mathbb {R} }
3379:{\displaystyle \mathbb {R} }
3357:{\displaystyle \mathbb {R} }
3328:{\displaystyle \mathbb {R} }
3313:. But it is common to write
3228:{\displaystyle \mathbb {R} }
3148:{\displaystyle \mathbb {R} }
1902:. This property is known as
1631:{\displaystyle \mathbb {Z} }
727:{\displaystyle \mathbb {Z} }
702:{\displaystyle \mathbb {Z} }
665:{\displaystyle \mathbb {Z} }
250:, which culminated with the
240:representations of the group
7:
29241:
29147:Infinite dimensional groups
28610:University of Chicago Press
28294:, Oxford: Clarendon Press,
28157:(4th ed.), CRC Press,
27649:The Theory of Finite Groups
26993:, Oxford University Press,
26659:Romanowska & Smith 2002
25065:List of group theory topics
25042:
19790:is called a generator or a
17066:
16871:founded what is now called
16058:in the category of groups.
12778:in the category of groups.
11270:{\displaystyle X=\{x,y,z\}}
11229:The following sections use
9734:. It follows that for each
9549:
8983:counterpart, the theory of
8903:Disquisitiones Arithmeticae
8900:'s number-theoretical work
8794:, gave a criterion for the
7466:blue (below table center).
4020:(rotation by 90° clockwise)
2785:Such an element is unique (
1394:First example: the integers
1389:Definition and illustration
452:List of group theory topics
10:
29770:
28743:
28411:Cambridge University Press
28366:
28084:Shatz, Stephen S. (1972),
27986:Fourier Analysis on Groups
27954:Rosen, Kenneth H. (2000),
27905:; Smith, J. D. H. (2002),
27832:Naber, Gregory L. (2003),
27807:Geometric Invariant Theory
27580:Princeton University Press
27499:Cambridge University Press
27299:Dudek, WiesĆaw A. (2001),
27097:Group Theory and Chemistry
26909:Robinson, Derek John Scott
26647:Denecke & Wismath 2002
26280:Chancey & O'Brien 2021
25283:SeifertâVan Kampen theorem
23166:
22999:
22795:
22445:is abelian, isomorphic to
21565:objects. For example, the
21488:
21235:
21068:in this group are used in
20835:
20811:differential Galois theory
20540:
20534:
19320:
18923:{\displaystyle a\cdot b-1}
18805:, there exists an integer
18115:as the inverse element of
17733:
17478:{\displaystyle x\cdot 0=1}
17228:{\displaystyle a\cdot b=1}
17038:
17035:benefit from the concept.
17021:computational group theory
16811:
16494:{\displaystyle \ker \phi }
16395:{\displaystyle \ker \phi }
16065:
15103:serves as the identity of
14781:
13696:
13056:{\displaystyle h_{1}^{-1}}
12785:
12063:that maps each element of
11611:{\displaystyle (G,\cdot )}
11575:
11228:
11214:{\displaystyle (G,\cdot )}
10239:{\displaystyle e\cdot f=f}
9974:{\displaystyle x\cdot a=b}
9685:{\displaystyle a\cdot x=b}
8786:, extending prior work of
8767:
8763:
8280:: The identity element is
2946:{\displaystyle b\cdot a=e}
2912:{\displaystyle a\cdot b=e}
2847:, there exists an element
2773:{\displaystyle a\cdot e=a}
2737:{\displaystyle e\cdot a=a}
244:computational group theory
18:
29711:
29680:
29664:
29593:
29500:
29439:
29360:
29267:
29249:
29186:
29146:
29022:
28870:
28804:
28751:
28269:Gravitation and Cosmology
27763:Michler, Gerhard (2006),
27574:Kuipers, Jack B. (1999),
27470:Gollmann, Dieter (2011),
27153:Simple Groups of Lie Type
27032:2027/spo.5550190.p008.911
26719:Cook, Mariana R. (2009),
25349:The same is true for any
24593:
24502:
24472:
24442:
23406:and classifies all known
22915:(the group operation), a
22822:several infinite families
22365:Any group of prime order
22154:give a partial converse.
21501:finite number of elements
20809:. Another application is
20734:
20727:
20720:
20713:
20581:Symmetries in mathematics
18987:{\displaystyle \gcd(a,p)}
18669:it yields a remainder of
17779:defines any two elements
17756:Modular arithmetic for a
16808:Examples and applications
15951:first isomorphism theorem
15158:in the quotient group is
13768:. In symbolic terms, the
13692:
11025:{\displaystyle G=\{e,f\}}
9949:, the unique solution to
8945:Ferdinand Georg Frobenius
8774:The modern concept of an
7240:
4167:(horizontal reflection)
3808:, the operation is often
2330:", that combines any two
1584:is also an integer; this
260:finitely generated groups
147:geometric transformations
32:The manipulations of the
29050:Special orthogonal group
28434:University of St Andrews
27782:Milne, James S. (1980),
27399:: 239â50, archived from
27039:Bersuker, Isaac (2006),
26367:Fulton & Harris 1991
26192:, pp. 26, 29, §I.5.
26063:, p. 30, Chapter I.
25070:
24911:N/A, 1, 2, respectively
23151:, which plays a role in
23066:must not vary wildly if
23029:{\displaystyle g\cdot h}
22803:Computer algebra systems
22706:itself, questions about
22418:). Any group of order
21695:The order of an element
21005:special orthogonal group
20882:-coordinate by factor 2.
20785:tetrachloroplatinate(II)
19979:. Any cyclic group with
19631:{\displaystyle a\cdot a}
19441:of a particular element
17528:However, the set of all
17093:under addition, denoted
14942:{\displaystyle G\to G/N}
13800:, containing an element
12566:{\displaystyle \varphi }
11829:{\displaystyle \varphi }
11470:{\displaystyle f:X\to Y}
10110:{\displaystyle x\cdot a}
9839:{\displaystyle a\cdot x}
7497:{\displaystyle a\circ b}
4648:{\displaystyle b\circ a}
4625:is written symbolically
4552:of degree four, denoted
3843:geometric transformation
3832:{\displaystyle f\circ g}
3679:{\displaystyle a\cdot b}
3101:. The set is called the
3093:Formally, a group is an
3089:Notation and terminology
3050:and is commonly denoted
2975:is the identity element.
2626:There exists an element
2438:{\displaystyle a\cdot b}
570:Elementary abelian group
447:Glossary of group theory
29594:Algebraic constructions
29297:Algebraic number theory
28602:Mackey, George Whitelaw
27937:Oxford University Press
27864:Algebraic Number Theory
27516:Husain, Taqdir (1966),
27280:Dove, Martin T (2003),
27119:Linear Algebraic Groups
26790:Herstein, Israel Nathan
26760:Herstein, Israel Nathan
26144:, p. 292, §VIII.1.
26051:, pp. 56â67, §1.6.
24893:{\displaystyle a\neq 0}
24402:) and in the theory of
23364:Lorentz transformations
23179:differentiable manifold
23157:Ă©tale fundamental group
23135:and can be studied via
22950:existential quantifiers
22840:conjectures, proved by
21783:{\displaystyle a^{n}=e}
21631:for a suitable integer
21224:, especially (locally)
20301:{\displaystyle 3^{1}=3}
20078:{\displaystyle z^{n}=1}
19770:, etc. Such an element
19311:public-key cryptography
19108:is represented by
19064:is that represented by
18958:greatest common divisor
18550:{\displaystyle 1,2,3,4}
16980:{\displaystyle m\neq k}
16068:Presentation of a group
9546:inverse of an element.
9044:elementary group theory
8770:History of group theory
4206:(diagonal reflection)
2255:A group is a non-empty
1511:. For any two integers
81:mathematical structures
29337:Noncommutative algebra
29076:Exceptional Lie groups
28292:Codes and Cryptography
28207:Warner, Frank (1983),
27868:
27566:10.1098/rspa.1937.0142
27393:Compositio Mathematica
27042:The JahnâTeller Effect
26989:Awodey, Steve (2010),
25620:, p. 360, App. 2.
24908:0, 2, 1, respectively
24894:
24863:
24827:
24796:
24652:
24584:
24554:
24524:
24493:
24463:
24390:
24374:matches the target of
24366:
24342:
24311:
24263:
24239:
24217:
24177:
24146:
24118:
24094:
24046:
23998:
23950:
23419:Group-like structures
23388:quantum field theories
23329:
23309:
23289:
23269:
23231:
23211:
23153:infinite Galois theory
23100:
23080:
23060:
23059:{\displaystyle g^{-1}}
23030:
22997:
22996:(shown at the bottom).
22938:
22909:
22877:
22770:
22740:
22720:
22700:
22680:
22654:
22640:has a normal subgroup
22634:
22606:
22579:
22550:
22521:
22475:
22439:
22408:
22379:
22338:
22310:
22290:
22256:
22227:
22207:
22180:
22142:
22117:
22097:
22070:
22050:
22028:
22004:
21982:
21953:
21952:{\displaystyle \cdot }
21929:
21907:
21885:
21884:{\displaystyle \cdot }
21863:
21813:
21784:
21749:
21729:
21709:
21682:
21647:
21625:
21591:
21559:
21537:
21428:
21408:
21388:
21368:
21293:
21210:
21152:
21132:
21110:
21058:
21036:
20985:
20961:
20936:
20883:
20874:
20819:differential equations
20645:(2,3,7) triangle group
20635:
20611:
20572:
20504:
20468:
20446:
20419:
20382:
20345:
20302:
20267:
20245:
20217:
20197:
20165:rotation by 60°. From
20155:
20133:
20105:
20079:
20044:
20025:complex roots of unity
20015:
19993:
19971:
19949:
19929:
19880:
19784:
19762:
19664:
19663:{\displaystyle a^{-3}}
19632:
19604:
19577:
19455:
19430:
19421:
19392:
19372:
19345:
19301:
19265:
19230:
19176:
19124:
19102:
19080:
19058:
19038:
19010:
18988:
18956:and the fact that the
18952:can be found by using
18946:
18924:
18890:
18870:
18819:
18797:
18775:
18753:
18729:
18707:
18685:
18663:
18641:
18619:
18597:
18551:
18509:
18479:
18451:
18427:
18399:
18377:
18349:
18325:
18272:
18250:
18230:
18202:
18180:
18160:
18131:
18109:
18083:
18061:
18012:
17981:
17953:
17927:
17899:
17877:
17833:
17813:
17793:
17773:
17753:
17720:
17689:
17659:
17629:
17598:
17525:is still not a group.
17519:
17479:
17445:
17423:
17381:
17359:nonzero) are known as
17353:
17331:
17290:
17266:
17229:
17197:
17167:
17127:
17087:
17071:The group of integers
17019:knowledge obtained in
16998:geometric group theory
16981:
16953:
16933:
16899:
16881:topological properties
16846:
16827:
16785:
16685:yields an isomorphism
16677:
16655:
16618:
16527:
16495:
16467:
16396:
16368:
16339:
16319:
16292:
16272:
16243:
16209:
16187:
16158:
16129:
16102:
16046:
15983:
15940:
15907:
15878:
15841:
15763:
15652:
15612:
15592:
15550:
15529:
15508:
15485:
15464:
15443:
15420:
15399:
15378:
15377:{\displaystyle \cdot }
15355:
15300:
15276:
15252:
15220:
15152:
15127:
15097:
15066:
15032:
15009:
14986:
14963:
14943:
14909:
14881:
14824:
14802:
14766:
14676:
14645:
14589:
14567:
14536:
14432:
14412:
14390:
14368:
14345:
14323:
14283:
14261:
14227:
14207:
14187:
14165:
14141:
14117:
14058:
14005:
13979:
13953:
13928:
13873:
13816:
13792:
13760:
13738:
13718:
13681:
13620:
13593:
13564:
13535:
13513:
13493:
13473:
13447:
13425:
13396:
13374:
13354:
13334:
13290:
13266:
13250:for a nonempty subset
13234:
13153:
13152:{\displaystyle H\to G}
13125:
13103:
13081:
13057:
13020:
12978:
12956:
12929:
12900:
12878:
12856:
12832:
12810:
12766:
12742:
12720:
12668:
12611:
12589:
12567:
12545:
12525:
12503:
12465:
12445:
12413:
12391:
12371:
12319:
12299:
12279:
12225:
12184:
12145:
12113:
12077:
12057:
12018:
11989:
11967:
11947:
11881:
11830:
11805:
11783:
11763:
11743:
11676:
11644:
11612:
11537:
11515:
11491:
11471:
11437:
11415:
11395:
11394:{\displaystyle x\in X}
11367:
11343:
11319:
11294:
11271:
11215:
11182:
11160:
11136:
11090:
11046:
11045:{\displaystyle \cdot }
11026:
10967:
10784:
10336:
10292:
10272:
10271:{\displaystyle f^{-1}}
10240:
10206:
10161:
10135:
10117:is a bijection called
10111:
10085:
10065:
10064:{\displaystyle G\to G}
10037:
10013:
9975:
9941:
9919:
9894:
9868:
9840:
9814:
9794:
9793:{\displaystyle G\to G}
9768:
9748:
9726:
9686:
9652:
9632:
9610:
9588:
9568:
9533:
9290:
9270:
9250:
9232:Uniqueness of inverses
9216:
9176:
9156:
9124:
8987:, was first shaped by
8969:locally compact groups
8935:
8851:
8810:correspond to certain
8754:
8723:
8664:
8605:
8571:
8544:
8517:
8496:and the 180° rotation
8488:
8455:
8422:
8389:
8356:
8323:
8301:
8269:
8055:
7943:
7876:
7852:
7830:
7808:
7786:
7766:
7744:
7711:
7687:
7663:
7633:
7602:
7540:
7518:
7498:
7454:
7421:
7355:
7324:
7293:
7262:
7231:
7205:
7177:
7149:
7121:
7091:
7061:
7031:
7001:
6969:
6941:
6915:
6887:
6859:
6829:
6799:
6769:
6739:
6707:
6679:
6651:
6625:
6597:
6567:
6537:
6507:
6477:
6445:
6417:
6389:
6361:
6335:
6305:
6275:
6245:
6215:
6183:
6153:
6123:
6093:
6063:
6035:
6007:
5981:
5953:
5923:
5893:
5863:
5833:
5803:
5775:
5749:
5721:
5693:
5663:
5633:
5603:
5573:
5543:
5517:
5489:
5461:
5433:
5403:
5373:
5343:
5313:
5283:
5255:
5227:
5199:
5173:
5145:
5115:
5085:
5055:
5025:
4997:
4969:
4941:
4915:
4914:{\displaystyle \circ }
4892:
4857:
4795:
4762:
4729:
4691:
4669:
4649:
4619:
4599:
4577:
4542:
4513:
4480:
4447:
4418:), or through the two
4410:
4377:
4341:
4312:
4283:
4239:
4200:
4161:
4123:
4084:
4049:
4014:
3979:
3942:
3871:
3870:{\displaystyle \circ }
3833:
3785:
3763:
3743:
3723:
3680:
3652:
3627:
3626:{\displaystyle x^{-1}}
3595:
3573:
3545:
3520:
3498:
3469:
3435:
3406:
3380:
3358:
3329:
3303:
3263:
3229:
3205:
3177:
3149:
3076:
3075:{\displaystyle a^{-1}}
3044:
3016:
2994:
2969:
2947:
2913:
2881:
2861:
2839:
2817:
2774:
2738:
2702:
2680:
2660:
2640:
2612:
2546:
2522:
2498:
2474:
2439:
2409:
2393:to form an element of
2387:
2367:
2347:
2322:
2321:{\displaystyle \cdot }
2298:
2272:
2241:
2217:
2189:
2164:
2138:
2116:
2082:
2050:
2036:, there is an integer
2028:
1991:
1957:
1925:
1894:
1872:
1852:
1832:
1812:
1792:
1770:
1704:
1682:
1660:
1632:
1604:
1578:
1547:
1525:
1501:
986:Linear algebraic group
728:
703:
666:
256:geometric group theory
238:(that is, through the
202:
115:
41:
29574:Orthogonal complement
29347:Representation theory
29062:Special unitary group
28472:Mathematische Annalen
28363:Historical references
28110:Simons, Jack (2003),
28003:Seress, Ăkos (1997),
27157:John Wiley & Sons
26847:Undergraduate Algebra
25991:, p. 39, §II.12.
25884:, pp. 4â5, §1.2.
25530:absolute Galois group
25368:, p. 86, §III.1.
25087:under the operation "
24895:
24864:
24828:
24797:
24653:
24585:
24555:
24525:
24494:
24464:
24391:
24367:
24343:
24312:
24264:
24240:
24218:
24178:
24147:
24119:
24095:
24047:
23999:
23951:
23330:
23310:
23290:
23270:
23232:
23212:
23127:. These examples are
23101:
23081:
23061:
23031:
22978:
22954:computer-aided proofs
22939:
22910:
22878:
22786:JordanâHölder theorem
22771:
22741:
22721:
22701:
22681:
22679:{\displaystyle \{1\}}
22655:
22635:
22612:above is an example.
22607:
22605:{\displaystyle 2^{3}}
22580:
22556:; the dihedral group
22551:
22549:{\displaystyle p^{3}}
22522:
22476:
22440:
22438:{\displaystyle p^{2}}
22409:
22380:
22350:Finite abelian groups
22339:
22311:
22291:
22257:
22228:
22208:
22206:{\displaystyle r_{1}}
22181:
22143:
22118:
22098:
22071:
22051:
22029:
22005:
21983:
21981:{\displaystyle a^{n}}
21954:
21930:
21908:
21886:
21864:
21814:
21812:{\displaystyle a^{n}}
21785:
21750:
21730:
21710:
21683:
21648:
21626:
21592:
21560:
21538:
21446:exist in general for
21429:
21409:
21389:
21369:
21294:
21211:
21153:
21133:
21111:
21080:group representations
21076:Representation theory
21059:
21037:
20986:
20962:
20937:
20895:matrix multiplication
20875:
20853:
20842:Representation theory
20647:acts on a triangular
20636:
20612:
20562:
20505:
20469:
20447:
20420:
20383:
20346:
20303:
20273:is a generator since
20268:
20246:
20218:
20198:
20156:
20134:
20106:
20080:
20045:
20016:
19994:
19972:
19950:
19930:
19881:
19785:
19763:
19665:
19633:
19605:
19603:{\displaystyle a^{2}}
19578:
19456:
19422:
19420:{\displaystyle z^{2}}
19393:
19373:
19371:{\displaystyle z^{2}}
19346:
19330:
19302:
19266:
19231:
19177:
19125:
19103:
19081:
19059:
19039:
19011:
18989:
18947:
18925:
18891:
18871:
18820:
18798:
18776:
18754:
18730:
18708:
18686:
18664:
18642:
18620:
18598:
18552:
18510:
18480:
18452:
18428:
18400:
18378:
18350:
18333:For any prime number
18326:
18273:
18251:
18231:
18203:
18186:hours, it ends up on
18181:
18161:
18132:
18110:
18084:
18062:
18013:
17982:
17954:
17928:
17900:
17878:
17834:
17814:
17794:
17774:
17743:
17721:
17690:
17660:
17630:
17599:
17520:
17480:
17446:
17424:
17382:
17354:
17332:
17291:
17267:
17230:
17198:
17168:
17128:
17088:
16982:
16954:
16934:
16897:
16854:multiplicative groups
16847:
16821:
16786:
16678:
16676:{\displaystyle \phi }
16656:
16619:
16528:
16496:
16468:
16397:
16369:
16367:{\displaystyle f_{1}}
16340:
16320:
16318:{\displaystyle r_{1}}
16293:
16273:
16249:on two generators to
16244:
16210:
16208:{\displaystyle \phi }
16188:
16159:
16130:
16128:{\displaystyle r_{1}}
16103:
16047:
15984:
15941:
15908:
15879:
15842:
15764:
15653:
15613:
15593:
15551:
15530:
15509:
15486:
15465:
15444:
15421:
15400:
15379:
15356:
15301:
15277:
15253:
15221:
15153:
15135:, and the inverse of
15128:
15098:
15067:
15065:{\displaystyle (gh)N}
15033:
15010:
14987:
14964:
14949:sending each element
14944:
14910:
14882:
14825:
14803:
14767:
14677:
14646:
14590:
14568:
14537:
14433:
14413:
14391:
14369:
14346:
14324:
14284:
14262:
14260:{\displaystyle gH=Hg}
14228:
14208:
14188:
14173:. The left cosets of
14166:
14142:
14118:
14059:
14006:
13980:
13954:
13929:
13874:
13817:
13793:
13761:
13739:
13719:
13682:
13621:
13594:
13565:
13563:{\displaystyle r_{2}}
13536:
13514:
13494:
13474:
13448:
13426:
13397:
13375:
13355:
13335:
13291:
13267:
13235:
13154:
13126:
13104:
13082:
13065:, so the elements of
13058:
13021:
12979:
12957:
12955:{\displaystyle h_{2}}
12930:
12928:{\displaystyle h_{1}}
12901:
12884:must be contained in
12879:
12857:
12833:
12811:
12767:
12743:
12721:
12669:
12612:
12590:
12568:
12546:
12531:can be obtained from
12526:
12504:
12466:
12446:
12431:homomorphism. Groups
12414:
12392:
12372:
12320:
12300:
12280:
12233:, that is, such that
12226:
12185:
12146:
12114:
12078:
12058:
12019:
12002:identity homomorphism
11990:
11968:
11948:
11882:
11831:
11806:
11784:
11764:
11744:
11677:
11645:
11643:{\displaystyle (H,*)}
11613:
11538:
11516:
11492:
11472:
11438:
11416:
11396:
11368:
11344:
11320:
11295:
11272:
11216:
11183:
11161:
11137:
11091:
11047:
11027:
10968:
10785:
10337:
10293:
10273:
10248:) and a left inverse
10241:
10207:
10162:
10136:
10112:
10086:
10066:
10038:
10014:
9976:
9942:
9920:
9895:
9869:
9841:
9815:
9795:
9769:
9749:
9727:
9687:
9653:
9633:
9611:
9589:
9569:
9534:
9291:
9271:
9251:
9217:
9177:
9157:
9125:
9004:University of Chicago
8979:and many others. Its
8953:representation theory
8916:Fermat's Last Theorem
8885:founded the study of
8852:
8816:Augustin Louis Cauchy
8792:Joseph-Louis Lagrange
8755:
8724:
8665:
8606:
8572:
8570:{\displaystyle r_{1}}
8545:
8543:{\displaystyle r_{3}}
8518:
8516:{\displaystyle r_{2}}
8489:
8456:
8423:
8390:
8357:
8324:
8302:
8270:
8056:
7944:
7877:
7853:
7831:
7809:
7787:
7767:
7745:
7712:
7688:
7664:
7634:
7603:
7541:
7519:
7499:
7455:
7422:
7356:
7354:{\displaystyle r_{3}}
7325:
7323:{\displaystyle r_{2}}
7294:
7292:{\displaystyle r_{1}}
7263:
7232:
7206:
7204:{\displaystyle r_{2}}
7178:
7176:{\displaystyle r_{3}}
7150:
7148:{\displaystyle r_{1}}
7122:
7092:
7062:
7032:
7002:
6970:
6968:{\displaystyle r_{2}}
6942:
6916:
6914:{\displaystyle r_{1}}
6888:
6886:{\displaystyle r_{3}}
6860:
6830:
6800:
6770:
6740:
6708:
6706:{\displaystyle r_{1}}
6680:
6678:{\displaystyle r_{3}}
6652:
6626:
6624:{\displaystyle r_{2}}
6598:
6568:
6538:
6508:
6478:
6446:
6444:{\displaystyle r_{3}}
6418:
6416:{\displaystyle r_{1}}
6390:
6388:{\displaystyle r_{2}}
6362:
6336:
6306:
6276:
6246:
6216:
6184:
6154:
6124:
6094:
6064:
6062:{\displaystyle r_{2}}
6036:
6034:{\displaystyle r_{1}}
6008:
5982:
5980:{\displaystyle r_{3}}
5954:
5952:{\displaystyle r_{3}}
5924:
5894:
5864:
5834:
5804:
5802:{\displaystyle r_{1}}
5776:
5750:
5748:{\displaystyle r_{3}}
5722:
5720:{\displaystyle r_{2}}
5694:
5692:{\displaystyle r_{2}}
5664:
5634:
5604:
5574:
5544:
5518:
5516:{\displaystyle r_{3}}
5490:
5488:{\displaystyle r_{2}}
5462:
5460:{\displaystyle r_{1}}
5434:
5432:{\displaystyle r_{1}}
5404:
5374:
5344:
5314:
5284:
5282:{\displaystyle r_{3}}
5256:
5254:{\displaystyle r_{2}}
5228:
5226:{\displaystyle r_{1}}
5200:
5174:
5146:
5116:
5086:
5056:
5026:
5024:{\displaystyle r_{3}}
4998:
4996:{\displaystyle r_{2}}
4970:
4968:{\displaystyle r_{1}}
4942:
4916:
4893:
4858:
4796:
4763:
4730:
4728:{\displaystyle r_{3}}
4692:
4670:
4655:("apply the symmetry
4650:
4620:
4600:
4578:
4543:
4514:
4512:{\displaystyle r_{1}}
4481:
4448:
4411:
4378:
4342:
4340:{\displaystyle r_{3}}
4313:
4311:{\displaystyle r_{2}}
4284:
4282:{\displaystyle r_{1}}
4240:
4201:
4162:
4129:(vertical reflection)
4124:
4085:
4083:{\displaystyle r_{3}}
4050:
4048:{\displaystyle r_{2}}
4015:
4013:{\displaystyle r_{1}}
3985:(keeping it as it is)
3980:
3943:
3872:
3834:
3786:
3764:
3744:
3724:
3681:
3653:
3628:
3596:
3574:
3546:
3521:
3499:
3470:
3436:
3407:
3381:
3359:
3330:
3304:
3264:
3230:
3206:
3178:
3150:
3077:
3045:
3017:
2995:
2970:
2948:
2914:
2882:
2862:
2840:
2818:
2775:
2739:
2703:
2681:
2666:such that, for every
2661:
2641:
2613:
2547:
2523:
2499:
2475:
2440:
2410:
2388:
2368:
2348:
2323:
2299:
2273:
2233:
2218:
2190:
2165:
2139:
2117:
2115:{\displaystyle b+a=0}
2083:
2081:{\displaystyle a+b=0}
2051:
2029:
1992:
1990:{\displaystyle a+0=a}
1958:
1956:{\displaystyle 0+a=a}
1931:is any integer, then
1926:
1895:
1873:
1853:
1833:
1813:
1793:
1771:
1705:
1683:
1661:
1633:
1605:
1579:
1548:
1526:
1502:
729:
704:
667:
236:representation theory
116:
31:
29749:Algebraic structures
29672:Algebraic structures
29440:Algebraic structures
29425:Equivalence relation
29368:Algebraic expression
29159:Diffeomorphism group
29038:Special linear group
29032:General linear group
28545:Mathematics Magazine
28420:Robertson, Edmund F.
28382:, Providence, R.I.:
27197:Thurston, William P.
26744:Applied Group Theory
26168:, p. 26, §II.2.
26156:, p. 22, §II.1.
26027:, p. 45, §II.4.
26003:, p. 41, §II.4.
25979:, p. 19, §II.1.
25952:, p. 34, §II.3.
25911:, p. 17, §II.1.
25668:, p. 16, II.§1.
25563:Schwarzschild metric
25490:More precisely, the
25327:, p. 84, §II.1.
25091:", which means that
24878:
24845:
24814:
24783:
24603:
24572:
24542:
24512:
24481:
24451:
24400:fundamental groupoid
24380:
24356:
24329:
24319:. More generally, a
24277:
24253:
24229:
24189:
24167:
24136:
24108:
24058:
24010:
23962:
23938:
23408:elementary particles
23344:conserved quantities
23319:
23299:
23279:
23241:
23221:
23201:
23197:of the space of all
23090:
23070:
23040:
23014:
22922:
22887:
22867:
22752:
22730:
22710:
22690:
22664:
22644:
22624:
22589:
22560:
22533:
22487:
22449:
22422:
22389:
22369:
22322:
22300:
22266:
22237:
22217:
22190:
22161:
22132:
22107:
22087:
22060:
22040:
22018:
21994:
21965:
21943:
21919:
21897:
21875:
21823:
21796:
21761:
21739:
21719:
21699:
21690:equilateral triangle
21663:
21637:
21606:
21572:
21549:
21518:
21418:
21398:
21387:{\displaystyle \pm }
21378:
21303:
21252:
21166:
21142:
21122:
21091:
21048:
21012:
20975:
20951:
20904:
20899:general linear group
20864:
20838:General linear group
20750:icosahedral symmetry
20745:Buckminsterfullerene
20664:molecular symmetries
20625:
20601:
20480:
20458:
20454:, all the powers of
20436:
20396:
20359:
20316:
20279:
20257:
20229:
20207:
20203:is cyclic for prime
20173:
20145:
20117:
20095:
20056:
20034:
20005:
19983:
19961:
19939:
19892:
19798:
19774:
19676:
19644:
19616:
19587:
19465:
19445:
19404:
19382:
19355:
19335:
19277:
19240:
19188:
19138:
19114:
19092:
19070:
19048:
19022:
19000:
18963:
18936:
18902:
18880:
18829:
18809:
18787:
18765:
18743:
18719:
18697:
18675:
18653:
18631:
18609:
18565:
18523:
18493:
18469:
18441:
18411:
18389:
18367:
18357:, there is also the
18339:
18284:
18262:
18240:
18214:
18192:
18170:
18150:
18121:
18093:
18073:
18024:
17993:
17965:
17943:
17911:
17889:
17845:
17823:
17803:
17783:
17763:
17708:
17671:
17641:
17610:
17536:
17491:
17457:
17435:
17395:
17369:
17343:
17311:
17278:
17241:
17207:
17181:
17139:
17099:
17075:
16965:
16943:
16923:
16834:
16691:
16667:
16636:
16539:
16505:
16479:
16412:
16380:
16351:
16329:
16302:
16282:
16253:
16221:
16217:from the free group
16199:
16168:
16139:
16112:
16083:
15995:
15961:
15921:
15888:
15851:
15775:
15771:. Both the subgroup
15666:
15624:
15602:
15565:
15540:
15519:
15498:
15475:
15454:
15433:
15410:
15389:
15368:
15328:
15290:
15266:
15234:
15164:
15139:
15109:
15096:{\displaystyle eN=N}
15078:
15044:
15019:
14996:
14973:
14953:
14919:
14891:
14836:
14814:
14792:
14690:
14657:
14599:
14579:
14548:
14442:
14422:
14402:
14380:
14374:are either equal to
14355:
14335:
14304:
14273:
14239:
14217:
14197:
14177:
14155:
14131:
14073:
14022:
13995:
13969:
13943:
13885:
13830:
13806:
13782:
13750:
13728:
13708:
13634:
13605:
13574:
13547:
13525:
13503:
13483:
13463:
13437:
13415:
13404:. Knowing a group's
13386:
13364:
13344:
13302:
13280:
13256:
13168:
13137:
13115:
13093:
13071:
13032:
12990:
12968:
12939:
12912:
12890:
12868:
12846:
12822:
12800:
12756:
12732:
12678:
12641:
12601:
12579:
12557:
12535:
12515:
12481:
12455:
12435:
12403:
12381:
12329:
12309:
12289:
12237:
12196:
12155:
12123:
12091:
12085:inverse homomorphism
12067:
12028:
12024:is the homomorphism
12008:
11979:
11957:
11893:
11842:
11836:respect identities,
11820:
11795:
11773:
11753:
11688:
11654:
11622:
11590:
11527:
11505:
11481:
11449:
11427:
11405:
11379:
11357:
11333:
11309:
11284:
11237:
11231:mathematical symbols
11193:
11172:
11150:
11102:
11056:
11036:
10998:
10800:
10351:
10304:
10282:
10252:
10218:
10196:
10151:
10125:
10119:right multiplication
10095:
10075:
10049:
10027:
9987:
9953:
9931:
9909:
9884:
9858:
9824:
9804:
9778:
9758:
9738:
9700:
9664:
9642:
9622:
9600:
9578:
9558:
9303:
9280:
9260:
9240:
9188:
9166:
9146:
9054:
8898:Carl Friedrich Gauss
8828:
8780:polynomial equations
8735:
8676:
8617:
8586:
8554:
8527:
8500:
8469:
8436:
8403:
8370:
8341:
8313:
8286:
8065:
7953:
7888:
7866:
7842:
7820:
7798:
7776:
7756:
7725:
7701:
7677:
7653:
7614:
7552:
7530:
7508:
7482:
7462:, is highlighted in
7435:
7389:
7338:
7307:
7276:
7247:
7216:
7188:
7160:
7132:
7102:
7072:
7042:
7012:
6982:
6952:
6926:
6898:
6870:
6840:
6810:
6780:
6750:
6720:
6690:
6662:
6636:
6608:
6578:
6548:
6518:
6488:
6458:
6428:
6400:
6372:
6346:
6316:
6286:
6256:
6226:
6196:
6164:
6134:
6104:
6074:
6046:
6018:
5992:
5964:
5936:
5904:
5874:
5844:
5814:
5786:
5760:
5732:
5704:
5676:
5644:
5614:
5584:
5554:
5528:
5500:
5472:
5444:
5416:
5384:
5354:
5324:
5294:
5266:
5238:
5210:
5184:
5158:
5126:
5096:
5066:
5036:
5008:
4980:
4952:
4926:
4905:
4873:
4807:
4776:
4743:
4712:
4681:
4659:
4633:
4609:
4589:
4558:
4523:
4496:
4461:
4428:
4391:
4358:
4324:
4295:
4266:
4220:
4181:
4142:
4104:
4067:
4032:
3997:
3964:
3923:
3861:
3817:
3810:function composition
3775:
3753:
3733:
3695:
3664:
3639:
3607:
3585:
3563:
3555:multiplicative group
3532:
3510:
3488:
3445:
3416:
3394:
3388:multiplicative group
3368:
3346:
3317:
3273:
3239:
3217:
3192:
3161:
3137:
3118:mathematical objects
3056:
3034:
3006:
2984:
2959:
2925:
2891:
2871:
2851:
2829:
2807:
2789:). It is called the
2752:
2716:
2692:
2670:
2650:
2630:
2560:
2536:
2512:
2488:
2464:
2423:
2399:
2377:
2357:
2337:
2312:
2288:
2262:
2237:monster simple group
2207:
2176:
2170:and is denoted
2154:
2128:
2094:
2060:
2040:
2018:
1969:
1935:
1915:
1884:
1862:
1842:
1822:
1802:
1782:
1718:
1694:
1672:
1650:
1620:
1594:
1562:
1537:
1515:
1405:
1273:Group with operators
1216:Complemented lattice
1051:Algebraic structures
716:
691:
654:
190:polynomial equations
105:
29601:Composition algebra
29525:Inner product space
29503:multilinear algebra
29391:Polynomial function
29332:Multilinear algebra
29317:Homological algebra
29307:Commutative algebra
28984:Other finite groups
28771:Commutator subgroup
28625:Smith, David Eugene
28418:O'Connor, John J.;
27744:MathSciNet (2021),
27674:Lay, David (2003),
27588:1999qrsp.book.....K
27557:1937RSPSA.161..220J
27452:Classical Mechanics
27193:Conway, John Horton
26964:Aschbacher, Michael
26246:, pp. 197â202.
