2630:
2122:
2625:{\displaystyle {\begin{matrix}\mathrm {r} _{0}=\left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right),&\mathrm {r} _{1}=\left({\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}\right),&\mathrm {r} _{2}=\left({\begin{smallmatrix}-1&0\\0&-1\end{smallmatrix}}\right),&\mathrm {r} _{3}=\left({\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}\right),\\\mathrm {s} _{0}=\left({\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}}\right),&\mathrm {s} _{1}=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right),&\mathrm {s} _{2}=\left({\begin{smallmatrix}-1&0\\0&1\end{smallmatrix}}\right),&\mathrm {s} _{3}=\left({\begin{smallmatrix}0&-1\\-1&0\end{smallmatrix}}\right).\end{matrix}}}
5248:
5205:
36:
4027:
2969:
5264:
1391:
3598:
5229:
4940:
1411:
1027:
3927:
3920:
3913:
3904:
5937:, the group has mirrors at 18° intervals. The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect to the inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20 more outer automorphisms; e.g., multiplying rotations by 3.
2647:
3897:
3890:
3883:
3876:
3941:
3934:
4034:
4652:
114:
2047:
2964:{\displaystyle {\begin{aligned}\mathrm {r} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&-\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&\cos {\frac {2\pi k}{n}}\end{pmatrix}}\ \ {\text{and}}\\\mathrm {s} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&-\cos {\frac {2\pi k}{n}}\end{pmatrix}}.\end{aligned}}}
3270:
4479:
875:
4935:{\displaystyle {\begin{aligned}\mathrm {r} _{j}\,\mathrm {r} _{k}&=\mathrm {r} _{(j+k){\text{ mod }}n}\\\mathrm {r} _{j}\,\mathrm {s} _{k}&=\mathrm {s} _{(j+k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {r} _{k}&=\mathrm {s} _{(j-k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {s} _{k}&=\mathrm {r} _{(j-k){\text{ mod }}n}\end{aligned}}}
1836:
3040:
5574:
only half of the mirrors can be reached from one by these rotations. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon there are two sets of axes, each corresponding to a conjugacy class: those that pass through two vertices and those
4286:
2069:
1333:
3810:
2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the
2042:{\displaystyle \mathrm {r} _{i}\,\mathrm {r} _{j}=\mathrm {r} _{i+j},\quad \mathrm {r} _{i}\,\mathrm {s} _{j}=\mathrm {s} _{i+j},\quad \mathrm {s} _{i}\,\mathrm {r} _{j}=\mathrm {s} _{i-j},\quad \mathrm {s} _{i}\,\mathrm {s} _{j}=\mathrm {r} _{i-j}.}
5920:, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are all automorphisms. As abstract group there are in addition to these, 36
3571:
3265:{\displaystyle {\begin{aligned}\mathrm {D} _{n}&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs^{-1}=r^{-1}\right\rangle \\&=\left\langle r,s\mid r^{n}=s^{2}=(sr)^{2}=1\right\rangle .\end{aligned}}}
4474:{\displaystyle \mathrm {r} _{1}={\begin{bmatrix}\cos {2\pi \over n}&-\sin {2\pi \over n}\\\sin {2\pi \over n}&\cos {2\pi \over n}\end{bmatrix}}\qquad \mathrm {s} _{0}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}
1382:
of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a
4588:
1369:. Here, the first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left.
2652:
5955:= 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order).
4531:
4657:
3045:
1018:
The word "dihedral" comes from "di-" and "-hedron". The latter comes from the Greek word hédra, which means "face of a geometrical solid". Overall it thus refers to the two faces of a polygon.
4221:
5882:
The rotations are a normal subgroup; conjugation by a reflection changes the sign (direction) of the rotation, but otherwise leaves them unchanged. Thus automorphisms that multiply angles by
1406:
remain fixed in space (on the page) and do not themselves move when a symmetry operation (rotation or reflection) is done on the triangle (this matters when doing compositions of symmetries).
5105:
elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection.
5042:
in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.
5204:
4638:
5027:
1329:
elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
4271:
4972:
3448:
3370:
1363:
1188:
6158:
5740:
5715:
5691:
5666:
5447:
559:
534:
497:
3341:
3306:
1151:
5871:
even, they fall into two classes (those through two vertices and those through two faces), related by an outer automorphism, which can be represented by rotation by
3456:
5263:
5247:
3419:
1284:
1256:
1327:
1079:
3390:
1304:
1228:
1208:
1122:
1099:
1056:
6236:
6396:
861:
4964:
can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the
5970:) = 2 are 3, 4, and 6, and consequently, there are only three dihedral groups that are isomorphic to their own automorphism groups, namely
5621:
interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism).
4536:
6500:
5023:, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees.
5945:
1795:
is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not
5127:, i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation
5161:≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore, it is also called a
6780:
6636:
5351:
In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.
419:
6163:
6133:
4946:
4487:
369:
4179:
4141:
where we have a plane with respectively a point offset from the "center" of the "1-gon" and a "2-gon" or line segment).
854:
364:
5610:
even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group.
7127:
6357:
6331:
6302:
6276:
6220:
79:
57:
17:
6090:
with algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the
50:
5341:
5228:
6504:
4061:
780:
7132:
7122:
847:
4593:
5941:
2085:
464:
278:
7096:
5795:
It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by
4236:
6629:
6113:
196:
3424:
3346:
1339:
1164:
6389:
6244:
6153:
6148:
5017:
4153:
4015:
4000:
2110:
1423:
5834:
even, the rotation by 180° (reflection through the origin) is the non-trivial element of the center.
5830:
odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for
6691:
5368:
4090:
4057:
1792:
1379:
1158:
662:
396:
273:
161:
44:
5723:
5698:
5674:
5649:
5430:
542:
517:
480:
7091:
6083:
5643:
3311:
3032:
6976:
3566:{\displaystyle \mathrm {D} _{n}=\left\langle s,t\mid s^{2}=1,t^{2}=1,(st)^{n}=1\right\rangle .}
812:
602:
61:
6347:
6962:
6914:
6622:
6321:
6292:
6266:
5523:
5345:
3278:
2099:
2073:
1130:
686:
6938:
6932:
6696:
6452:
6212:
6138:
5508:
4161:
3395:
2103:
2095:
2080:
If we center the regular polygon at the origin, then elements of the dihedral group act as
1430:
1210:
is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If
626:
614:
232:
166:
8:
6671:
6645:
6120:
5590:
odd, each reflection, together with the identity, form a subgroup of order 2, which is a
4274:
3799:
1261:
1233:
1124:
1101:
1031:
923:
919:
907:
201:
96:
6456:
1309:
1061:
6871:
6687:
6681:
6475:
6440:
6374:
5921:
5913:
5630:
5618:
3375:
2577:
2516:
2458:
2397:
2334:
2270:
2209:
2151:
2053:
1289:
1213:
1193:
1107:
1084:
1041:
186:
158:
4042:
6794:
6595:
6578:
6575:
6559:
6556:
6540:
6537:
6518:
6480:
6353:
6327:
6298:
6272:
6216:
6183:
5314:
4988:
591:
434:
328:
7033:
757:
7101:
7038:
7013:
7005:
6997:
6989:
6981:
6968:
6950:
6944:
6521:
6470:
6460:
6186:
6098:
5527:
5239:
5123:
4992:
3846:
3812:
3647:
988:
946:
742:
734:
726:
718:
710:
698:
638:
578:
568:
410:
352:
227:
5334:
is even the center has two elements, namely the identity and the element r (with D
7054:
6888:
6757:
6666:
5551:
4230:
4111:
4060:
which keep the origin fixed. These groups form one of the two series of discrete
4026:
4007:
3712:
2981:
915:
826:
819:
805:
762:
650:
573:
403:
317:
257:
137:
5534:) has as normal subgroup order-2 subgroups generated by a reflection (flip) in D
1001:
refers to this same dihedral group. This article uses the geometric convention,
7059:
6854:
6847:
6840:
6833:
6805:
6772:
6742:
6737:
6727:
6676:
6510:
6143:
6087:
5591:
5570:
there are rotations in the group between every pair of mirrors, while for even
5150:
4956:
is generated by the rotation r of 180 degrees, and the reflection s across the
4107:
2052:
In all cases, addition and subtraction of subscripts are to be performed using
1390:
879:
833:
769:
459:
439:
376:
262:
237:
222:
176:
153:
7116:
7069:
7028:
6956:
6877:
6862:
6810:
6762:
6717:
5579:
5278:
5274:
5215:
5013:
4226:
In geometric terms: in the mirror a rotation looks like an inverse rotation.
3684:
3597:
3586:
1796:
752:
674:
508:
381:
247:
7023:
6823:
6785:
6732:
6722:
6712:
6653:
6484:
6297:, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 98,
6123:
are family of finite groups with similar properties to the dihedral groups.
5348:
by −1, it is clear that it commutes with any linear transformation).
3829:
3623:
1419:
1384:
931:
927:
607:
306:
295:
242:
217:
212:
171:
142:
105:
6465:
1410:
6752:
6747:
5190:
5182:
5003:
1457:
denote reflections across the three lines shown in the adjacent picture.
1286:
axes of symmetry connecting opposite vertices. In either case, there are
899:
7064:
5186:
3636:
3612:
2081:
774:
502:
3940:
3933:
3926:
3919:
3912:
3903:
1445:
denote counterclockwise rotations by 120° and 240° respectively, and s
1026:
6924:
6600:
6583:
6564:
6545:
6526:
6191:
3896:
3889:
3882:
3875:
1366:
883:
595:
6078:
There are several important generalizations of the dihedral groups:
5818:); which automorphisms are inner and outer depends on the parity of
6661:
6609:
5415:
5118:
5039:
5002:> 2 the operations of rotation and reflection in general do not
4583:{\displaystyle \mathrm {s} _{j}=\mathrm {r} _{j}\,\mathrm {s} _{0}}
4079:
4043:
The dihedral group as symmetry group in 2D and rotation group in 3D
4033:
3708:
1791:
results in a rotation of 120°. The order of elements denoting the
1154:
950:
942:
935:
911:
132:
6116:
includes both of the examples above, as well as many other groups.
2068:
1336:
The following picture shows the effect of the sixteen elements of
6091:
5811:
5773:
5488:
5392:
5026:
891:
474:
388:
5214:
symmetry – Imperial Seal of Japan, representing eightfold
1258:
axes of symmetry connecting the midpoints of opposite sides and
6106:
113:
6390:"Automorphism groups for semidirect products of cyclic groups"
5257:
symmetry — The Naval Jack of the
Republic of China (White Sun)
4310:
2984:, expressing a counterclockwise rotation through an angle of
2127:
5219:
5137:
972:
874:
4971:
1332:
6592:
6573:
6554:
6535:
6614:
4099:
lines through the origin, making angles of multiples of
6516:
6181:
5181:, referring to the proper symmetry groups of a regular
5165:(Greek: solid with two faces), which explains the name
1414:
The composition of these two reflections is a rotation.
5038:
Thus, beyond their obvious application to problems of
4437:
2836:
2679:
6109:, also has similar properties to the dihedral groups.
5726:
5701:
5677:
5652:
5433:
4655:
4596:
4539:
4526:{\displaystyle \mathrm {r} _{j}=\mathrm {r} _{1}^{j}}
4490:
4289:
4239:
4182:
3459:
3427:
3398:
3378:
3349:
3314:
3281:
3043:
3001:
is a reflection across a line that makes an angle of
2650:
2125:
1839:
1342:
1312:
1292:
1264:
1236:
1216:
1196:
1167:
1133:
1110:
1087:
1064:
1044:
926:. Dihedral groups are among the simplest examples of
545:
520:
483:
5578:
Algebraically, this is an instance of the conjugate
5562:
is even. If we think of the isometries of a regular
5558:
is odd, but they fall into two conjugacy classes if
4056:, and a common way to visualize it, is the group of
3601:
Example subgroups from a hexagonal dihedral symmetry
2116:
can be represented by the following eight matrices:
1830:, with composition given by the following formulae:
6377:. Dept of Mathematics, University of South Florida.
5545:
5522:) is the smallest example of a group that is not a
5196:
5155:
regular polygon embedded in three-dimensional space
5933:has 10 inner automorphisms. As 2D isometry group D
5901:
5734:
5709:
5685:
5660:
5441:
4934:
4632:
4582:
4525:
4473:
4265:
4216:{\displaystyle \mathrm {srs} =\mathrm {r} ^{-1}\,}
4215:
3565:
3442:
3413:
3384:
3364:
3335:
3300:
3264:
2963:
2624:
2041:
1357:
1321:
1298:
1278:
1250:
1222:
1202:
1182:
1145:
1116:
1093:
1073:
1050:
553:
528:
491:
890:, a dihedral symmetry, the same as for a regular
7114:
5099:listed elements are rotations and the remaining
941:The notation for the dihedral group differs in
6387:
5503:) is the sum of the positive divisors of
5454:. Therefore, the total number of subgroups of
6630:
6323:Abstract Algebra: Structures and Applications
5924:; e.g., multiplying angles of rotation by 2.
5841:odd, the inner automorphism group has order 2
3775:is trivial, whereas for other even values of
1422:shows the effect of composition in the group
855:
6207:Dummit, David S.; Foote, Richard M. (2004).
4627:
4603:
6206:
5994:
6637:
6623:
2635:In general, the matrices for elements of D
2102:. This is an example of a (2-dimensional)
862:
848:
6474:
6464:
6388:Sommer-Simpson, Jasha (2 November 2013).
6345:
6229:
5856:) the inner automorphism group has order
5728:
5703:
5679:
5654:
5538:, but these subgroups are not normal in D
5435:
4876:
4808:
4740:
4672:
4567:
4212:
2002:
1952:
1902:
1852:
547:
522:
485:
80:Learn how and when to remove this message
6352:. Oxford University Press. p. 195.
6315:
6313:
5946:multiplicative group of integers modulo
5867:odd, all reflections are conjugate; for
5358:twice an odd number, the abstract group
5034:is nonabelian (x-axis is vertical here).
5025:
4970:
3596:
3592:
2067:
2063:
1409:
1389:
1331:
1025:
873:
43:This article includes a list of general
27:Group of symmetries of a regular polygon
6271:, Oxford University Press, p. 95,
6264:
2109:For example, the elements of the group
2088:. This lets us represent elements of D
14:
7115:
6781:Classification of finite simple groups
6441:"Automorphisms of the Dihedral Groups"
6438:
6319:
5291:The properties of the dihedral groups
5279:National flag of the Republic of India
4633:{\displaystyle j\in \{1,\ldots ,n-1\}}
3031:can also be defined as the group with
420:Classification of finite simple groups
6618:
6593:
6574:
6555:
6536:
6517:
6310:
6182:
6164:Dihedral symmetry in three dimensions
5624:
5140:which is also of abstract group type
930:, and they play an important role in
6372:
6290:
6134:Coordinate rotations and reflections
4947:coordinate rotations and reflections
4640:we can write the product rules for D
3421:. This substitution also shows that
3018:
2072:The symmetries of this pentagon are
29:
5598:is the maximum power of 2 dividing
24:
6073:
4898:
4879:
4866:
4830:
4811:
4798:
4762:
4743:
4730:
4694:
4675:
4662:
4570:
4557:
4542:
4508:
4493:
4419:
4292:
4266:{\displaystyle e^{2\pi i \over n}}
4199:
4190:
4187:
4184:
3462:
3430:
3352:
3050:
2814:
2657:
2558:
2497:
2439:
2378:
2315:
2251:
2190:
2132:
2020:
2005:
1992:
1970:
1955:
1942:
1920:
1905:
1892:
1870:
1855:
1842:
1394:The lines of reflection labelled S
1378:As with any geometric object, the
1373:
1345:
1170:
49:it lacks sufficient corresponding
25:
7144:
6494:
6402:from the original on 2016-08-06.
6139:Cycle index of the dihedral group
5326:consists only of the identity if
5313:is even or odd. For example, the
3802:of dihedral groups consist of an
2576:
2515:
2457:
2396:
2333:
2269:
2208:
2150:
6294:Glimpses of Algebra and Geometry
5999:The inner automorphism group of
5546:Conjugacy classes of reflections
5530:subgroups (which are normal in D
5518:The dihedral group of order 8 (D
5262:
5246:
5227:
5203:
5197:Examples of 2D dihedral symmetry
4032:
4025:
3939:
3932:
3925:
3918:
3911:
3902:
3895:
3888:
3881:
3874:
3768:The inner automorphism group of
3443:{\displaystyle \mathrm {D} _{n}}
3365:{\displaystyle \mathrm {D} _{n}}
1358:{\displaystyle \mathrm {D} _{8}}
1183:{\displaystyle \mathrm {D} _{n}}
971:refers to the symmetries of the
112:
34:
6432:
6265:Cameron, Peter Jephson (1998),
5940:Compare the values 6 and 4 for
5902:Examples of automorphism groups
5340:as a subgroup of O(2), this is
5136:is also used for a subgroup of
4416:
2076:of the plane as a vector space.
1989:
1939:
1889:
6505:Wolfram Demonstrations Project
6381:
6366:
6339:
6284:
6258:
6200:
6175:
6159:Dihedral symmetry groups in 3D
4915:
4903:
4847:
4835:
4779:
4767:
4711:
4699:
4062:point groups in two dimensions
3765:is too large to be a subgroup.
3540:
3530:
3235:
3225:
3120:
3114:
3096:
3090:
781:Infinite dimensional Lie group
13:
1:
6610:Dihedral groups on GroupNames
6439:Miller, GA (September 1942).
6169:
5575:that pass through two sides.
5467: ≥ 1), is equal to
5286:
4106:with each other. This is the
4047:An example of abstract group
1013:
6501:Dihedral Group n of Order 2n
5879:(half the minimal rotation).
5735:{\displaystyle \mathbb {Z} }
5710:{\displaystyle \mathbb {Z} }
5686:{\displaystyle \mathbb {Z} }
5661:{\displaystyle \mathbb {Z} }
5487:) is the number of positive
5442:{\displaystyle \mathbb {Z} }
4127:; this extends to the cases
3687:dihedral groups. Otherwise,
1787:followed by the reflection s
1459:
554:{\displaystyle \mathbb {Z} }
529:{\displaystyle \mathbb {Z} }
492:{\displaystyle \mathbb {Z} }
7:
7048:Infinite dimensional groups
6346:Humphreys, John F. (1996).
6237:"Dihedral Groups: Notation"
6127:
6114:generalized dihedral groups
5108:So far, we have considered
4280:In matrix form, by setting
3336:{\displaystyle r=s\cdot sr}
1161:make up the dihedral group
1021:
279:List of group theory topics
10:
7149:
6644:
6105:the symmetry group of the
4058:Euclidean plane isometries
1783:, because the reflection s
7087:
7047:
6923:
6771:
6705:
6652:
6326:, CRC Press, p. 71,
6154:Dihedral group of order 8
6149:Dihedral group of order 6
5617:even there is instead an
4979:(x-axis is vertical here)
3666:are exceptional in that:
3308:, we obtain the relation
2641:have the following form:
2098:, with composition being
7128:Finite reflection groups
6951:Special orthogonal group
6445:Proc Natl Acad Sci U S A
6349:A Course in Group Theory
6320:Lovett, Stephen (2015),
5995:Inner automorphism group
5942:Euler's totient function
5916:. As 2D isometry group D
5776:function, the number of
5550:All the reflections are
4960:-axis. The elements of D
3585:belongs to the class of
397:Elementary abelian group
274:Glossary of group theory
6375:"Groups of small order"
6268:Introduction to Algebra
6084:infinite dihedral group
5554:to each other whenever
5367:is isomorphic with the
3301:{\displaystyle s^{2}=1}
1802:In general, the group D
1437:denotes the identity; r
1146:{\displaystyle n\geq 3}
1038:A regular polygon with
64:more precise citations.