26015:, p. 12, §I.2.
25680:, p. 54, §2.6.
25644:, p. 40, §2.2.
25498:, pp. 105â113.
25300:singular cohomology
24862:{\displaystyle 1/a}
24436:
23420:
23108:topological groups,
22838:monstrous moonshine
22769:{\displaystyle G/N}
22337:{\displaystyle p-1}
22285:
22157:The dihedral group
21988:corresponds to the
21450:and higher. In the
20778:octahedral symmetry
20555:Symmetry in physics
20244:{\displaystyle p=5}
20192:
20132:{\displaystyle n=6}
19398:are not a power of
19296:
19037:{\displaystyle p=5}
18876:that is, such that
18693:. The primality of
18508:{\displaystyle p=5}
18487:. For example, for
18426:{\displaystyle p-1}
18229:{\displaystyle 9+4}
18108:{\displaystyle n-a}
17980:{\displaystyle n-1}
17926:{\displaystyle n-1}
17688:{\displaystyle b/a}
17658:{\displaystyle a/b}
17196:{\displaystyle a=2}
17008:and number theory.
16908:equivalence classes
16875:by introducing the
16626:. This is called a
16406:; examples include
16135:and the reflection
15361:
15251:{\displaystyle G/N}
15126:{\displaystyle G/N}
14908:{\displaystyle G/N}
14595:is normal, because
14093:
13159:is a homomorphism.
13052:
11572:Group homomorphisms
9852:left multiplication
8955:of finite groups),
8879:projective geometry
8613:, as, for example,
4898:
3951:
3885:Two figures in the
3855:automorphism groups
3176:{\displaystyle a+b}
1590:property says that
1577:{\displaystyle a+b}
1327:Composition algebra
1087:Quasigroup and loop
360:Group homomorphisms
270:Algebraic structure
29381:Quadratic equation
29312:Elementary algebra
29280:Algebraic geometry
29014:Rubik's Cube group
28971:Baby monster group
28781:Group homomorphism
28697:Weisstein, Eric W.
28670:Dover Publications
28485:10.1007/BF01443322
28441:Curtis, Charles W.
28227:Weibel, Charles A.
28130:Notices of the AMS
28032:Serre, Jean-Pierre
27698:Mac Lane, Saunders
27494:Algebraic Topology
27447:Goldstein, Herbert
26946:Dover Publications
26931:Special references
26687:General references
26282:, pp. 15, 16.
26229:Conway et al. 2001
26039:, p. 9, §I.2.
25940:, p. 7, §I.2.
25896:, p. 3, §I.1.
25520:See, for example,
25423:discrete logarithm
25421:protocol uses the
25338:field of fractions
25251:áœÎŒÏÏâthe same and
25222:See, for example,
25057:Mathematics portal
24890:
24859:
24826:{\displaystyle -a}
24823:
24795:{\displaystyle -a}
24792:
24648:
24580:
24550:
24520:
24489:
24459:
24434:
24386:
24362:
24341:{\displaystyle fg}
24338:
24307:
24259:
24235:
24213:
24173:
24142:
24128:Grothendieck group
24114:
24090:
24042:
23994:
23946:
23858:Commutative monoid
23418:
23404:fundamental forces
23380:special relativity
23325:
23305:
23285:
23265:
23227:
23207:
23183:looks locally like
23096:
23076:
23056:
23026:
23008:topological spaces
22998:
22971:Topological groups
22934:
22905:
22873:
22805:have been used to
22766:
22736:
22716:
22696:
22676:
22650:
22630:
22602:
22575:
22546:
22517:
22471:
22435:
22416:Lagrange's theorem
22414:(a consequence of
22404:
22375:
22334:
22306:
22286:
22269:
22252:
22223:
22203:
22176:
22138:
22113:
22093:
22081:Lagrange's Theorem
22066:
22046:
22024:
22000:
21978:
21949:
21925:
21903:
21881:
21859:
21855:
21843:
21809:
21780:
21745:
21725:
21705:
21678:
21643:
21621:
21587:
21555:
21533:
21495:A group is called
21424:
21404:
21384:
21364:
21289:
21247:quadratic equation
21222:topological groups
21206:
21148:
21128:
21106:
21054:
21032:
20981:
20957:
20932:
20884:
20870:
20676:quantum mechanical
20631:
20607:
20573:
20543:Molecular symmetry
20500:
20464:
20442:
20415:
20378:
20341:
20298:
20263:
20241:
20223:: for example, if
20213:
20193:
20176:
20151:
20129:
20101:
20075:
20040:
20011:
19989:
19967:
19945:
19925:
19876:
19780:
19758:
19660:
19628:
19600:
19573:
19451:
19431:
19417:
19388:
19368:
19341:
19297:
19280:
19261:
19226:
19172:
19120:
19098:
19076:
19054:
19034:
19006:
18984:
18942:
18920:
18886:
18866:
18815:
18793:
18771:
18749:
18725:
18703:
18681:
18659:
18649:: when divided by
18637:
18618:{\displaystyle 16}
18615:
18593:
18547:
18505:
18475:
18447:
18423:
18395:
18373:
18345:
18321:
18280:" or, in symbols,
18271:{\displaystyle 12}
18268:
18246:
18226:
18198:
18176:
18156:
18127:
18105:
18079:
18057:
18008:
17977:
17949:
17923:
17895:
17873:
17829:
17809:
17789:
17769:
17754:
17736:Modular arithmetic
17730:Modular arithmetic
17716:
17685:
17655:
17625:
17594:
17515:
17475:
17441:
17419:
17377:
17349:
17327:
17286:
17262:
17260:
17225:
17193:
17163:
17123:
17083:
17063:also form groups.
17006:algebraic geometry
16977:
16949:
16929:
16900:
16873:algebraic topology
16842:
16828:
16814:Examples of groups
16781:
16673:
16651:
16614:
16523:
16491:
16463:
16392:
16364:
16335:
16315:
16288:
16268:
16239:
16205:
16183:
16154:
16125:
16098:
16042:
15979:
15936:
15915:semidirect product
15903:
15874:
15837:
15759:
15648:
15608:
15588:
15546:
15525:
15504:
15481:
15460:
15439:
15416:
15395:
15374:
15351:
15323:
15318:universal property
15296:
15272:
15248:
15216:
15151:{\displaystyle gN}
15148:
15123:
15093:
15062:
15031:{\displaystyle hN}
15028:
15008:{\displaystyle gN}
15005:
14985:{\displaystyle gN}
14982:
14959:
14939:
14915:for which the map
14905:
14877:
14820:
14798:
14762:
14672:
14641:
14585:
14563:
14532:
14428:
14408:
14386:
14367:{\displaystyle gR}
14364:
14341:
14319:
14279:
14257:
14223:
14203:
14183:
14161:
14137:
14113:
14076:
14054:
14001:
13975:
13949:
13924:
13869:
13812:
13788:
13756:
13734:
13714:
13677:
13628:, and the element
13616:
13589:
13560:
13531:
13509:
13489:
13469:
13443:
13421:
13392:
13370:
13350:
13330:
13286:
13262:
13230:
13149:
13121:
13099:
13077:
13053:
13035:
13016:
12974:
12952:
12925:
12896:
12874:
12852:
12828:
12806:
12762:
12738:
12726:for some subgroup
12716:
12664:
12628:category of groups
12607:
12585:
12563:
12541:
12521:
12499:
12461:
12441:
12409:
12387:
12367:
12315:
12295:
12275:
12221:
12180:
12141:
12119:is a homomorphism
12109:
12087:of a homomorphism
12073:
12053:
12014:
11985:
11963:
11943:
11877:
11826:
11801:
11779:
11759:
11739:
11672:
11640:
11608:
11578:Group homomorphism
11533:
11511:
11487:
11467:
11433:
11411:
11391:
11363:
11339:
11315:
11290:
11267:
11211:
11178:
11156:
11132:
11086:
11042:
11032:with the operator
11022:
10963:
10961:
10780:
10778:
10332:
10288:
10268:
10236:
10202:
10157:
10131:
10107:
10081:
10061:
10033:
10009:
9971:
9937:
9915:
9890:
9864:
9836:
9810:
9790:
9764:
9744:
9722:
9682:
9648:
9628:
9606:
9584:
9564:
9529:
9527:
9296:as inverses. Then
9286:
9266:
9246:
9212:
9172:
9152:
9120:
8847:
8750:
8731:. In other words,
8719:
8660:
8601:
8567:
8540:
8513:
8484:
8451:
8418:
8385:
8364:, the reflections
8352:
8319:
8297:
8265:
8263:
8051:
7939:
7872:
7848:
7826:
7804:
7782:
7762:
7740:
7707:
7683:
7659:
7629:
7598:
7536:
7514:
7494:
7450:
7417:
7351:
7320:
7289:
7258:
7227:
7201:
7173:
7145:
7117:
7087:
7057:
7027:
6997:
6965:
6937:
6911:
6883:
6855:
6825:
6795:
6765:
6735:
6703:
6675:
6647:
6621:
6593:
6563:
6533:
6503:
6473:
6441:
6413:
6385:
6357:
6331:
6301:
6271:
6241:
6211:
6179:
6149:
6119:
6089:
6059:
6031:
6003:
5977:
5949:
5919:
5889:
5859:
5829:
5799:
5771:
5745:
5717:
5689:
5659:
5629:
5599:
5569:
5539:
5513:
5485:
5457:
5429:
5399:
5369:
5339:
5309:
5279:
5251:
5223:
5195:
5169:
5141:
5111:
5081:
5051:
5021:
4993:
4965:
4937:
4911:
4888:
4865:
4853:
4791:
4758:
4725:
4687:
4665:
4645:
4627:from right to left
4615:
4595:
4573:
4538:
4509:
4476:
4443:
4406:
4373:
4337:
4308:
4279:
4255:identity operation
4235:
4196:
4157:
4119:
4080:
4055:(rotation by 180°)
4045:
4010:
3975:
3938:
3916:
3867:
3851:permutation groups
3829:
3781:
3759:
3739:
3719:
3676:
3651:{\displaystyle ab}
3648:
3623:
3591:
3569:
3544:{\displaystyle -x}
3541:
3516:
3494:
3465:
3431:
3402:
3376:
3354:
3325:
3299:
3259:
3225:
3204:{\displaystyle ab}
3201:
3173:
3145:
3072:
3040:
3012:
2990:
2965:
2943:
2909:
2877:
2857:
2835:
2813:
2770:
2734:
2698:
2676:
2656:
2636:
2608:
2542:
2518:
2494:
2470:
2435:
2405:
2383:
2363:
2343:
2318:
2294:
2268:
2213:
2188:{\displaystyle -a}
2185:
2160:
2134:
2112:
2078:
2046:
2024:
2012:For every integer
1987:
1953:
1921:
1890:
1868:
1848:
1828:
1808:
1788:
1766:
1700:
1678:
1656:
1628:
1600:
1574:
1543:
1521:
1497:
836:Special orthogonal
724:
699:
662:
543:Lagrange's theorem
175:special relativity
111:
89:addition operation
42:
38:Rubik's Cube group
29729:
29728:
29651:Symmetric algebra
29621:Geometric algebra
29401:Linear inequality
29352:Universal algebra
29285:Algebraic variety
29209:
29208:
28894:Alternating group
28679:978-0-486-45868-7
28653:978-0-486-60269-1
28463:von Dyck, Walther
28454:978-0-8218-2677-5
28393:978-0-8218-0288-5
28347:978-0-691-14034-6
28323:978-0-691-02374-8
28301:978-0-19-853287-3
28240:978-0-521-55987-4
28218:978-0-387-90894-6
28164:978-1-4822-4582-0
28121:978-0-521-53047-7
28095:978-0-691-08017-8
28076:978-0-88385-511-9
28049:978-0-387-90190-9
27965:978-0-201-87073-2
27946:978-0-19-280723-6
27920:978-981-02-4942-7
27903:Romanowska, A. B.
27879:978-3-540-65399-8
27843:978-0-486-43235-9
27816:978-3-540-56963-3
27795:978-0-691-08238-7
27774:978-0-521-86625-5
27737:978-0-486-43830-6
27713:978-0-387-98403-2
27689:978-0-201-70970-4
27658:978-0-387-40510-0
27631:978-0-8176-3688-3
27597:978-0-691-05872-6
27527:978-0-89874-193-3
27508:978-0-521-79540-1
27481:978-0-470-74115-3
27472:Computer Security
27431:978-0-387-97495-8
27368:978-0-19-883298-0
27342:978-0-471-01670-0
27272:978-1-58488-254-1
27241:978-3-540-52977-4
27185:978-0-691-22534-0
27166:978-0-471-50683-6
27132:978-0-387-97370-8
27106:978-0-486-67355-4
27000:978-0-19-958736-0
26955:978-0-486-62342-9
26922:978-0-387-94461-6
26889:Ledermann, Walter
26869:Ledermann, Walter
26860:978-0-387-22025-3
26827:978-0-387-95385-4
26794:Topics in Algebra
26773:978-0-13-374562-7
26732:978-0-691-13951-7
26711:978-0-13-468960-9
25588:, p. 26, §2.
25417:For example, the
25304:classifying space
25101:is an element of
25040:
25039:
24595:Integers modulo 3
24389:{\displaystyle g}
24365:{\displaystyle f}
24262:{\displaystyle x}
24238:{\displaystyle G}
24176:{\displaystyle x}
24145:{\displaystyle M}
24117:{\displaystyle M}
24052:produces a group
23922:
23921:
23340:Noether's theorem
23328:{\displaystyle n}
23308:{\displaystyle n}
23288:{\displaystyle A}
23230:{\displaystyle n}
23210:{\displaystyle n}
23137:harmonic analysis
23116:topological field
23099:{\displaystyle h}
23079:{\displaystyle g}
23002:Topological group
22946:nullary operation
22876:{\displaystyle G}
22853:extension problem
22846:modular functions
22842:Richard Borcherds
22739:{\displaystyle N}
22719:{\displaystyle G}
22699:{\displaystyle G}
22653:{\displaystyle N}
22633:{\displaystyle G}
22378:{\displaystyle p}
22309:{\displaystyle p}
22226:{\displaystyle R}
22141:{\displaystyle G}
22116:{\displaystyle H}
22096:{\displaystyle G}
22069:{\displaystyle a}
22049:{\displaystyle n}
22027:{\displaystyle a}
22003:{\displaystyle n}
21928:{\displaystyle a}
21906:{\displaystyle n}
21852:
21828:
21826:
21748:{\displaystyle n}
21728:{\displaystyle G}
21708:{\displaystyle a}
21646:{\displaystyle N}
21558:{\displaystyle N}
21452:quadratic formula
21440:quartic equations
21427:{\displaystyle -}
21407:{\displaystyle +}
21359:
21348:
21151:{\displaystyle n}
21131:{\displaystyle G}
21070:computer graphics
21066:Rotation matrices
21057:{\displaystyle n}
20984:{\displaystyle n}
20960:{\displaystyle n}
20873:{\displaystyle x}
20791:
20790:
20691:Curie temperature
20634:{\displaystyle X}
20610:{\displaystyle X}
20467:{\displaystyle a}
20445:{\displaystyle a}
20266:{\displaystyle 3}
20216:{\displaystyle p}
20163:counter-clockwise
20161:corresponds to a
20154:{\displaystyle z}
20104:{\displaystyle n}
20043:{\displaystyle z}
20014:{\displaystyle n}
19992:{\displaystyle n}
19970:{\displaystyle 1}
19948:{\displaystyle 1}
19792:primitive element
19783:{\displaystyle a}
19454:{\displaystyle a}
19391:{\displaystyle z}
19344:{\displaystyle z}
19309:, are crucial to
19123:{\displaystyle 2}
19101:{\displaystyle 3}
19079:{\displaystyle 4}
19057:{\displaystyle 4}
19009:{\displaystyle 1}
18954:BĂ©zout's identity
18945:{\displaystyle b}
18889:{\displaystyle p}
18818:{\displaystyle b}
18796:{\displaystyle p}
18781:not divisible by
18774:{\displaystyle a}
18752:{\displaystyle 1}
18728:{\displaystyle p}
18706:{\displaystyle p}
18684:{\displaystyle 1}
18662:{\displaystyle 5}
18640:{\displaystyle 1}
18625:is equivalent to
18559:. In this group,
18478:{\displaystyle p}
18450:{\displaystyle p}
18398:{\displaystyle 1}
18376:{\displaystyle p}
18348:{\displaystyle p}
18249:{\displaystyle 1}
18201:{\displaystyle 1}
18179:{\displaystyle 4}
18159:{\displaystyle 9}
18130:{\displaystyle a}
18082:{\displaystyle 0}
17952:{\displaystyle 0}
17898:{\displaystyle 0}
17832:{\displaystyle n}
17812:{\displaystyle b}
17792:{\displaystyle a}
17772:{\displaystyle n}
17532:rational numbers
17444:{\displaystyle x}
17352:{\displaystyle b}
17322:
17259:
17002:hyperbolic groups
16952:{\displaystyle m}
16932:{\displaystyle k}
16912:smoothly deformed
16904:topological space
16877:fundamental group
16338:{\displaystyle f}
16291:{\displaystyle r}
16035:
15953:implies that any
15884:are abelian, but
15847:and the quotient
15611:{\displaystyle R}
15559:
15558:
15549:{\displaystyle R}
15528:{\displaystyle U}
15507:{\displaystyle U}
15484:{\displaystyle U}
15463:{\displaystyle R}
15442:{\displaystyle R}
15419:{\displaystyle U}
15398:{\displaystyle R}
15299:{\displaystyle N}
15275:{\displaystyle G}
14962:{\displaystyle g}
14823:{\displaystyle G}
14801:{\displaystyle N}
14588:{\displaystyle R}
14431:{\displaystyle R}
14418:is an element of
14411:{\displaystyle g}
14389:{\displaystyle R}
14344:{\displaystyle R}
14282:{\displaystyle H}
14226:{\displaystyle G}
14206:{\displaystyle g}
14186:{\displaystyle H}
14164:{\displaystyle H}
14140:{\displaystyle H}
14018:. The first case
14004:{\displaystyle G}
13978:{\displaystyle G}
13952:{\displaystyle H}
13815:{\displaystyle g}
13791:{\displaystyle H}
13759:{\displaystyle g}
13737:{\displaystyle H}
13717:{\displaystyle H}
13534:{\displaystyle S}
13512:{\displaystyle G}
13492:{\displaystyle S}
13472:{\displaystyle S}
13446:{\displaystyle G}
13424:{\displaystyle S}
13411:Given any subset
13395:{\displaystyle H}
13373:{\displaystyle h}
13353:{\displaystyle g}
13340:for all elements
13289:{\displaystyle G}
13265:{\displaystyle H}
13124:{\displaystyle H}
13102:{\displaystyle G}
13080:{\displaystyle H}
12977:{\displaystyle H}
12899:{\displaystyle H}
12877:{\displaystyle G}
12855:{\displaystyle G}
12831:{\displaystyle G}
12809:{\displaystyle H}
12765:{\displaystyle G}
12741:{\displaystyle H}
12703:
12610:{\displaystyle H}
12588:{\displaystyle G}
12544:{\displaystyle G}
12524:{\displaystyle H}
12464:{\displaystyle H}
12444:{\displaystyle G}
12412:{\displaystyle H}
12390:{\displaystyle h}
12318:{\displaystyle G}
12298:{\displaystyle g}
12076:{\displaystyle G}
12017:{\displaystyle G}
11988:{\displaystyle G}
11966:{\displaystyle a}
11804:{\displaystyle G}
11782:{\displaystyle b}
11762:{\displaystyle a}
11749:for all elements
11536:{\displaystyle Y}
11514:{\displaystyle X}
11490:{\displaystyle f}
11436:{\displaystyle X}
11421:is an element of
11414:{\displaystyle x}
11366:{\displaystyle z}
11342:{\displaystyle y}
11318:{\displaystyle x}
11293:{\displaystyle X}
11181:{\displaystyle e}
11159:{\displaystyle e}
10957:
10936:
10909:
10860:
10774:
10753:
10690:
10615:
10518:
10421:
10291:{\displaystyle f}
10278:for each element
10205:{\displaystyle e}
10160:{\displaystyle a}
10143:right translation
10134:{\displaystyle a}
10084:{\displaystyle x}
10036:{\displaystyle a}
9940:{\displaystyle b}
9918:{\displaystyle a}
9905:Similarly, given
9893:{\displaystyle a}
9867:{\displaystyle a}
9813:{\displaystyle x}
9767:{\displaystyle G}
9747:{\displaystyle a}
9651:{\displaystyle G}
9631:{\displaystyle x}
9609:{\displaystyle G}
9587:{\displaystyle b}
9567:{\displaystyle a}
9523:
9509:
9501:
9480:
9472:
9464:
9437:
9398:
9390:
9382:
9343:
9335:
9289:{\displaystyle c}
9269:{\displaystyle b}
9249:{\displaystyle a}
9175:{\displaystyle f}
9155:{\displaystyle e}
9008:Daniel Gorenstein
8908:Leopold Kronecker
8322:{\displaystyle a}
7875:{\displaystyle a}
7851:{\displaystyle c}
7829:{\displaystyle b}
7807:{\displaystyle c}
7785:{\displaystyle b}
7765:{\displaystyle a}
7710:{\displaystyle c}
7686:{\displaystyle b}
7662:{\displaystyle a}
7539:{\displaystyle b}
7517:{\displaystyle a}
7470:
7469:
4690:{\displaystyle a}
4668:{\displaystyle b}
4618:{\displaystyle b}
4598:{\displaystyle a}
4250:
4249:
3784:{\displaystyle G}
3762:{\displaystyle b}
3742:{\displaystyle a}
3729:for all elements
3594:{\displaystyle x}
3572:{\displaystyle 1}
3519:{\displaystyle x}
3497:{\displaystyle 0}
3043:{\displaystyle a}
3015:{\displaystyle b}
2993:{\displaystyle a}
2968:{\displaystyle e}
2880:{\displaystyle G}
2860:{\displaystyle b}
2838:{\displaystyle G}
2816:{\displaystyle a}
2701:{\displaystyle G}
2679:{\displaystyle a}
2659:{\displaystyle G}
2639:{\displaystyle e}
2545:{\displaystyle G}
2521:{\displaystyle c}
2497:{\displaystyle b}
2473:{\displaystyle a}
2451:, are satisfied:
2408:{\displaystyle G}
2386:{\displaystyle G}
2366:{\displaystyle b}
2346:{\displaystyle a}
2297:{\displaystyle G}
2271:{\displaystyle G}
2245:Richard Borcherds
2216:{\displaystyle +}
2163:{\displaystyle a}
2137:{\displaystyle b}
2049:{\displaystyle b}
2027:{\displaystyle a}
1924:{\displaystyle a}
1893:{\displaystyle c}
1871:{\displaystyle b}
1851:{\displaystyle a}
1831:{\displaystyle c}
1811:{\displaystyle b}
1791:{\displaystyle a}
1703:{\displaystyle c}
1681:{\displaystyle b}
1659:{\displaystyle a}
1644:For all integers
1603:{\displaystyle +}
1546:{\displaystyle b}
1524:{\displaystyle a}
1386:
1385:
1045:
1044:
620:
619:
502:Alternating group
459:
458:
114:{\displaystyle 1}
29761:
29719:
29718:
29606:Exterior algebra
29275:Abstract algebra
29236:
29229:
29222:
29213:
29212:
29201:Abstract algebra
29138:Quaternion group
29068:Symplectic group
29044:Orthogonal group
28738:
28731:
28724:
28715:
28714:
28710:
28709:
28682:
28656:
28634:
28620:
28596:
28576:
28534:
28518:Joseph Liouville
28515:
28506:Galois, Ăvariste
28500:
28495:, archived from
28457:
28436:
28413:
28396:
28358:
28326:
28304:
28282:
28264:Weinberg, Steven
28259:
28221:
28202:
28193:
28167:
28146:
28145:
28143:10.1090/noti1689
28124:
28106:
28079:
28060:
28041:
28026:
28009:
27998:
27976:
27949:
27923:
27911:World Scientific
27898:
27871:
27860:Neukirch, JĂŒrgen
27854:
27827:
27798:
27785:Ătale Cohomology
27777:
27759:
27758:
27756:
27751:
27740:
27716:
27692:
27669:
27642:
27623:
27608:
27569:
27568:
27551:(905): 220â235,
27530:
27511:
27484:
27465:
27442:
27404:
27379:
27345:
27326:
27309:
27304:
27294:
27275:
27252:
27225:
27216:
27188:
27169:
27149:Carter, Roger W.