6977:Exceptional Lie groups
5736:
5711:
5687:
5662:
5443:
5035:
4980:
4975:The four elements of D
4936:
4634:
4584:
4527:
4475:
4267:
4217:
4089:about the origin, and
3602:
3567:
3444:
3415:
3386:
3366:
3337:
3302:
3266:
2965:
2626:
2082:linear transformations
2077:
2074:linear transformations
2043:
1429:(the symmetries of an
1415:
1407:
1370:
1359:
1323:
1300:
1280:
1252:
1224:
1204:
1184:
1147:
1118:
1095:
1081:different symmetries:
1075:
1052:
1035:
895:
813:Linear algebraic group
555:
530:
493:
6963:Special unitary group
6466:10.1073/pnas.28.9.368
6213:John Wiley & Sons
5737:
5712:
5688:
5663:
5642:is isomorphic to the
5444:
5346:scalar multiplication
5277:, as depicted on the
5240:The Red Star of David
5151:proper symmetry group
5029:
4974:
4937:
4635:
4585:
4528:
4476:
4268:
4218:
4173:of order 2 such that
3600:
3593:Small dihedral groups
3568:
3450:has the presentation
3445:
3416:
3414:{\displaystyle t:=sr}
3387:
3367:
3338:
3303:
3267:
2966:
2627:
2100:matrix multiplication
2071:
2064:Matrix representation
2044:
1413:
1393:
1360:
1335:
1324:
1306:axes of symmetry and
1301:
1281:
1253:
1225:
1205:
1185:
1153:here. The associated
1148:
1125:reflection symmetries
1119:
1102:rotational symmetries
1096:
1076:
1053:
1029:
877:
556:
531:
494:
7133:Properties of groups
7123:Euclidean symmetries
7060:Diffeomorphism group
6939:Special linear group
6933:General linear group
6291:Toth, Gabor (2006),
6121:quasidihedral groups
5724:
5699:
5675:
5650:
5509:list of small groups
5431:
4952:The dihedral group D
4653:
4594:
4537:
4488:
4287:
4237:
4233:: multiplication by
4180:
3756:, for these values,
3457:
3425:
3396:
3376:
3347:
3312:
3279:
3041:
2648:
2123:
2104:group representation
1837:
1431:equilateral triangle
1340:
1310:
1290:
1262:
1234:
1214:
1194:
1165:
1131:
1108:
1085:
1062:
1042:
1034:of a regular hexagon
543:
518:
481:
6885:Other finite groups
6672:Commutator subgroup
6596:"Dihedral Group D6"
6579:"Dihedral Group D5"
6560:"Dihedral Group D4"
6541:"Dihedral Group D3"
6457:1942PNAS...28..368M
6241:Math Images Project
5958:The only values of
5922:outer automorphisms
5914:inner automorphisms
5890:) are outer unless
5511:for the cases
5427:, and one subgroup
4522:
4275:complex conjugation
3820:
3806:-element cycle and
3343:. It follows that
3275:Using the relation
1279:{\displaystyle n/2}
1251:{\displaystyle n/2}
1230:is even, there are
1127:. Usually, we take
980:, a group of order
187:Group homomorphisms
97:Algebraic structure
6915:Rubik's Cube group
6872:Baby monster group
6682:Group homomorphism
6576:Weisstein, Eric W.
6557:Weisstein, Eric W.
6538:Weisstein, Eric W.
6519:Weisstein, Eric W.
6184:Weisstein, Eric W.
6008:is isomorphic to:
5732:
5707:
5683:
5658:
5631:automorphism group
5625:Automorphism group
5619:outer automorphism
5439:
5307:depend on whether
5061:can be written as
5036:
5016:; for example, in
4981:
4932:
4930:
4630:
4580:
4523:
4506:
4471:
4465:
4410:
4263:
4213:
3818:
3603:
3563:
3440:
3411:
3382:
3362:
3333:
3298:
3262:
3260:
2961:
2959:
2948:
2791:
2622:
2620:
2609:
2608:
2545:
2544:
2484:
2483:
2426:
2425:
2363:
2362:
2302:
2301:
2238:
2237:
2177:
2176:
2078:
2054:modular arithmetic
2039:
1416:
1408:
1371:
1355:
1322:{\displaystyle 2n}
1319:
1296:
1276:
1248:
1220:
1200:
1180:
1143:
1114:
1091:
1074:{\displaystyle 2n}
1071:
1048:
1036:
896:
663:Special orthogonal
551:
526:
489:
370:Lagrange's theorem
7110:
7109:
6795:Alternating group
6503:by Shawn Dudzik,
5849:even (other than
5526:. Any of its two
5273:symmetry –
5238:symmetry –
5179:icosahedral group
4921:
4853:
4785:
4717:
4406:
4383:
4358:
4332:
4260:
4169:and a reflection
4040:
4039:
3995:
3994:
3385:{\displaystyle s}
3019:Other definitions
2944:
2915:
2887:
2861:
2806:
2802:
2799:
2787:
2761:
2733:
2704:
1766:
1765:
1299:{\displaystyle n}
1223:{\displaystyle n}
1203:{\displaystyle n}
1117:{\displaystyle n}
1094:{\displaystyle n}
1051:{\displaystyle n}
918:, which includes
872:
871:
447:
446:
329:Alternating group
286:
285:
90:
89:
82:
18:Dihedral symmetry
16:(Redirected from
7140:
7102:Abstract algebra
7039:Quaternion group
6969:Symplectic group
6945:Orthogonal group
6639:
6632:
6625:
6616:
6615:
6606:
6605:
6589:
6588:
6570:
6569:
6551:
6550:
6532:
6531:
6522:"Dihedral Group"
6489:
6488:
6478:
6468:
6436:
6430:
6429:
6401:
6394:
6385:
6379:
6378:
6373:Pedersen, John.
6370:
6364:
6363:
6343:
6337:
6336:
6317:
6308:
6307:
6288:
6282:
6281:
6262:
6256:
6255:
6253:
6252:
6243:. Archived from
6233:
6227:
6226:
6211:(3rd ed.).
6209:Abstract Algebra
6204:
6198:
6197:
6196:
6187:"Dihedral Group"
6179:
6099:orthogonal group
6068:
6051:
6044:
6038:
6019:
6007:
5990:
5983:
5976:
5932:
5911:
5896:
5855:
5787:
5759:
5741:
5739:
5738:
5733:
5731:
5716:
5714:
5713:
5708:
5706:
5692:
5690:
5689:
5684:
5682:
5667:
5665:
5664:
5659:
5657:
5641:
5605:
5597:
5592:Sylow 2-subgroup
5528:Klein four-group
5515: ≤ 8.
5475:) + σ(
5462:
5448:
5446:
5445:
5440:
5438:
5426:
5406:
5388:. Generally, if
5387:
5380:
5366:
5325:
5312:
5306:
5299:
5266:
5250:
5231:
5207:
5148:
5135:
5126:
5116:
5104:
5098:
5092:
5088:
5084:
5080:
5076:
5072:
5068:
5064:
5060:
5051:
4993:Klein four-group
4941:
4939:
4938:
4933:
4931:
4927:
4926:
4922:
4919:
4901:
4888:
4887:
4882:
4875:
4874:
4869:
4859:
4858:
4854:
4851:
4833:
4820:
4819:
4814:
4807:
4806:
4801:
4791:
4790:
4786:
4783:
4765:
4752:
4751:
4746:
4739:
4738:
4733:
4723:
4722:
4718:
4715:
4697:
4684:
4683:
4678:
4671:
4670:
4665:
4639:
4637:
4636:
4631:
4589:
4587:
4586:
4581:
4579:
4578:
4573:
4566:
4565:
4560:
4551:
4550:
4545:
4532:
4530:
4529:
4524:
4521:
4516:
4511:
4502:
4501:
4496:
4480:
4478:
4477:
4472:
4470:
4469:
4428:
4427:
4422:
4415:
4414:
4407:
4402:
4394:
4384:
4379:
4371:
4359:
4354:
4346:
4333:
4328:
4320:
4301:
4300:
4295:
4272:
4270:
4269:
4264:
4262:
4261:
4256:
4245:
4222:
4220:
4219:
4214:
4211:
4210:
4202:
4193:
4172:
4168:
4159:
4151:
4140:
4133:
4126:
4119:
4105:
4098:
4088:
4082:of multiples of
4078:
4072:
4055:
4036:
4029:
3997:
3996:
3943:
3936:
3929:
3922:
3915:
3906:
3899:
3892:
3885:
3878:
3821:
3817:
3813:identity element
3793:
3780:
3774:
3764:
3755:
3748:
3741:
3729:
3722:
3706:
3695:
3682:
3675:
3665:
3658:
3648:Klein four-group
3645:
3634:
3621:
3610:
3584:
3572:
3570:
3569:
3564:
3559:
3555:
3548:
3547:
3520:
3519:
3501:
3500:
3471:
3470:
3465:
3449:
3447:
3446:
3441:
3439:
3438:
3433:
3420:
3418:
3417:
3412:
3391:
3389:
3388:
3383:
3372:is generated by
3371:
3369:
3368:
3363:
3361:
3360:
3355:
3342:
3340:
3339:
3334:
3307:
3305:
3304:
3299:
3291:
3290:
3271:
3269:
3268:
3263:
3261:
3254:
3250:
3243:
3242:
3221:
3220:
3208:
3207:
3175:
3171:
3167:
3166:
3165:
3150:
3149:
3059:
3058:
3053:
3030:
3010:
2994:
2970:
2968:
2967:
2962:
2960:
2953:
2952:
2945:
2940:
2929:
2916:
2911:
2900:
2888:
2883:
2872:
2862:
2857:
2846:
2823:
2822:
2817:
2807:
2804:
2800:
2797:
2796:
2795:
2788:
2783:
2772:
2762:
2757:
2746:
2734:
2729:
2718:
2705:
2700:
2689:
2666:
2665:
2660:
2631:
2629:
2628:
2623:
2621:
2614:
2610:
2567:
2566:
2561:
2550:
2546:
2506:
2505:
2500:
2489:
2485:
2448:
2447:
2442:
2431:
2427:
2387:
2386:
2381:
2368:
2364:
2324:
2323:
2318:
2307:
2303:
2260:
2259:
2254:
2243:
2239:
2199:
2198:
2193:
2182:
2178:
2141:
2140:
2135:
2048:
2046:
2045:
2040:
2035:
2034:
2023:
2014:
2013:
2008:
2001:
2000:
1995:
1985:
1984:
1973:
1964:
1963:
1958:
1951:
1950:
1945:
1935:
1934:
1923:
1914:
1913:
1908:
1901:
1900:
1895:
1885:
1884:
1873:
1864:
1863:
1858:
1851:
1850:
1845:
1782:
1460:
1364:
1362:
1361:
1356:
1354:
1353:
1348:
1328:
1326:
1325:
1320:
1305:
1303:
1302:
1297:
1285:
1283:
1282:
1277:
1272:
1257:
1255:
1254:
1249:
1244:
1229:
1227:
1226:
1221:
1209:
1207:
1206:
1201:
1189:
1187:
1186:
1181:
1179:
1178:
1173:
1152:
1150:
1149:
1144:
1123:
1121:
1120:
1115:
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1057:
1055:
1054:
1049:
1030:The six axes of
1009:
1000:
989:abstract algebra
986:
977:
970:
961:
947:abstract algebra
864:
857:
850:
806:Algebraic groups
579:Hyperbolic group
569:Arithmetic group
560:
558:
557:
552:
550:
535:
533:
532:
527:
525:
498:
496:
495:
490:
488:
411:Schur multiplier
365:Cauchy's theorem
353:Quaternion group
301:
300:
127:
126:
116:
103:
92:
91:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
7148:
7147:
7143:
7142:
7141:
7139:
7138:
7137:
7113:
7112:
7111:
7106:
7083:
7055:Conformal group
7043:
7017:
7009:
7001:
6993:
6985:
6919:
6911:
6898:
6889:Symmetric group
6868:
6858:
6851:
6844:
6837:
6829:
6820:
6816:
6806:Sporadic groups
6800:
6791:
6773:Discrete groups
6767:
6758:Wallpaper group
6738:Solvable groups
6706:Types of groups
6701:
6667:Normal subgroup
6648:
6643:
6594:Davis, Declan.
6497:
6492:
6437:
6433:
6419:if and only if
6418:
6412:
6399:
6392:
6386:
6382:
6371:
6367:
6360:
6344:
6340:
6334:
6318:
6311:
6305:
6289:
6285:
6279:
6263:
6259:
6250:
6248:
6235:
6234:
6230:
6223:
6205:
6201:
6180:
6176:
6172:
6130:
6076:
6074:Generalizations
6063:
6059:
6053:
6046:
6040:
6037:
6033:
6027:
6018:
6012:
6006:
6000:
5997:
5989:
5985:
5984:(order 8), and
5982:
5978:
5975:
5971:
5936:
5931:
5927:
5919:
5910:
5906:
5904:
5891:
5850:
5781:
5727:
5725:
5722:
5721:
5702:
5700:
5697:
5696:
5694:
5678:
5676:
5673:
5672:
5653:
5651:
5648:
5647:
5640:
5634:
5627:
5599:
5595:
5548:
5541:
5537:
5533:
5521:
5461:
5455:
5453:
5434:
5432:
5429:
5428:
5425:
5419:
5405:
5399:
5386:
5382:
5379:
5372:
5365:
5359:
5339:
5330:is odd, but if
5324:
5318:
5308:
5301:
5298:
5292:
5289:
5282:
5272:
5267:
5258:
5256:
5251:
5242:
5237:
5232:
5223:
5213:
5208:
5199:
5193:respectively).
5169:(in analogy to
5147:
5141:
5134:
5128:
5122:
5115:
5109:
5100:
5094:
5090:
5086:
5082:
5078:
5074:
5070:
5066:
5062:
5059:
5053:
5046:
5033:
5021:
5011:
4986:
4978:
4963:
4955:
4929:
4928:
4920: mod
4918:
4902:
4897:
4896:
4889:
4883:
4878:
4877:
4870:
4865:
4864:
4861:
4860:
4852: mod
4850:
4834:
4829:
4828:
4821:
4815:
4810:
4809:
4802:
4797:
4796:
4793:
4792:
4784: mod
4782:
4766:
4761:
4760:
4753:
4747:
4742:
4741:
4734:
4729:
4728:
4725:
4724:
4716: mod
4714:
4698:
4693:
4692:
4685:
4679:
4674:
4673:
4666:
4661:
4660:
4656:
4654:
4651:
4650:
4645:
4595:
4592:
4591:
4574:
4569:
4568:
4561:
4556:
4555:
4546:
4541:
4540:
4538:
4535:
4534:
4517:
4512:
4507:
4497:
4492:
4491:
4489:
4486:
4485:
4464:
4463:
4455:
4449:
4448:
4443:
4433:
4432:
4423:
4418:
4417:
4409:
4408:
4395:
4393:
4385:
4372:
4370:
4361:
4360:
4347:
4345:
4334:
4321:
4319:
4306:
4305:
4296:
4291:
4290:
4288:
4285:
4284:
4246:
4244:
4240:
4238:
4235:
4234:
4231:complex numbers
4203:
4198:
4197:
4183:
4181:
4178:
4177:
4170:
4164:
4157:
4150:
4144:
4135:
4128:
4121:
4115:
4112:regular polygon
4100:
4094:
4083:
4074:
4071:
4065:
4054:
4048:
4045:
4019:
4012:
4004:
3991:
3987:
3983:
3977:
3971:
3965:
3959:
3955:
3951:
3869:
3863:
3857:
3850:
3844:
3840:
3833:
3827:
3792:
3788:
3782:
3776:
3773:
3769:
3763:
3757:
3750:
3743:
3731:
3724:
3721:
3715:
3713:symmetric group
3705:
3699:
3696:is non-abelian.
3694:
3688:
3681:
3677:
3674:
3670:
3664:
3660:
3657:
3653:
3644:
3640:
3633:
3629:
3620:
3616:
3609:
3605:
3595:
3583:
3577:
3576:In particular,
3543:
3539:
3515:
3511:
3496:
3492:
3479:
3475:
3466:
3461:
3460:
3458:
3455:
3454:
3434:
3429:
3428:
3426:
3423:
3422:
3397:
3394:
3393:
3377:
3374:
3373:
3356:
3351:
3350:
3348:
3345:
3344:
3313:
3310:
3309:
3286:
3282:
3280:
3277:
3276:
3259:
3258:
3238:
3234:
3216:
3212:
3203:
3199:
3186:
3182:
3173:
3172:
3158:
3154:
3142:
3138:
3071:
3067:
3060:
3054:
3049:
3048:
3044:
3042:
3039:
3038:
3029:
3023:
3021:
3002:
3000:
2985:
2982:rotation matrix
2979:
2958:
2957:
2947:
2946:
2930:
2928:
2917:
2901:
2899:
2890:
2889:
2873:
2871:
2863:
2847:
2845:
2832:
2831:
2824:
2818:
2813:
2812:
2809:
2808:
2803:
2790:
2789:
2773:
2771:
2763:
2747:
2745:
2736:
2735:
2719:
2717:
2706:
2690:
2688:
2675:
2674:
2667:
2661:
2656:
2655:
2651:
2649:
2646:
2645:
2640:
2619:
2618:
2607:
2606:
2601:
2592:
2591:
2583:
2575:
2571:
2562:
2557:
2556:
2554:
2543:
2542:
2537:
2531:
2530:
2525:
2514:
2510:
2501:
2496:
2495:
2493:
2482:
2481:
2476:
2470:
2469:
2464:
2456:
2452:
2443:
2438:
2437:
2435:
2424:
2423:
2415:
2409:
2408:
2403:
2395:
2391:
2382:
2377:
2376:
2373:
2372:
2361:
2360:
2355:
2346:
2345:
2340:
2332:
2328:
2319:
2314:
2313:
2311:
2300:
2299:
2291:
2285:
2284:
2279:
2268:
2264:
2255:
2250:
2249:
2247:
2236:
2235:
2230:
2224:
2223:
2215:
2207:
2203:
2194:
2189:
2188:
2186:
2175:
2174:
2169:
2163:
2162:
2157:
2149:
2145:
2136:
2131:
2130:
2126:
2124:
2121:
2120:
2114:
2093:
2066:
2024:
2019:
2018:
2009:
2004:
2003:
1996:
1991:
1990:
1974:
1969:
1968:
1959:
1954:
1953:
1946:
1941:
1940:
1924:
1919:
1918:
1909:
1904:
1903:
1896:
1891:
1890:
1874:
1869:
1868:
1859:
1854:
1853:
1846:
1841:
1840:
1838:
1835:
1834:
1829:
1822:
1818:
1811:
1807:
1790:
1786:
1781:
1777:
1773:
1769:
1762:
1756:
1750:
1744:
1738:
1732:
1726:
1718:
1712:
1706:
1700:
1694:
1688:
1682:
1674:
1668:
1662:
1656:
1650:
1644:
1638:
1630:
1624:
1618:
1612:
1606:
1600:
1594:
1586:
1580:
1574:
1568:
1562:
1556:
1550:
1542:
1536:
1530:
1524:
1518:
1512:
1506:
1498:
1492:
1486:
1480:
1474:
1468:
1456:
1452:
1448:
1444:
1440:
1436:
1427:
1405:
1401:
1397:
1376:
1374:Group structure
1349:
1344:
1343:
1341:
1338:
1337:
1311:
1308:
1307:
1291:
1288:
1287:
1268:
1263:
1260:
1259:
1240:
1235:
1232:
1231:
1215:
1212:
1211:
1195:
1192:
1191:
1174:
1169:
1168:
1166:
1163:
1162:
1132:
1129:
1128:
1109:
1106:
1105:
1086:
1083:
1082:
1063:
1060:
1059:
1043:
1040:
1039:
1024:
1016:
1008:
1002:
999:
992:
981:
973:
969:
963:
960:
954:
916:regular polygon
889:
868:
839:
838:
827:Abelian variety
820:Reductive group
808:
798:
797:
796:
795:
746:
738:
730:
722:
714:
687:Special unitary
598:
584:
583:
565:
564:
546:
544:
541:
540:
521:
519:
516:
515:
484:
482:
479:
478:
470:
469:
460:Discrete groups
449:
448:
404:Frobenius group
349:
336:
325:
318:Symmetric group
314:
298:
288:
287:
138:Normal subgroup
124:
104:
95:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
7146:
7136:
7135:
7130:
7125:
7108:
7107:
7105:
7104:
7099:
7094:
7088:
7085:
7084:
7082:
7081:
7078:
7075:
7072:
7067:
7062:
7057:
7051:
7049:
7045:
7044:
7042:
7041:
7036:
7034:Poincaré group
7031:
7026:
7020:
7019:
7015:
7011:
7007:
7003:
6999:
6995:
6991:
6987:
6983:
6979:
6973:
6972:
6966:
6960:
6954:
6948:
6942:
6936:
6929:
6927:
6921:
6920:
6918:
6917:
6912:
6907:
6902:Dihedral group
6899:
6894:
6886:
6882:
6881:
6875:
6869:
6866:
6860:
6856:
6849:
6842:
6835:
6830:
6827:
6821:
6818:
6814:
6808:
6802:
6801:
6798:
6792:
6789:
6783:
6777:
6775:
6769:
6768:
6766:
6765:
6760:
6755:
6750:
6745:
6743:Symmetry group
6740:
6735:
6730:
6728:Infinite group
6725:
6720:
6718:Abelian groups
6715:
6709:
6707:
6703:
6702:
6700:
6699:
6694:
6692:direct product
6684:
6679:
6677:Quotient group
6674:
6669:
6664:
6658:
6656:
6650:
6649:
6642:
6641:
6634:
6627:
6619:
6613:
6612:
6607:
6590:
6571:
6552:
6533:
6514:
6511:Dihedral group
6508:
6496:
6495:External links
6493:
6491:
6490:
6431:
6414:
6408:
6405:Corollary 7.3.