27143:
27109:
27090:
27081:
27055:
27035:
27034:
27024:
27003:
26984:
26972:
26958:
26925:
26903:
26883:
26863:
26838:
26804:
26784:
26764:Abstract Algebra
26754:
26735:
26714:
26674:
26668:
26662:
26656:
26650:
26644:
26638:
26632:
26626:
26620:
26614:
26608:
26602:
26596:
26590:
26584:
26578:
26572:
26566:
26560:
26554:
26548:
26542:
26536:
26530:
26524:
26518:
26512:
26506:
26500:
26494:
26488:
26482:
26472:
26466:
26460:
26454:
26448:
26442:
26436:
26430:
26424:
26418:
26412:
26406:
26400:
26394:
26388:
26382:
26376:
26370:
26364:
26355:
26349:
26343:
26337:
26331:
26325:
26319:
26313:
26307:
26301:
26295:
26289:
26283:
26277:
26271:
26265:
26259:
26253:
26247:
26241:
26235:
26226:
26220:
26214:
26208:
26202:
26193:
26187:
26181:
26175:
26169:
26163:
26157:
26151:
26145:
26139:
26133:
26127:
26121:
26115:
26109:
26103:
26097:
26082:
26076:
26070:
26064:
26058:
26052:
26046:
26040:
26034:
26028:
26022:
26016:
26010:
26004:
25998:
25992:
25986:
25980:
25974:
25968:
25962:
25953:
25947:
25941:
25935:
25924:
25918:
25912:
25906:
25897:
25891:
25885:
25879:
25873:
25867:
25861:
25855:
25849:
25843:
25837:
25831:
25825:
25819:
25813:
25807:
25801:
25795:
25789:
25783:
25777:
25771:
25765:
25759:
25753:
25747:
25741:
25735:
25729:
25723:
25717:
25711:
25705:
25699:
25693:
25687:
25681:
25675:
25669:
25663:
25657:
25651:
25645:
25639:
25633:
25627:
25621:
25615:
25609:
25595:
25589:
25583:
25566:
25559:
25553:
25543:
25537:
25534:Ă©tale cohomology
25518:
25512:
25505:
25499:
25488:
25482:
25475:Frucht's theorem
25467:
25461:
25459:
25453:
25447:
25436:
25430:
25415:
25409:
25400:
25394:
25375:
25369:
25363:
25357:
25347:
25341:
25334:
25328:
25317:
25311:
25296:group cohomology
25292:
25286:
25279:
25273:
25266:
25260:
25257:Schwartzman 1994
25255:âstructure. See
25245:
25239:
25220:
25214:
25212:
25202:
25192:
25186:
25185:
25183:
25182:
25177:
25174:
25163:
25157:
25146:
25140:
25135:. See Lang
25134:
25128:
25124:
25118:
25112:
25106:
25100:
25090:
25081:
25059:
25054:
25053:
24901:
24899:
24897:
24896:
24891:
24870:
24868:
24866:
24865:
24860:
24855:
24834:
24832:
24830:
24829:
24824:
24803:
24801:
24799:
24798:
24793:
24659:
24657:
24655:
24654:
24649:
24623:
24615:
24610:
24591:
24589:
24587:
24586:
24581:
24579:
24561:
24559:
24557:
24556:
24551:
24549:
24531:
24529:
24527:
24526:
24521:
24519:
24504:Rational numbers
24500:
24498:
24496:
24495:
24490:
24488:
24470:
24468:
24466:
24465:
24460:
24458:
24437:
24433:
24428:
24423:
24419:
24413:
24397:
24395:
24393:
24392:
24387:
24373:
24371:
24369:
24368:
24363:
24349:
24347:
24345:
24344:
24339:
24318:
24316:
24314:
24313:
24308:
24270:
24268:
24266:
24265:
24260:
24246:
24244:
24242:
24241:
24236:
24222:
24220:
24219:
24214:
24184:
24182:
24180:
24179:
24174:
24161:with one object
24153:
24151:
24149:
24148:
24143:
24125:
24123:
24121:
24120:
24115:
24101:
24099:
24097:
24096:
24091:
24068:
24051:
24049:
24048:
24043:
24020:
24005:
24003:
24001:
24000:
23995:
23972:
23955:
23953:
23952:
23947:
23945:
23421:
23417:
23334:
23332:
23331:
23326:
23314:
23312:
23311:
23306:
23294:
23292:
23291:
23286:
23274:
23272:
23271:
23266:
23236:
23234:
23233:
23228:
23216:
23214:
23213:
23208:
23124:
23105:
23103:
23102:
23097:
23085:
23083:
23082:
23077:
23065:
23063:
23062:
23057:
23055:
23054:
23035:
23033:
23032:
23027:
22943:
22941:
22940:
22935:
22914:
22912:
22911:
22906:
22882:
22880:
22879:
22874:
22777:
22775:
22773:
22772:
22767:
22762:
22745:
22743:
22742:
22737:
22725:
22723:
22722:
22717:
22705:
22703:
22702:
22697:
22685:
22683:
22682:
22677:
22659:
22657:
22656:
22651:
22639:
22637:
22636:
22631:
22611:
22609:
22608:
22603:
22601:
22600:
22584:
22582:
22581:
22576:
22574:
22573:
22568:
22555:
22553:
22552:
22547:
22545:
22544:
22528:
22526:
22524:
22523:
22518:
22516:
22515:
22510:
22501:
22500:
22495:
22480:
22478:
22477:
22472:
22470:
22469:
22468:
22467:
22457:
22444:
22442:
22441:
22436:
22434:
22433:
22413:
22411:
22410:
22405:
22403:
22402:
22397:
22384:
22382:
22381:
22376:
22345:
22343:
22341:
22340:
22335:
22315:
22313:
22312:
22307:
22295:
22293:
22292:
22287:
22284:
22279:
22274:
22261:
22259:
22258:
22253:
22251:
22250:
22249:
22232:
22230:
22229:
22224:
22212:
22210:
22209:
22204:
22202:
22201:
22185:
22183:
22182:
22177:
22175:
22174:
22169:
22149:
22147:
22145:
22144:
22139:
22122:
22120:
22119:
22114:
22102:
22100:
22099:
22094:
22075:
22073:
22072:
22067:
22055:
22053:
22052:
22047:
22035:
22033:
22031:
22030:
22025:
22011:
22009:
22007:
22006:
22001:
21987:
21985:
21984:
21979:
21977:
21976:
21960:
21958:
21956:
21955:
21950:
21936:
21934:
21932:
21931:
21926:
21912:
21910:
21909:
21904:
21892:
21890:
21888:
21887:
21882:
21868:
21866:
21865:
21860:
21854:
21853:
21850:
21844:
21839:
21818:
21816:
21815:
21810:
21808:
21807:
21791:
21789:
21787:
21786:
21781:
21773:
21772:
21754:
21752:
21751:
21746:
21734:
21732:
21731:
21726:
21714:
21712:
21711:
21706:
21687:
21685:
21684:
21679:
21677:
21676:
21671:
21657:Cayley's theorem
21654:
21652:
21650:
21649:
21644:
21630:
21628:
21627:
21622:
21620:
21619:
21614:
21596:
21594:
21593:
21588:
21586:
21585:
21580:
21564:
21562:
21561:
21556:
21544:
21542:
21540:
21539:
21534:
21532:
21531:
21526:
21510:symmetric groups
21471:field extensions
21433:
21431:
21430:
21425:
21413:
21411:
21410:
21405:
21393:
21391:
21390:
21385:
21373:
21371:
21370:
21365:
21360:
21358:
21350:
21349:
21335:
21334:
21325:
21313:
21298:
21296:
21295:
21290:
21267:
21266:
21215:
21213:
21212:
21207:
21202:
21188:
21157:
21155:
21154:
21149:
21137:
21135:
21134:
21129:
21117:
21115:
21113:
21112:
21107:
21105:
21104:
21099:
21063:
21061:
21060:
21055:
21043:
21041:
21039:
21038:
21033:
21022:
20992:
20990:
20988:
20987:
20982:
20968:
20966:
20964:
20963:
20958:
20942:consists of all
20941:
20939:
20938:
20933:
20928:
20914:
20881:
20879:
20877:
20876:
20871:
20846:Character theory
20826:invariant theory
20803:error correction
20739:
20732:
20725:
20718:
20711:
20710:
20706:Goldstone bosons
20683:phase transition
20653:hyperbolic plane
20642:
20640:
20638:
20637:
20632:
20618:
20616:
20614:
20613:
20608:
20511:
20509:
20507:
20506:
20501:
20490:
20473:
20471:
20470:
20465:
20453:
20451:
20449:
20448:
20443:
20426:
20424:
20422:
20421:
20416:
20408:
20407:
20389:
20387:
20385:
20384:
20379:
20371:
20370:
20352:
20350:
20348:
20347:
20342:
20328:
20327:
20309:
20307:
20305:
20304:
20299:
20291:
20290:
20272:
20270:
20269:
20264:
20252:
20250:
20248:
20247:
20242:
20222:
20220:
20219:
20214:
20202:
20200:
20199:
20194:
20191:
20186:
20181:
20160:
20158:
20157:
20152:
20140:
20138:
20136:
20135:
20130:
20110:
20108:
20107:
20102:
20086:
20084:
20082:
20081:
20076:
20068:
20067:
20049:
20047:
20046:
20041:
20022:
20020:
20018:
20017:
20012:
19998:
19996:
19995:
19990:
19978:
19976:
19974:
19973:
19968:
19954:
19952:
19951:
19946:
19934:
19932:
19931:
19926:
19915:
19907:
19902:
19885:
19883:
19882:
19877:
19789:
19787:
19786:
19781:
19769:
19767:
19765:
19764:
19759:
19757:
19756:
19723:
19722:
19707:
19706:
19691:
19690:
19669:
19667:
19666:
19661:
19659:
19658:
19639:
19637:
19635:
19634:
19629:
19609:
19607:
19606:
19601:
19599:
19598:
19582:
19580:
19579:
19574:
19563:
19562:
19550:
19549:
19531:
19530:
19518:
19517:
19502:
19501:
19486:
19485:
19460:
19458:
19457:
19452:
19428:
19426:
19424:
19423:
19418:
19416:
19415:
19397:
19395:
19394:
19389:
19377:
19375:
19374:
19369:
19367:
19366:
19350:
19348:
19347:
19342:
19308:
19306:
19304:
19303:
19298:
19295:
19290:
19285:
19270:
19268:
19267:
19262:
19260:
19252:
19247:
19235:
19233:
19232:
19227:
19225:
19221:
19214:
19200:
19183:
19181:
19179:
19178:
19173:
19171:
19170:
19131:
19129:
19127:
19126:
19121:
19107:
19105:
19104:
19099:
19087:
19085:
19083:
19082:
19077:
19063:
19061:
19060:
19055:
19043:
19041:
19040:
19035:
19017:
19015:
19013:
19012:
19007:
18993:
18991:
18990:
18985:
18951:
18949:
18948:
18943:
18931:
18929:
18927:
18926:
18921:
18895:
18893:
18892:
18887:
18875:
18873:
18872:
18867:
18862:
18824:
18822:
18821:
18816:
18804:
18802:
18800:
18799:
18794:
18780:
18778:
18777:
18772:
18760:
18758:
18756:
18755:
18750:
18736:
18734:
18732:
18731:
18726:
18712:
18710:
18709:
18704:
18692:
18690:
18688:
18687:
18682:
18668:
18666:
18665:
18660:
18648:
18646:
18644:
18643:
18638:
18624:
18622:
18621:
18616:
18604:
18602:
18600:
18599:
18594:
18592:
18591:
18558:
18556:
18554:
18553:
18548:
18516:
18514:
18512:
18511:
18506:
18486:
18484:
18482:
18481:
18476:
18458:
18456:
18454:
18453:
18448:
18434:
18432:
18430:
18429:
18424:
18404:
18402:
18401:
18396:
18384:
18382:
18380:
18379:
18374:
18356:
18354:
18352:
18351:
18346:
18330:
18328:
18327:
18322:
18317:
18279:
18277:
18275:
18274:
18269:
18255:
18253:
18252:
18247:
18236:is congruent to
18235:
18233:
18232:
18227:
18209:
18207:
18205:
18204:
18199:
18185:
18183:
18182:
18177:
18166:and is advanced
18165:
18163:
18162:
18157:
18138:
18136:
18134:
18133:
18128:
18114:
18112:
18111:
18106:
18088:
18086:
18085:
18080:
18068:
18066:
18064:
18063:
18058:
18047:
18039:
18034:
18017:
18015:
18014:
18009:
18007:
18006:
18001:
17988:
17986:
17984:
17983:
17978:
17958:
17956:
17955:
17950:
17934:
17932:
17930:
17929:
17924:
17904:
17902:
17901:
17896:
17884:
17882:
17880:
17879:
17874:
17872:
17838:
17836:
17835:
17830:
17818:
17816:
17815:
17810:
17798:
17796:
17795:
17790:
17778:
17776:
17775:
17770:
17751:
17748: 12. Here,
17725:
17723:
17722:
17717:
17715:
17696:
17694:
17692:
17691:
17686:
17681:
17664:
17662:
17661:
17656:
17651:
17636:
17634:
17632:
17631:
17626:
17624:
17623:
17618:
17603:
17601:
17600:
17595:
17593:
17589:
17576:
17557:
17543:
17524:
17522:
17521:
17516:
17514:
17510:
17503:
17486:
17484:
17482:
17481:
17476:
17450:
17448:
17447:
17442:
17430:
17428:
17426:
17425:
17420:
17418:
17414:
17407:
17388:
17386:
17384:
17383:
17378:
17376:
17361:rational numbers
17358:
17356:
17355:
17350:
17336:
17334:
17333:
17328:
17323:
17315:
17295:
17293:
17292:
17287:
17285:
17273:
17271:
17269:
17268:
17263:
17261:
17252:
17235:in this case is
17234:
17232:
17231:
17226:
17202:
17200:
17199:
17194:
17172:
17170:
17169:
17164:
17162:
17158:
17151:
17134:
17132:
17130:
17129:
17124:
17122:
17118:
17111:
17092:
17090:
17089:
17084:
17082:
17033:computer science
16988:
16986:
16984:
16983:
16978:
16958:
16956:
16955:
16950:
16938:
16936:
16935:
16930:
16858:abstract algebra
16851:
16849:
16848:
16843:
16841:
16792:
16790:
16788:
16787:
16782:
16780:
16779:
16774:
16756:
16755:
16731:
16730:
16718:
16717:
16684:
16682:
16680:
16679:
16674:
16660:
16658:
16657:
16652:
16650:
16649:
16644:
16625:
16623:
16621:
16620:
16615:
16604:
16603:
16579:
16578:
16566:
16565:
16532:
16530:
16529:
16524:
16500:
16498:
16497:
16492:
16474:
16472:
16470:
16469:
16464:
16462:
16461:
16437:
16436:
16424:
16423:
16401:
16399:
16398:
16393:
16375:
16373:
16371:
16370:
16365:
16363:
16362:
16344:
16342:
16341:
16336:
16324:
16322:
16321:
16316:
16314:
16313:
16297:
16295:
16294:
16289:
16277:
16275:
16274:
16269:
16267:
16266:
16261:
16248:
16246:
16245:
16240:
16216:
16214:
16212:
16211:
16206:
16192:
16190:
16189:
16184:
16182:
16181:
16176:
16163:
16161:
16160:
16155:
16153:
16152:
16151:
16134:
16132:
16131:
16126:
16124:
16123:
16107:
16105:
16104:
16099:
16097:
16096:
16091:
16076:, in many ways.
16053:
16051:
16049:
16048:
16043:
16037:
16036:
16034:
16029:
16024:
16011:
15988:
15986:
15985:
15980:
15945:
15943:
15942:
15937:
15935:
15934:
15929:
15912:
15910:
15909:
15904:
15902:
15901:
15896:
15883:
15881:
15880:
15875:
15870:
15865:
15864:
15859:
15846:
15844:
15843:
15838:
15833:
15832:
15820:
15819:
15807:
15806:
15794:
15770:
15768:
15766:
15765:
15760:
15746:
15745:
15744:
15731:
15730:
15729:
15710:
15709:
15708:
15692:
15691:
15690:
15659:
15657:
15655:
15654:
15649:
15644:
15643:
15642:
15617:
15615:
15614:
15609:
15597:
15595:
15594:
15589:
15584:
15579:
15578:
15573:
15555:
15553:
15552:
15547:
15534:
15532:
15531:
15526:
15513:
15511:
15510:
15505:
15490:
15488:
15487:
15482:
15469:
15467:
15466:
15461:
15448:
15446:
15445:
15440:
15425:
15423:
15422:
15417:
15404:
15402:
15401:
15396:
15383:
15381:
15380:
15375:
15362:
15360:
15358:
15357:
15352:
15347:
15342:
15341:
15336:
15322:
15307:
15305:
15303:
15302:
15297:
15283:
15281:
15279:
15278:
15273:
15259:
15257:
15255:
15254:
15249:
15244:
15227:
15225:
15223:
15222:
15217:
15212:
15208:
15207:
15188:
15187:
15157:
15155:
15154:
15149:
15134:
15132:
15130:
15129:
15124:
15119:
15102:
15100:
15099:
15094:
15073:
15071:
15069:
15068:
15063:
15037:
15035:
15034:
15029:
15014:
15012:
15011:
15006:
14991:
14989:
14988:
14983:
14968:
14966:
14965:
14960:
14948:
14946:
14945:
14940:
14935:
14914:
14912:
14911:
14906:
14901:
14886:
14884:
14883:
14878:
14846:
14831:
14829:
14827:
14826:
14821:
14807:
14805:
14804:
14799:
14773:
14771:
14769:
14768:
14763:
14758:
14757:
14756:
14740:
14739:
14738:
14722:
14721:
14720:
14704:
14703:
14702:
14683:
14681:
14679:
14678:
14673:
14671:
14670:
14665:
14650:
14648:
14647:
14642:
14640:
14639:
14638:
14613:
14612:
14611:
14594:
14592:
14591:
14586:
14575:). The subgroup
14574:
14572:
14570:
14569:
14564:
14562:
14561:
14556:
14541:
14539:
14538:
14533:
14528:
14527:
14526:
14513:
14512:
14511:
14498:
14497:
14496:
14483:
14482:
14481:
14462:
14461:
14460:
14437:
14435:
14434:
14429:
14417:
14415:
14414:
14409:
14397:
14395:
14393:
14392:
14387:
14373:
14371:
14370:
14365:
14350:
14348:
14347:
14342:
14330:
14328:
14326:
14325:
14320:
14318:
14317:
14312:
14289:is said to be a
14288:
14286:
14285:
14280:
14268:
14266:
14264:
14263:
14258:
14232:
14230:
14229:
14224:
14212:
14210:
14209:
14204:
14192:
14190:
14189:
14184:
14172:
14170:
14168:
14167:
14162:
14148:
14146:
14144:
14143:
14138:
14124:
14122:
14120:
14119:
14114:
14106:
14105:
14092:
14084:
14063:
14061:
14060:
14055:
14050:
14049:
14034:
14033:
14010:
14008:
14007:
14002:
13986:
13984:
13982:
13981:
13976:
13958:
13956:
13955:
13950:
13935:
13933:
13931:
13930:
13925:
13878:
13876:
13875:
13870:
13823:
13821:
13819:
13818:
13813:
13799:
13797:
13795:
13794:
13789:
13767:
13765:
13763:
13762:
13757:
13743:
13741:
13740:
13735:
13723:
13721:
13720:
13715:
13688:
13686:
13684:
13683:
13678:
13676:
13675:
13663:
13662:
13661:
13648:
13647:
13646:
13627:
13625:
13623:
13622:
13617:
13615:
13598:
13596:
13595:
13590:
13588:
13587:
13586:
13569:
13567:
13566:
13561:
13559:
13558:
13542:
13540:
13538:
13537:
13532:
13518:
13516:
13515:
13510:
13498:
13496:
13495:
13490:
13478:
13476:
13475:
13470:
13454:
13452:
13450:
13449:
13444:
13430:
13428:
13427:
13422:
13403:
13401:
13399:
13398:
13393:
13379:
13377:
13376:
13371:
13359:
13357:
13356:
13351:
13339:
13337:
13336:
13331:
13317:
13316:
13297:
13295:
13293:
13292:
13287:
13273:
13271:
13269:
13268:
13263:
13241:
13239:
13237:
13236:
13231:
13226:
13225:
13213:
13212:
13200:
13199:
13187:
13158:
13156:
13155:
13150:
13132:
13130:
13128:
13127:
13122:
13108:
13106:
13105:
13100:
13088:
13086:
13084:
13083:
13078:
13064:
13062:
13060:
13059:
13054:
13051:
13043:
13025:
13023:
13022:
13017:
13015:
13014:
13002:
13001:
12985:
12983:
12981:
12980:
12975:
12961:
12959:
12958:
12953:
12951:
12950:
12934:
12932:
12931:
12926:
12924:
12923:
12907:
12905:
12903:
12902:
12897:
12883:
12881:
12880:
12875:
12863:
12861:
12859:
12858:
12853:
12839:
12837:
12835:
12834:
12829:
12815:
12813:
12812:
12807:
12773:
12771:
12769:
12768:
12763:
12749:
12747:
12745:
12744:
12739:
12725:
12723:
12722:
12717:
12705:
12704:
12702:
12697:
12692:
12688:
12673:
12671:
12670:
12665:
12657:
12618:
12616:
12614:
12613:
12608:
12594:
12592:
12591:
12586:
12574:
12572:
12570:
12569:
12564:
12550:
12548:
12547:
12542:
12530:
12528:
12527:
12522:
12511:. In this case,
12510:
12508:
12506:
12505:
12500:
12470:
12468:
12467:
12462:
12450:
12448:
12447:
12442:
12420:
12418:
12416:
12415:
12410:
12396:
12394:
12393:
12388:
12376:
12374:
12373:
12368:
12360:
12359:
12341:
12340:
12324:
12322:
12321:
12316:
12304:
12302:
12301:
12296:
12284:
12282:
12281:
12276:
12268:
12267:
12249:
12248:
12232:
12230:
12228:
12227:
12222:
12220:
12219:
12189:
12187:
12186:
12181:
12179:
12178:
12150:
12148:
12147:
12142:
12118:
12116:
12115:
12110:
12082:
12080:
12079:
12074:
12062:
12060:
12059:
12054:
12040:
12039:
12023:
12021:
12020:
12015:
11996:
11994:
11992:
11991:
11986:
11972:
11970:
11969:
11964:
11952:
11950:
11949:
11944:
11942:
11941:
11914:
11913:
11889:, and inverses,
11888:
11886:
11884:
11883:
11878:
11876:
11875:
11860:
11859:
11835:
11833:
11832:
11827:
11812:
11810:
11808:
11807:
11802:
11788:
11786:
11785:
11780:
11768:
11766:
11765:
11760:
11748:
11746:
11745:
11740:
11681:
11679:
11678:
11673:
11649:
11647:
11646:
11641:
11617:
11615:
11614:
11609:
11552:, function, and
11544:
11542:
11540:
11539:
11534:
11520:
11518:
11517:
11512:
11496:
11494:
11493:
11488:
11476:
11474:
11473:
11468:
11444:
11442:
11440:
11439:
11434:
11420:
11418:
11417:
11412:
11400:
11398:
11397:
11392:
11374:
11372:
11370:
11369:
11364:
11350:
11348:
11346:
11345:
11340:
11326:
11324:
11322:
11321:
11316:
11299:
11297:
11296:
11291:
11276:
11274:
11273:
11268:
11220:
11218:
11217:
11212:
11187:
11185:
11184:
11179:
11167:
11165:
11163:
11162:
11157:
11143:
11141:
11139:
11138:
11133:
11095:
11093:
11092:
11087:
11051:
11049:
11048:
11043:
11031:
11029:
11028:
11023:
10972:
10970:
10969:
10964:
10962:
10958:
10955:
10952:
10941:
10937:
10934:
10931:
10914:
10910:
10907:
10904:
10893:
10892:
10865:
10861:
10858:
10855:
10844:
10843:
10789:
10787:
10786:
10781:
10779:
10775:
10772:
10769:
10758:
10754:
10751:
10748:
10746:
10745:
10730:
10729:
10717:
10716:
10695:
10691:
10688:
10685:
10680:
10679:
10655:
10654:
10642:
10641:
10620:
10616:
10613:
10610:
10605:
10604:
10580:
10579:
10558:
10557:
10545:
10544:
10523:
10519:
10516:
10513:
10508:
10507:
10480:
10479:
10464:
10463:
10451:
10450:
10426:
10422:
10419:
10416:
10411:
10410:
10376:
10375:
10343:
10341:
10339:
10338:
10333:
10319:
10318:
10297:
10295:
10294:
10289:
10277:
10275:
10274:
10269:
10267:
10266:
10247:
10245:
10243:
10242:
10237:
10211:
10209:
10208:
10203:
10187:one-sided axioms
10168:
10166:
10164:
10163:
10158:
10140:
10138:
10137:
10132:
10116:
10114:
10113:
10108:
10090:
10088:
10087:
10082:
10070:
10068:
10067:
10062:
10044:
10042:
10040:
10039:
10034:
10020:
10018:
10016:
10015:
10010:
10008:
10007:
9980:
9978:
9977:
9972:
9948:
9946:
9944:
9943:
9938:
9924:
9922:
9921:
9916:
9901:
9899:
9897:
9896:
9891:
9876:left translation
9873:
9871:
9870:
9865:
9845:
9843:
9842:
9837:
9819:
9817:
9816:
9811:
9799:
9797:
9796:
9791:
9773:
9771:
9770:
9765:
9753:
9751:
9750:
9745:
9733:
9731:
9729:
9728:
9723:
9715:
9714:
9693:
9691:
9689:
9688:
9683:
9658:to the equation
9657:
9655:
9654:
9649:
9637:
9635:
9634:
9629:
9617:
9615:
9613:
9612:
9607:
9593:
9591:
9590:
9585:
9573:
9571:
9570:
9565:
9538:
9536:
9535:
9530:
9528:
9524:
9521:
9510:
9507:
9502:
9499:
9496:
9485:
9481:
9478:
9473:
9470:
9465:
9462:
9459:
9442:
9438:
9435:
9432:
9403:
9399:
9396:
9391:
9388:
9383:
9380:
9377:
9348:
9344:
9341:
9336:
9333:
9330:
9295:
9293:
9292:
9287:
9275:
9273:
9272:
9267:
9255:
9253:
9252:
9247:
9223:
9221:
9219:
9218:
9213:
9181:
9179:
9178:
9173:
9161:
9159:
9158:
9153:
9129:
9127:
9126:
9121:
9012:John G. Thompson
8989:Claude Chevalley
8985:algebraic groups
8949:William Burnside
8941:Walther von Dyck
8938:
8871:Erlangen program
8856:
8854:
8853:
8848:
8840:
8839:
8760:is not abelian.
8759:
8757:
8756:
8751:
8749:
8748:
8743:
8730:
8728:
8726:
8725:
8720:
8718:
8717:
8716:
8703:
8702:
8701:
8688:
8687:
8669:
8667:
8666:
8661:
8659:
8658:
8657:
8644:
8643:
8631:
8630:
8629:
8612:
8610:
8608:
8607:
8602:
8600:
8599:
8594:
8576:
8574:
8573:
8568:
8566:
8565:
8549:
8547:
8546:
8541:
8539:
8538:
8522:
8520:
8519:
8514:
8512:
8511:
8495:
8493:
8491:
8490:
8485:
8483:
8482:
8481:
8462:
8460:
8458:
8457:
8452:
8450:
8449:
8448:
8429:
8427:
8425:
8424:
8419:
8417:
8416:
8415:
8396:
8394:
8392:
8391:
8386:
8384:
8383:
8382:
8363:
8361:
8359:
8358:
8353:
8351:
8328:
8326:
8325:
8320:
8308:
8306:
8304:
8303:
8298:
8296:
8278:Identity element
8274:
8272:
8271:
8266:
8264:
8257:
8256:
8244:
8243:
8242:
8229:
8228:
8227:
8207:
8206:
8194:
8193:
8192:
8176:
8175:
8174:
8160:
8159:
8147:
8146:
8134:
8133:
8117:
8116:
8101:
8100:
8099:
8086:
8085:
8084:
8060:
8058:
8057:
8052:
8047:
8046:
8034:
8033:
8032:
8016:
8015:
8014:
8001:
8000:
7985:
7984:
7983:
7970:
7969:
7968:
7948:
7946:
7945:
7940:
7883:
7881:
7879:
7878:
7873:
7859:
7857:
7855:
7854:
7849:
7835:
7833:
7832:
7827:
7815:
7813:
7811:
7810:
7805:
7791:
7789:
7788:
7783:
7771:
7769:
7768:
7763:
7751:
7749:
7747:
7746:
7741:
7739:
7738:
7733:
7718:
7716:
7714:
7713:
7708:
7694:
7692:
7690:
7689:
7684:
7670:
7668:
7666:
7665:
7660:
7640:
7638:
7636:
7635:
7630:
7628:
7627:
7626:
7607:
7605:
7604:
7599:
7594:
7593:
7592:
7579:
7578:
7577:
7564:
7563:
7547:
7545:
7543:
7542:
7537:
7523:
7521:
7520:
7515:
7503:
7501:
7500:
7495:
7476:Binary operation
7465:
7461:
7459:
7457:
7456:
7451:
7449:
7448:
7447:
7428:
7426:
7424:
7423:
7418:
7416:
7415:
7403:
7402:
7401:
7382:
7378:
7370:
7362:
7360:
7358:
7357:
7352:
7350:
7349:
7331:
7329:
7327:
7326:
7321:
7319:
7318:
7300:
7298:
7296:
7295:
7290:
7288:
7287:
7269:
7267:
7265:
7264:
7259:
7257:
7236:
7234:
7233:
7228:
7226:
7210:
7208:
7207:
7202:
7200:
7199:
7182:
7180:
7179:
7174:
7172:
7171:
7154:
7152:
7151:
7146:
7144:
7143:
7126:
7124:
7123:
7118:
7116:
7115:
7114:
7096:
7094:
7093:
7088:
7086:
7085:
7084:
7066:
7064:
7063:
7058:
7056:
7055:
7054:
7036:
7034:
7033:
7028:
7026:
7025:
7024:
7006:
7004:
7003:
6998:
6996:
6995:
6994:
6974:
6972:
6971:
6966:
6964:
6963:
6946:
6944:
6943:
6938:
6936:
6920:
6918:
6917:
6912:
6910:
6909:
6892:
6890:
6889:
6884:
6882:
6881:
6864:
6862:
6861:
6856:
6854:
6853:
6852:
6834:
6832:
6831:
6826:
6824:
6823:
6822:
6804:
6802:
6801:
6796:
6794:
6793:
6792:
6774:
6772:
6771:
6766:
6764:
6763:
6762:
6744:
6742:
6741:
6736:
6734:
6733:
6732:
6712:
6710:
6709:
6704:
6702:
6701:
6684:
6682:
6681:
6676:
6674:
6673:
6656:
6654:
6653:
6648:
6646:
6630:
6628:
6627:
6622:
6620:
6619:
6602:
6600:
6599:
6594:
6592:
6591:
6590:
6572:
6570:
6569:
6564:
6562:
6561:
6560:
6542:
6540:
6539:
6534:
6532:
6531:
6530:
6512:
6510:
6509:
6504:
6502:
6501:
6500:
6482:
6480:
6479:
6474:
6472:
6471:
6470:
6450:
6448:
6447:
6442:
6440:
6439:
6422:
6420:
6419:
6414:
6412:
6411:
6394:
6392:
6391:
6386:
6384:
6383:
6366:
6364:
6363:
6358:
6356:
6340:
6338:
6337:
6332:
6330:
6329:
6328:
6310:
6308:
6307:
6302:
6300:
6299:
6298:
6280:
6278:
6277:
6272:
6270:
6269:
6268:
6250:
6248:
6247:
6242:
6240:
6239:
6238:
6220:
6218:
6217:
6212:
6210:
6209:
6208:
6188:
6186:
6185:
6180:
6178:
6177:
6176:
6158:
6156:
6155:
6150:
6148:
6147:
6146:
6128:
6126:
6125:
6120:
6118:
6117:
6116:
6098:
6096:
6095:
6090:
6088:
6087:
6086:
6068:
6066:
6065:
6060:
6058:
6057:
6040:
6038:
6037:
6032:
6030:
6029:
6012:
6010:
6009:
6004:
6002:
5986:
5984:
5983:
5978:
5976:
5975:
5958:
5956:
5955:
5950:
5948:
5947:
5928:
5926:
5925:
5920:
5918:
5917:
5916:
5898:
5896:
5895:
5890:
5888:
5887:
5886:
5868:
5866:
5865:
5860:
5858:
5857:
5856:
5838:
5836:
5835:
5830:
5828:
5827:
5826:
5808:
5806:
5805:
5800:
5798:
5797:
5780:
5778:
5777:
5772:
5770:
5754:
5752:
5751:
5746:
5744:
5743:
5726:
5724:
5723:
5718:
5716:
5715:
5698:
5696:
5695:
5690:
5688:
5687:
5668:
5666:
5665:
5660:
5658:
5657:
5656:
5638:
5636:
5635:
5630:
5628:
5627:
5626:
5608:
5606:
5605:
5600:
5598:
5597:
5596:
5578:
5576:
5575:
5570:
5568:
5567:
5566:
5548:
5546:
5545:
5540:
5538:
5522:
5520:
5519:
5514:
5512:
5511:
5494:
5492:
5491:
5486:
5484:
5483:
5466:
5464:
5463:
5458:
5456:
5455:
5438:
5436:
5435:
5430:
5428:
5427:
5408:
5406:
5405:
5400:
5398:
5397:
5396:
5378:
5376:
5375:
5370:
5368:
5367:
5366:
5348:
5346:
5345:
5340:
5338:
5337:
5336:
5318:
5316:
5315:
5310:
5308:
5307:
5306:
5288:
5286:
5285:
5280:
5278:
5277:
5260:
5258:
5257:
5252:
5250:
5249:
5232:
5230:
5229:
5224:
5222:
5221:
5204:
5202:
5201:
5196:
5194:
5178:
5176:
5175:
5170:
5168:
5150:
5148:
5147:
5142:
5140:
5139:
5138:
5120:
5118:
5117:
5112:
5110:
5109:
5108:
5090:
5088:
5087:
5082:
5080:
5079:
5078:
5060:
5058:
5057:
5052:
5050:
5049:
5048:
5030:
5028:
5027:
5022:
5020:
5019:
5002:
5000:
4999:
4994:
4992:
4991:
4974:
4972:
4971:
4966:
4964:
4963:
4946:
4944:
4943:
4938:
4936:
4920:
4918:
4917:
4912:
4899:
4897:
4895:
4894:
4889:
4887:
4886:
4881:
4864:
4862:
4860:
4859:
4854:
4849:
4848:
4847:
4834:
4833:
4821:
4820:
4819:
4802:
4800:
4798:
4797:
4792:
4790:
4789:
4788:
4769:
4767:
4765:
4764:
4759:
4757:
4756:
4755:
4736:
4734:
4732:
4731:
4726:
4724:
4723:
4698:
4696:
4694:
4693:
4688:
4674:
4672:
4671:
4666:
4654:
4652:
4651:
4646:
4624:
4622:
4621:
4616:
4604:
4602:
4601:
4596:
4584:
4582:
4580:
4579:
4574:
4572:
4571:
4566:
4547:
4545:
4544:
4539:
4537:
4536:
4535:
4518:
4516:
4515:
4510:
4508:
4507:
4487:
4485:
4483:
4482:
4477:
4475:
4474:
4473:
4454:
4452:
4450:
4449:
4444:
4442:
4441:
4440:
4417:
4415:
4413:
4412:
4407:
4405:
4404:
4403:
4384:
4382:
4380:
4379:
4374:
4372:
4371:
4370:
4348:
4346:
4344:
4343:
4338:
4336:
4335:
4317:
4315:
4314:
4309:
4307:
4306:
4290:
4288:
4286:
4285:
4280:
4278:
4277:
4244:
4242:
4241:
4236:
4234:
4233:
4232:
4215:
4205:
4203:
4202:
4197:
4195:
4194:
4193:
4176:
4166:
4164:
4163:
4158:
4156:
4155:
4154:
4137:
4128:
4126:
4125:
4120:
4118:
4117:
4116:
4099:
4089:
4087:
4086:
4081:
4079:
4078:
4062:
4054:
4052:
4051:
4046:
4044:
4043:
4027:
4019:
4017:
4016:
4011:
4009:
4008:
3992:
3984:
3982:
3981:
3976:
3974:
3959:
3952:
3949:
3947:
3945:
3944:
3939:
3937:
3936:
3931:
3915:
3876:
3874:
3873:
3868:
3840:
3838:
3836:
3835:
3830:
3792:
3790:
3788:
3787:
3782:
3768:
3766:
3765:
3760:
3748:
3746:
3745:
3740:
3728:
3726:
3725:
3720:
3687:
3685:
3683:
3682:
3677:
3657:
3655:
3654:
3649:
3634:
3632:
3630:
3629:
3624:
3622:
3621:
3600:
3598:
3597:
3592:
3580:
3578:
3576:
3575:
3570:
3552:
3550:
3548:
3547:
3542:
3525:
3523:
3522:
3517:
3505:
3503:
3501:
3500:
3495:
3474:
3472:
3471:
3466:
3452:
3440:
3438:
3437:
3432:
3430:
3429:
3424:
3411:
3409:
3408:
3403:
3401:
3385:
3383:
3382:
3377:
3375:
3363:
3361:
3360:
3355:
3353:
3334:
3332:
3331:
3326:
3324:
3308:
3306:
3305:
3300:
3283:
3269:is a group, and
3268:
3266:
3265:
3260:
3249:
3234:
3232:
3231:
3226:
3224:
3212:
3210:
3208:
3207:
3202:
3182:
3180:
3179:
3174:
3156:
3154:
3152:
3151:
3146:
3144:
3083:
3081:
3079:
3078:
3073:
3071:
3070:
3049:
3047:
3046:
3041:
3026:); it is called
3021:
3019:
3018:
3013:
3001:
2999:
2997:
2996:
2991:
2974:
2972:
2971:
2966:
2954:
2952:
2950:
2949:
2944:
2918:
2916:
2915:
2910:
2886:
2884:
2883:
2878:
2866:
2864:
2863:
2858:
2846:
2844:
2842:
2841:
2836:
2822:
2820:
2819:
2814:
2791:identity element
2781:
2779:
2777:
2776:
2771:
2745:
2743:
2741:
2740:
2735:
2709:
2707:
2705:
2704:
2699:
2685:
2683:
2682:
2677:
2665:
2663:
2662:
2657:
2645:
2643:
2642:
2637:
2623:Identity element
2619:
2617:
2615:
2614:
2609:
2553:
2551:
2549:
2548:
2543:
2529:
2527:
2525:
2524:
2519:
2505:
2503:
2501:
2500:
2495:
2481:
2479:
2477:
2476:
2471:
2446:
2444:
2442:
2441:
2436:
2416:
2414:
2412:
2411:
2406:
2392:
2390:
2389:
2384:
2372:
2370:
2369:
2364:
2352:
2350:
2349:
2344:
2329:
2327:
2325:
2324:
2319:
2306:, here denoted "
2305:
2303:
2301:
2300:
2295:
2280:binary operation
2278:together with a
2277:
2275:
2274:
2269:
2251:
2224:
2222:
2220:
2219:
2214:
2196:
2194:
2192:
2191:
2186:
2169:
2167:
2166:
2161:
2143:
2141:
2140:
2135:
2123:
2121:
2119:
2118:
2113:
2087:
2085:
2084:
2079:
2055:
2053:
2052:
2047:
2035:
2033:
2031:
2030:
2025:
2006:identity element
1998:
1996:
1994:
1993:
1988:
1962:
1960:
1959:
1954:
1930:
1928:
1927:
1922:
1901:
1899:
1897:
1896:
1891:
1877:
1875:
1874:
1869:
1857:
1855:
1854:
1849:
1837:
1835:
1834:
1829:
1817:
1815:
1814:
1809:
1797:
1795:
1794:
1789:
1777:
1775:
1773:
1772:
1767:
1711:
1709:
1707:
1706:
1701:
1687:
1685:
1684:
1679:
1667:
1665:
1663:
1662:
1657:
1639:
1637:
1635:
1634:
1629:
1627:
1612:binary operation
1609:
1607:
1606:
1601:
1583:
1581:
1580:
1575:
1554:
1552:
1550:
1549:
1544:
1530:
1528:
1527:
1522:
1506:
1504:
1503:
1498:
1412:
1378:
1371:
1364:
1153:Commutative ring
1082:Rack and quandle
1047:
1046:
1037:
1030:
1023:
979:Algebraic groups
752:Hyperbolic group
742:Arithmetic group
733:
731:
730:
725:
723:
708:
706:
705:
700:
698:
671:
669:
668:
663:
661:
584:Schur multiplier
538:Cauchy's theorem
526:Quaternion group
474:
473:
300:
299:
289:
276:
265:
264:
258:, which studies
205:
192:, starting with
163:particle physics
132:polynomial roots
128:geometric shapes
122:
120:
118:
117:
112:
95:group, which is
70:identity element
29769:
29768:
29764:
29763:
29762:
29760:
29759:
29758:
29734:
29733:
29730:
29725:
29707:
29676:
29660:
29641:Quotient object
29631:Polynomial ring
29589:
29550:Linear subspace
29502:
29496:
29435:
29377:Linear equation
29356:
29302:Category theory
29263:
29245:
29240:
29210:
29205:
29182:
29154:Conformal group
29142:
29116:
29108:
29100:
29092:
29084:
29018:
29010:
28997:
28988:Symmetric group
28967:
28957:
28950:
28943:
28936:
28928:
28919:
28915:
28905:Sporadic groups
28899:
28890:
28872:Discrete groups
28866:
28857:Wallpaper group
28837:Solvable groups
28805:Types of groups
28800:
28766:Normal subgroup
28747:
28742:
28691:
28686:
28680:
28654:
28558:10.2307/2690312
28540:Kleiner, Israel
28525:Jordan, Camille
28455:
28394:
28371:
28365:
28348:
28324:
28302:
28280:
28241:
28219:
28191:10.2307/1990375
28165:
28122:
28096:
28077:
28050:
28007:
27996:
27966:
27947:
27921:
27880:
27844:
27817:
27796:
27775:
27754:
27752:
27738:
27722:Magnus, Wilhelm
27714:
27690:
27659:
27632:
27598:
27528:
27509:
27482:
27463:
27432:
27410:Fulton, William
27369:
27343:
27307:
27302:
27292:
27273:
27242:
27214:math.MG/9911185
27186:
27167:
27133:
27123:Springer-Verlag
27107:
27053:
27001:
26991:Category Theory
26970:
26956:
26933:
26923:
26861:
26851:Springer-Verlag
26828:
26774:
26733:
26712:
26689:
26683:
26678:
26677:
26669:
26665:
26657:
26653:
26645:
26641:
26633:
26629:
26621:
26617:
26609:
26605:
26597:
26593:
26585:
26581:
26573:
26569:
26561:
26557:
26549:
26545:
26537:
26533:
26525:
26521:
26513:
26509:
26503:Aschbacher 2004
26501:
26497:
26489:
26485:
26473:
26469:
26461:
26457:
26449:
26445:
26437:
26433:
26425:
26421:
26413:
26409:
26405:, p. viii.