6395:. p. 13.
6380:
6365:
6358:
6338:
6332:
6309:
6303:
6283:
6277:
6257:
6228:
6221:
6199:
6173:
6171:
6168:
6167:
6166:
6161:
6156:
6151:
6146:
6144:Dicyclic group
6141:
6136:
6129:
6126:
6125:
6124:
6117:
6112:The family of
6110:
6095:
6088:infinite group
6075:
6072:
6071:
6070:
6061:
6055:
6035:
6029:
6025:
6014:
6002:
5996:
5993:
5987:
5980:
5973:
5934:
5929:
5917:
5908:
5903:
5900:
5899:
5898:
5880:
5861:
5835:
5760:and has order
5730:
5705:
5681:
5656:
5636:
5626:
5623:
5566:-gon: for odd
5547:
5544:
5539:
5535:
5531:
5519:
5457:
5449:
5437:
5421:
5401:
5384:
5374:
5369:direct product
5361:
5344:; since it is
5335:
5320:
5294:
5288:
5285:
5284:
5283:
5270:
5268:
5261:
5259:
5254:
5252:
5245:
5243:
5235:
5233:
5226:
5224:
5211:
5209:
5202:
5198:
5195:
5167:dihedral group
5143:
5130:
5111:
5055:
5031:
5019:
5007:
4984:
4976:
4961:
4953:
4943:
4942:
4925:
4917:
4914:
4911:
4908:
4905:
4900:
4895:
4892:
4890:
4886:
4881:
4873:
4868:
4863:
4862:
4857:
4849:
4846:
4843:
4840:
4837:
4832:
4827:
4824:
4822:
4818:
4813:
4805:
4800:
4795:
4794:
4789:
4781:
4778:
4775:
4772:
4769:
4764:
4759:
4756:
4754:
4750:
4745:
4737:
4732:
4727:
4726:
4721:
4713:
4710:
4707:
4704:
4701:
4696:
4691:
4688:
4686:
4682:
4677:
4669:
4664:
4659:
4658:
4641:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4577:
4572:
4564:
4559:
4554:
4549:
4544:
4520:
4515:
4510:
4505:
4500:
4495:
4482:
4481:
4468:
4462:
4459:
4456:
4454:
4451:
4450:
4447:
4444:
4442:
4439:
4438:
4436:
4431:
4426:
4421:
4413:
4405:
4401:
4398:
4392:
4389:
4386:
4382:
4378:
4375:
4369:
4366:
4363:
4362:
4357:
4353:
4350:
4344:
4341:
4338:
4335:
4331:
4327:
4324:
4318:
4315:
4312:
4311:
4309:
4304:
4299:
4294:
4259:
4255:
4252:
4249:
4243:
4224:
4223:
4209:
4206:
4201:
4196:
4192:
4189:
4186:
4156:by a rotation
4146:
4108:symmetry group
4067:
4050:
4044:
4041:
4038:
4037:
4030:
4022:
4021:
4017:
4013:
4010:
4002:
3993:
3992:
3989:
3985:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3953:
3949:
3945:
3944:
3937:
3930:
3923:
3916:
3908:
3907:
3900:
3893:
3886:
3879:
3871:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3848:
3842:
3838:
3835:
3831:
3825:
3796:
3795:
3790:
3784:
3771:
3766:
3759:
3717:
3701:
3697:
3690:
3679:
3672:
3662:
3655:
3642:
3631:
3618:
3607:
3594:
3591:
3587:Coxeter groups
3579:
3574:
3573:
3562:
3558:
3554:
3551:
3546:
3542:
3538:
3535:
3532:
3529:
3526:
3523:
3518:
3514:
3510:
3507:
3504:
3499:
3495:
3491:
3488:
3485:
3482:
3478:
3474:
3469:
3464:
3437:
3432:
3410:
3407:
3404:
3401:
3381:
3359:
3354:
3332:
3329:
3326:
3323:
3320:
3317:
3297:
3294:
3289:
3285:
3273:
3272:
3257:
3253:
3249:
3246:
3241:
3237:
3233:
3230:
3227:
3224:
3219:
3215:
3211:
3206:
3202:
3198:
3195:
3192:
3189:
3185:
3181:
3178:
3176:
3174:
3170:
3164:
3161:
3157:
3153:
3148:
3145:
3141:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3070:
3066:
3063:
3061:
3057:
3052:
3047:
3046:
3025:
3020:
3017:
2996:
2975:
2972:
2971:
2956:
2951:
2943:
2939:
2936:
2933:
2927:
2924:
2921:
2918:
2914:
2910:
2907:
2904:
2898:
2895:
2892:
2891:
2886:
2882:
2879:
2876:
2870:
2867:
2864:
2860:
2856:
2853:
2850:
2844:
2841:
2838:
2837:
2835:
2830:
2827:
2825:
2821:
2816:
2811:
2810:
2794:
2786:
2782:
2779:
2776:
2770:
2767:
2764:
2760:
2756:
2753:
2750:
2744:
2741:
2738:
2737:
2732:
2728:
2725:
2722:
2716:
2713:
2710:
2707:
2703:
2699:
2696:
2693:
2687:
2684:
2681:
2680:
2678:
2673:
2670:
2668:
2664:
2659:
2654:
2653:
2636:
2633:
2632:
2617:
2613:
2605:
2602:
2600:
2597:
2594:
2593:
2590:
2587:
2584:
2582:
2579:
2578:
2574:
2570:
2565:
2560:
2555:
2553:
2549:
2541:
2538:
2536:
2533:
2532:
2529:
2526:
2524:
2521:
2518:
2517:
2513:
2509:
2504:
2499:
2494:
2492:
2488:
2480:
2477:
2475:
2472:
2471:
2468:
2465:
2463:
2460:
2459:
2455:
2451:
2446:
2441:
2436:
2434:
2430:
2422:
2419:
2416:
2414:
2411:
2410:
2407:
2404:
2402:
2399:
2398:
2394:
2390:
2385:
2380:
2375:
2374:
2371:
2367:
2359:
2356:
2354:
2351:
2348:
2347:
2344:
2341:
2339:
2336:
2335:
2331:
2327:
2322:
2317:
2312:
2310:
2306:
2298:
2295:
2292:
2290:
2287:
2286:
2283:
2280:
2278:
2275:
2272:
2271:
2267:
2263:
2258:
2253:
2248:
2246:
2242:
2234:
2231:
2229:
2226:
2225:
2222:
2219:
2216:
2214:
2211:
2210:
2206:
2202:
2197:
2192:
2187:
2185:
2181:
2173:
2170:
2168:
2165:
2164:
2161:
2158:
2156:
2153:
2152:
2148:
2144:
2139:
2134:
2129:
2128:
2112:
2089:
2065:
2062:
2050:
2049:
2038:
2033:
2030:
2027:
2022:
2017:
2012:
2007:
1999:
1994:
1988:
1983:
1980:
1977:
1972:
1967:
1962:
1957:
1949:
1944:
1938:
1933:
1930:
1927:
1922:
1917:
1912:
1907:
1899:
1894:
1888:
1883:
1880:
1877:
1872:
1867:
1862:
1857:
1849:
1844:
1824:
1820:
1813:
1809:
1808:has elements r
1803:
1788:
1784:
1779:
1775:
1771:
1764:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1720:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1676:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1632:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1588:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1544:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1500:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1454:
1450:
1446:
1442:
1438:
1434:
1425:
1418:The following
1403:
1399:
1395:
1375:
1372:
1352:
1347:
1318:
1315:
1295:
1275:
1271:
1267:
1247:
1243:
1239:
1219:
1199:
1177:
1172:
1142:
1139:
1136:
1113:
1090:
1070:
1067:
1047:
1023:
1020:
1015:
1012:
1004:
994:
965:
956:
904:dihedral group
887:
880:symmetry group
870:
869:
867:
866:
859:
852:
844:
841:
840:
837:
836:
834:Elliptic curve
830:
829:
823:
822:
816:
815:
809:
804:
803:
800:
799:
794:
793:
790:
787:
783:
779:
778:
777:
772:
770:Diffeomorphism
766:
765:
760:
755:
749:
748:
744:
740:
736:
732:
728:
724:
720:
716:
712:
707:
706:
695:
694:
683:
682:
671:
670:
659:
658:
647:
646:
635:
634:
627:Special linear
623:
622:
615:General linear
611:
610:
605:
599:
590:
589:
586:
585:
582:
581:
576:
571:
563:
562:
549:
537:
524:
511:
509:Modular groups
507:
506:
505:
500:
487:
471:
468:
467:
462:
456:
455:
454:
451:
450:
445:
444:
443:
442:
437:
432:
429:
423:
422:
416:
415:
414:
413:
407:
406:
400:
399:
394:
385:
384:
382:Hall's theorem
379:
377:Sylow theorems
373:
372:
367:
359:
358:
357:
356:
350:
345:
342:Dihedral group
338:
337:
332:
326:
321:
315:
310:
299:
294:
293:
290:
289:
284:
283:
282:
281:
276:
268:
267:
266:
265:
260:
255:
250:
245:
240:
235:
233:multiplicative
230:
225:
220:
215:
207:
206:
205:
204:
199:
191:
190:
182:
181:
180:
179:
177:Wreath product
174:
169:
164:
162:direct product
156:
154:Quotient group
148:
147:
146:
145:
140:
135:
125:
122:
121:
118:
117:
109:
108:
88:
87:
42:
40:
33:
26:
9:
6:
4:
3:
2:
7145:
7134:
7131:
7129:
7126:
7124:
7121:
7120:
7118:
7103:
7100:
7098:
7095:
7093:
7090:
7089:
7086:
7079:
7076:
7073:
7071:
7070:Quantum group
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7052:
7050:
7046:
7040:
7037:
7035:
7032:
7030:
7029:Lorentz group
7027:
7025:
7022:
7021:
7018:
7012:
7010:
7004:
7002:
6996:
6994:
6988:
6986:
6980:
6978:
6975:
6974:
6970:
6967:
6964:
6961:
6958:
6957:Unitary group
6955:
6952:
6949:
6946:
6943:
6940:
6937:
6934:
6931:
6930:
6928:
6926:
6922:
6916:
6913:
6910:
6906:
6903:
6900:
6897:
6893:
6890:
6887:
6884:
6883:
6879:
6878:Monster group
6876:
6873:
6870:
6864:
6863:Fischer group
6861:
6859:
6852:
6845:
6838:
6832:Janko groups
6831:
6825:
6822:
6812:
6811:Mathieu group
6809:
6807:
6804:
6803:
6796:
6793:
6787:
6784:
6782:
6779:
6778:
6776:
6774:
6770:
6764:
6763:Trivial group
6761:
6759:
6756:
6754:
6751:
6749:
6746:
6744:
6741:
6739:
6736:
6734:
6733:Simple groups
6731:
6729:
6726:
6724:
6723:Cyclic groups
6721:
6719:
6716:
6714:
6713:Finite groups
6711:
6710:
6708:
6704:
6698:
6695:
6693:
6689:
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6659:
6657:
6655:
6654:Basic notions
6651:
6647:
6640:
6635:
6633:
6628:
6626:
6621:
6620:
6617:
6611:
6608:
6603:
6602:
6597:
6591:
6586:
6585:
6580:
6577:
6572:
6567:
6566:
6561:
6558:
6553:
6548:
6547:
6542:
6539:
6534:
6529:
6528:
6523:
6520:
6515:
6513:at Groupprops
6512:
6509:
6506:
6502:
6499:
6498:
6486:
6482:
6477:
6472:
6467:
6462:
6458:
6454:
6451:(9): 368–71.
6450:
6446:
6442:
6435:
6428:
6426:
6422:
6417:
6411:
6406:
6398:
6391:
6384:
6376:
6369:
6361:
6359:9780198534594
6355:
6351:
6350:
6342:
6335:
6333:9781482248913
6329:
6325:
6324:
6316:
6314:
6306:
6304:9780387224558
6300:
6296:
6295:
6287:
6280:
6278:9780198501954
6274:
6270:
6269:
6261:
6247:on 2016-03-20
6246:
6242:
6238:
6232:
6224:
6222:0-471-43334-9
6218:
6214:
6210:
6203:
6194:
6193:
6188:
6185:
6178:
6174:
6165:
6162:
6160:
6157:
6155:
6152:
6150:
6147:
6145:
6142:
6140:
6137:
6135:
6132:
6131:
6122:
6118:
6115:
6111:
6108:
6104:
6100:
6096:
6093:
6089:
6085:
6081:
6080:
6079:
6067:
6058:
6049:
6045:is even (for
6043:
6032:
6026:
6023:
6017:
6011:
6010:
6009:
6005:
5992:
5969:
5965:
5961:
5956:
5954:
5950:
5949:
5943:
5938:
5925:
5923:
5915:
5894:
5889:
5885:
5881:
5878:
5874:
5870:
5866:
5862:
5859:
5853:
5848:
5844:
5840:
5836:
5833:
5829:
5825:
5824:
5823:
5821:
5817:
5813:
5810:
5806:
5802:
5798:
5793:
5791:
5785:
5779:
5775:
5771:
5767:
5763:
5757:
5753:
5749:
5745:
5720:
5671:
5645:
5639:
5632:
5622:
5620:
5616:
5611:
5609:
5606:), while for
5603:
5593:
5589:
5585:
5581:
5580:Sylow theorem
5576:
5573:
5569:
5565:
5561:
5557:
5553:
5543:
5529:
5525:
5516:
5514:
5510:
5506:
5502:
5498:
5494:
5490:
5486:
5482:
5478:
5474:
5470:
5466:
5460:
5452:
5424:
5417:
5414:
5410:
5404:
5397:
5394:
5391:
5377:
5370:
5364:
5357:
5352:
5349:
5347:
5343:
5338:
5333:
5329:
5323:
5316:
5311:
5304:
5297:
5280:
5276:
5275:Ashoka Chakra
5265:
5260:
5249:
5244:
5241:
5230:
5225:
5221:
5218:with sixteen
5217:
5216:chrysanthemum
5206:
5201:
5200:
5194:
5192:
5188:
5184:
5180:
5176:
5172:
5168:
5164:
5160:
5156:
5152:
5146:
5139:
5133:
5125:
5120:
5114:
5106:
5103:
5097:
5058:
5050:
5043:
5041:
5028:
5024:
5022:
5015:
5010:
5005:
5001:
4996:
4994:
4990:
4973:
4969:
4967:
4959:
4950:
4948:
4923:
4912:
4909:
4906:
4893:
4891:
4884:
4871:
4855:
4844:
4841:
4838:
4825:
4823:
4816:
4803:
4787:
4776:
4773:
4770:
4757:
4755:
4748:
4735:
4719:
4708:
4705:
4702:
4689:
4687:
4680:
4667:
4649:
4648:
4647:
4644:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4600:
4597:
4575:
4562:
4552:
4547:
4518:
4513:
4503:
4498:
4484:and defining
4466:
4460:
4457:
4452:
4445:
4440:
4434:
4429:
4424:
4411:
4403:
4399:
4396:
4390:
4387:
4380:
4376:
4373:
4367:
4364:
4355:
4351:
4348:
4342:
4339:
4336:
4329:
4325:
4322:
4316:
4313:
4307:
4302:
4297:
4283:
4282:
4281:
4278:
4276:
4257:
4253:
4250:
4247:
4241:
4232:
4227:
4207:
4204:
4194:
4176:
4175:
4174:
4167:
4163:
4155:
4149:
4142:
4138:
4131:
4124:
4118:
4113:
4109:
4104:
4097:
4092:
4087:
4081:
4077:
4070:
4063:
4059:
4053:
4035:
4031:
4028:
4024:
4023:
4020:
4014:
4009:
4005:
3999:
3998:
3979:
3973:
3967:
3961:
3947:
3946:
3942:
3938:
3935:
3931:
3928:
3924:
3921:
3917:
3914:
3910:
3909:
3905:
3901:
3898:
3894:
3891:
3887:
3884:
3880:
3877:
3873:
3872:
3865:
3859:
3853:
3851:
3836:
3834:
3823:
3822:
3819:Cycle graphs
3816:
3814:
3809:
3805:
3801:
3787:
3779:
3767:
3762:
3753:
3746:
3739:
3735:
3727:
3720:
3714:
3710:
3704:
3698:
3693:
3686:
3683:are the only
3669:
3668:
3667:
3651:
3649:
3638:
3627:
3625:
3614:
3599:
3590:
3588:
3582:
3560:
3556:
3552:
3549:
3544:
3536:
3533:
3527:
3524:
3521:
3516:
3512:
3508:
3505:
3502:
3497:
3493:
3489:
3486:
3483:
3480:
3476:
3472:
3467:
3453:
3452:
3451:
3435:
3408:
3405:
3402:
3399:
3379:
3357:
3330:
3327:
3324:
3321:
3318:
3315:
3295:
3292:
3287:
3283:
3255:
3251:
3247:
3244:
3239:
3231:
3228:
3222:
3217:
3213:
3209:
3204:
3200:
3196:
3193:
3190:
3187:
3183:
3179:
3177:
3168:
3162:
3159:
3155:
3151:
3146:
3143:
3139:
3135:
3132:
3129:
3126:
3123:
3117:
3111:
3108:
3105:
3102:
3099:
3093:
3087:
3084:
3081:
3078:
3075:
3072:
3068:
3064:
3062:
3055:
3037:
3036:
3035:
3034:
3028:
3016:
3014:
3009:
3005:
2999:
2993:
2989:
2983:
2978:
2954:
2949:
2941:
2937:
2934:
2931:
2925:
2922:
2919:
2912:
2908:
2905:
2902:
2896:
2893:
2884:
2880:
2877:
2874:
2868:
2865:
2858:
2854:
2851:
2848:
2842:
2839:
2833:
2828:
2826:
2819:
2792:
2784:
2780:
2777:
2774:
2768:
2765:
2758:
2754:
2751:
2748:
2742:
2739:
2730:
2726:
2723:
2720:
2714:
2711:
2708:
2701:
2697:
2694:
2691:
2685:
2682:
2676:
2671:
2669:
2662:
2644:
2643:
2642:
2639:
2615:
2611:
2603:
2598:
2595:
2588:
2585:
2580:
2572:
2568:
2563:
2551:
2547:
2539:
2534:
2527:
2522:
2519:
2511:
2507:
2502:
2490:
2486:
2478:
2473:
2466:
2461:
2453:
2449:
2444:
2432:
2428:
2420:
2417:
2412:
2405:
2400:
2392:
2388:
2383:
2369:
2365:
2357:
2352:
2349:
2342:
2337:
2329:
2325:
2320:
2308:
2304:
2296:
2293:
2288:
2281:
2276:
2273:
2265:
2261:
2256:
2244:
2240:
2232:
2227:
2220:
2217:
2212:
2204:
2200:
2195:
2183:
2179:
2171:
2166:
2159:
2154:
2146:
2142:
2137:
2119:
2118:
2117:
2115:
2107:
2105:
2101:
2097:
2092:
2087:
2083:
2075:
2070:
2061:
2059:
2056:with modulus
2055:
2036:
2031:
2028:
2025:
2015:
2010:
1997:
1986:
1981:
1978:
1975:
1965:
1960:
1947:
1936:
1931:
1928:
1925:
1915:
1910:
1897:
1886:
1881:
1878:
1875:
1865:
1860:
1847:
1833:
1832:
1831:
1827:
1816:
1806:
1800:
1798:
1794:
1768:For example,
1758:
1752:
1746:
1740:
1734:
1728:
1722:
1721:
1714:
1708:
1702:
1696:
1690:
1684:
1678:
1677:
1670:
1664:
1658:
1652:
1646:
1640:
1634:
1633:
1626:
1620:
1614:
1608:
1602:
1596:
1590:
1589:
1582:
1576:
1570:
1564:
1558:
1552:
1546:
1545:
1538:
1532:
1526:
1520:
1514:
1508:
1502:
1501:
1494:
1488:
1482:
1476:
1470:
1464:
1462:
1461:
1458:
1432:
1428:
1421:
1412:
1392:
1388:
1386:
1381:
1368:
1350:
1334:
1330:
1316:
1313:
1293:
1273:
1269:
1265:
1245:
1241:
1237:
1217:
1197:
1175:
1160:
1156:
1140:
1137:
1134:
1126:
1111:
1103:
1088:
1068:
1065:
1045:
1033:
1028:
1019:
1011:
1007:
998:
990:
985:
979:
976:
968:
959:
952:
948:
944:
939:
937:
933:
929:
928:finite groups
925:
921:
917:
913:
909:
905:
901:
893:
885:
881:
876:
865:
860:
858:
853:
851:
846:
845:
843:
842:
835:
832:
831:
828:
825:
824:
821:
818:
817:
814:
811:
810:
807:
802:
801:
791:
788:
785:
784:
782:
776:
773:
771:
768:
767:
764:
761:
759:
756:
754:
751:
750:
747:
741:
739:
733:
731:
725:
723:
717:
715:
709:
708:
704:
700:
697:
696:
692:
688:
685:
684:
680:
676:
673:
672:
668:
664:
661:
660:
656:
652:
649:
648:
644:
640:
637:
636:
632:
628:
625:
624:
620:
616:
613:
612:
609:
606:
604:
601:
600:
597:
593:
588:
587:
580:
577:
575:
572:
570:
567:
566:
538:
513:
512:
510:
504:
501:
476:
473:
472:
466:
463:
461:
458:
457:
453:
452:
441:
438:
436:
433:
430:
427:
426:
425:
424:
421:
418:
417:
412:
409:
408:
405:
402:
401:
398:
395:
393:
391:
387:
386:
383:
380:
378:
375:
374:
371:
368:
366:
363:
362:
361:
360:
354:
351:
348:
343:
340:
339:
335:
330:
327:
324:
319:
316:
313:
308:
305:
304:
303:
302:
297:
296:Finite groups
292:
291:
280:
277:
275:
272:
271:
270:
269:
264:
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
210:
209:
208:
203:
200:
198:
195:
194:
193:
192:
189:
188:
184:
183:
178:
175:
173:
170:
168:
165:
163:
160:
157:
155:
152:
151:
150:
149:
144:
141:
139:
136:
134:
131:
130:
129:
128:
123:Basic notions
120:
119:
115:
111:
110:
107:
102:
98:
94:
93:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
7097:Applications
7024:Circle group
6908:
6904:
6901:
6895:
6891:
6824:Conway group
6786:Cyclic group
6599:
6582:
6563:
6544:
6525:
6448:
6444:
6434:
6424:
6420:
6415:
6409:
6404:
6403:
6383:
6368:
6348:
6341:
6322:
6293:
6286:
6267:
6260:
6249:. Retrieved
6245:the original
6240:
6231:
6208:
6202:
6190:
6177:
6102:
6077:
6065:
6056:
6047:
6041:
6030:
6021:
6015:
6003:
5998:
5991:(order 12).
5967:
5963:
5959:
5957:
5952:
5947:
5939:
5926:
5905:
5892:
5887:
5886:(coprime to
5883:
5876:
5872:
5868:
5864:
5857:
5851:
5846:
5842:
5838:
5831:
5827:
5819:
5815:
5808:
5804:
5800:
5796:
5794:
5789:
5783:
5777:
5769:
5765:
5761:
5755:
5751:
5747:
5743:
5718:
5669:
5637:
5628:
5614:
5612:
5607:
5601:
5587:
5583:
5577:
5571:
5567:
5563:
5559:
5555:
5549:
5517:
5512:
5504:
5500:
5496:
5492:
5484:
5480:
5476:
5472:
5468:
5464:
5458:
5450:
5422:
5412:
5408:
5402:
5395:
5389:
5375:
5362:
5355:
5353:
5350:
5336:
5331:
5327:
5321:
5309:
5302:
5295:
5290:
5178:
5174:
5170:
5166:
5162:
5158:
5154:
5144:
5131:
5112:
5107:
5101:
5095:
5093:. The first
5056:
5052:elements of
5048:
5044:
5037:
5008:
4999:
4997:
4982:
4965:
4957:
4951:
4944:
4642:
4483:
4279:
4229:In terms of
4228:
4225:
4165:
4147:
4143:
4136:
4129:
4122:
4116:
4102:
4095:
4085:
4075:
4073:consists of
4068:
4051:
4046:
3807:
3803:
3800:cycle graphs
3797:
3785:
3777:
3760:
3751:
3744:
3737:
3733:
3725:
3718:
3702:
3691:
3652:
3628:
3626:of order 2.
3624:cyclic group
3604:
3580:
3575:
3274:
3033:presentation
3026:
3022:
3012:
3007:
3003:
2997:
2991:
2987:
2976:
2973:
2637:
2634:
2108:
2090:
2079:
2057:
2051:
1825:
1814:
1804:
1801:
1767:
1420:Cayley table
1417:
1385:finite group
1377:
1037:
1017:
1005:
996:
983:
974:
966:
957:
940:
932:group theory
903:
897:
702:
690:
678:
666:
654:
642:
630:
618:
389:
346:
341:
333:
322:
311:
307:Cyclic group
252:
185:
172:Free product
143:Group action
106:Group theory
101:Group theory
100:
76:
67:
48:
6753:Point group
6748:Space group
5977:(order 6),
5788:coprime to
5772:is Euler's
5693:, i.e., to
5191:icosahedron
5183:tetrahedron
5171:tetrahedral
4120:sides (for
4091:reflections
1797:commutative
1793:composition
1380:composition
1159:reflections
924:reflections
900:mathematics
592:Topological
431:alternating
62:introducing
7117:Categories
7065:Loop group
6925:Lie groups
6697:direct sum
6251:2016-06-11
6170:References
5962:for which
5845:, and for
5586:odd): for
5287:Properties
5187:octahedron
5175:octahedral
4989:isomorphic
3781:, this is
3637:isomorphic
3613:isomorphic
1058:sides has
1032:reflection
1014:Definition
912:symmetries
699:Symplectic
639:Orthogonal
596:Lie groups
503:Free group
228:continuous
167:Direct sum
70:April 2015
45:references
6601:MathWorld
6584:MathWorld
6565:MathWorld
6546:MathWorld
6527:MathWorld
6192:MathWorld
5837:Thus for
5786:− 1
5768:), where
5644:holomorph
5552:conjugate
5479:), where
5416:subgroups
5342:inversion
5089:, ... ,
4945:(Compare
4910:−
4842:−
4622:−
4613:…
4601:∈
4458:−
4400:π
4391:
4377:π
4368:
4352:π
4343:
4337:−
4326:π
4317:
4251:π
4205:−
4154:generated
4080:rotations
3490:∣
3325:⋅
3197:∣
3160:−
3144:−
3112:
3088:
3082:∣
3011:with the
2935:π
2926:
2920:−
2906:π
2897:
2878:π
2869:
2852:π
2843:
2778:π
2769:
2752:π
2743:
2724:π
2715:
2709:−
2695:π
2686:
2596:−
2586:−
2520:−
2418:−
2350:−
2294:−
2274:−
2218:−
2029:−
1979:−
1367:stop sign
1155:rotations
1138:≥
920:rotations
884:snowflake
763:Conformal
651:Euclidean
258:nilpotent
6662:Subgroup
6485:16588559
6397:Archived
6128:See also
6092:integers
5782:1, ...,
5489:divisors
5418:of type
5163:dihedron
5119:subgroup
5117:to be a
5073:, ... ,
5040:symmetry
3730:. Since
3709:subgroup
3557:⟩
3477:⟨
3252:⟩
3184:⟨
3169:⟩
3069:⟨
2096:matrices
1828:−1
1823:, ..., s
1817:−1
1812:, ..., r
1022:Elements
951:geometry
943:geometry
936:geometry
758:Poincaré
603:Solenoid
475:Integers
465:Lattices
440:sporadic
435:Lie type
263:solvable
253:dihedral
238:additive
223:infinite
133:Subgroup
7092:History
6476:1078492
6453:Bibcode
6024:is odd;
5912:has 18
5812:coprime
5807:), for
5774:totient
5758:) = 1}
5524:T-group
5507:. See
5398:, then
5393:divides
5014:abelian
5012:is not
5004:commute
4991:to the
4968:-axis.