26401:
26397:
26389:
26385:
26377:
26373:
26365:
26358:
26350:
26346:
26338:
26334:
26326:
26322:
26314:
26310:
26302:
26298:
26290:
26286:
26278:
26274:
26266:
26262:
26254:
26250:
26242:
26238:
26227:
26223:
26215:
26211:
26203:
26196:
26188:
26184:
26176:
26172:
26164:
26160:
26152:
26148:
26140:
26136:
26128:
26124:
26116:
26112:
26104:
26100:
26083:
26079:
26071:
26067:
26059:
26055:
26047:
26043:
26035:
26031:
26023:
26019:
26011:
26007:
25999:
25995:
25987:
25983:
25975:
25971:
25963:
25956:
25948:
25944:
25936:
25927:
25919:
25915:
25907:
25900:
25892:
25888:
25880:
25876:
25868:
25864:
25856:
25852:
25844:
25840:
25832:
25828:
25820:
25816:
25808:
25804:
25796:
25792:
25784:
25780:
25772:
25768:
25760:
25756:
25748:
25744:
25736:
25732:
25724:
25720:
25712:
25708:
25700:
25696:
25688:
25684:
25676:
25672:
25664:
25660:
25652:
25648:
25640:
25636:
25628:
25624:
25616:
25612:
25596:
25592:
25584:
25580:
25575:
25570:
25569:
25560:
25556:
25544:
25540:
25519:
25515:
25509:Aschbacher 2004
25506:
25502:
25489:
25485:
25468:
25464:
25455:
25449:
25439:
25437:
25433:
25416:
25412:
25401:
25397:
25391:simple algebras
25376:
25372:
25359:
25353:
25348:
25344:
25335:
25331:
25318:
25314:
25293:
25289:
25285:for an example.
25280:
25276:
25267:
25263:
25246:
25242:
25221:
25217:
25204:
25194:
25188:
25178:
25175:
25170:
25169:
25167:
25166:
25164:
25160:
25154:MathSciNet 2021
25147:
25143:
25130:
25126:
25120:
25114:
25108:
25102:
25092:
25088:
25082:
25078:
25073:
25055:
25048:
25045:
24879:
24876:
24875:
24873:
24871:
24851:
24846:
24843:
24842:
24840:
24815:
24812:
24811:
24809:
24784:
24781:
24780:
24778:
24619:
24611:
24606:
24604:
24601:
24600:
24598:
24597:
24575:
24573:
24570:
24569:
24567:
24564:Complex numbers
24562:
24545:
24543:
24540:
24539:
24537:
24532:
24515:
24513:
24510:
24509:
24507:
24484:
24482:
24479:
24478:
24476:
24454:
24452:
24449:
24448:
24446:
24444:Natural numbers
24426:
24421:
24417:
24411:
24381:
24378:
24377:
24375:
24357:
24354:
24353:
24351:
24330:
24327:
24326:
24324:
24278:
24275:
24274:
24272:
24254:
24251:
24250:
24248:
24230:
24227:
24226:
24224:
24190:
24187:
24186:
24168:
24165:
24164:
24162:
24137:
24134:
24133:
24131:
24109:
24106:
24105:
24103:
24064:
24059:
24056:
24055:
24053:
24016:
24011:
24008:
24007:
23968:
23963:
23960:
23959:
23957:
23941:
23939:
23936:
23935:
23933:natural numbers
23416:
23414:Generalizations
23374:âas a model of
23368:Minkowski space
23320:
23317:
23316:
23300:
23297:
23296:
23280:
23277:
23276:
23242:
23239:
23238:
23222:
23219:
23218:
23202:
23199:
23198:
23171:
23165:
23131:, so they have
23129:locally compact
23120:
23091:
23088:
23087:
23071:
23068:
23067:
23047:
23043:
23041:
23038:
23037:
23015:
23012:
23011:
23004:
22973:
22923:
22920:
22919:
22917:unary operation
22888:
22885:
22884:
22868:
22865:
22864:
22861:
22826:sporadic groups
22800:
22794:
22758:
22753:
22750:
22749:
22747:
22731:
22728:
22727:
22711:
22708:
22707:
22691:
22688:
22687:
22665:
22662:
22661:
22645:
22642:
22641:
22625:
22622:
22621:
22618:
22596:
22592:
22590:
22587:
22586:
22569:
22564:
22563:
22561:
22558:
22557:
22540:
22536:
22534:
22531:
22530:
22511:
22506:
22505:
22496:
22491:
22490:
22488:
22485:
22484:
22482:
22463:
22459:
22458:
22453:
22452:
22450:
22447:
22446:
22429:
22425:
22423:
22420:
22419:
22398:
22393:
22392:
22390:
22387:
22386:
22370:
22367:
22366:
22352:
22323:
22320:
22319:
22317:
22301:
22298:
22297:
22280:
22275:
22270:
22267:
22264:
22263:
22245:
22244:
22240:
22238:
22235:
22234:
22218:
22215:
22214:
22197:
22193:
22191:
22188:
22187:
22170:
22165:
22164:
22162:
22159:
22158:
22133:
22130:
22129:
22127:
22108:
22105:
22104:
22088:
22085:
22084:
22061:
22058:
22057:
22041:
22038:
22037:
22019:
22016:
22015:
22013:
21995:
21992:
21991:
21989:
21972:
21968:
21966:
21963:
21962:
21944:
21941:
21940:
21938:
21920:
21917:
21916:
21914:
21898:
21895:
21894:
21876:
21873:
21872:
21870:
21849:
21845:
21829:
21827:
21824:
21821:
21820:
21803:
21799:
21797:
21794:
21793:
21768:
21764:
21762:
21759:
21758:
21756:
21740:
21737:
21736:
21720:
21717:
21716:
21700:
21697:
21696:
21672:
21667:
21666:
21664:
21661:
21660:
21655:, according to
21638:
21635:
21634:
21632:
21615:
21610:
21609:
21607:
21604:
21603:
21581:
21576:
21575:
21573:
21570:
21569:
21550:
21547:
21546:
21527:
21522:
21521:
21519:
21516:
21515:
21513:
21493:
21487:
21475:splitting field
21419:
21416:
21415:
21399:
21396:
21395:
21379:
21376:
21375:
21351:
21330:
21326:
21324:
21314:
21312:
21304:
21301:
21300:
21262:
21258:
21253:
21250:
21249:
21240:
21234:
21198:
21181:
21167:
21164:
21163:
21143:
21140:
21139:
21123:
21120:
21119:
21100:
21095:
21094:
21092:
21089:
21088:
21086:
21084:Euclidean space
21049:
21046:
21045:
21015:
21013:
21010:
21009:
21007:
20976:
20973:
20972:
20970:
20952:
20949:
20948:
20946:
20924:
20907:
20905:
20902:
20901:
20865:
20862:
20861:
20859:
20848:
20836:Main articles:
20834:
20815:antiderivatives
20776:
20774:
20770:
20760:
20748:
20672:crystallography
20626:
20623:
20622:
20620:
20602:
20599:
20598:
20596:
20576:Symmetry groups
20557:
20539:
20533:
20531:Symmetry groups
20486:
20481:
20478:
20477:
20475:
20459:
20456:
20455:
20437:
20434:
20433:
20431:
20403:
20399:
20397:
20394:
20393:
20391:
20366:
20362:
20360:
20357:
20356:
20354:
20323:
20319:
20317:
20314:
20313:
20311:
20286:
20282:
20280:
20277:
20276:
20274:
20258:
20255:
20254:
20230:
20227:
20226:
20224:
20208:
20205:
20204:
20187:
20182:
20177:
20174:
20171:
20170:
20146:
20143:
20142:
20118:
20115:
20114:
20112:
20096:
20093:
20092:
20063:
20059:
20057:
20054:
20053:
20051:
20035:
20032:
20031:
20029:complex numbers
20006:
20003:
20002:
20000:
19984:
19981:
19980:
19962:
19959:
19958:
19956:
19940:
19937:
19936:
19911:
19903:
19898:
19893:
19890:
19889:
19799:
19796:
19795:
19775:
19772:
19771:
19749:
19745:
19715:
19711:
19699:
19695:
19683:
19679:
19677:
19674:
19673:
19671:
19651:
19647:
19645:
19642:
19641:
19617:
19614:
19613:
19611:
19594:
19590:
19588:
19585:
19584:
19558:
19554:
19545:
19541:
19526:
19522:
19510:
19506:
19494:
19490:
19478:
19474:
19466:
19463:
19462:
19446:
19443:
19442:
19411:
19407:
19405:
19402:
19401:
19399:
19383:
19380:
19379:
19362:
19358:
19356:
19353:
19352:
19336:
19333:
19332:
19325:
19319:
19291:
19286:
19281:
19278:
19275:
19274:
19272:
19256:
19248:
19243:
19241:
19238:
19237:
19204:
19196:
19195:
19191:
19189:
19186:
19185:
19166:
19162:
19139:
19136:
19135:
19133:
19115:
19112:
19111:
19109:
19093:
19090:
19089:
19071:
19068:
19067:
19065:
19049:
19046:
19045:
19023:
19020:
19019:
19001:
18998:
18997:
18995:
18964:
18961:
18960:
18937:
18934:
18933:
18903:
18900:
18899:
18897:
18896:evenly divides
18881:
18878:
18877:
18847:
18830:
18827:
18826:
18810:
18807:
18806:
18788:
18785:
18784:
18782:
18766:
18763:
18762:
18744:
18741:
18740:
18738:
18720:
18717:
18716:
18714:
18698:
18695:
18694:
18676:
18673:
18672:
18670:
18654:
18651:
18650:
18632:
18629:
18628:
18626:
18610:
18607:
18606:
18587:
18583:
18566:
18563:
18562:
18560:
18524:
18521:
18520:
18518:
18494:
18491:
18490:
18488:
18470:
18467:
18466:
18464:
18463:of division by
18442:
18439:
18438:
18436:
18412:
18409:
18408:
18406:
18390:
18387:
18386:
18368:
18365:
18364:
18362:
18340:
18337:
18336:
18334:
18302:
18285:
18282:
18281:
18263:
18260:
18259:
18257:
18241:
18238:
18237:
18215:
18212:
18211:
18193:
18190:
18189:
18187:
18171:
18168:
18167:
18151:
18148:
18147:
18122:
18119:
18118:
18116:
18094:
18091:
18090:
18074:
18071:
18070:
18043:
18035:
18030:
18025:
18022:
18021:
18019:
18002:
17997:
17996:
17994:
17991:
17990:
17966:
17963:
17962:
17960:
17944:
17941:
17940:
17912:
17909:
17908:
17906:
17890:
17887:
17886:
17857:
17846:
17843:
17842:
17840:
17824:
17821:
17820:
17804:
17801:
17800:
17784:
17781:
17780:
17764:
17761:
17760:
17749:
17746:addition modulo
17738:
17732:
17711:
17709:
17706:
17705:
17677:
17672:
17669:
17668:
17666:
17647:
17642:
17639:
17638:
17619:
17614:
17613:
17611:
17608:
17607:
17605:
17572:
17565:
17561:
17547:
17539:
17537:
17534:
17533:
17499:
17498:
17494:
17492:
17489:
17488:
17458:
17455:
17454:
17452:
17436:
17433:
17432:
17403:
17402:
17398:
17396:
17393:
17392:
17390:
17372:
17370:
17367:
17366:
17364:
17344:
17341:
17340:
17314:
17312:
17309:
17308:
17302:
17281:
17279:
17276:
17275:
17250:
17242:
17239:
17238:
17236:
17208:
17205:
17204:
17182:
17179:
17178:
17147:
17146:
17142:
17140:
17137:
17136:
17107:
17106:
17102:
17100:
17097:
17096:
17094:
17078:
17076:
17073:
17072:
17069:
17041:
16966:
16963:
16962:
16960:
16944:
16941:
16940:
16924:
16921:
16920:
16837:
16835:
16832:
16831:
16824:wallpaper group
16816:
16810:
16775:
16770:
16769:
16751:
16747:
16726:
16722:
16713:
16709:
16692:
16689:
16688:
16686:
16668:
16665:
16664:
16662:
16645:
16640:
16639:
16637:
16634:
16633:
16599:
16595:
16574:
16570:
16561:
16557:
16540:
16537:
16536:
16534:
16506:
16503:
16502:
16480:
16477:
16476:
16457:
16453:
16432:
16428:
16419:
16415:
16413:
16410:
16409:
16407:
16381:
16378:
16377:
16358:
16354:
16352:
16349:
16348:
16346:
16330:
16327:
16326:
16309:
16305:
16303:
16300:
16299:
16283:
16280:
16279:
16262:
16257:
16256:
16254:
16251:
16250:
16222:
16219:
16218:
16200:
16197:
16196:
16194:
16177:
16172:
16171:
16169:
16166:
16165:
16147:
16146:
16142:
16140:
16137:
16136:
16119:
16115:
16113:
16110:
16109:
16092:
16087:
16086:
16084:
16081:
16080:
16070:
16064:
16030:
16025:
16023:
16022:
16007:
15996:
15993:
15992:
15990:
15962:
15959:
15958:
15946:is an example.
15930:
15925:
15924:
15922:
15919:
15918:
15897:
15892:
15891:
15889:
15886:
15885:
15866:
15860:
15855:
15854:
15852:
15849:
15848:
15828:
15824:
15815:
15811:
15802:
15798:
15787:
15776:
15773:
15772:
15740:
15739:
15735:
15725:
15724:
15720:
15704:
15703:
15699:
15686:
15685:
15681:
15667:
15664:
15663:
15661:
15638:
15637:
15633:
15625:
15622:
15621:
15619:
15603:
15600:
15599:
15580:
15574:
15569:
15568:
15566:
15563:
15562:
15541:
15538:
15537:
15520:
15517:
15516:
15499:
15496:
15495:
15476:
15473:
15472:
15455:
15452:
15451:
15434:
15431:
15430:
15411:
15408:
15407:
15390:
15387:
15386:
15369:
15366:
15365:
15343:
15337:
15332:
15331:
15329:
15326:
15325:
15308:", is called a
15291:
15288:
15287:
15285:
15267:
15264:
15263:
15261:
15240:
15235:
15232:
15231:
15229:
15200:
15196:
15192:
15180:
15176:
15165:
15162:
15161:
15159:
15140:
15137:
15136:
15115:
15110:
15107:
15106:
15104:
15079:
15076:
15075:
15045:
15042:
15041:
15039:
15020:
15017:
15016:
14997:
14994:
14993:
14974:
14971:
14970:
14954:
14951:
14950:
14931:
14920:
14917:
14916:
14897:
14892:
14889:
14888:
14842:
14837:
14834:
14833:
14815:
14812:
14811:
14809:
14793:
14790:
14789:
14786:
14780:
14778:Quotient groups
14752:
14751:
14747:
14734:
14733:
14729:
14716:
14715:
14711:
14698:
14697:
14693:
14691:
14688:
14687:
14685:
14666:
14661:
14660:
14658:
14655:
14654:
14652:
14634:
14633:
14629:
14607:
14606:
14602:
14600:
14597:
14596:
14580:
14577:
14576:
14557:
14552:
14551:
14549:
14546:
14545:
14543:
14522:
14521:
14517:
14507:
14506:
14502:
14492:
14491:
14487:
14477:
14476:
14472:
14456:
14455:
14451:
14443:
14440:
14439:
14423:
14420:
14419:
14403:
14400:
14399:
14381:
14378:
14377:
14375:
14356:
14353:
14352:
14336:
14333:
14332:
14313:
14308:
14307:
14305:
14302:
14301:
14299:
14292:normal subgroup
14274:
14271:
14270:
14240:
14237:
14236:
14234:
14218:
14215:
14214:
14198:
14195:
14194:
14178:
14175:
14174:
14156:
14153:
14152:
14150:
14132:
14129:
14128:
14126:
14101:
14097:
14085:
14080:
14074:
14071:
14070:
14068:
14045:
14041:
14029:
14025:
14023:
14020:
14019:
13996:
13993:
13992:
13987:; that is, the
13970:
13967:
13966:
13964:
13944:
13941:
13940:
13937:
13936:, respectively.
13886:
13883:
13882:
13880:
13831:
13828:
13827:
13807:
13804:
13803:
13801:
13783:
13780:
13779:
13777:
13751:
13748:
13747:
13745:
13729:
13726:
13725:
13709:
13706:
13705:
13701:
13695:
13671:
13667:
13657:
13656:
13652:
13642:
13641:
13637:
13635:
13632:
13631:
13629:
13608:
13606:
13603:
13602:
13600:
13582:
13581:
13577:
13575:
13572:
13571:
13554:
13550:
13548:
13545:
13544:
13526:
13523:
13522:
13520:
13504:
13501:
13500:
13484:
13481:
13480:
13464:
13461:
13460:
13455:, the subgroup
13438:
13435:
13434:
13432:
13416:
13413:
13412:
13387:
13384:
13383:
13381:
13365:
13362:
13361:
13345:
13342:
13341:
13309:
13305:
13303:
13300:
13299:
13281:
13278:
13277:
13275:
13257:
13254:
13253:
13251:
13221:
13217:
13208:
13204:
13195:
13191:
13180:
13169:
13166:
13165:
13163:
13138:
13135:
13134:
13116:
13113:
13112:
13110:
13094:
13091:
13090:
13072:
13069:
13068:
13066:
13044:
13039:
13033:
13030:
13029:
13027:
13010:
13006:
12997:
12993:
12991:
12988:
12987:
12969:
12966:
12965:
12963:
12946:
12942:
12940:
12937:
12936:
12919:
12915:
12913:
12910:
12909:
12908:, and whenever
12891:
12888:
12887:
12885:
12869:
12866:
12865:
12847:
12844:
12843:
12841:
12823:
12820:
12819:
12817:
12801:
12798:
12797:
12790:
12784:
12757:
12754:
12753:
12751:
12733:
12730:
12729:
12727:
12698:
12693:
12691:
12690:
12681:
12679:
12676:
12675:
12650:
12642:
12639:
12638:
12602:
12599:
12598:
12596:
12580:
12577:
12576:
12558:
12555:
12554:
12552:
12536:
12533:
12532:
12516:
12513:
12512:
12482:
12479:
12478:
12476:
12456:
12453:
12452:
12436:
12433:
12432:
12404:
12401:
12400:
12398:
12382:
12379:
12378:
12355:
12354:
12336:
12335:
12330:
12327:
12326:
12310:
12307:
12306:
12290:
12287:
12286:
12263:
12262:
12244:
12243:
12238:
12235:
12234:
12215:
12211:
12197:
12194:
12193:
12191:
12174:
12170:
12156:
12153:
12152:
12124:
12121:
12120:
12092:
12089:
12088:
12068:
12065:
12064:
12035:
12031:
12029:
12026:
12025:
12009:
12006:
12005:
11980:
11977:
11976:
11974:
11958:
11955:
11954:
11934:
11930:
11906:
11902:
11894:
11891:
11890:
11871:
11867:
11855:
11851:
11843:
11840:
11839:
11837:
11821:
11818:
11817:
11814:
11796:
11793:
11792:
11790:
11774:
11771:
11770:
11754:
11751:
11750:
11689:
11686:
11685:
11655:
11652:
11651:
11623:
11620:
11619:
11591:
11588:
11587:
11580:
11574:
11566:quotient groups
11546:
11528:
11525:
11524:
11522:
11506:
11503:
11502:
11482:
11479:
11478:
11450:
11447:
11446:
11445:. The notation
11428:
11425:
11424:
11422:
11406:
11403:
11402:
11380:
11377:
11376:
11358:
11355:
11354:
11352:
11334:
11331:
11330:
11328:
11310:
11307:
11306:
11304:
11285:
11282:
11281:
11238:
11235:
11234:
11227:
11194:
11191:
11190:
11173:
11170:
11169:
11151:
11148:
11147:
11145:
11103:
11100:
11099:
11097:
11057:
11054:
11053:
11037:
11034:
11033:
10999:
10996:
10995:
10990:identity and a
10960:
10959:
10956:(left identity)
10954:
10951:
10939:
10938:
10935:(right inverse)
10933:
10930:
10912:
10911:
10908:(associativity)
10906:
10903:
10885:
10881:
10863:
10862:
10857:
10854:
10836:
10832:
10816:
10803:
10801:
10798:
10797:
10777:
10776:
10771:
10768:
10756:
10755:
10752:(left identity)
10750:
10747:
10738:
10734:
10722:
10718:
10709:
10705:
10693:
10692:
10687:
10684:
10672:
10668:
10647:
10643:
10634:
10630:
10618:
10617:
10614:(associativity)
10612:
10609:
10597:
10593:
10572:
10568:
10550:
10546:
10537:
10533:
10521:
10520:
10515:
10512:
10500:
10496:
10472:
10468:
10456:
10452:
10443:
10439:
10424:
10423:
10420:(left identity)
10418:
10415:
10403:
10399:
10377:
10368:
10364:
10354:
10352:
10349:
10348:
10311:
10307:
10305:
10302:
10301:
10299:
10283:
10280:
10279:
10259:
10255:
10253:
10250:
10249:
10219:
10216:
10215:
10213:
10197:
10194:
10193:
10175:
10152:
10149:
10148:
10146:
10126:
10123:
10122:
10096:
10093:
10092:
10076:
10073:
10072:
10071:that maps each
10050:
10047:
10046:
10045:, the function
10028:
10025:
10024:
10022:
10000:
9996:
9988:
9985:
9984:
9982:
9954:
9951:
9950:
9932:
9929:
9928:
9926:
9910:
9907:
9906:
9885:
9882:
9881:
9879:
9859:
9856:
9855:
9850:; it is called
9825:
9822:
9821:
9805:
9802:
9801:
9800:that maps each
9779:
9776:
9775:
9774:, the function
9759:
9756:
9755:
9739:
9736:
9735:
9707:
9703:
9701:
9698:
9697:
9695:
9665:
9662:
9661:
9659:
9643:
9640:
9639:
9623:
9620:
9619:
9601:
9598:
9597:
9595:
9579:
9576:
9575:
9559:
9556:
9555:
9554:Given elements
9552:
9526:
9525:
9520:
9506:
9498:
9495:
9483:
9482:
9477:
9469:
9461:
9458:
9440:
9439:
9436:(associativity)
9434:
9431:
9401:
9400:
9395:
9389: and
9387:
9379:
9376:
9346:
9345:
9340:
9332:
9329:
9313:
9306:
9304:
9301:
9300:
9281:
9278:
9277:
9261:
9258:
9257:
9241:
9238:
9237:
9234:
9189:
9186:
9185:
9183:
9167:
9164:
9163:
9147:
9144:
9143:
9140:
9055:
9052:
9051:
9046:. For example,
9040:
8971:was studied by
8951:(who worked on
8835:
8831:
8829:
8826:
8825:
8784:Ăvariste Galois
8772:
8766:
8744:
8739:
8738:
8736:
8733:
8732:
8712:
8711:
8707:
8697:
8696:
8692:
8683:
8679:
8677:
8674:
8673:
8671:
8653:
8652:
8648:
8639:
8635:
8625:
8624:
8620:
8618:
8615:
8614:
8595:
8590:
8589:
8587:
8584:
8583:
8581:
8561:
8557:
8555:
8552:
8551:
8534:
8530:
8528:
8525:
8524:
8507:
8503:
8501:
8498:
8497:
8477:
8476:
8472:
8470:
8467:
8466:
8464:
8444:
8443:
8439:
8437:
8434:
8433:
8431:
8411:
8410:
8406:
8404:
8401:
8400:
8398:
8378:
8377:
8373:
8371:
8368:
8367:
8365:
8344:
8342:
8339:
8338:
8336:
8333:Inverse element
8314:
8311:
8310:
8289:
8287:
8284:
8283:
8281:
8262:
8261:
8252:
8248:
8238:
8237:
8233:
8223:
8222:
8218:
8211:
8202:
8198:
8188:
8187:
8183:
8170:
8169:
8165:
8162:
8161:
8155:
8151:
8142:
8138:
8129:
8125:
8118:
8112:
8108:
8095:
8094:
8090:
8080:
8079:
8075:
8068:
8066:
8063:
8062:
8042:
8038:
8028:
8027:
8023:
8010:
8009:
8005:
7996:
7992:
7979:
7978:
7974:
7964:
7963:
7959:
7954:
7951:
7950:
7889:
7886:
7885:
7867:
7864:
7863:
7861:
7843:
7840:
7839:
7837:
7821:
7818:
7817:
7799:
7796:
7795:
7793:
7777:
7774:
7773:
7757:
7754:
7753:
7734:
7729:
7728:
7726:
7723:
7722:
7720:
7702:
7699:
7698:
7696:
7678:
7675:
7674:
7672:
7654:
7651:
7650:
7648:
7622:
7621:
7617:
7615:
7612:
7611:
7609:
7588:
7587:
7583:
7573:
7572:
7568:
7559:
7555:
7553:
7550:
7549:
7548:. For example,
7531:
7528:
7527:
7525:
7509:
7506:
7505:
7483:
7480:
7479:
7463:
7443:
7442:
7438:
7436:
7433:
7432:
7430:
7429:, the symmetry
7411:
7407:
7397:
7396:
7392:
7390:
7387:
7386:
7384:
7380:
7376:
7368:
7345:
7341:
7339:
7336:
7335:
7333:
7314:
7310:
7308:
7305:
7304:
7302:
7283:
7279:
7277:
7274:
7273:
7271:
7250:
7248:
7245:
7244:
7242:
7219:
7217:
7214:
7213:
7195:
7191:
7189:
7186:
7185:
7167:
7163:
7161:
7158:
7157:
7139:
7135:
7133:
7130:
7129:
7110:
7109:
7105:
7103:
7100:
7099:
7080:
7079:
7075:
7073:
7070:
7069:
7050:
7049:
7045:
7043:
7040:
7039:
7020:
7019:
7015:
7013:
7010:
7009:
6990:
6989:
6985:
6983:
6980:
6979:
6959:
6955:
6953:
6950:
6949:
6929:
6927:
6924:
6923:
6905:
6901:
6899:
6896:
6895:
6877:
6873:
6871:
6868:
6867:
6848:
6847:
6843:
6841:
6838:
6837:
6818:
6817:
6813:
6811:
6808:
6807:
6788:
6787:
6783:
6781:
6778:
6777:
6758:
6757:
6753:
6751:
6748:
6747:
6728:
6727:
6723:
6721:
6718:
6717:
6697:
6693:
6691:
6688:
6687:
6669:
6665:
6663:
6660:
6659:
6639:
6637:
6634:
6633:
6615:
6611:
6609:
6606:
6605:
6586:
6585:
6581:
6579:
6576:
6575:
6556:
6555:
6551:
6549:
6546:
6545:
6526:
6525:
6521:
6519:
6516:
6515:
6496:
6495:
6491:
6489:
6486:
6485:
6466:
6465:
6461:
6459:
6456:
6455:
6435:
6431:
6429:
6426:
6425:
6407:
6403:
6401:
6398:
6397:
6379:
6375:
6373:
6370:
6369:
6349:
6347:
6344:
6343:
6324:
6323:
6319:
6317:
6314:
6313:
6294:
6293:
6289:
6287:
6284:
6283:
6264:
6263:
6259:
6257:
6254:
6253:
6234:
6233:
6229:
6227:
6224:
6223:
6204:
6203:
6199:
6197:
6194:
6193:
6172:
6171:
6167:
6165:
6162:
6161:
6142:
6141:
6137:
6135:
6132:
6131:
6112:
6111:
6107:
6105:
6102:
6101:
6082:
6081:
6077:
6075:
6072:
6071:
6053:
6049:
6047:
6044:
6043:
6025:
6021:
6019:
6016:
6015:
5995:
5993:
5990:
5989:
5971:
5967:
5965:
5962:
5961:
5943:
5939:
5937:
5934:
5933:
5912:
5911:
5907:
5905:
5902:
5901:
5882:
5881:
5877:
5875:
5872:
5871:
5852:
5851:
5847:
5845:
5842:
5841:
5822:
5821:
5817:
5815:
5812:
5811:
5793:
5789:
5787:
5784:
5783:
5763:
5761:
5758:
5757:
5739:
5735:
5733:
5730:
5729:
5711:
5707:
5705:
5702:
5701:
5683:
5679:
5677:
5674:
5673:
5652:
5651:
5647:
5645:
5642:
5641:
5622:
5621:
5617:
5615:
5612:
5611:
5592:
5591:
5587:
5585:
5582:
5581:
5562:
5561:
5557:
5555:
5552:
5551:
5531:
5529:
5526:
5525:
5507:
5503:
5501:
5498:
5497:
5479:
5475:
5473:
5470:
5469:
5451:
5447:
5445:
5442:
5441:
5423:
5419:
5417:
5414:
5413:
5392:
5391:
5387:
5385:
5382:
5381:
5362:
5361:
5357:
5355:
5352:
5351:
5332:
5331:
5327:
5325:
5322:
5321:
5302:
5301:
5297:
5295:
5292:
5291:
5273:
5269:
5267:
5264:
5263:
5245:
5241:
5239:
5236:
5235:
5217:
5213:
5211:
5208:
5207:
5187:
5185:
5182:
5181:
5161:
5159:
5156:
5155:
5134:
5133:
5129:
5127:
5124:
5123:
5104:
5103:
5099:
5097:
5094:
5093:
5074:
5073:
5069:
5067:
5064:
5063:
5044:
5043:
5039:
5037:
5034:
5033:
5015:
5011:
5009:
5006:
5005:
4987:
4983:
4981:
4978:
4977:
4959:
4955:
4953:
4950:
4949:
4929:
4927:
4924:
4923:
4906:
4903:
4902:
4882:
4877:
4876:
4874:
4871:
4870:
4843:
4842:
4838:
4829:
4825:
4815:
4814:
4810:
4808:
4805:
4804:
4784:
4783:
4779:
4777:
4774:
4773:
4771:
4751:
4750:
4746:
4744:
4741:
4740:
4738:
4719:
4715:
4713:
4710:
4709:
4707:
4682:
4679:
4678:
4676:
4660:
4657:
4656:
4634:
4631:
4630:
4610:
4607:
4606:
4590:
4587:
4586:
4567:
4562:
4561:
4559:
4556:
4555:
4553:
4531:
4530:
4526:
4524:
4521:
4520:
4503:
4499:
4497:
4494:
4493:
4469:
4468:
4464:
4462:
4459:
4458:
4456:
4436:
4435:
4431:
4429:
4426:
4425:
4423:
4399:
4398:
4394:
4392:
4389:
4388:
4386:
4366:
4365:
4361:
4359:
4356:
4355:
4353:
4349:, respectively;
4331:
4327:
4325:
4322:
4321:
4319:
4302:
4298:
4296:
4293:
4292:
4273:
4269:
4267:
4264:
4263:
4261:
4228:
4227:
4223:
4221:
4218:
4217:
4216:
4189:
4188:
4184:
4182:
4179:
4178:
4177:
4150:
4149:
4145:
4143:
4140:
4139:
4138:
4112:
4111:
4107:
4105:
4102:
4101:
4100:
4074:
4070:
4068:
4065:
4064:
4063:
4039:
4035:
4033:
4030:
4029:
4028:
4004:
4000:
3998:
3995:
3994:
3993:
3967:
3965:
3962:
3961:
3960:
3932:
3927:
3926:
3924:
3921:
3920:
3918:
3883:
3862:
3859:
3858:
3818:
3815:
3814:
3812:
3776:
3773:
3772:
3770:
3754:
3751:
3750:
3734:
3731:
3730:
3696:
3693:
3692:
3665:
3662:
3661:
3659:
3640:
3637:
3636:
3614:
3610:
3608:
3605:
3604:
3602:
3586:
3583:
3582:
3564:
3561:
3560:
3558:
3533:
3530:
3529:
3527:
3511:
3508:
3507:
3489:
3486:
3485:
3483:
3448:
3446:
3443:
3442:
3425:
3420:
3419:
3417:
3414:
3413:
3397:
3395:
3392:
3391:
3371:
3369:
3366:
3365:
3349:
3347:
3344:
3343:
3320:
3318:
3315:
3314:
3279:
3274:
3271:
3270:
3245:
3240:
3237:
3236:
3220:
3218:
3215:
3214:
3193:
3190:
3189:
3187:
3162:
3159:
3158:
3140:
3138:
3135:
3134:
3132:
3107:group operation
3091:
3063:
3059:
3057:
3054:
3053:
3051:
3035:
3032:
3031:
3007:
3004:
3003:
2985:
2982:
2981:
2979:
2960:
2957:
2956:
2926:
2923:
2922:
2920:
2892:
2889:
2888:
2872:
2869:
2868:
2852:
2849:
2848:
2830:
2827:
2826:
2824:
2808:
2805:
2804:
2800:Inverse element
2797:) of the group.
2795:neutral element
2753:
2750:
2749:
2747:
2717:
2714:
2713:
2711:
2693:
2690:
2689:
2687:
2671:
2668:
2667:
2651:
2648:
2647:
2631:
2628:
2627:
2561:
2558:
2557:
2555:
2537:
2534:
2533:
2531:
2513:
2510:
2509:
2507:
2489:
2486:
2485:
2483:
2465:
2462:
2461:
2459:
2424:
2421:
2420:
2418:
2400:
2397:
2396:
2394:
2378:
2375:
2374:
2358:
2355:
2354:
2338:
2335:
2334:
2313:
2310:
2309:
2307:
2289:
2286:
2285:
2283:
2263:
2260:
2259:
2253:
2243:
2231:
2208:
2205:
2204:
2202:
2177:
2174:
2173:
2171:
2155:
2152:
2151:
2150:of the integer
2147:inverse element
2129:
2126:
2125:
2095:
2092:
2091:
2089:
2061:
2058:
2057:
2041:
2038:
2037:
2019:
2016:
2015:
2013:
1970:
1967:
1966:
1964:
1936:
1933:
1932:
1916:
1913:
1912:
1885:
1882:
1881:
1879:
1863:
1860:
1859:
1843:
1840:
1839:
1823:
1820:
1819:
1803:
1800:
1799:
1783:
1780:
1779:
1719:
1716:
1715:
1713:
1695:
1692:
1691:
1689:
1673:
1670:
1669:
1651:
1648:
1647:
1645:
1623:
1621:
1618:
1617:
1615:
1595:
1592:
1591:
1563:
1560:
1559:
1538:
1535:
1534:
1532:
1516:
1513:
1512:
1408:
1406:
1403:
1402:
1396:
1391:
1382:
1353:
1352:
1351:
1322:Non-associative
1304:
1293:
1292:
1282:
1262:
1251:
1250:
1239:Map of lattices
1235:
1231:Boolean algebra
1226:Heyting algebra
1200:
1189:
1188:
1182:
1163:Integral domain
1127:
1116:
1115:
1109:
1063:
1041:
1012:
1011:
1000:Abelian variety
993:Reductive group
981:
971:
970:
969:
968:
919:
911:
903:
895:
887:
860:Special unitary
771:
757:
756:
738:
737:
719:
717:
714:
713:
694:
692:
689:
688:
657:
655:
652:
651:
643:
642:
633:Discrete groups
622:
621:
577:Frobenius group
522:
509:
498:
491:Symmetric group
487:
471:
461:
460:
311:Normal subgroup
297:
277:
268:
228:quotient groups
194:Ăvariste Galois
106:
103:
102:
100:
74:inverse element
24:
17:
12:
11:
5:
29767:
29757:
29756:
29751:
29746:
29727:
29726:
29724:
29723:
29712:
29709:
29708:
29706:
29705:
29700:
29695:
29693:Linear algebra
29690:
29684:
29682:
29678:
29677:
29675:
29674:
29668:
29666:
29662:
29661:
29659:
29658:
29656:Tensor algebra
29653:
29648:
29645:Quotient group
29638:
29628:
29618:
29608:
29603:
29597:
29595:
29591:
29590:
29588:
29587:
29582:
29577:
29567:
29564:Euclidean norm
29557:
29547:
29537:
29532:
29522:
29517:
29512:
29506:
29504:
29498:
29497:
29495:
29494:
29484:
29474:
29464:
29454:
29443:
29441:
29437:
29436:
29434:
29433:
29428:
29418:
29415:Multiplication
29404:
29394:
29384:
29370:
29364:
29362:
29361:Basic concepts
29358:
29357:
29355:
29354:
29349:
29344:
29339:
29334:
29329:
29327:Linear algebra
29324:
29319:
29314:
29309:
29304:
29299:
29294:
29293:
29292:
29287:
29277:
29271:
29269:
29265:
29264:
29262:
29261:
29256:
29250:
29247:
29246:
29239:
29238:
29231:
29224:
29216:
29207:
29206:
29204:
29203:
29198:
29193:
29187:
29184:
29183:
29181:
29180:
29177:
29174:
29171:
29166:
29161:
29156:
29150:
29148:
29144:
29143:
29141:
29140:
29135:
29133:Poincaré group
29130:
29125:
29119:
29118:
29114:
29110:
29106:
29102:
29098:
29094:
29090:
29086:
29082:
29078:
29072:
29071:
29065:
29059:
29053:
29047:
29041:
29035:
29028:
29026:
29020:
29019:
29017:
29016:
29011:
29006:
29001:Dihedral group
28998:
28993:
28985:
28981:
28980:
28974:
28968:
28965:
28959:
28955:
28948:
28941:
28934:
28929:
28926:
28920:
28917:
28913:
28907:
28901:
28900:
28897:
28891:
28888:
28882:
28876:
28874:
28868:
28867:
28865:
28864:
28859:
28854:
28849:
28844:
28842:Symmetry group
28839:
28834:
28829:
28827:Infinite group
28824:
28819:
28817:Abelian groups
28814:
28808:
28806:
28802:
28801:
28799:
28798:
28793:
28791:direct product
28783:
28778:
28776:Quotient group
28773:
28768:
28763:
28757:
28755:
28749:
28748:
28741:
28740:
28733:
28726:
28718:
28712:
28711:
28690:
28689:External links
28687:
28685:
28684:
28678:
28658:
28652:
28636:
28621:
28598:
28578:
28552:(4): 195â215,
28536:
28521:
28502:
28459:
28453:
28437:
28415:
28401:Cayley, Arthur
28397:
28392:
28364:
28361:
28360:
28359:
28346:
28328:
28322:
28306:
28300:
28288:Welsh, Dominic
28284:
28278:
28260:
28239:
28223:
28217:
28204:
28184:(2): 345â371,
28172:Suzuki, Michio
28168:
28163:
28147:
28125:
28120:
28107:
28094:
28081:
28075:
28062:
28048:
28028:
28018:(6): 671â679,
28000:
27994:
27978:
27964:
27951:
27945:
27925:
27919:
27899:
27878:
27856:
27842:
27829:
27815:
27803:Mumford, David
27799:
27794:
27779:
27773:
27760:
27741:
27736:
27718:
27712:
27694:
27688:
27680:Addison-Wesley
27671:
27657:
27644:
27630:
27610:
27596:
27571:
27531:
27526:
27513:
27507:
27489:Hatcher, Allen
27485:
27480:
27467:
27461:
27443:
27430:
27406:
27381:
27367:
27346:
27341:
27328:
27296:
27290:
27277:
27271:
27254:
27240:
27227:
27207:(2): 475â507,
27189:
27184:
27171:
27165:
27145:
27131:
27111:
27105:
27092:
27057:
27051:
27036:
27004:
26999:
26986:
26960:
26954:
26932:
26929:
26928:
26927:
26921:
26905:
26885:
26865:
26859:
26839:
26826:
26806:
26786:
26772:
26756:
26736:
26731:
26716:
26710:
26694:Artin, Michael
26688:
26685:
26684:
26682:
26679:
26676:
26675:
26663:
26651:
26639:
26627:
26615:
26603:
26599:Goldstein 1980
26591:
26579:
26567:
26555:
26543:
26531:
26519:
26507:
26505:, p. 737.
26495:
26483:
26467:
26455:
26443:
26431:
26419:
26407:
26395:
26383:
26371:
26356:
26344:
26332:
26320:
26308:
26296:
26284:
26272:
26270:, p. 228.
26260:
26248:
26236:
26221:
26209:
26194:
26182:
26170:
26158:
26146:
26134:
26122:
26120:, Chapter VII.
26110:
26098:
26077:
26065:
26053:
26041:
26029:
26017:
26005:
25993:
25989:Ledermann 1973
25981:
25969:
25954:
25942:
25925:
25913:
25898:
25894:Ledermann 1973
25886:
25882:Ledermann 1953
25874:
25862:
25850:
25838:
25826:
25814:
25802:
25790:
25788:, p. 204.
25778:
25766:
25754:
25742:
25740:, p. 202.
25730:
25718:
25706:
25694:
25682:
25670:
25658:
25646:
25634:
25622:
25610:
25590:
25577:
25576:
25574:
25571:
25568:
25567:
25554:
25538:
25526:simple modules
25513:
25500:
25483:
25462:
25431:
25419:DiffieâHellman
25410:
25395:
25370:
25342:
25329:
25312:
25294:An example is
25287:
25274:
25261:
25259:, p. 108.
25240:
25215:
25158:
25141:
25075:
25074:
25072:
25069:
25068:
25067:
25061:
25060:
25044:
25041:
25038:
25037:
25032:
25027:
25022:
25017:
25012:
25007:
25002:
24997:
24992:
24987:
24983:
24982:
24979:
24976:
24973:
24970:
24967:
24964:
24961:
24958:
24955:
24952:
24948:
24947:
24944:
24941:
24938:
24935:
24932:
24929:
24926:
24923:
24920:
24917:
24913:
24912:
24909:
24906:
24903:
24889:
24886:
24883:
24858:
24854:
24850:
24838:
24835:
24822:
24819:
24807:
24804:
24791:
24788:
24776:
24773:
24770:
24766:
24765:
24762:
24759:
24756:
24753:
24750:
24747:
24744:
24741:
24738:
24735:
24731:
24730:
24727:
24724:
24721:
24718:
24715:
24712:
24709:
24706:
24703:
24700:
24696:
24695:
24692:
24689:
24686:
24683:
24680:
24677:
24674:
24671:
24668:
24665:
24661:
24660:
24647:
24644:
24641:
24638:
24635:
24632:
24629:
24626:
24622:
24618:
24614:
24609:
24592:
24578:
24548:
24518:
24501:
24487:
24471:
24457:
24441:
24385:
24361:
24337:
24334:
24306:
24303:
24300:
24297:
24294:
24291:
24288:
24285:
24282:
24258:
24234:
24212:
24209:
24206:
24203:
24200:
24197:
24194:
24172:
24159:small category
24141:
24113:
24089:
24086:
24083:
24080:
24077:
24074:
24071:
24067:
24063:
24041:
24038:
24035:
24032:
24029:
24026:
24023:
24019:
24015:
23993:
23990:
23987:
23984:
23981:
23978:
23975:
23971:
23967:
23944:
23920:
23919:
23916:
23913:
23910:
23907:
23904:
23898:
23897:
23894:
23891:
23888:
23885:
23882:
23876:
23875:
23872:
23869:
23866:
23863:
23860:
23854:
23853:
23850:
23847:
23844:
23841:
23838:
23832:
23831:
23828:
23825:
23822:
23819:
23816:
23809:
23808:
23805:
23802:
23799:
23796:
23793:
23786:
23785:
23782:
23779:
23776:
23773:
23770:
23763:
23762:
23759:
23756:
23753:
23750:
23747:
23741:
23740:
23737:
23734:
23731:
23728:
23725:
23718:
23717:
23714:
23711:
23708:
23705:
23702:
23696:
23695:
23692:
23689:
23686:
23683:
23680:
23673:
23672:
23669:
23666:
23663:
23660:
23657:
23651:
23650:
23647:
23644:
23641:
23638:
23635:
23628:
23627:
23624:
23621:
23618:
23615:
23612:
23606:
23605:
23602:
23599:
23596:
23593:
23590:
23583:
23582:
23579:
23576:
23573:
23570:
23567:
23561:
23560:
23557:
23554:
23551:
23548:
23545:
23538:
23537:
23534:
23531:
23528:
23525:
23522:
23516:
23515:
23512:
23509:
23506:
23503:
23500:
23498:Small category
23494:
23493:
23490:
23487:
23484:
23481:
23478:
23472:
23471:
23468:
23465:
23462:
23459:
23456:
23450:
23449:
23444:
23439:
23434:
23429:
23424:
23415:
23412:
23400:Standard Model
23384:Poincaré group
23324:
23304:
23284:
23264:
23261:
23258:
23255:
23252:
23249:
23246:
23226:
23206:
23167:Main article:
23164:
23161:
23149:Krull topology
23095:
23075:
23053:
23050:
23046:
23025:
23022:
23019:
23000:Main article:
22972:
22969:
22933:
22930:
22927:
22904:
22901:
22898:
22895:
22892:
22872:
22860:
22857:
22832:is called the
22830:sporadic group
22796:Main article:
22793:
22790:
22765:
22761:
22757:
22735:
22715:
22695:
22675:
22672:
22669:
22649:
22629:
22617:
22614:
22599:
22595:
22572:
22567:
22543:
22539:
22514:
22509:
22504:
22499:
22494:
22466:
22462:
22456:
22432:
22428:
22401:
22396:
22374:
22351:
22348:
22333:
22330:
22327:
22305:
22283:
22278:
22273:
22248:
22243:
22222:
22200:
22196:
22173:
22168:
22152:Sylow theorems
22137:
22112:
22092:
22065:
22045:
22023:
21999:
21975:
21971:
21948:
21924:
21902:
21880:
21858:
21848:
21842:
21838:
21835:
21832:
21806:
21802:
21779:
21776:
21771:
21767:
21744:
21724:
21704:
21675:
21670:
21642:
21618:
21613:
21584:
21579:
21554:
21530:
21525:
21489:Main article:
21486:
21483:
21473:formed as the
21423:
21403:
21383:
21363:
21357:
21354:
21347:
21344:
21341:
21338:
21333:
21329:
21323:
21320:
21317:
21311:
21308:
21288:
21285:
21282:
21279:
21276:
21273:
21270:
21265:
21261:
21257:
21236:Main article:
21233:
21230:
21226:compact groups
21205:
21201:
21197:
21194:
21191:
21187:
21184:
21180:
21177:
21174:
21171:
21147:
21127:
21103:
21098:
21053:
21031:
21028:
21025:
21021:
21018:
20980:
20956:
20931:
20927:
20923:
20920:
20917:
20913:
20910:
20893:together with
20869:
20833:
20830:
20795:Mathieu groups
20789:
20788:
20781:
20772:
20768:
20762:
20758:
20752:
20741:
20740:
20733:
20726:
20719:
20658:In chemistry,
20630:
20606:
20537:Symmetry group
20535:Main article:
20532:
20529:
20499:
20496:
20493:
20489:
20485:
20463:
20441:
20414:
20411:
20406:
20402:
20377:
20374:
20369:
20365:
20340:
20337:
20334:
20331:
20326:
20322:
20297:
20294:
20289:
20285:
20262:
20240:
20237:
20234:
20212:
20190:
20185:
20180:
20150:
20128:
20125:
20122:
20100:
20074:
20071:
20066:
20062:
20039:
20010:
19988:
19966:
19944:
19924:
19921:
19918:
19914:
19910:
19906:
19901:
19897:
19888:In the groups
19875:
19872:
19869:
19866:
19863:
19860:
19857:
19854:
19851:
19848:
19845:
19842:
19839:
19836:
19833:
19830:
19827:
19824:
19821:
19818:
19815:
19812:
19809:
19806:
19803:
19779:
19755:
19752:
19748:
19744:
19741:
19738:
19735:
19732:
19729:
19726:
19721:
19718:
19714:
19710:
19705:
19702:
19698:
19694:
19689:
19686:
19682:
19657:
19654:
19650:
19627:
19624:
19621:
19597:
19593:
19572:
19569:
19566:
19561:
19557:
19553:
19548:
19544:
19540:
19537:
19534:
19529:
19525:
19521:
19516:
19513:
19509:
19505:
19500:
19497:
19493:
19489:
19484:
19481:
19477:
19473:
19470:
19450:
19414:
19410:
19387:
19365:
19361:
19340:
19321:Main article:
19318:
19315:
19294:
19289:
19284:
19259:
19255:
19251:
19246:
19224:
19220:
19217:
19213:
19210:
19207:
19203:
19199:
19194:
19169:
19165:
19161:
19158:
19155:
19152:
19149:
19146:
19143:
19119:
19097:
19075:
19053:
19033:
19030:
19027:
19018:. In the case
19005:
18983:
18980:
18977:
18974:
18971:
18968:
18941:
18932:. The inverse
18919:
18916:
18913:
18910:
18907:
18885:
18865:
18861:
18858:
18854:
18851:
18846:
18843:
18840:
18837:
18834:
18814:
18792:
18770:
18748:
18724:
18702:
18680:
18658:
18636:
18614:
18590:
18586:
18582:
18579:
18576:
18573:
18570:
18546:
18543:
18540:
18537:
18534:
18531:
18528:
18504:
18501:
18498:
18474:
18446:
18422:
18419:
18416:
18394:
18372:
18344:
18320:
18316:
18313:
18309:
18306:
18301:
18298:
18295:
18292:
18289:
18267:
18245:
18225:
18222:
18219:
18197:
18175:
18155:
18126:
18104:
18101:
18098:
18078:
18056:
18053:
18050:
18046:
18042:
18038:
18033:
18029:
18005:
18000:
17976:
17973:
17970:
17948:
17937:representative
17922:
17919:
17916:
17894:
17871:
17868:
17864:
17861:
17856:
17853:
17850:
17828:
17808:
17788:
17768:
17734:Main article:
17731:
17728:
17714:
17684:
17680:
17676:
17654:
17650:
17646:
17622:
17617:
17592:
17588:
17585:
17582:
17579:
17575:
17571:
17568:
17564:
17560:
17556:
17553:
17550:
17546:
17542:
17513:
17509:
17506:
17502:
17497:
17474:
17471:
17468:
17465:
17462:
17440:
17417:
17413:
17410:
17406:
17401:
17375:
17348:
17326:
17321:
17318:
17301:
17298:
17284:
17258:
17255:
17249:
17246:
17224:
17221:
17218:
17215:
17212:
17192:
17189:
17186:
17161:
17157:
17154:
17150:
17145:
17121:
17117:
17114:
17110:
17105:
17081:
17068:
17065:
17040:
17037:
16991:winding number
16976:
16973:
16970:
16948:
16928:
16869:Henri Poincaré
16840:
16812:Main article:
16809:
16806:
16802:discrete group
16778:
16773:
16768:
16765:
16762:
16759:
16754:
16750:
16746:
16743:
16740:
16737:
16734:
16729:
16725:
16721:
16716:
16712:
16708:
16705:
16702:
16699:
16696:
16672:
16648:
16643:
16613:
16610:
16607:
16602:
16598:
16594:
16591:
16588:
16585:
16582:
16577:
16573:
16569:
16564:
16560:
16556:
16553:
16550:
16547:
16544:
16522:
16519:
16516:
16513:
16510:
16490:
16487:
16484:
16460:
16456:
16452:
16449:
16446:
16443:
16440:
16435:
16431:
16427:
16422:
16418:
16391:
16388:
16385:
16376:. Elements in
16361:
16357:
16334:
16312:
16308:
16287:
16265:
16260:
16238:
16235:
16232:
16229:
16226:
16204:
16180:
16175:
16150:
16145:
16122:
16118:
16095:
16090:
16066:Main article:
16063:
16060:
16041:
16033:
16028:
16020:
16017:
16014:
16010:
16006:
16003:
16000:
15978:
15975:
15972:
15969:
15966:
15933:
15928:
15917:construction;
15900:
15895:
15873:
15869:
15863:
15858:
15836:
15831:
15827:
15823:
15818:
15814:
15810:
15805:
15801:
15797:
15793:
15790:
15786:
15783:
15780:
15758:
15755:
15752:
15749:
15743:
15738:
15734:
15728:
15723:
15719:
15716:
15713:
15707:
15702:
15698:
15695:
15689:
15684:
15680:
15677:
15674:
15671:
15647:
15641:
15636:
15632:
15629:
15607:
15587:
15583:
15577:
15572:
15557:
15556:
15545:
15535:
15524:
15514:
15503:
15492:
15491:
15480:
15470:
15459:
15449:
15438:
15427:
15426:
15415:
15405:
15394:
15384:
15373:
15350:
15346:
15340:
15335:
15310:quotient group
15295:
15271:
15247:
15243:
15239:
15215:
15211:
15206:
15203:
15199:
15195:
15191:
15186:
15183:
15179:
15175:
15172:
15169:
15147:
15144:
15122:
15118:
15114:
15092:
15089:
15086:
15083:
15061:
15058:
15055:
15052:
15049:
15027:
15024:
15004:
15001:
14981:
14978:
14958:
14938:
14934:
14930:
14927:
14924:
14904:
14900:
14896:
14876:
14873:
14870:
14867:
14864:
14861:
14858:
14855:
14852:
14849:
14845:
14841:
14819:
14797:
14784:Quotient group
14782:Main article:
14779:
14776:
14761:
14755:
14750:
14746:
14743:
14737:
14732:
14728:
14725:
14719:
14714:
14710:
14707:
14701:
14696:
14669:
14664:
14637:
14632:
14628:
14625:
14622:
14619:
14616:
14610:
14605:
14584:
14560:
14555:
14531:
14525:
14520:
14516:
14510:
14505:
14501:
14495:
14490:
14486:
14480:
14475:
14471:
14468:
14465:
14459:
14454:
14450:
14447:
14427:
14407:
14385:
14363:
14360:
14340:
14316:
14311:
14278:
14256:
14253:
14250:
14247:
14244:
14222:
14202:
14182:
14160:
14136:
14112:
14109:
14104:
14100:
14096:
14091:
14088:
14083:
14079:
14066:precisely when
14053:
14048:
14044:
14040:
14037:
14032:
14028:
14000:
13974:
13948:
13923:
13920:
13917:
13914:
13911:
13908:
13905:
13902:
13899:
13896:
13893:
13890:
13868:
13865:
13862:
13859:
13856:
13853:
13850:
13847:
13844:
13841:
13838:
13835:
13826:
13811:
13787:
13755:
13733:
13713:
13697:Main article:
13694:
13691:
13674:
13670:
13666:
13660:
13655:
13651:
13645:
13640:
13614:
13611:
13585:
13580:
13557:
13553:
13530:
13508:
13488:
13468:
13442:
13420:
13391:
13369:
13349:
13329:
13326:
13323:
13320:
13315:
13312:
13308:
13285:
13261:
13229:
13224:
13220:
13216:
13211:
13207:
13203:
13198:
13194:
13190:
13186:
13183:
13179:
13176:
13173:
13148:
13145:
13142:
13120:
13109:restricted to
13098:
13076:
13050:
13047:
13042:
13038:
13013:
13009:
13005:
13000:
12996:
12986:, then so are
12973:
12949:
12945:
12922:
12918:
12895:
12873:
12851:
12827:
12805:
12792:Informally, a
12786:Main article:
12783:
12780:
12761:
12737:
12715:
12712:
12709:
12701:
12696:
12687:
12684:
12663:
12660:
12656:
12653:
12649:
12646:
12606:
12584:
12562:
12540:
12520:
12498:
12495:
12492:
12489:
12486:
12460:
12440:
12408:
12386:
12366:
12363:
12358:
12353:
12350:
12347:
12344:
12339:
12334:
12325:and such that
12314:
12294:
12274:
12271:
12266:
12261:
12258:
12255:
12252:
12247:
12242:
12218:
12214:
12210:
12207:
12204:
12201:
12177:
12173:
12169:
12166:
12163:
12160:
12140:
12137:
12134:
12131:
12128:
12108:
12105:
12102:
12099:
12096:
12083:to itself. An
12072:
12052:
12049:
12046:
12043:
12038:
12034:
12013:
11984:
11962:
11940:
11937:
11933:
11929:
11926:
11923:
11920:
11917:
11912:
11909:
11905:
11901:
11898:
11874:
11870:
11866:
11863:
11858:
11854:
11850:
11847:
11825:
11800:
11778:
11758:
11738:
11735:
11732:
11729:
11726:
11723:
11720:
11717:
11714:
11711:
11708:
11705:
11702:
11699:
11696:
11693:
11684:
11671:
11668:
11665:
11662:
11659:
11650:is a function
11639:
11636:
11633:
11630:
11627:
11607:
11604:
11601:
11598:
11595:
11576:Main article:
11573:
11570:
11532:
11521:an element of
11510:
11486:
11466:
11463:
11460:
11457:
11454:
11432:
11410:
11401:to state that
11390:
11387:
11384:
11362:
11338:
11314:
11289:
11266:
11263:
11260:
11257:
11254:
11251:
11248:
11245:
11242:
11226:
11225:Basic concepts
11223:
11210:
11207:
11204:
11201:
11198:
11177:
11155:
11131:
11128:
11125:
11122:
11119:
11116:
11113:
11110:
11107:
11085:
11082:
11079:
11076:
11073:
11070:
11067:
11064:
11061:
11041:
11021:
11018:
11015:
11012:
11009:
11006:
11003:
10974:
10973:
10953:
10950:
10947:
10944:
10942:
10940:
10932:
10929:
10926:
10923:
10920:
10917:
10915:
10913:
10905:
10902:
10899:
10896:
10891:
10888:
10884:
10880:
10877:
10874:
10871:
10868:
10866:
10864:
10859:(left inverse)
10856:
10853:
10850:
10847:
10842:
10839:
10835:
10831:
10828:
10825:
10822:
10819:
10817:
10815:
10812:
10809:
10806:
10805:
10791:
10790:
10773:(left inverse)
10770:
10767:
10764:
10761:
10759:
10757:
10749:
10744:
10741:
10737:
10733:
10728:
10725:
10721:
10715:
10712:
10708:
10704:
10701:
10698:
10696:
10694:
10689:(left inverse)
10686:
10683:
10678:
10675:
10671:
10667:
10664:
10661:
10658:
10653:
10650:
10646:
10640:
10637:
10633:
10629:
10626:
10623:
10621:
10619:
10611:
10608:
10603:
10600:
10596:
10592:
10589:
10586:
10583:
10578:
10575:
10571:
10567:
10564:
10561:
10556:
10553:
10549:
10543:
10540:
10536:
10532:
10529:
10526:
10524:
10522:
10517:(left inverse)
10514:
10511:
10506:
10503:
10499:
10495:
10492:
10489:
10486:
10483:
10478:
10475:
10471:
10467:
10462:
10459:
10455:
10449:
10446:
10442:
10438:
10435:
10432:
10429:
10427:
10425:
10417:
10414:
10409:
10406:
10402:
10398:
10395:
10392:
10389:
10386:
10383:
10380:
10378:
10374:
10371:
10367:
10363:
10360:
10357:
10356:
10331:
10328:
10325:
10322:
10317:
10314:
10310:
10287:
10265:
10262:
10258:
10235:
10232:
10229:
10226:
10223:
10201:
10174:
10171:
10156:
10130:
10106:
10103:
10100:
10080:
10060:
10057:
10054:
10032:
10006:
10003:
9999:
9995:
9992:
9970:
9967:
9964:
9961:
9958:
9936:
9914:
9889:
9863:
9835:
9832:
9829:
9809:
9789:
9786:
9783:
9763:
9743:
9721:
9718:
9713:
9710:
9706:
9681:
9678:
9675:
9672:
9669:
9647:
9627:
9605:
9583:
9563:
9551:
9548:
9540:
9539:
9519:
9516:
9513:
9505:
9497:
9494:
9491:
9488:
9486:
9484:
9476:
9468:
9460:
9457:
9454:
9451:
9448:
9445:
9443:
9441:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9412:
9409:
9406:
9404:
9402:
9394:
9386:
9378:
9375:
9372:
9369:
9366:
9363:
9360:
9357:
9354:
9351:
9349:
9347:
9339:
9331:
9328:
9325:
9322:
9319:
9316:
9314:
9312:
9309:
9308:
9285:
9265:
9245:
9233:
9230:
9211:
9208:
9205:
9202:
9199:
9196:
9193:
9171:
9151:
9139:
9136:
9119:
9116:
9113:
9110:
9107:
9104:
9101:
9098:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9039:
9036:
9032:examples below
8957:Richard Brauer
8931:Camille Jordan
8918:by developing
8846:
8843:
8838:
8834:
8800:symmetry group
8776:abstract group
8768:Main article:
8765:
8762:
8747:
8742:
8715:
8710:
8706:
8700:
8695:
8691:
8686:
8682:
8656:
8651:
8647:
8642:
8638:
8634:
8628:
8623:
8598:
8593:
8564:
8560:
8537:
8533:
8510:
8506:
8480:
8475:
8447:
8442:
8414:
8409:
8381:
8376:
8350:
8347:
8318:
8295:
8292:
8260:
8255:
8251:
8247:
8241:
8236:
8232:
8226:
8221:
8217:
8214:
8212:
8210:
8205:
8201:
8197:
8191:
8186:
8182:
8179:
8173:
8168:
8164:
8163:
8158:
8154:
8150:
8145:
8141:
8137:
8132:
8128:
8124:
8121:
8119:
8115:
8111:
8107:
8104:
8098:
8093:
8089:
8083:
8078:
8074:
8071:
8070:
8050:
8045:
8041:
8037:
8031:
8026:
8022:
8019:
8013:
8008:
8004:
7999:
7995:
7991:
7988:
7982:
7977:
7973:
7967:
7962:
7958:
7938:
7935:
7932:
7929:
7926:
7923:
7920:
7917:
7914:
7911:
7908:
7905:
7902:
7899:
7896:
7893:
7871:
7847:
7825:
7803:
7781:
7761:
7737:
7732:
7706:
7682:
7658:
7625:
7620:
7597:
7591:
7586:
7582:
7576:
7571:
7567:
7562:
7558:
7535:
7513:
7493:
7490:
7487:
7468:
7467:
7446:
7441:
7414:
7410:
7406:
7400:
7395:
7348:
7344:
7317:
7313:
7286:
7282:
7256:
7253:
7238:
7237:
7225:
7222:
7211:
7198:
7194:
7183:
7170:
7166:
7155:
7142:
7138:
7127:
7113:
7108:
7097:
7083:
7078:
7067:
7053:
7048:
7037:
7023:
7018:
7007:
6993:
6988:
6976:
6975:
6962:
6958:
6947:
6935:
6932:
6921:
6908:
6904:
6893:
6880:
6876:
6865:
6851:
6846:
6835:
6821:
6816:
6805:
6791:
6786:
6775:
6761:
6756:
6745:
6731:
6726:
6714:
6713:
6700:
6696:
6685:
6672:
6668:
6657:
6645:
6642:
6631:
6618:
6614:
6603:
6589:
6584:
6573:
6559:
6554:
6543:
6529:
6524:
6513:
6499:
6494:
6483:
6469:
6464:
6452:
6451:
6438:
6434:
6423:
6410:
6406:
6395:
6382:
6378:
6367:
6355:
6352:
6341:
6327:
6322:
6311:
6297:
6292:
6281:
6267:
6262:
6251:
6237:
6232:
6221:
6207:
6202:
6190:
6189:
6175:
6170:
6159:
6145:
6140:
6129:
6115:
6110:
6099:
6085:
6080:
6069:
6056:
6052:
6041:
6028:
6024:
6013:
6001:
5998:
5987:
5974:
5970:
5959:
5946:
5942:
5930:
5929:
5915:
5910:
5899:
5885:
5880:
5869:
5855:
5850:
5839:
5825:
5820:
5809:
5796:
5792:
5781:
5769:
5766:
5755:
5742:
5738:
5727:
5714:
5710:
5699:
5686:
5682:
5670:
5669:
5655:
5650:
5639:
5625:
5620:
5609:
5595:
5590:
5579:
5565:
5560:
5549:
5537:
5534:
5523:
5510:
5506:
5495:
5482:
5478:
5467:
5454:
5450:
5439:
5426:
5422:
5410:
5409:
5395:
5390:
5379:
5365:
5360:
5349:
5335:
5330:
5319:
5305:
5300:
5289:
5276:
5272:
5261:
5248:
5244:
5233:
5220:
5216:
5205:
5193:
5190:
5179:
5167:
5164:
5152:
5151:
5137:
5132:
5121:
5107:
5102:
5091:
5077:
5072:
5061:
5047:
5042:
5031:
5018:
5014:
5003:
4990:
4986:
4975:
4962:
4958:
4947:
4935:
4932:
4921:
4910:
4885:
4880:
4852:
4846:
4841:
4837:
4832:
4828:
4824:
4818:
4813:
4787:
4782:
4754:
4749:
4722:
4718:
4686:
4664:
4644:
4641:
4638:
4614:
4594:
4570:
4565:
4550:dihedral group
4534:
4529:
4506:
4502:
4490:
4489:
4472:
4467:
4439:
4434:
4402:
4397:
4369:
4364:
4350:
4334:
4330:
4305:
4301:
4276:
4272:
4258:
4248:
4247:
4231:
4226:
4208:
4192:
4187:
4169:
4153:
4148:
4130:
4115:
4110:
4092:
4091:
4077:
4073:
4056:
4042:
4038:
4021:
4007:
4003:
3986:
3973:
3970:
3935:
3930:
3882:
3879:
3866:
3828:
3825:
3822:
3780:
3758:
3738:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3675:
3672:
3669:
3647:
3644:
3620:
3617:
3613:
3590:
3568:
3540:
3537:
3515:
3493:
3480:additive group
3464:
3461:
3458:
3455:
3451:
3428:
3423:
3400:
3374:
3352:
3340:additive group
3323:
3298:
3295:
3292:
3289:
3286:
3282:
3278:
3258:
3255:
3252:
3248:
3244:
3223:
3200:
3197:
3185:multiplication
3172:
3169:
3166:
3143:
3122:abuse notation
3103:underlying set
3090:
3087:
3086:
3085:
3069:
3066:
3062:
3039:
3011:
3002:, the element
2989:
2976:
2964:
2942:
2939:
2936:
2933:
2930:
2908:
2905:
2902:
2899:
2896:
2876:
2856:
2834:
2812:
2801:
2798:
2793:(or sometimes
2783:
2769:
2766:
2763:
2760:
2757:
2733:
2730:
2727:
2724:
2721:
2697:
2675:
2655:
2635:
2624:
2621:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2541:
2517:
2493:
2469:
2456:
2434:
2431:
2428:
2404:
2382:
2362:
2342:
2317:
2293:
2267:
2232:
2230:
2227:
2212:
2199:
2198:
2184:
2181:
2159:
2144:is called the
2133:
2124:. The integer
2111:
2108:
2105:
2102:
2099:
2077:
2074:
2071:
2068:
2065:
2045:
2023:
2010:
2003:is called the
1986:
1983:
1980:
1977:
1974:
1952:
1949:
1946:
1943:
1940:
1920:
1909:
1889:
1867:
1858:to the sum of
1847:
1827:
1807:
1787:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1699:
1677:
1655:
1626:
1599:
1573:
1570:
1567:
1542:
1520:
1507:together with
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1411:
1395:
1392:
1390:
1387:
1384:
1383:
1381:
1380:
1373:
1366:
1358:
1355:
1354:
1350:
1349:
1344:
1339:
1334:
1329:
1324:
1319:
1313:
1312:
1311:
1305:
1299:
1298:
1295:
1294:
1291:
1290:
1287:Linear algebra
1281:
1280:
1275:
1270:
1264:
1263:
1257:
1256:
1253:
1252:
1249:
1248:
1245:Lattice theory
1241:
1234:
1233:
1228:
1223:
1218:
1213:
1208:
1202:
1201:
1195:
1194:
1191:
1190:
1181:
1180:
1175:
1170:
1165:
1160:
1155:
1150:
1145:
1140:
1135:
1129:
1128:
1122:
1121:
1118:
1117:
1108:
1107:
1102:
1097:
1091:
1090:
1089:
1084:
1079:
1070:
1064:
1058:
1057:
1054:
1053:
1043:
1042:
1040:
1039:
1032:
1025:
1017:
1014:
1013:
1010:
1009:
1007:Elliptic curve
1003:
1002:
996:
995:
989:
988:
982:
977:
976:
973:
972:
967:
966:
963:
960:
956:
952:
951:
950:
945:
943:Diffeomorphism
939:
938:
933:
928:
922:
921:
917:
913:
909:
905:
901:
897:
893:
889:
885:
880:
879:
868:
867:
856:
855:
844:
843:
832:
831:
820:
819:
808:
807:
800:Special linear
796:
795:
788:General linear
784:
783:
778:
772:
763:
762:
759:
758:
755:
754:
749:
744:
736:
735:
722:
710:
697:
684:
682:Modular groups
680:
679:
678:
673:
660:
644:
641:
640:
635:
629:
628:
627:
624:
623:
618:
617:
616:
615:
610:
605:
602:
596:
595:
589:
588:
587:
586:
580:
579:
573:
572:
567:
558:
557:
555:Hall's theorem
552:
550:Sylow theorems
546:
545:
540:
532:
531:
530:
529:
523:
518:
515:Dihedral group
511:
510:
505:
499:
494:
488:
483:
472:
467:
466:
463:
462:
457:
456:
455:
454:
449:
441:
440:
439:
438:
433:
428:
423:
418:
413:
408:
406:multiplicative
403:
398:
393:
388:
380:
379:
378:
377:
372:
364:
363:
355:
354:
353:
352:
350:Wreath product
347:
342:
337:
335:direct product
329:
327:Quotient group
321:
320:
319:
318:
313:
308:
298:
295:
294:
291:
290:
282:
281:
167:Poincaré group
159:Standard Model
151:symmetry group
110:
15:
9:
6:
4:
3:
2:
29766:
29755:
29752:
29750:
29747:
29745:
29742:
29741:
29739:
29732:
29722:
29714:
29713:
29710:
29704:
29701:
29699:
29696:
29694:
29691:
29689:
29686:
29685:
29683:
29679:
29673:
29670:
29669:
29667:
29663:
29657:
29654:
29652:
29649:
29646:
29642:
29639:
29636:
29632:
29629:
29626:
29622:
29619:
29616:
29612:
29609:
29607:
29604:
29602:
29599:
29598:
29596:
29592:
29586:
29583:
29581:
29578:
29575:
29571:
29570:Orthogonality
29568:
29565:
29561:
29558:
29555:
29551:
29548:
29545:
29541:
29538:
29536:
29535:Hilbert space
29533:
29530:
29526:
29523:
29521:
29518:
29516:
29513:
29511:
29508:
29507:
29505:
29499:
29492:
29488:
29485:
29482:
29478:
29475:
29472:
29468:
29465:
29462:
29458:
29455:
29452:
29448:
29445:
29444:
29442:
29438:
29432:
29429:
29426:
29422:
29419:
29416:
29412:
29408:
29405:
29402:
29398:
29395:
29392:
29388:
29385:
29382:
29378:
29374:
29371:
29369:
29366:
29365:
29363:
29359:
29353:
29350:
29348:
29345:
29343:
29340:
29338:
29335:
29333:
29330:
29328:
29325:
29323:
29320:
29318:
29315:
29313:
29310:
29308:
29305:
29303:
29300:
29298:
29295:
29291:
29288:
29286:
29283:
29282:
29281:
29278:
29276:
29273:
29272:
29270:
29266:
29260:
29257:
29255:
29252:
29251:
29248:
29244:
29237:
29232:
29230:
29225:
29223:
29218:
29217:
29214:
29202:
29199:
29197:
29194:
29192:
29189:
29188:
29185:
29178:
29175:
29172:
29170:
29169:Quantum group
29167:
29165:
29162:
29160:
29157:
29155:
29152:
29151:
29149:
29145:
29139:
29136:
29134:
29131:
29129:
29128:Lorentz group
29126:
29124:
29121:
29120:
29117:
29111:
29109:
29103:
29101:
29095:
29093:
29087:
29085:
29079:
29077:
29074:
29073:
29069:
29066:
29063:
29060:
29057:
29056:Unitary group
29054:
29051:
29048:
29045:
29042:
29039:
29036:
29033:
29030:
29029:
29027:
29025:
29021:
29015:
29012:
29009:
29005:
29002:
28999:
28996:
28992:
28989:
28986:
28983:
28982:
28978:
28977:Monster group
28975:
28972:
28969:
28963:
28962:Fischer group
28960:
28958:
28951:
28944:
28937:
28931:Janko groups
28930:
28924:
28921:
28911:
28910:Mathieu group
28908:
28906:
28903:
28902:
28895:
28892:
28886:
28883:
28881:
28878:
28877:
28875:
28873:
28869:
28863:
28862:Trivial group
28860:
28858:
28855:
28853:
28850:
28848:
28845:
28843:
28840:
28838:
28835:
28833:
28832:Simple groups
28830:
28828:
28825:
28823:
28822:Cyclic groups
28820:
28818:
28815:
28813:
28812:Finite groups
28810:
28809:
28807:
28803:
28797:
28794:
28792:
28788:
28784:
28782:
28779:
28777:
28774:
28772:
28769:
28767:
28764:
28762:
28759:
28758:
28756:
28754:
28753:Basic notions
28750:
28746:
28739:
28734:
28732:
28727:
28725:
28720:
28719:
28716:
28708:
28707:
28702:
28698:
28693:
28692:
28681:
28675:
28671:
28667:
28663:
28662:Wussing, Hans
28659:
28655:
28649:
28645:
28641:
28640:Weyl, Hermann
28637:
28632:
28631:
28626:
28622:
28619:
28615:
28611:
28607:
28603:
28599:
28595:
28591:
28587:
28583:
28579:
28575:
28571:
28567:
28563:
28559:
28555:
28551:
28547:
28546:
28541:
28537:
28532:
28531:
28526:
28522:
28519:
28513:
28512:
28507:
28503:
28499:on 2014-02-22
28498:
28494:
28490:
28486:
28482:
28478:
28475:(in German),
28474:
28473:
28468:
28464:
28460:
28456:
28450:
28446:
28442:
28438:
28435:
28431:
28430:
28425:
28421:
28416:
28412:
28408:
28407:
28402:
28398:
28395:
28389:
28385:
28381:
28377:
28376:Borel, Armand
28373:
28372:
28370:
28357:
28353:
28349:
28343:
28339:
28338:
28333:
28329:
28325:
28319:
28315:
28311:
28310:Weyl, Hermann
28307:
28303:
28297:
28293:
28289:
28285:
28281:
28279:0-471-92567-5
28275:
28271:
28270:
28265:
28261:
28258:
28254:
28250:
28246:
28242:
28236:
28232:
28228:
28224:
28220:
28214:
28210:
28205:
28201:
28197:
28192:
28187:
28183:
28179:
28178:
28173:
28169:
28166:
28160:
28156:
28155:Galois Theory
28152:
28148:
28144:
28139:
28135:
28131:
28126:
28123:
28117:
28113:
28108:
28105:
28101:
28097:
28091:
28087:
28082:
28078:
28072:
28068:
28063:
28059:
28055:
28051:
28045:
28040:
28039:
28033:
28029:
28025:
28021:
28017:
28013:
28006:
28001:
27997:
27995:0-471-52364-X
27991:
27987:
27983:
27982:Rudin, Walter
27979:
27975:
27971:
27967:
27961:
27957:
27952:
27948:
27942:
27938:
27934:
27930:
27926:
27922:
27916:
27912:
27908:
27904:
27900:
27897:
27893:
27889:
27885:
27881:
27875:
27870:
27865:
27861:
27857:
27853:
27849:
27845:
27839:
27835:
27830:
27826:
27822:
27818:
27812:
27808:
27804:
27800:
27797:
27791:
27787:
27786:
27780:
27776:
27770:
27766:
27761:
27750:
27749:
27742:
27739:
27733:
27729:
27728:
27723:
27719:
27715:
27709:
27705:
27704:
27699:
27695:
27691:
27685:
27681:
27677:
27672:
27668:
27664:
27660:
27654:
27650:
27645:
27641:
27637:
27633:
27627:
27622:
27621:
27615:
27611:
27607:
27603:
27599:
27593:
27589:
27585:
27581:
27577:
27572:
27567:
27562:
27558:
27554:
27550:
27546:
27545:
27540:
27536:
27532:
27529:
27523:
27519:
27514:
27510:
27504:
27500:
27496:
27495:
27490:
27486:
27483:
27477:
27473:
27468:
27464:
27462:0-201-02918-9
27458:
27454:
27453:
27448:
27444:
27441:
27437:
27433:
27427:
27423:
27419:
27415:
27411:
27407:
27403:on 2008-12-01
27402:
27398:
27395:(in German),
27394:
27390:
27386:
27382:
27378:
27374:
27370:
27364:
27360:
27356:
27352:
27347:
27344:
27338:
27334:
27329:
27325:
27321:
27317:
27313:
27306:
27297:
27293:
27291:0-19-850678-3
27287:
27283:
27278:
27274:
27268:
27264:
27260:
27255:
27251:
27247:
27243:
27237:
27233:
27228:
27224:
27220:
27215:
27210:
27206:
27202:
27198:
27194:
27190:
27187:
27181:
27177:
27172:
27168:
27162:
27158:
27154:
27150:
27146:
27142:
27138:
27134:
27128:
27124:
27120:
27116:
27115:Borel, Armand
27112:
27108:
27102:
27098:
27093:
27089:
27085:
27080:
27075:
27071:
27067:
27063:
27058:
27054:
27052:0-521-82212-2
27048:
27044:
27043:
27037:
27033:
27028:
27023:
27018:
27014:
27010:
27005:
27002:
26996:
26992:
26987:
26982:
26978:
26977:
26969:
26965:
26961:
26957:
26951:
26947:
26943:
26942:Galois Theory
26939:
26935:
26934:
26924:
26918:
26914:
26910:
26906:
26902:
26898:
26894:
26890:
26886:
26882:
26878:
26874:
26870:
26866:
26862:
26856:
26852:
26848:
26844:
26840:
26837:
26833:
26829:
26823:
26819:
26815:
26811:
26807:
26803:
26799:
26795:
26791:
26787:
26783:
26779:
26775:
26769:
26765:
26761:
26757:
26753:
26749:
26745:
26741:
26737:
26734:
26728:
26724:
26723:
26717:
26713:
26707:
26703:
26702:Prentice Hall
26699:
26695:
26691:
26690:
26672:
26667:
26660:
26655:
26648:
26643:
26636:
26631:
26624:
26619:
26612:
26611:Weinberg 1972
26607:
26600:
26595:
26588:
26583:
26576:
26571:
26564:
26559:
26552:
26547:
26540:
26539:Neukirch 1999
26535:
26528:
26523:
26516:
26511:
26504:
26499:
26492:
26487:
26480:
26476:
26471:
26464:
26459:
26452:
26447:
26440:
26435:
26428:
26423:
26416:
26411:
26404:
26403:Robinson 1996
26399:
26392:
26387:
26380:
26375:
26368:
26363:
26361:
26353:
26348:
26341:
26336:
26329:
26324:
26317:
26312:
26306:, p. 82.
26305:
26300:
26293:
26288:
26281:
26276:
26269:
26264:
26257:
26252:
26245:
26240:
26234:
26230:
26225:
26218:
26213:
26206:
26201:
26199:
26191:
26186:
26179:
26174:
26167:
26162:
26155:
26150:
26143:
26138:
26131:
26126:
26119:
26114:
26107:
26102:
26095:
26094:Neukirch 1999
26091:
26090:Picard groups
26087:
26084:For example,
26081:
26074:
26069:
26062:
26057:
26050:
26045:
26038:
26033:
26026:
26021:
26014:
26009:
26002:
25997:
25990:
25985:
25978:
25973:
25966:
25965:Mac Lane 1998
25961:
25959:
25951:
25946:
25939:
25934:
25932:
25930:
25923:, p. 40.
25922:
25917:
25910:
25905:
25903:
25895:
25890:
25883:
25878:
25871:
25866:
25859:
25854:
25847:
25842:
25835:
25830:
25823:
25822:von Dyck 1882
25818:
25811:
25806:
25799:
25794:
25787:
25782:
25775:
25770:
25763:
25758:
25751:
25746:
25739:
25734:
25727:
25722:
25715:
25710:
25703:
25698:
25691:
25686:
25679:
25678:Herstein 1975
25674:
25667:
25662:
25655:
25650:
25643:
25638:
25632:, p. 24.
25631:
25626:
25619:
25614:
25607:
25603:
25599:
25594:
25587:
25586:Herstein 1975
25582:
25578:
25564:
25558:
25551:
25547:
25542:
25535:
25531:
25527:
25523:
25522:Schur's Lemma
25517:
25510:
25504:
25497:
25493:
25487:
25480:
25476:
25472:
25466:
25458:
25452:
25446:
25442:
25435:
25428:
25427:Gollmann 2011
25424:
25420:
25414:
25407:
25406:
25405:Prime element
25399:
25392:
25388:
25384:
25380:
25374:
25367:
25362:
25356:
25352:
25346:
25339:
25333:
25326:
25322:
25316:
25309:
25305:
25301:
25297:
25291:
25284:
25278:
25271:
25265:
25258:
25254:
25250:
25244:
25237:
25236:Herstein 1975
25233:
25232:Herstein 1996
25229:
25225:
25219:
25211:
25207:
25201:
25197:
25191:
25181:
25173:
25162:
25155:
25151:
25145:
25138:
25133:
25123:
25117:
25111:
25105:
25099:
25095:
25086:
25080:
25076:
25066:
25063:
25062:
25058:
25052:
25047:
25036:
25033:
25031:
25030:abelian group
25028:
25026:
25023:
25021:
25018:
25016:
25013:
25011:
25010:abelian group
25008:
25006:
25003:
25001:
25000:abelian group
24998:
24996:
24993:
24991:
24988:
24985:
24984:
24980:
24977:
24974:
24971:
24968:
24965:
24962:
24959:
24956:
24953:
24950:
24949:
24945:
24942:
24939:
24936:
24933:
24930:
24927:
24924:
24921:
24918:
24915:
24914:
24910:
24907:
24904:
24887:
24884:
24881:
24856:
24852:
24848:
24839:
24836:
24820:
24817:
24808:
24805:
24789:
24786:
24777:
24774:
24771:
24768:
24767:
24763:
24760:
24757:
24754:
24751:
24748:
24745:
24742:
24739:
24736:
24733:
24732:
24728:
24725:
24722:
24719:
24716:
24713:
24710:
24707:
24704:
24701:
24698:
24697:
24693:
24690:
24687:
24684:
24681:
24678:
24675:
24672:
24669:
24666:
24663:
24662:
24642:
24639:
24636:
24633:
24630:
24624:
24616:
24612:
24596:
24565:
24535:
24505:
24475:
24445:
24439:
24438:
24432:
24430:
24415:
24407:
24405:
24401:
24383:
24359:
24335:
24332:
24322:
24304:
24301:
24295:
24292:
24289:
24283:
24280:
24256:
24232:
24207:
24204:
24201:
24195:
24192:
24170:
24160:
24155:
24139:
24129:
24111:
24084:
24081:
24075:
24069:
24036:
24033:
24027:
24021:
23988:
23985:
23979:
23973:
23934:
23930:
23925:
23917:
23914:
23911:
23908:
23905:
23903:
23902:Abelian group
23900:
23899:
23895:
23892:
23889:
23886:
23883:
23881:
23878:
23877:
23873:
23870:
23867:
23864:
23861:
23859:
23856:
23855:
23851:
23848:
23845:
23842:
23839:
23837:
23834:
23833:
23829:
23826:
23823:
23820:
23817:
23815:
23811:
23810:
23806:
23803:
23800:
23797:
23794:
23792:
23788:
23787:
23783:
23780:
23777:
23774:
23771:
23769:
23765:
23764:
23760:
23757:
23754:
23751:
23748:
23746:
23743:
23742:
23738:
23735:
23732:
23729:
23726:
23724:
23720:
23719:
23715:
23712:
23709:
23706:
23703:
23701:
23698:
23697:
23693:
23690:
23687:
23684:
23681:
23679:
23675:
23674:
23670:
23667:
23664:
23661:
23658:
23656:
23653:
23652:
23648:
23645:
23642:
23639:
23636:
23634:
23630:
23629:
23625:
23622:
23619:
23616:
23613:
23611:
23608:
23607:
23603:
23600:
23597:
23594:
23591:
23589:
23585:
23584:
23580:
23577:
23574:
23571:
23568:
23566:
23563:
23562:
23558:
23555:
23552:
23549:
23546:
23544:
23540:
23539:
23535:
23532:
23529:
23526:
23523:
23521:
23518:
23517:
23513:
23510:
23507:
23504:
23501:
23499:
23496:
23495:
23491:
23488:
23485:
23482:
23479:
23477:
23474:
23473:
23469:
23466:
23463:
23460:
23457:
23455:
23454:Partial magma
23452:
23451:
23448:
23445:
23443:
23440:
23438:
23435:
23433:
23430:
23428:
23425:
23423:
23422:
23411:
23409:
23405:
23401:
23397:
23393:
23389:
23385:
23381:
23377:
23373:
23369:
23365:
23361:
23357:
23353:
23349:
23345:
23341:
23336:
23322:
23302:
23282:
23262:
23259:
23256:
23250:
23224:
23204:
23196:
23191:
23189:
23184:
23180:
23176:
23170:
23160:
23158:
23154:
23150:
23146:
23142:
23138:
23134:
23133:Haar measures
23130:
23126:
23125:-adic numbers
23123:
23117:
23113:
23109:
23093:
23073:
23051:
23048:
23044:
23023:
23020:
23017:
23009:
23003:
22995:
22991:
22986:
22985:complex plane
22982:
22977:
22968:
22966:
22962:
22957:
22955:
22951:
22947:
22931:
22925:
22918:
22902:
22896:
22893:
22890:
22870:
22856:
22854:
22849:
22847:
22843:
22839:
22835:
22834:monster group
22831:
22827:
22823:
22819:
22814:
22812:
22808:
22804:
22799:
22789:
22787:
22783:
22782:
22763:
22759:
22755:
22733:
22713:
22693:
22670:
22647:
22627:
22620:When a group
22616:Simple groups
22613:
22597:
22593:
22570:
22541:
22537:
22512:
22502:
22497:
22464:
22460:
22430:
22426:
22417:
22399:
22372:
22363:
22361:
22357:
22347:
22331:
22328:
22325:
22303:
22281:
22276:
22241:
22220:
22198:
22194:
22171:
22155:
22153:
22135:
22126:the order of
22125:
22110:
22090:
22082:
22077:
22063:
22043:
22021:
21997:
21973:
21969:
21946:
21922:
21900:
21878:
21856:
21851: factors
21846:
21840:
21836:
21833:
21830:
21804:
21800:
21777:
21774:
21769:
21765:
21742:
21722:
21702:
21693:
21691:
21673:
21658:
21640:
21616:
21600:
21582:
21568:
21552:
21528:
21512:
21511:
21506:
21502:
21498:
21492:
21485:Finite groups
21482:
21480:
21476:
21472:
21468:
21467:Galois theory
21463:
21461:
21457:
21453:
21449:
21445:
21441:
21437:
21421:
21401:
21381:
21361:
21355:
21352:
21345:
21342:
21339:
21336:
21331:
21327:
21321:
21318:
21315:
21309:
21306:
21299:are given by
21286:
21283:
21280:
21277:
21274:
21271:
21268:
21263:
21259:
21255:
21248:
21244:
21243:Galois groups
21239:
21232:Galois groups
21229:
21227:
21223:
21217:
21195:
21192:
21175:
21172:
21169:
21161:
21145:
21125:
21101:
21085:
21081:
21077:
21073:
21071:
21067:
21051:
21026:
21006:
21002:
21001:
21000:linear groups
20996:
20995:matrix groups
20978:
20954:
20945:
20921:
20918:
20900:
20896:
20892:
20888:
20887:Matrix groups
20867:
20857:
20852:
20847:
20843:
20839:
20829:
20827:
20824:
20820:
20816:
20812:
20808:
20804:
20800:
20799:coding theory
20796:
20786:
20782:
20779:
20766:
20763:
20756:
20753:
20751:
20746:
20743:
20742:
20738:
20731:
20724:
20717:
20712:
20709:
20707:
20703:
20698:
20696:
20692:
20688:
20687:ferroelectric
20684:
20679:
20677:
20673:
20669:
20665:
20661:
20656:
20654:
20650:
20646:
20628:
20604:
20594:
20590:
20586:
20582:
20577:
20570:
20566:
20561:
20556:
20552:
20548:
20544:
20538:
20528:
20526:
20522:
20518:
20513:
20494:
20491:
20461:
20439:
20428:
20412:
20409:
20404:
20400:
20375:
20372:
20367:
20363:
20338:
20335:
20332:
20329:
20324:
20320:
20295:
20292:
20287:
20283:
20260:
20238:
20235:
20232:
20210:
20188:
20183:
20168:
20164:
20148:
20126:
20123:
20120:
20098:
20091:on a regular
20090:
20072:
20069:
20064:
20060:
20037:
20030:
20026:
20008:
19986:
19964:
19942:
19919:
19916:
19908:
19904:
19886:
19873:
19870:
19867:
19864:
19861:
19858:
19855:
19852:
19849:
19846:
19843:
19840:
19837:
19834:
19828:
19825:
19819:
19813:
19810:
19804:
19801:
19793:
19777:
19753:
19750:
19742:
19739:
19736:
19733:
19730:
19724:
19719:
19716:
19712:
19708:
19703:
19700:
19696:
19692:
19687:
19684:
19680:
19655:
19652:
19648:
19625:
19622:
19619:
19595:
19591:
19570:
19567:
19564:
19559:
19555:
19551:
19546:
19542:
19538:
19535:
19532:
19527:
19523:
19519:
19514:
19511:
19507:
19503:
19498:
19495:
19491:
19487:
19482:
19479:
19475:
19471:
19468:
19448:
19440:
19436:
19412:
19408:
19385:
19363:
19359:
19338:
19329:
19324:
19317:Cyclic groups
19314:
19312:
19292:
19287:
19253:
19249:
19222:
19218:
19215:
19211:
19208:
19205:
19201:
19192:
19167:
19159:
19156:
19153:
19150:
19147:
19144:
19141:
19117:
19095:
19073:
19051:
19031:
19028:
19025:
19003:
18978:
18975:
18972:
18959:
18955:
18939:
18917:
18914:
18911:
18908:
18905:
18883:
18863:
18856:
18852:
18844:
18841:
18838:
18835:
18832:
18812:
18790:
18768:
18746:
18722:
18700:
18678:
18656:
18634:
18612:
18588:
18580:
18577:
18574:
18571:
18568:
18544:
18541:
18538:
18535:
18532:
18529:
18526:
18502:
18499:
18496:
18472:
18462:
18444:
18420:
18417:
18414:
18392:
18370:
18360:
18342:
18331:
18318:
18311:
18307:
18299:
18296:
18293:
18290:
18287:
18265:
18243:
18223:
18220:
18217:
18195:
18173:
18153:
18145:
18140:
18124:
18102:
18099:
18096:
18076:
18051:
18048:
18040:
18036:
18003:
17974:
17971:
17968:
17946:
17938:
17920:
17917:
17914:
17892:
17866:
17862:
17854:
17851:
17848:
17826:
17806:
17786:
17766:
17759:
17747:
17742:
17737:
17727:
17703:
17698:
17682:
17678:
17674:
17652:
17648:
17644:
17620:
17590:
17586:
17583:
17580:
17577:
17569:
17566:
17562:
17558:
17554:
17551:
17548:
17544:
17531:
17526:
17511:
17507:
17504:
17495:
17472:
17469:
17466:
17463:
17460:
17438:
17415:
17411:
17408:
17399:
17362:
17346:
17337:
17324:
17319:
17316:
17307:
17297:
17256:
17253:
17247:
17244:
17222:
17219:
17216:
17213:
17210:
17190:
17187:
17184:
17176:
17159:
17155:
17152:
17143:
17119:
17115:
17112:
17103:
17064:
17062:
17058:
17057:vector spaces
17054:
17050:
17046:
17036:
17034:
17030:
17026:
17022:
17018:
17017:algorithmical
17014:
17009:
17007:
17003:
16999:
16994:
16992:
16974:
16971:
16968:
16946:
16926:
16918:
16913:
16909:
16905:
16896:
16892:
16890:
16886:
16882:
16878:
16874:
16870:
16866:
16861:
16859:
16855:
16825:
16820:
16815:
16805:
16803:
16799:
16794:
16776:
16760:
16757:
16752:
16744:
16741:
16738:
16732:
16727:
16723:
16719:
16714:
16710:
16706:
16703:
16700:
16697:
16670:
16646:
16631:
16630:
16608:
16605:
16600:
16592:
16589:
16586:
16580:
16575:
16571:
16567:
16562:
16558:
16554:
16551:
16548:
16545:
16517:
16514:
16511:
16488:
16485:
16482:
16458:
16450:
16447:
16444:
16438:
16433:
16429:
16425:
16420:
16416:
16405:
16389:
16386:
16383:
16359:
16355:
16332:
16310:
16306:
16285:
16263:
16233:
16230:
16227:
16202:
16178:
16143:
16120:
16116:
16093:
16077:
16075:
16069:
16062:Presentations
16059:
16057:
16039:
16031:
16018:
16015:
16012:
16008:
16004:
15998:
15976:
15970:
15967:
15964:
15957:homomorphism
15956:
15952:
15947:
15931:
15916:
15898:
15871:
15867:
15861:
15829:
15825:
15821:
15816:
15812:
15808:
15803:
15799:
15795:
15781:
15778:
15756:
15753:
15750:
15736:
15732:
15721:
15714:
15711:
15700:
15696:
15693:
15682:
15678:
15675:
15672:
15669:
15645:
15634:
15630:
15627:
15605:
15585:
15581:
15575:
15543:
15536:
15522:
15515:
15501:
15494:
15493:
15478:
15471:
15457:
15450:
15436:
15429:
15428:
15413:
15406:
15392:
15385:
15371:
15364:
15363:
15348:
15344:
15338:
15321:
15319:
15315:
15311:
15293:
15269:
15245:
15241:
15237:
15213:
15209:
15204:
15201:
15197:
15193:
15189:
15184:
15181:
15173:
15170:
15145:
15142:
15120:
15116:
15112:
15090:
15087:
15084:
15081:
15059:
15053:
15050:
15025:
15022:
15002:
14999:
14979:
14976:
14956:
14936:
14932:
14928:
14922:
14902:
14898:
14894:
14871:
14868:
14865:
14862:
14859:
14856:
14850:
14847:
14843:
14839:
14817:
14795:
14788:Suppose that
14785:
14775:
14759:
14748:
14744:
14741:
14730:
14726:
14723:
14712:
14708:
14705:
14694:
14667:
14630:
14626:
14623:
14620:
14617:
14614:
14603:
14582:
14558:
14518:
14514:
14503:
14499:
14488:
14484:
14473:
14466:
14463:
14452:
14448:
14445:
14425:
14405:
14383:
14361:
14358:
14338:
14314:
14296:
14294:
14293:
14276:
14254:
14251:
14248:
14245:
14242:
14220:
14200:
14180:
14158:
14134:
14110:
14107:
14102:
14098:
14094:
14089:
14086:
14081:
14077:
14067:
14051:
14046:
14042:
14038:
14035:
14030:
14026:
14017:
14014:
13998:
13990:
13972:
13962:
13946:
13918:
13915:
13912:
13909:
13906:
13903:
13900:
13894:
13891:
13888:
13863:
13860:
13857:
13854:
13851:
13848:
13845:
13839:
13836:
13833:
13825:
13809:
13785:
13775:
13771:
13753:
13731:
13711:
13700:
13690:
13672:
13668:
13664:
13653:
13649:
13638:
13578:
13555:
13551:
13528:
13506:
13486:
13466:
13458:
13440:
13418:
13409:
13407:
13389:
13367:
13347:
13327:
13324:
13321:
13318:
13313:
13310:
13306:
13283:
13259:
13249:
13245:
13244:subgroup test
13222:
13218:
13214:
13209:
13205:
13201:
13196:
13192:
13188:
13174:
13171:
13160:
13146:
13140:
13118:
13096:
13074:
13048:
13045:
13040:
13036:
13011:
13007:
13003:
12998:
12994:
12971:
12947:
12943:
12920:
12916:
12893:
12871:
12849:
12825:
12803:
12795:
12789:
12779:
12777:
12776:monomorphisms
12759:
12735:
12713:
12707:
12699:
12685:
12682:
12661:
12654:
12651:
12647:
12644:
12637:homomorphism
12636:
12631:
12629:
12625:
12620:
12604:
12582:
12560:
12538:
12518:
12496:
12490:
12487:
12484:
12474:
12458:
12438:
12430:
12426:
12425:
12406:
12384:
12364:
12361:
12348:
12342:
12332:
12312:
12292:
12272:
12269:
12256:
12250:
12240:
12216:
12212:
12208:
12205:
12202:
12199:
12175:
12171:
12167:
12164:
12161:
12158:
12138:
12132:
12129:
12126:
12106:
12100:
12097:
12094:
12086:
12070:
12050:
12044:
12041:
12036:
12032:
12011:
12003:
11998:
11982:
11960:
11938:
11935:
11927:
11921:
11918:
11910:
11907:
11903:
11896:
11872:
11868:
11864:
11856:
11852:
11845:
11823:
11798:
11776:
11756:
11733:
11727:
11724:
11718:
11712:
11709:
11703:
11700:
11697:
11691:
11683:
11669:
11663:
11660:
11657:
11634:
11631:
11628:
11602:
11599:
11596:
11586:from a group
11585:
11579:
11569:
11567:
11563:
11562:homomorphisms
11559:
11555:
11551:
11530:
11508:
11500:
11484:
11464:
11458:
11455:
11452:
11430:
11408:
11388:
11385:
11382:
11360:
11336:
11312:
11303:
11287:
11280:
11261:
11258:
11255:
11252:
11249:
11243:
11240:
11232:
11222:
11205:
11202:
11199:
11175:
11153:
11129:
11126:
11123:
11120:
11117:
11114:
11111:
11108:
11105:
11083:
11080:
11077:
11074:
11071:
11068:
11065:
11062:
11059:
11039:
11016:
11013:
11010:
11004:
11001:
10993:
10989:
10984:
10981:
10979:
10948:
10945:
10943:
10927:
10924:
10921:
10918:
10916:
10900:
10897:
10889:
10886:
10882:
10878:
10875:
10869:
10867:
10848:
10845:
10840:
10837:
10833:
10826:
10823:
10820:
10818:
10813:
10810:
10807:
10796:
10795:
10794:
10765:
10762:
10760:
10742:
10739:
10735:
10731:
10726:
10723:
10713:
10710:
10706:
10699:
10697:
10676:
10673:
10669:
10665:
10662:
10656:
10651:
10648:
10638:
10635:
10631:
10624:
10622:
10601:
10598:
10594:
10590:
10584:
10581:
10576:
10573:
10569:
10559:
10554:
10551:
10541:
10538:
10534:
10527:
10525:
10504:
10501:
10497:
10493:
10490:
10484:
10476:
10473:
10469:
10465:
10460:
10457:
10447:
10444:
10440:
10430:
10428:
10407:
10404:
10400:
10396:
10393:
10387:
10384:
10381:
10379:
10372:
10369:
10365:
10361:
10358:
10347:
10346:
10345:
10329:
10326:
10323:
10320:
10315:
10312:
10308:
10285:
10263:
10260:
10256:
10233:
10230:
10227:
10224:
10221:
10199:
10190:
10188:
10185:. From these
10184:
10183:left inverses
10180:
10179:left identity
10170:
10154:
10144:
10128:
10120:
10104:
10101:
10098:
10078:
10058:
10052:
10030:
10004:
10001:
9997:
9993:
9990:
9968:
9965:
9962:
9959:
9956:
9934:
9912:
9903:
9887:
9877:
9861:
9853:
9849:
9833:
9830:
9827:
9807:
9787:
9781:
9761:
9741:
9719:
9716:
9711:
9708:
9704:
9679:
9676:
9673:
9670:
9667:
9645:
9625:
9603:
9581:
9561:
9547:
9545:
9517:
9514:
9511:
9503:
9492:
9489:
9487:
9474:
9466:
9455:
9452:
9449:
9446:
9444:
9428:
9425:
9419:
9416:
9413:
9407:
9405:
9392:
9384:
9370:
9367:
9364:
9358:
9355:
9352:
9350:
9337:
9326:
9323:
9320:
9317:
9315:
9310:
9299:
9298:
9297:
9283:
9263:
9243:
9229:
9227:
9209:
9206:
9203:
9200:
9197:
9194:
9191:
9169:
9149:
9135:
9133:
9114:
9111:
9108:
9102:
9099:
9096:
9093:
9090:
9084:
9081:
9078:
9072:
9069:
9066:
9063:
9060:
9057:
9049:
9045:
9035:
9033:
9029:
9025:
9021:
9017:
9013:
9009:
9005:
9000:
8998:
8994:
8990:
8986:
8982:
8978:
8974:
8970:
8966:
8962:
8958:
8954:
8950:
8946:
8942:
8937:
8932:
8927:
8925:
8924:prime numbers
8921:
8917:
8913:
8909:
8905:
8904:
8899:
8895:
8894:number theory
8890:
8888:
8884:
8880:
8876:
8872:
8868:
8863:
8861:
8857:
8844:
8841:
8836:
8832:
8821:
8820:Arthur Cayley
8817:
8813:
8809:
8805:
8801:
8797:
8793:
8789:
8788:Paolo Ruffini
8785:
8781:
8777:
8771:
8761:
8745:
8708:
8704:
8693:
8689:
8684:
8680:
8649:
8645:
8640:
8636:
8632:
8621:
8596:
8578:
8562:
8558:
8535:
8531:
8508:
8504:
8473:
8440:
8407:
8374:
8334:
8330:
8316:
8279:
8275:
8258:
8253:
8249:
8245:
8234:
8230:
8219:
8215:
8213:
8203:
8199:
8195:
8184:
8177:
8166:
8156:
8152:
8148:
8143:
8139:
8135:
8130:
8126:
8122:
8120:
8113:
8109:
8105:
8091:
8087:
8076:
8043:
8039:
8035:
8024:
8017:
8006:
8002:
7997:
7993:
7989:
7975:
7971:
7960:
7949:For example,
7936:
7930:
7927:
7924:
7918:
7915:
7912:
7909:
7906:
7900:
7897:
7894:
7869:
7845:
7823:
7801:
7779:
7759:
7735:
7704:
7680:
7656:
7646:
7645:Associativity
7642:
7618:
7595:
7584:
7580:
7569:
7565:
7560:
7556:
7533:
7511:
7491:
7488:
7485:
7477:
7473:
7439:
7412:
7408:
7404:
7393:
7374:
7366:
7346:
7342:
7315:
7311:
7284:
7280:
7241:The elements
7239:
7212:
7196:
7192:
7184:
7168:
7164:
7156:
7140:
7136:
7128:
7106:
7098:
7076:
7068:
7046:
7038:
7016:
7008:
6986:
6978:
6977:
6960:
6956:
6948:
6922:
6906:
6902:
6894:
6878:
6874:
6866:
6844:
6836:
6814:
6806:
6784:
6776:
6754:
6746:
6724:
6716:
6715:
6698:
6694:
6686:
6670:
6666:
6658:
6632:
6616:
6612:
6604:
6582:
6574:
6552:
6544:
6522:
6514:
6492:
6484:
6462:
6454:
6453:
6436:
6432:
6424:
6408:
6404:
6396:
6380:
6376:
6368:
6342:
6320:
6312:
6290:
6282:
6260:
6252:
6230:
6222:
6200:
6192:
6191:
6168:
6160:
6138:
6130:
6108:
6100:
6078:
6070:
6054:
6050:
6042:
6026:
6022:
6014:
5988:
5972:
5968:
5960:
5944:
5940:
5932:
5931:
5908:
5900:
5878:
5870:
5848:
5840:
5818:
5810:
5794:
5790:
5782:
5756:
5740:
5736:
5728:
5712:
5708:
5700:
5684:
5680:
5672:
5671:
5648:
5640:
5618:
5610:
5588:
5580:
5558:
5550:
5524:
5508:
5504:
5496:
5480:
5476:
5468:
5452:
5448:
5440:
5424:
5420:
5412:
5411:
5388:
5380:
5358:
5350:
5328:
5320:
5298:
5290:
5274:
5270:
5262:
5246:
5242:
5234:
5218:
5214:
5206:
5180:
5154:
5153:
5130:
5122:
5100:
5092:
5070:
5062:
5040:
5032:
5016:
5012:
5004:
4988:
4984:
4976:
4960:
4956:
4948:
4922:
4908:
4901:
4900:
4883:
4868:
4863:
4850:
4839:
4835:
4830:
4826:
4822:
4811:
4780:
4747:
4720:
4716:
4705:
4700:
4684:
4662:
4642:
4639:
4636:
4628:
4612:
4592:
4568:
4551:
4527:
4504:
4500:
4465:
4432:
4421:
4395:
4362:
4351:
4332:
4328:
4303:
4299:
4274:
4270:
4259:
4256:
4252:
4251:
4246:
4224:
4214:
4209:
4207:
4185:
4175:
4170:
4168:
4146:
4136:
4131:
4108:
4098:
4094:
4093:
4075:
4071:
4061:
4057:
4040:
4036:
4026:
4022:
4005:
4001:
3991:
3987:
3958:
3954:
3953:
3933:
3914:
3912:
3908:
3904:
3900:
3896:
3892:
3888:
3878:
3864:
3857:, the symbol
3856:
3852:
3848:
3844:
3826:
3823:
3820:
3811:
3807:
3802:
3800:
3799:abelian group
3796:
3778:
3756:
3736:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3689:
3673:
3670:
3667:
3645:
3642:
3618:
3615:
3611:
3588:
3566:
3556:
3538:
3535:
3513:
3491:
3481:
3476:
3459:
3453:
3426:
3412:is the group
3390:of the field
3389:
3342:of the field
3341:
3336:
3312:
3293:
3290:
3287:
3284:
3253:
3250:
3198:
3195:
3186:
3170:
3167:
3164:
3131:
3126:
3123:
3119:
3114:
3112:
3108:
3104:
3100:
3096:
3067:
3064:
3060:
3037:
3029:
3025:
3009:
2987:
2977:
2962:
2940:
2937:
2934:
2931:
2928:
2906:
2903:
2900:
2897:
2894:
2874:
2854:
2832:
2810:
2802:
2799:
2796:
2792:
2788:
2784:
2767:
2764:
2761:
2758:
2755:
2731:
2728:
2725:
2722:
2719:
2695:
2673:
2653:
2633:
2625:
2622:
2602:
2599:
2596:
2590:
2587:
2584:
2581:
2578:
2572:
2569:
2566:
2539:
2515:
2491:
2467:
2457:
2455:Associativity
2454:
2453:
2452:
2450:
2432:
2429:
2426:
2402:
2380:
2360:
2340:
2333:
2315:
2291:
2281:
2265:
2258:
2252:
2250:
2246:
2240:
2238:
2226:
2210:
2182:
2179:
2157:
2149:
2148:
2131:
2109:
2106:
2103:
2100:
2097:
2075:
2072:
2069:
2066:
2063:
2043:
2021:
2011:
2008:
2007:
2002:
1984:
1981:
1978:
1975:
1972:
1950:
1947:
1944:
1941:
1938:
1918:
1910:
1907:
1906:
1905:associativity
1887:
1865:
1845:
1825:
1805:
1785:
1760:
1757:
1754:
1748:
1745:
1742:
1739:
1736:
1730:
1727:
1724:
1697:
1675:
1653:
1643:
1642:
1641:
1613:
1597:
1589:
1588:
1571:
1568:
1565:
1558:
1540:
1518:
1510:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1413:
1401:
1379:
1374:
1372:
1367:
1365:
1360:
1359:
1357:
1356:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1314:
1310:
1307:
1306:
1302:
1297:
1296:
1289:
1288:
1284:
1283:
1279:
1276:
1274:
1271:
1269:
1266:
1265:
1260:
1255:
1254:
1247:
1246:
1242:
1240:
1237:
1236:
1232:
1229:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1203:
1198:
1193:
1192:
1187:
1186:
1179:
1176:
1174:
1173:Division ring
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1130:
1125:
1120:
1119:
1114:
1113:
1106:
1103:
1101:
1098:
1096:
1095:Abelian group
1093:
1092:
1088:
1085:
1083:
1080:
1078:
1074:
1071:
1069:
1066:
1065:
1061:
1056:
1055:
1052:
1049:
1048:
1038:
1033:
1031:
1026:
1024:
1019:
1018:
1016:
1015:
1008:
1005:
1004:
1001:
998:
997:
994:
991:
990:
987:
984:
983:
980:
975:
974:
964:
961:
958:
957:
955:
949:
946:
944:
941:
940:
937:
934:
932:
929:
927:
924:
923:
920:
914:
912:
906:
904:
898:
896:
890:
888:
882:
881:
877:
873:
870:
869:
865:
861:
858:
857:
853:
849:
846:
845:
841:
837:
834:
833:
829:
825:
822:
821:
817:
813:
810:
809:
805:
801:
798:
797:
793:
789:
786:
785:
782:
779:
777:
774:
773:
770:
766:
761:
760:
753:
750:
748:
745:
743:
740:
739:
711:
686:
685:
683:
677:
674:
649:
646:
645:
639:
636:
634:
631:
630:
626:
625:
614:
611:
609:
606:
603:
600:
599:
598:
597:
594:
591:
590:
585:
582:
581:
578:
575:
574:
571:
568:
566:
564:
560:
559:
556:
553:
551:
548:
547:
544:
541:
539:
536:
535:
534:
533:
527:
524:
521:
516:
513:
512:
508:
503:
500:
497:
492:
489:
486:
481:
478:
477:
476:
475:
470:
469:Finite groups
465:
464:
453:
450:
448:
445:
444:
443:
442:
437:
434:
432:
429:
427:
424:
422:
419:
417:
414:
412:
409:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
383:
382:
381:
376:
373:
371:
368:
367:
366:
365:
362:
361:
357:
356:
351:
348:
346:
343:
341:
338:
336:
333:
330:
328:
325:
324:
323:
322:
317:
314:
312:
309:
307:
304:
303:
302:
301:
296:Basic notions
293:
292:
288:
284:
283:
280:
275:
271:
267:
266:
263:
261:
257:
253:
249:
248:finite groups
245:
241:
237:
233:
232:simple groups
229:
225:
221:
217:
216:number theory
213:
209:
204:
199:
195:
191:
186:
184:
180:
176:
172:
168:
164:
160:
156:
152:
148:
144:
140:
135:
133:
129:
124:
108:
98:
94:
90:
86:
82:
77:
75:
71:
67:
63:
59:
55:
51:
47:
39:
35:
30:
26:
22:
29744:Group theory
29731:
29698:Order theory
29688:Field theory
29554:Affine space
29487:Vector space
29456:
29342:Order theory
29196:Applications
29123:Circle group
29007:
29003:
28994:
28990:
28923:Conway group
28885:Cyclic group
28744:
28704:
28668:, New York:
28665:
28643:
28629:
28605:
28585:
28549:
28543:
28529:
28510:
28497:the original
28476:
28470:
28444:
28427:
28405:
28379:
28336:
28313:
28291:
28268:
28230:
28208:
28181:
28175:
28154:
28151:Stewart, Ian
28133:
28129:
28111:
28085:
28066:
28037:
28015:
28011:
27985:
27955:
27932:
27906:
27863:
27833:
27806:
27784:
27764:
27753:, retrieved
27747:
27726:
27701:
27675:
27648:
27619:
27614:Kuga, Michio
27575:
27548:
27542:
27517:
27493:
27471:
27450:
27417:
27401:the original
27396:
27392:
27350:
27332:
27315:
27311:
27305:-ary groups"
27281:
27258:
27231:
27204:
27200:
27175:
27155:, New York:
27152:
27118:
27096:
27069:
27065:
27041:
27015:(2): 64â75,
27012:
27008:
26990:
26983:(7): 736â740
26980:
26974:
26944:, New York:
26941:
26912:
26892:
26872:
26846:
26813:
26793:
26763:
26743:
26721:
26697:
26666:
26654:
26642:
26630:
26618:
26606:
26594:
26582:
26570:
26558:
26546:
26534:
26522:
26510:
26498:
26486:
26470:
26458:
26451:Stewart 2015
26446:
26434:
26422:
26410:
26398:
26386:
26374:
26352:Kuipers 1999
26347:
26335:
26323:
26311:
26299:
26287:
26275:
26263:
26251:
26239:
26224:
26212:
26185:
26173:
26161:
26149:
26137:
26125:
26113:
26101:
26086:class groups
26080:
26068:
26061:Hatcher 2002
26056:
26044:
26032:
26020:
26008:
25996:
25984:
25972:
25945:
25916:
25889:
25877:
25870:Solomon 2018
25865:
25853:
25841:
25829:
25817:
25805:
25798:Wussing 2007
25793:
25786:Kleiner 1986
25781:
25769:
25762:Wussing 2007
25757:
25745:
25738:Kleiner 1986
25733:
25721:
25709:
25702:Kleiner 1986
25697:
25690:Wussing 2007
25685:
25673:
25661:
25649:
25637:
25625:
25613:
25593:
25581:
25557:
25541:
25516:
25503:
25486:
25465:
25456:
25450:
25444:
25440:
25434:
25413:
25403:
25398:
25373:
25360:
25354:
25345:
25332:
25315:
25290:
25277:
25264:
25243:
25218:
25209:
25205:
25199:
25195:
25189:
25179:
25171:
25161:
25144:
25131:
25121:
25115:
25109:
25103:
25097:
25093:
25084:
25079:
24951:Commutative
24916:Associative
24534:Real numbers
24408:
24156:
23926:
23923:
23879:
23789:Associative
23766:Commutative
23721:Commutative
23678:unital magma
23676:Commutative
23655:Unital magma
23631:Commutative
23586:Commutative
23541:Commutative
23476:Semigroupoid
23442:Cancellation
23396:gauge theory
23337:
23192:
23174:
23172:
23121:
23107:
23005:
22961:group object
22958:
22862:
22850:
22817:
22815:
22801:
22779:
22619:
22364:
22353:
22156:
22078:
22012:th power of
21694:
21508:
21499:if it has a
21496:
21494:
21491:Finite group
21464:
21443:
21242:
21241:
21238:Galois group
21218:
21075:
21074:
21064:dimensions.
20998:
20994:
20898:
20885:
20797:are used in
20792:
20699:
20680:
20668:space groups
20660:point groups
20657:
20575:
20574:
20514:
20429:
20169:, the group
20167:field theory
19887:
19435:cyclic group
19434:
19432:
19323:Cyclic group
18994:equals
18332:
18141:
17757:
17755:
17699:
17529:
17527:
17338:
17303:
17174:
17070:
17042:
17013:Cryptography
17010:
16995:
16959:times (with
16901:
16862:
16829:
16798:Cayley graph
16795:
16629:presentation
16627:
16403:
16078:
16071:
16056:epimorphisms
15948:
15560:
15314:factor group
15313:
15309:
15228:. The group
15074:, the coset
14787:
14297:
14290:
14016:intersection
13938:
13773:
13769:
13702:
13410:
13161:
12962:are both in
12793:
12791:
12632:
12621:
12595:is true for
12472:
12422:
12084:
12001:
11999:
11815:
11584:homomorphism
11583:
11581:
11547:
11277:to denote a
10991:
10987:
10985:
10982:
10975:
10792:
10191:
10186:
10176:
10142:
10118:
9904:
9875:
9851:
9553:
9543:
9541:
9235:
9225:
9141:
9043:
9041:
9034:illustrate.
9001:
8997:Jacques Tits
8993:Armand Borel
8973:Hermann Weyl
8928:
8912:Ernst Kummer
8901:
8891:
8864:
8860:finite group
8823:
8812:permutations
8808:Galois group
8773:
8579:
8332:
8331:
8277:
8276:
7644:
7643:
7475:
7474:
7471:
4867:Cayley table
4704:Cayley table
4701:
4626:
4491:
4210:
4171:
4132:
3903:translations
3884:
3803:
3690:
3554:
3479:
3477:
3387:
3339:
3337:
3213:. Formally,
3130:real numbers
3127:
3115:
3110:
3106:
3102:
3099:group axioms
3095:ordered pair
3092:
3027:
2794:
2790:
2449:group axioms
2448:
2254:
2248:
2242:
2234:
2200:
2145:
2004:
1903:
1585:
1397:
1347:Hopf algebra
1285:
1278:Vector space
1243:
1183:
1112:Group theory
1110:
1075: /
1067:
1059:
875:
863:
851:
839:
827:
815:
803:
791:
562:
519:
506:
495:
484:
480:Cyclic group
358:
345:Free product
316:Group action
279:Group theory
274:Group theory
273:
220:group theory
212:Galois group
197:
187:
179:Point groups
136:
125:
78:
68:, it has an
49:
43:
34:Rubik's Cube
25:
21:Group theory
29703:Ring theory
29665:Topic lists
29625:Multivector
29611:Free object
29529:Dot product
29515:Determinant
29501:Linear and
28852:Point group
28847:Space group
28582:Lie, Sophus
28479:(1): 1â44,
27929:Ronan, Mark
27730:, Courier,
27414:Harris, Joe
26938:Artin, Emil
26843:Lang, Serge
26810:Lang, Serge
26740:Hall, G. G.
26575:Warner 1983
26527:Husain 1966
26515:Awodey 2010
26292:Simons 2003
26233:Bishop 1993
26231:. See also
26106:Seress 1997
25846:Mackey 1976
25834:Curtis 2003
25810:Jordan 1870
25750:Cayley 1889
25726:Galois 1908
25479:Frucht 1939
25358:instead of
25308:Weibel 1994
25270:Suzuki 1951
25025:quasi-group
25015:quasi-group
23447:Commutative
23432:Associative
23372:gravitation
23295:denotes an
23195:open subset
23141:local field
22990:small piece
22981:unit circle
22811:classifying
22660:other than
22316:have order
21819:represents
21715:in a group
21456:solvability
21160:dimensional
20889:consist of
20823:(geometric)
20585:geometrical
20551:Point group
20547:Space group
20050:satisfying
20027:, given by
19670:stands for
18825:such that
16865:associating
16402:are called
15260:, read as "
13519:containing
13431:of a group
13274:of a group
13246:provides a
12796:is a group
12471:are called
12424:isomorphism
12004:of a group
11618:to a group
11300:containing
11052:satisfying
10021:. For each
9594:of a group
9132:parentheses
9016:Walter Feit
8977:Ălie Cartan
8965:Issai Schur
8910:. In 1847,
8867:Felix Klein
8796:solvability
3899:reflections
3795:commutative
3658:instead of
3601:is denoted
3526:is denoted
3028:the inverse
3022:is unique (
1332:Lie algebra
1317:Associative
1221:Total order
1211:Semilattice
1185:Ring theory
765:Topological
604:alternating
66:associative
46:mathematics
29738:Categories
29681:Glossaries
29635:Polynomial
29615:Free group
29540:Linear map
29397:Inequality
29164:Loop group
29024:Lie groups
28796:direct sum
28367:See also:
27896:0956.11021
27539:Teller, E.
27385:Frucht, R.
27261:, London:
26681:References
26671:Dudek 2001
26623:Naber 2003
26587:Borel 1991
26563:Milne 1980
26551:Shatz 1972
26491:Ronan 2007
26475:Artin 2018
26415:Artin 1998
26391:Rudin 1990
26379:Serre 1977
26316:Welsh 1989
26205:Ellis 2019
26130:Rosen 2000
25921:Artin 2018
25858:Borel 2001
25714:Smith 1906
25642:Artin 2018
25429:, §15.3.2.
25150:MathSciNet
25107:for every
24986:Structure
24664:Operation
24429:-ary group
23814:quasigroup
23791:quasigroup
23633:quasigroup
23610:Quasigroup
23163:Lie groups
23145:adele ring
21913:copies of
21755:such that
20944:invertible
20807:CD players
20589:analytical
20569:hyperbolic
20541:See also:
20525:commutator
17451:such that
16889:continuity
16074:free group
15955:surjective
13776:cosets of
12473:isomorphic
12151:such that
11682:such that
10298:(that is,
10212:(that is,
9256:have both
9024:Aschbacher
8887:Lie groups
8883:Sophus Lie
8875:hyperbolic
3907:symmetries
3235:is a set,
2887:such that
2710:, one has
2554:, one has
2417:, denoted
2229:Definition
2056:such that
1712:, one has
872:Symplectic
812:Orthogonal
769:Lie groups
676:Free group
401:continuous
340:Direct sum
155:Lie groups
143:symmetries
29407:Operation
28706:MathWorld
28642:(1950) ,
28520:in 1843).
28493:179178038
28356:768477138
27335:, Wiley,
27318:: 15â36,
27263:CRC Press
26479:Lang 2002
26439:Lang 2002
26427:Lang 2002
26294:, §4.2.1.
26256:Dove 2003
26244:Weyl 1950
26217:Weyl 1952
26190:Lang 2002
26178:Lang 2005
26166:Lang 2005
26154:Lang 2005
26142:Lang 2005
26118:Lang 2005
26037:Lang 2002
26025:Lang 2005
26013:Lang 2002
26001:Lang 2005
25977:Lang 2005
25950:Lang 2005
25938:Lang 2002
25909:Lang 2005
25800:, §I.3.4.
25764:, §III.2.
25666:Lang 2005
25654:Lang 2002
25630:Cook 2009
25618:Lang 2005
25598:Hall 1967
25573:Citations
25496:Kuga 1993
25492:monodromy
25379:Lang 2002
25366:Lang 2005
25325:Lang 2002
25228:Lang 2005
25224:Lang 2002
24885:≠
24818:−
24787:−
24734:Identity
24435:Examples
24302:≃
24284:
24271:in which
24196:
24085:⋅
24070:∖
24037:⋅
24022:∖
23989:⋅
23974:∖
23918:Required
23896:Unneeded
23874:Required
23852:Unneeded
23830:Required
23807:Unneeded
23784:Required
23768:semigroup
23761:Unneeded
23745:Semigroup
23739:Required
23716:Unneeded
23694:Required
23671:Unneeded
23649:Required
23626:Unneeded
23604:Required
23581:Unneeded
23559:Required
23536:Unneeded
23514:Unneeded
23492:Unneeded
23470:Unneeded
23376:spacetime
23360:mechanics
23257:≠
23175:Lie group
23169:Lie group
23049:−
23021:⋅
22994:real line
22965:morphisms
22929:→
22900:→
22894:×
22585:of order
22503:×
22329:−
22282:×
21947:⋅
21879:⋅
21841:⏟
21834:⋯
21599:factorial
21442:, but do
21422:−
21382:±
21337:−
21322:±
21316:−
21179:→
21170:ρ
20662:describe
20410:≡
20373:≡
20336:≡
20189:×
19871:…
19838:−
19826:−
19811:−
19802:…
19751:−
19740:⋅
19734:⋅
19717:−
19709:⋅
19701:−
19693:⋅
19685:−
19653:−
19623:⋅
19568:…
19512:−
19496:−
19480:−
19469:…
19293:×
19219:⋅
19202:∖
19157:≡
19145:⋅
18915:−
18909:⋅
18842:≡
18836:⋅
18578:≡
18572:⋅
18461:remainder
18418:−
18297:≡
18100:−
17972:−
17918:−
17852:≡
17750:9 + 4 ⥠1
17621:×
17584:≠
17578:∣
17570:∈
17545:∖
17508:⋅
17464:⋅
17412:⋅
17306:fractions
17300:Rationals
17214:⋅
17156:⋅
17045:rationals
17029:chemistry
16972:≠
16885:proximity
16767:→
16764:⟩
16742:⋅
16707:∣
16695:⟨
16671:ϕ
16612:⟩
16590:⋅
16555:∣
16543:⟨
16521:⟩
16509:⟨
16489:ϕ
16486:
16448:⋅
16404:relations
16390:ϕ
16387:
16237:⟩
16225:⟨
16203:ϕ
16032:∼
16027:→
16019:ϕ
16016:
16002:→
15974:→
15965:ϕ
15733:⋅
15697:⋅
15673:⋅
15372:⋅
15202:−
15182:−
14926:→
14869:∈
14863:∣
14108:∈
14095:⋅
14087:−
13961:partition
13916:∈
13910:∣
13904:⋅
13861:∈
13855:∣
13849:⋅
13665:⋅
13457:generated
13406:subgroups
13325:∈
13319:⋅
13311:−
13144:→
13046:−
13004:⋅
12782:Subgroups
12711:↪
12700:∼
12695:→
12659:→
12645:ϕ
12635:injective
12561:φ
12494:→
12485:φ
12429:bijective
12343:ψ
12333:φ
12251:φ
12241:ψ
12213:ι
12206:ψ
12203:∘
12200:φ
12172:ι
12165:φ
12162:∘
12159:ψ
12136:→
12127:ψ
12104:→
12095:φ
12048:→
12033:ι
11936:−
11922:φ
11908:−
11897:φ
11846:φ
11824:φ
11728:φ
11725:∗
11713:φ
11701:⋅
11692:φ
11667:→
11658:φ
11635:∗
11603:⋅
11558:subgroups
11462:→
11386:∈
11206:⋅
11121:⋅
11109:⋅
11075:⋅
11063:⋅
11040:⋅
10978:semigroup
10925:⋅
10898:⋅
10887:−
10879:⋅
10846:⋅
10838:−
10827:⋅
10811:⋅
10740:−
10732:⋅
10724:−
10711:−
10674:−
10666:⋅
10657:⋅
10649:−
10636:−
10599:−
10591:⋅
10582:⋅
10574:−
10560:⋅
10552:−
10539:−
10502:−
10494:⋅
10485:⋅
10474:−
10466:⋅
10458:−
10445:−
10405:−
10397:⋅
10388:⋅
10370:−
10362:⋅
10321:⋅
10313:−
10261:−
10225:⋅
10102:⋅
10056:→
10002:−
9994:⋅
9960:⋅
9848:bijection
9831:⋅
9785:→
9717:⋅
9709:−
9694:, namely
9671:⋅
9453:⋅
9426:⋅
9417:⋅
9368:⋅
9359:⋅
9324:⋅
9201:⋅
9112:⋅
9103:⋅
9091:⋅
9082:⋅
9067:⋅
9061:⋅
8981:algebraic
8889:in 1884.
8833:θ
8690:∘
8633:∘
8231:∘
8196:∘
8178:∘
8136:∘
8106:∘
8088:∘
8036:∘
8018:∘
7990:∘
7972:∘
7928:∘
7919:∘
7907:∘
7898:∘
7566:∘
7489:∘
7405:∘
4909:∘
4823:∘
4640:∘
4605:and then
4420:diagonals
3895:rotations
3891:congruent
3865:∘
3824:∘
3806:functions
3714:⋅
3702:⋅
3671:⋅
3616:−
3536:−
3454:∖
3427:×
3294:⋅
3111:group law
3065:−
3024:see below
2978:For each
2932:⋅
2898:⋅
2803:For each
2787:see below
2759:⋅
2723:⋅
2600:⋅
2591:⋅
2579:⋅
2570:⋅
2430:⋅
2316:⋅
2180:−
1878:and
1688:and
1492:…
1453:−
1444:−
1435:−
1426:−
1420:…
1342:Bialgebra
1148:Near-ring
1105:Lie group
1073:Semigroup
936:Conformal
824:Euclidean
431:nilpotent
242:) and of
224:subgroups
200:(French:
181:describe
171:spacetime
87:with the
58:operation
36:form the
29754:Symmetry
29721:Category
29431:Variable
29421:Relation
29411:Addition
29387:Function
29373:Equation
29322:K-theory
28761:Subgroup
28664:(2007),
28627:(1906),
28604:(1976),
28584:(1973),
28527:(1870),
28465:(1882),
28443:(2003),
28403:(1889),
28378:(2001),
28334:(2010),
28314:Symmetry
28312:(1952),
28290:(1989),
28266:(1972),
28257:36131259
28229:(1994),
28153:(2015),
28136:(6): 1,
28034:(1977),
27984:(1990),
27931:(2007),
27862:(1999),
27700:(1998),
27616:(1993),
27535:Jahn, H.
27491:(2002),
27449:(1980),
27416:(1991),
27387:(1939),
27151:(1989),
27117:(1991),
26966:(2004),
26940:(1998),
26911:(1996),
26891:(1973),
26871:(1953),
26845:(2005),
26812:(2002),
26792:(1975),
26762:(1996),
26742:(1967),
26696:(2018),
26635:Zee 2010
26453:, §12.1.
26340:Lay 2003
26268:Zee 2010
25774:Lie 1973
25448:, where
25281:See the
25043:See also
24769:Inverse
24474:Integers
24321:groupoid
23915:Required
23912:Required
23909:Required
23906:Required
23893:Required
23890:Required
23887:Required
23884:Required
23871:Unneeded
23868:Required
23865:Required
23862:Required
23849:Unneeded
23846:Required
23843:Required
23840:Required
23827:Required
23824:Unneeded
23821:Required
23818:Required
23804:Required
23801:Unneeded
23798:Required
23795:Required
23781:Unneeded
23778:Unneeded
23775:Required
23772:Required
23758:Unneeded
23755:Unneeded
23752:Required
23749:Required
23736:Required
23733:Required
23730:Unneeded
23727:Required
23713:Required
23710:Required
23707:Unneeded
23704:Required
23691:Unneeded
23688:Required
23685:Unneeded
23682:Required
23668:Unneeded
23665:Required
23662:Unneeded
23659:Required
23646:Required
23643:Unneeded
23640:Unneeded
23637:Required
23623:Required
23620:Unneeded
23617:Unneeded
23614:Required
23601:Unneeded
23598:Unneeded
23595:Unneeded
23592:Required
23578:Unneeded
23575:Unneeded
23572:Unneeded
23569:Required
23556:Required
23553:Required
23550:Required
23547:Unneeded
23543:Groupoid
23533:Required
23530:Required
23527:Required
23524:Unneeded
23520:Groupoid
23511:Unneeded
23508:Required
23505:Required
23502:Unneeded
23489:Unneeded
23486:Unneeded
23483:Required
23480:Unneeded
23467:Unneeded
23464:Unneeded
23461:Unneeded
23458:Unneeded
23437:Identity
23348:Rotation
23335:matrix.
21792:, where
21448:degree 5
21394:sign by
20891:matrices
20775:features
20747:displays
20666:, while
20089:vertices
18256:"modulo
17702:division
17067:Integers
17061:algebras
16883:such as
16278:sending
14269:), then
14233:satisfy
14064:happens
12794:subgroup
12788:Subgroup
12686:′
12655:′
12624:category
12377:for all
12285:for all
11953:for all
11499:function
11302:elements
11233:such as
9550:Division
9048:repeated
8939:(1870).
8869:'s 1872
7365:subgroup
3849:groups,
3847:symmetry
3845:groups,
2955:, where
2458:For all
2332:elements
1509:addition
1400:integers
1178:Lie ring
1143:Semiring
931:Poincaré
776:Solenoid
648:Integers
638:Lattices
613:sporadic
608:Lie type
436:solvable
426:dihedral
411:additive
396:infinite
306:Subgroup
139:geometry
93:infinite
91:form an
85:integers
56:with an
29633: (
29623: (
29489: (
29479: (
29469: (
29459: (
29449: (
29259:History
29254:Outline
29243:Algebra
29191:History
28701:"Group"
28618:0396826
28594:0392459
28574:0863090
28566:2690312
28332:Zee, A.
28249:1269324
28200:1990375
28104:0347778
28058:0450380
28024:1452069
27974:1739433
27888:1697859
27852:2044239
27825:1304906
27667:2014408
27640:1199112
27606:1670862
27584:Bibcode
27553:Bibcode
27440:1153249
27377:3971587
27324:1876783
27250:1075994
27223:1865535
27141:1102012
27088:1826989
27072:: 1â4,
27009:Arkivoc
26881:0054593
26836:1878556
26814:Algebra
26802:0356988
26782:1375019
26752:0219593
26698:Algebra
26517:, §4.1.
26465:, p. 3.
25606:applied
25383:torsion
25310:, §8.2.
25302:of its
25187:unless
25184:
25168:
25085:closure
24900:
24874:
24869:
24841:
24833:
24810:
24802:
24779:
24699:Closed
24658:
24599:
24590:
24568:
24560:
24538:
24530:
24508:
24499:
24477:
24469:
24447:
24396:
24376:
24372:
24352:
24348:
24325:
24317:
24273:
24269:
24249:
24245:
24225:
24183:
24163:
24152:
24132:
24124:
24104:
24100:
24054:
24004:
23958:
23427:Closure
22983:in the
22836:. The
22776:
22748:
22527:
22483:
22356:product
22344:
22318:
22148:
22128:
22124:divides
22034:
22014:
22010:
21990:
21959:
21939:
21937:. (If "
21935:
21915:
21891:
21871:
21790:
21757:
21653:
21633:
21543:
21514:
21465:Modern
21116:
21087:
21042:
21008:
20991:
20971:
20967:
20947:
20880:
20860:
20856:vectors
20755:Ammonia
20651:of the
20641:
20621:
20617:
20597:
20567:of the
20510:
20476:
20452:
20432:
20425:
20392:
20388:
20355:
20351:
20312:
20308:
20275:
20251:
20225:
20139:
20113:
20085:
20052:
20021:
20001:
19977:
19957:
19768:
19672:
19638:
19612:
19427:
19400:
19307:
19273:
19182:
19134:
19130:
19110:
19086:
19066:
19016:
18996:
18930:
18898:
18803:
18783:
18759:
18739:
18735:
18715:
18691:
18671:
18647:
18627:
18603:
18561:
18557:
18519:
18515:
18489:
18485:
18465:
18457:
18437:
18433:
18407:
18383:
18363:
18361:modulo
18355:
18335:
18278:
18258:
18208:
18188:
18137:
18117:
18069:, with
18067:
18020:
17987:
17961:
17933:
17907:
17883:
17841:
17758:modulus
17695:
17667:
17635:
17606:
17530:nonzero
17485:
17453:
17429:
17391:
17387:
17365:
17272:
17237:
17133:
17095:
17053:modules
17039:Numbers
17025:physics
16987:
16961:
16791:
16687:
16683:
16663:
16624:
16535:
16473:
16408:
16374:
16347:
16215:
16195:
16052:
15991:
15769:
15662:
15658:
15620:
15306:
15286:
15284:modulo
15282:
15262:
15258:
15230:
15226:
15160:
15133:
15105:
15072:
15040:
14830:
14810:
14772:
14686:
14682:
14653:
14573:
14544:
14396:
14376:
14329:
14300:
14267:
14235:
14171:
14151:
14147:
14127:
14123:
14069:
13985:
13965:
13959:form a
13934:
13881:
13822:
13802:
13798:
13778:
13766:
13746:
13687:
13630:
13626:
13601:
13541:
13521:
13453:
13433:
13402:
13382:
13296:
13276:
13272:
13252:
13240:
13164:
13131:
13111:
13087:
13067:
13063:
13028:
12984:
12964:
12906:
12886:
12862:
12842:
12838:
12818:
12772:
12752:
12748:
12728:
12617:
12597:
12573:
12553:
12509:
12477:
12419:
12399:
12231:
12192:
11995:
11975:
11887:
11838:
11811:
11791:
11543:
11523:
11443:
11423:
11373:
11353:
11349:
11329:
11325:
11305:
11166:
11146:
11142:
11098:
10342:
10300:
10246:
10214:
10167:
10147:
10043:
10023:
10019:
9983:
9947:
9927:
9900:
9880:
9732:
9696:
9692:
9660:
9616:
9596:
9222:
9184:
8802:of its
8764:History
8729:
8672:
8611:
8582:
8494:
8465:
8461:
8432:
8428:
8399:
8395:
8366:
8362:
8337:
8307:
8282:
7882:
7862:
7858:
7838:
7814:
7794:
7750:
7721:
7717:
7697:
7693:
7673:
7669:
7649:
7639:
7610:
7546:
7526:
7460:
7431:
7427:
7385:
7363:form a
7361:
7334:
7330:
7303:
7299:
7272:
7268:
7243:
4801:
4772:
4768:
4739:
4735:
4708:
4697:
4677:
4583:
4554:
4486:
4457:
4453:
4424:
4416:
4387:
4383:
4354:
4347:
4320:
4289:
4262:
3948:
3919:
3839:
3813:
3791:
3771:
3686:
3660:
3633:
3603:
3579:
3559:
3551:
3528:
3504:
3484:
3211:
3188:
3155:
3133:
3109:or the
3082:
3052:
3000:
2980:
2953:
2921:
2845:
2825:
2780:
2748:
2744:
2712:
2708:
2688:
2618:
2556:
2552:
2532:
2528:
2508:
2504:
2484:
2480:
2460:
2445:
2419:
2415:
2395:
2328:
2308:
2304:
2284:
2223:
2203:
2195:
2172:
2122:
2090:
2034:
2014:
1997:
1965:
1900:
1880:
1776:
1714:
1710:
1690:
1666:
1646:
1638:
1616:
1587:closure
1553:
1533:
1309:Algebra
1301:Algebra
1206:Lattice
1197:Lattice
926:Lorentz
848:Unitary
747:Lattice
687:PSL(2,
421:abelian
332:(Semi-)
121:
101:
99:called
29647:, ...)
29617:, ...)
29544:Matrix
29491:Vector
29481:theory
29471:theory
29467:Module
29461:theory
29451:theory
29290:Scheme
28966:22..24
28918:22..24
28914:11..12
28745:Groups
28676:
28650:
28616:
28592:
28572:
28564:
28491:
28451:
28390:
28354:
28344:
28320:
28298:
28276:
28255:
28247:
28237:
28215:
28198:
28161:
28118:
28102:
28092:
28073:
28056:
28046:
28022:
27992:
27972:
27962:
27943:
27917:
27894:
27886:
27876:
27850:
27840:
27823:
27813:
27792:
27771:
27755:14 May
27734:
27710:
27686:
27665:
27655:
27638:
27628:
27604:
27594:
27524:
27505:
27478:
27459:
27438:
27428:
27375:
27365:
27339:
27322:
27288:
27269:
27248:
27238:
27221:
27182:
27163:
27139:
27129:
27103:
27086:
27049:
26997:
26952:
26919:
26901:795613
26899:
26879:
26857:
26834:
26824:
26800:
26780:
26770:
26750:
26729:
26708:
26092:; see
25473:; see
25454:is in
25425:. See
25387:module
25364:. See
25323:, see
25306:, see
25035:monoid
25020:monoid
25005:monoid
24995:monoid
24990:monoid
24566:
24536:
24506:
24404:stacks
23931:. The
23929:monoid
23836:Monoid
23275:where
23188:smooth
22818:simple
22809:. But
22781:simple
22150:. The
21497:finite
21138:on an
20897:. The
20844:, and
20765:Cubane
20695:phonon
20649:tiling
20565:tiling
20553:, and
20521:center
20390:, and
19610:means
19583:where
19439:powers
14832:, and
13824:, are
13693:Cosets
12626:, the
11564:, and
11550:subset
11477:means
11351:, and
7464:
7381:
7377:
7369:
7332:, and
3911:square
3901:, and
3853:, and
2247:,
1555:, the
1337:Graded
1268:Module
1259:Module
1158:Domain
1077:Monoid
781:Circle
712:SL(2,
601:cyclic
565:-group
416:cyclic
391:finite
386:simple
370:kernel
203:groupe
165:. The
29585:Trace
29510:Basis
29457:Group
29447:Field
29268:Areas
29179:Sp(â)
29176:SU(â)
29070:Sp(n)
29064:SU(n)
29052:SO(n)
29040:SL(n)
29034:GL(n)
28787:Semi-
28562:JSTOR
28489:S2CID
28196:JSTOR
28008:(PDF)
27907:Modes
27308:(PDF)
27209:arXiv
26971:(PDF)
25546:Up to
25471:graph
25385:of a
25351:field
25321:units
25253:ÎŒÎżÏÏÎź
25249:Greek
25071:Notes
23880:Group
23588:magma
23565:Magma
23352:space
23006:Some
21893:" to
21505:order
21460:roots
21436:cubic
20700:Such
20571:plane
19132:, as
18144:clock
17049:rings
14398:, if
14013:empty
13989:union
13774:right
13699:Coset
12421:. An
11497:is a
11375:, or
10992:right
9846:is a
9028:proof
8922:into
8804:roots
7373:coset
3887:plane
3311:field
3309:is a
1610:is a
1303:-like
1261:-like
1199:-like
1168:Field
1126:-like
1100:Magma
1068:Group
1062:-like
1060:Group
965:Sp(â)
962:SU(â)
375:image
208:roots
198:group
79:Many
52:is a
50:group
29580:Rank
29560:Norm
29477:Ring
29173:O(â)
29058:U(n)
29046:O(n)
28927:1..3
28674:ISBN
28648:ISBN
28449:ISBN
28388:ISBN
28352:OCLC
28342:ISBN
28318:ISBN
28296:ISBN
28274:ISBN
28253:OCLC
28235:ISBN
28213:ISBN
28159:ISBN
28116:ISBN
28090:ISBN
28071:ISBN
28044:ISBN
27990:ISBN
27960:ISBN
27941:ISBN
27915:ISBN
27874:ISBN
27838:ISBN
27811:ISBN
27790:ISBN
27769:ISBN
27757:2021
27732:ISBN
27708:ISBN
27684:ISBN
27653:ISBN
27626:ISBN
27592:ISBN
27522:ISBN
27503:ISBN
27476:ISBN
27457:ISBN
27426:ISBN
27363:ISBN
27337:ISBN
27286:ISBN
27267:ISBN
27236:ISBN
27180:ISBN
27161:ISBN
27127:ISBN
27101:ISBN
27047:ISBN
27013:2015
26995:ISBN
26950:ISBN
26917:ISBN
26897:OCLC
26855:ISBN
26822:ISBN
26768:ISBN
26727:ISBN
26706:ISBN
26088:and
25604:and
25602:pure
25561:See
25389:and
25234:and
25148:The
25137:2002
25113:and
24981:Yes
24978:Yes
24972:Yes
24966:Yes
24963:Yes
24960:Yes
24957:Yes
24954:Yes
24946:Yes
24943:Yes
24937:Yes
24931:Yes
24928:Yes
24925:Yes
24922:Yes
24919:Yes
24905:N/A
24837:N/A
24806:N/A
24775:N/A
24772:N/A
24758:N/A
24752:N/A
24729:Yes
24726:Yes
24720:Yes
24717:Yes
24714:Yes
24711:Yes
24708:Yes
24705:Yes
24702:Yes
24440:Set
24414:-ary
23723:loop
23700:Loop
23356:time
23354:and
23315:-by-
23217:-by-
23086:and
23036:and
22979:The
22746:and
22686:and
21438:and
20969:-by-
20854:Two
20783:The
20757:, NH
20523:and
17799:and
17059:and
17031:and
16906:are
16887:and
16325:and
15949:The
15618:and
15598:are
15015:and
13879:and
13772:and
13770:left
13570:and
13360:and
13026:and
12935:and
12451:and
12190:and
12000:The
11769:and
11096:and
10988:left
10181:and
9925:and
9574:and
9276:and
9162:and
9014:and
9002:The
8995:and
8963:and
8947:and
8877:and
8790:and
8670:but
8550:and
7836:and
7772:and
7695:and
7524:and
4455:and
4385:and
4318:and
4253:the
3909:. A
3889:are
3749:and
3338:The
3183:and
2919:and
2746:and
2353:and
2088:and
2001:Zero
1963:and
1531:and
1133:Ring
1124:Ring
959:O(â)
948:Loop
767:and
230:and
145:and
130:and
62:pair
48:, a
28554:doi
28481:doi
28186:doi
28138:doi
27892:Zbl
27561:doi
27549:161
27355:doi
27074:doi
27027:hdl
27017:doi
25532:on
25203:or
25119:in
24975:No
24969:No
24940:No
24934:No
24723:No
24431:.
24281:Hom
24193:Hom
24130:of
23378:in
23245:det
23143:or
22481:or
21444:not
21414:or
20997:or
20593:act
20587:or
20023:th
19164:mod
18967:gcd
18853:mod
18585:mod
18405:to
18308:mod
18018:or
17959:to
17905:to
17863:mod
17665:is
17487:),
17175:not
17173:do
16793:.
16632:of
16483:ker
16384:ker
16345:to
16298:to
16013:ker
15312:or
15038:is
14969:to
14774:.)
14298:In
14213:in
13963:of
13459:by
13380:in
12750:of
12633:An
12397:in
12305:in
11973:in
11789:in
11279:set
10145:by
10141:or
10121:by
10091:to
9981:is
9878:by
9874:or
9854:by
9820:to
9754:in
9638:in
9544:the
9226:the
8959:'s
8933:'s
8822:'s
7719:of
4869:of
4629:as
3769:in
3030:of
2867:in
2823:in
2686:in
2646:in
2530:in
2373:of
2282:on
2257:set
1911:If
1798:to
1614:on
1557:sum
1138:Rng
874:Sp(
862:SU(
838:SO(
802:SL(
790:GL(
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161:of
137:In
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28925:Co
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26853:,
26832:MR
26830:,
26816:,
26798:MR
26778:MR
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26359:^
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15857:D
15835:}
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15822:,
15817:2
15813:r
15809:,
15804:1
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15789:i
15785:{
15782:=
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15757:R
15754:=
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15748:)
15742:v
15737:f
15727:v
15722:f
15718:(
15715:=
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15706:v
15701:f
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15688:v
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15679:=
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15670:U
15646:R
15640:v
15635:f
15631:=
15628:U
15606:R
15586:R
15582:/
15576:4
15571:D
15544:R
15523:U
15502:U
15479:U
15458:R
15437:R
15414:U
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15345:/
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15294:N
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15242:/
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15214:N
15210:)
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15190:=
15185:1
15178:)
15174:N
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15168:(
15146:N
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15117:/
15113:G
15091:N
15088:=
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15060:N
15057:)
15054:h
15051:g
15048:(
15026:N
15023:h
15003:N
15000:g
14980:N
14977:g
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14937:N
14933:/
14929:G
14923:G
14903:N
14899:/
14895:G
14875:}
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14860:N
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14854:{
14851:=
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14624:=
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14530:}
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14470:{
14467:=
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14249:=
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14090:1
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14039:=
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13947:H
13922:}
13919:H
13913:h
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13901:h
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13895:=
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13843:{
13840:=
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13834:g
13810:g
13786:H
13754:g
13732:H
13712:H
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13669:r
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13650:=
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13613:d
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13178:{
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12736:H
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12708:H
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12652:G
12648::
12605:H
12583:G
12539:G
12519:H
12497:H
12491:G
12488::
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12439:G
12407:H
12385:h
12365:h
12362:=
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12352:)
12349:h
12346:(
12338:(
12313:G
12293:g
12273:g
12270:=
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12260:)
12257:g
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12217:H
12209:=
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12168:=
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12133:H
12130::
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12101:G
12098::
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12051:G
12045:G
12042::
12037:G
12012:G
11983:G
11961:a
11939:1
11932:)
11928:a
11925:(
11919:=
11916:)
11911:1
11904:a
11900:(
11873:H
11869:1
11865:=
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11857:G
11853:1
11849:(
11813:.
11799:G
11777:b
11757:a
11737:)
11734:b
11731:(
11722:)
11719:a
11716:(
11710:=
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11704:b
11698:a
11695:(
11670:H
11664:G
11661::
11638:)
11632:,
11629:H
11626:(
11606:)
11600:,
11597:G
11594:(
11545:.
11531:Y
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11485:f
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11313:x
11288:X
11265:}
11262:z
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11256:y
11253:,
11250:x
11247:{
11244:=
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11209:)
11203:,
11200:G
11197:(
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11154:e
11130:f
11127:=
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11118:f
11115:=
11112:f
11106:e
11084:e
11081:=
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11072:f
11069:=
11066:e
11060:e
11020:}
11017:f
11014:,
11011:e
11008:{
11005:=
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10949:f
10946:=
10928:f
10922:e
10919:=
10901:f
10895:)
10890:1
10883:f
10876:f
10873:(
10870:=
10852:)
10849:f
10841:1
10834:f
10830:(
10824:f
10821:=
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10808:f
10766:e
10763:=
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10727:1
10720:)
10714:1
10707:f
10703:(
10700:=
10682:)
10677:1
10670:f
10663:e
10660:(
10652:1
10645:)
10639:1
10632:f
10628:(
10625:=
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10602:1
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10588:)
10585:f
10577:1
10570:f
10566:(
10563:(
10555:1
10548:)
10542:1
10535:f
10531:(
10528:=
10510:)
10505:1
10498:f
10491:f
10488:(
10482:)
10477:1
10470:f
10461:1
10454:)
10448:1
10441:f
10437:(
10434:(
10431:=
10413:)
10408:1
10401:f
10394:f
10391:(
10385:e
10382:=
10373:1
10366:f
10359:f
10330:e
10327:=
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10316:1
10309:f
10286:f
10264:1
10257:f
10234:f
10231:=
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10222:e
10200:e
10155:a
10129:a
10105:a
10099:x
10079:x
10059:G
10053:G
10031:a
10005:1
9998:a
9991:b
9969:b
9966:=
9963:a
9957:x
9935:b
9913:a
9888:a
9862:a
9834:x
9828:a
9808:x
9788:G
9782:G
9762:G
9742:a
9720:b
9712:1
9705:a
9680:b
9677:=
9674:x
9668:a
9646:G
9626:x
9604:G
9582:b
9562:a
9522:)
9518:c
9515:=
9512:b
9504:e
9500:(
9493:c
9490:=
9479:)
9475:a
9467:b
9463:(
9456:c
9450:e
9447:=
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9423:)
9420:a
9414:b
9411:(
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9381:(
9374:)
9371:c
9365:a
9362:(
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9334:(
9327:e
9321:b
9318:=
9311:b
9284:c
9264:b
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9210:f
9207:=
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9150:e
9118:)
9115:c
9109:b
9106:(
9100:a
9097:=
9094:c
9088:)
9085:b
9079:a
9076:(
9073:=
9070:c
9064:b
9058:a
8845:1
8842:=
8837:n
8746:4
8741:D
8714:d
8709:f
8705:=
8699:h
8694:f
8685:1
8681:r
8655:c
8650:f
8646:=
8641:1
8637:r
8627:h
8622:f
8597:4
8592:D
8563:1
8559:r
8536:3
8532:r
8509:2
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8413:v
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8380:h
8375:f
8349:d
8346:i
8317:a
8294:d
8291:i
8259:.
8254:1
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8246:=
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8225:d
8220:f
8216:=
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8204:2
8200:r
8190:v
8185:f
8181:(
8172:d
8167:f
8157:1
8153:r
8149:=
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8140:r
8131:3
8127:r
8123:=
8114:2
8110:r
8103:)
8097:v
8092:f
8082:d
8077:f
8073:(
8049:)
8044:2
8040:r
8030:v
8025:f
8021:(
8012:d
8007:f
8003:=
7998:2
7994:r
7987:)
7981:v
7976:f
7966:d
7961:f
7957:(
7937:,
7934:)
7931:c
7925:b
7922:(
7916:a
7913:=
7910:c
7904:)
7901:b
7895:a
7892:(
7870:a
7846:c
7824:b
7802:c
7780:b
7760:a
7736:4
7731:D
7705:c
7681:b
7657:a
7624:c
7619:f
7596:,
7590:c
7585:f
7581:=
7575:h
7570:f
7561:3
7557:r
7534:b
7512:a
7492:b
7486:a
7445:d
7440:f
7413:3
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7224:d
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7112:h
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6613:r
6588:d
6583:f
6558:v
6553:f
6528:c
6523:f
6498:h
6493:f
6468:h
6463:f
6437:3
6433:r
6409:1
6405:r
6381:2
6377:r
6354:d
6351:i
6326:c
6321:f
6296:h
6291:f
6266:d
6261:f
6236:v
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6169:f
6144:h
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6114:c
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6084:d
6079:f
6055:2
6051:r
6027:1
6023:r
6000:d
5997:i
5973:3
5969:r
5945:3
5941:r
5914:d
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5849:f
5824:h
5819:f
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5768:d
5765:i
5741:3
5737:r
5713:2
5709:r
5685:2
5681:r
5654:h
5649:f
5624:v
5619:f
5594:d
5589:f
5564:c
5559:f
5536:d
5533:i
5509:3
5505:r
5481:2
5477:r
5453:1
5449:r
5425:1
5421:r
5394:c
5389:f
5364:d
5359:f
5334:h
5329:f
5304:v
5299:f
5275:3
5271:r
5247:2
5243:r
5219:1
5215:r
5192:d
5189:i
5166:d
5163:i
5136:c
5131:f
5106:d
5101:f
5076:h
5071:f
5046:v
5041:f
5017:3
5013:r
4989:2
4985:r
4961:1
4957:r
4934:d
4931:i
4884:4
4879:D
4851:.
4845:d
4840:f
4836:=
4831:3
4827:r
4817:h
4812:f
4786:d
4781:f
4753:h
4748:f
4721:3
4717:r
4685:a
4663:b
4643:a
4637:b
4613:b
4593:a
4569:4
4564:D
4533:h
4528:f
4505:1
4501:r
4471:c
4466:f
4438:d
4433:f
4422:(
4401:h
4396:f
4368:v
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4333:3
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4300:r
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4230:c
4225:f
4191:d
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4076:3
4072:r
4041:2
4037:r
4006:1
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3972:d
3969:i
3934:4
3929:D
3827:g
3821:f
3779:G
3757:b
3737:a
3717:a
3711:b
3708:=
3705:b
3699:a
3674:b
3668:a
3646:b
3643:a
3619:1
3612:x
3589:x
3567:1
3539:x
3514:x
3492:0
3463:}
3460:0
3457:{
3450:R
3422:R
3399:R
3373:R
3351:R
3322:R
3297:)
3291:,
3288:+
3285:,
3281:R
3277:(
3257:)
3254:+
3251:,
3247:R
3243:(
3222:R
3199:b
3196:a
3171:b
3168:+
3165:a
3142:R
3084:.
3068:1
3061:a
3038:a
3010:b
2988:a
2963:e
2941:e
2938:=
2935:a
2929:b
2907:e
2904:=
2901:b
2895:a
2875:G
2855:b
2833:G
2811:a
2782:.
2768:a
2765:=
2762:e
2756:a
2732:a
2729:=
2726:a
2720:e
2696:G
2674:a
2654:G
2634:e
2620:.
2606:)
2603:c
2597:b
2594:(
2588:a
2585:=
2582:c
2576:)
2573:b
2567:a
2564:(
2540:G
2516:c
2492:b
2468:a
2433:b
2427:a
2403:G
2381:G
2361:b
2341:a
2292:G
2266:G
2211:+
2197:.
2183:a
2158:a
2132:b
2110:0
2107:=
2104:a
2101:+
2098:b
2076:0
2073:=
2070:b
2067:+
2064:a
2044:b
2022:a
1985:a
1982:=
1979:0
1976:+
1973:a
1951:a
1948:=
1945:a
1942:+
1939:0
1919:a
1908:.
1888:c
1866:b
1846:a
1826:c
1806:b
1786:a
1764:)
1761:c
1758:+
1755:b
1752:(
1749:+
1746:a
1743:=
1740:c
1737:+
1734:)
1731:b
1728:+
1725:a
1722:(
1698:c
1676:b
1654:a
1625:Z
1598:+
1572:b
1569:+
1566:a
1541:b
1519:a
1495:}
1489:,
1486:4
1483:,
1480:3
1477:,
1474:2
1471:,
1468:1
1465:,
1462:0
1459:,
1456:1
1450:,
1447:2
1441:,
1438:3
1432:,
1429:4
1423:,
1417:{
1414:=
1410:Z
1377:e
1370:t
1363:v
1036:e
1029:t
1022:v
918:8
916:E
910:7
908:E
902:6
900:E
894:4
892:F
886:2
884:G
878:)
876:n
866:)
864:n
854:)
852:n
842:)
840:n
830:)
828:n
818:)
816:n
806:)
804:n
794:)
792:n
734:)
721:Z
709:)
696:Z
672:)
659:Z
650:(
563:p
528:Q
520:n
517:D
507:n
504:A
496:n
493:S
485:n
482:Z
109:1
40:.
23:.
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