4093:across
3711:of the
3685:abelian
3015:-axis.
3004:πk
2988:πk
2084:of the
1402:, and S
906:is the
892:hexagon
753:Lorentz
675:Unitary
574:Lattice
514:PSL(2,
248:abelian
159:(Semi-)
58:improve
6867:22..24
6819:22..24
6815:11..12
6646:Groups
6483:
6473:
6356:
6330:
6301:
6275:
6219:
6107:circle
6101:O(2),
6086:is an
5944:, the
5315:center
5220:petals
5189:, and
5149:: the
3646:, the
3622:, the
2801:
2798:
608:Circle
539:SL(2,
428:cyclic
392:-group
243:cyclic
218:finite
213:simple
197:kernel
47:, but
7080:Sp(∞)
7077:SU(∞)
6971:Sp(n)
6965:SU(n)
6953:SO(n)
6941:SL(n)
6935:GL(n)
6688:Semi-
6427:) = 2
6413:) = D
6407:Aut(D
6400:(PDF)
6393:(PDF)
6103:i.e.,
5742:) = {
5596:2 = 2
5582:(for
5300:with
5153:of a
5138:SO(3)
5006:and D
4162:order
4114:with
4110:of a
4101:180°/
4084:360°/
3742:for
3736:>
3707:is a
2980:is a
2086:plane
1819:and s
1453:and s
1441:and r
1433:). r
1365:on a
1190:. If
987:. In
949:. In
914:of a
908:group
882:of a
792:Sp(∞)
789:SU(∞)
202:image
7074:O(∞)
6959:U(n)
6947:O(n)
6828:1..3
6481:PMID
6354:ISBN
6328:ISBN
6299:ISBN
6273:ISBN
6217:ISBN
6119:The
6097:The
6082:The
5951:for
5895:= ±1
5863:For
5826:For
5695:Hol(
5629:The
5613:For
5495:and
5407:has
5381:and
5354:For
5269:2D D
5253:2D D
5234:2D D
5210:2D D
5177:and
5157:(if
5124:O(2)
5045:The
4998:For
4590:for
4533:and
4273:and
4134:and
3798:The
3723:for
3676:and
3659:and
3392:and
2995:. s
1157:and
1104:and
978:-gon
945:and
934:and
922:and
902:, a
886:is D
878:The
786:O(∞)
775:Loop
594:and
6471:PMC
6461:doi
6060:/ Z
6050:= 2
6039:if
6034:/ Z
6020:if
5854:= 2
5814:to
5780:in
5750:| (
5646:of
5633:of
5604:= 2
5491:of
5378:/ 2
5371:of
5317:of
5305:≥ 3
5121:of
5083:r s
4987:is
4949:.)
4646:as
4388:cos
4365:sin
4340:sin
4314:cos
4160:of
4152:is
4139:= 2
4132:= 1
4125:≥ 3
3988:× Z
3984:= D
3956:× Z
3952:= D
3841:= Z
3789:/ Z
3754:= 2
3749:or
3747:= 1
3728:≥ 3
3639:to
3635:is
3615:to
3611:is
3109:ord
3085:ord
2923:cos
2894:sin
2866:sin
2840:cos
2805:and
2766:cos
2740:sin
2712:sin
2683:cos
2094:as
1778:= r
1449:, s
1398:, S
964:Dih
962:or
910:of
898:In
701:Sp(
689:SU(
665:SO(
629:SL(
617:GL(
7119::
6853:,
6846:,
6839:,
6826:Co
6817:,M
6690:)
6598:.
6581:.
6562:.
6543:.
6524:.
6479:.
6469:.
6459:.
6449:28
6447:.
6443:.
6312:^
6239:.
6215:.
6189:.
6069:).
6064:=
6052:,
5935:10
5930:10
5822:.
5799:(2
5792:.
5762:nϕ
5754:,
5746:+
5744:ax
5542:.
5281:.
5271:24
5255:12
5212:16
5185:,
5173:,
5091:rs
5087:rs
5085:,
5081:,
5077:,
5069:,
5065:,
4995:.
4277:.
4064:.
4006:=
3982:10
3845:=
3828:=
3815:.
3650:.
3589:.
3403::=
2106:.
2060:.
1799:.
1387:.
1010:.
991:,
953:,
938:.
677:U(
653:E(
641:O(
99:→
7016:8
7014:E
7008:7
7006:E
7000:6
6998:E
6992:4
6990:F
6984:2
6982:G
6909:n
6905:D
6896:n
6892:S
6880:M
6874:B
6865:F
6857:4
6855:J
6850:3
6848:J
6843:2
6841:J
6836:1
6834:J
6813:M
6799:n
6797:A
6790:n
6788:Z
6686:(
6638:e
6631:t
6624:v
6604:.
6587:.
6568:.
6549:.
6530:.
6507:.
6487:.
6463::
6455::
6425:n
6423:(
6421:φ
6416:n
6410:n
6362:.
6254:.
6225:.
6195:.
6094:.
6066:1
6062:2
6057:2
6054:D
6048:n
6042:n
6036:2
6031:n
6028:D
6022:n
6016:n
6013:D
6004:n
6001:D
5988:6
5986:D
5981:4
5979:D
5974:3
5972:D
5968:n
5966:(
5964:φ
5960:n
5953:n
5948:n
5928:D
5918:9
5909:9
5907:D
5897:.
5893:k
5888:n
5884:k
5877:n
5875:/
5873:π
5869:n
5865:n
5860:.
5858:n
5852:n
5847:n
5843:n
5839:n
5832:n
5828:n
5820:n
5816:n
5809:k
5805:n
5803:/
5801:π
5797:k
5790:n
5784:n
5778:k
5770:ϕ
5766:n
5764:(
5756:n
5752:a
5748:b
5729:Z
5719:n
5717:/
5704:Z
5680:Z
5670:n
5668:/
5655:Z
5638:n
5635:D
5615:n
5608:n
5602:n
5600:2
5594:(
5588:n
5584:n
5572:n
5568:n
5564:n
5560:n
5556:n
5540:4
5536:4
5532:4
5520:4
5513:n
5505:n
5501:n
5499:(
5497:σ
5493:n
5485:n
5483:(
5481:d
5477:n
5473:n
5471:(
5469:d
5465:n
5463:(
5459:n
5456:D
5451:m
5436:Z
5423:m
5420:D
5413:m
5411:/
5409:n
5403:n
5400:D
5396:n
5390:m
5385:2
5383:Z
5376:n
5373:D
5363:n
5360:D
5356:n
5337:n
5332:n
5328:n
5322:n
5319:D
5310:n
5303:n
5296:n
5293:D
5236:6
5222:.
5159:n
5145:n
5142:D
5132:n
5129:D
5113:n
5110:D
5102:n
5096:n
5079:s
5075:r
5071:r
5067:r
5063:e
5057:n
5054:D
5049:n
5047:2
5032:4
5030:D
5020:4
5018:D
5009:n
5000:n
4985:2
4983:D
4977:2
4966:y
4962:2
4958:x
4954:2
4924:n
4916:)
4913:k
4907:j
4904:(
4899:r
4894:=
4885:k
4880:s
4872:j
4867:s
4856:n
4848:)
4845:k
4839:j
4836:(
4831:s
4826:=
4817:k
4812:r
4804:j
4799:s
4788:n
4780:)
4777:k
4774:+
4771:j
4768:(
4763:s
4758:=
4749:k
4744:s
4736:j
4731:r
4720:n
4712:)
4709:k
4706:+
4703:j
4700:(
4695:r
4690:=
4681:k
4676:r
4668:j
4663:r
4643:n
4628:}
4625:1
4619:n
4616:,
4610:,
4607:1
4604:{
4598:j
4576:0
4571:s
4563:j
4558:r
4553:=
4548:j
4543:s
4519:j
4514:1
4509:r
4504:=
4499:j
4494:r
4467:]
4461:1
4453:0
4446:0
4441:1
4435:[
4430:=
4425:0
4420:s
4412:]
4404:n
4397:2
4381:n
4374:2
4356:n
4349:2
4330:n
4323:2
4308:[
4303:=
4298:1
4293:r
4258:n
4254:i
4248:2
4242:e
4208:1
4200:r
4195:=
4191:s
4188:r
4185:s
4171:s
4166:n
4158:r
4148:n
4145:D
4137:n
4130:n
4123:n
4117:n
4103:n
4096:n
4086:n
4076:n
4069:n
4066:D
4052:n
4049:D
4018:4
4016:D
4011:3
4008:S
4003:3
4001:D
3990:2
3986:5
3980:D
3976:9
3974:D
3970:8
3968:D
3964:7
3962:D
3958:2
3954:3
3950:6
3948:D
3868:5
3866:D
3862:4
3860:D
3856:3
3854:D
3849:4
3847:K
3843:2
3839:2
3837:D
3832:2
3830:Z
3826:1
3824:D
3808:n
3804:n
3794:.
3791:2
3786:n
3783:D
3778:n
3772:2
3770:D
3761:n
3758:D
3752:n
3745:n
3740:!
3738:n
3734:n
3732:2
3726:n
3719:n
3716:S
3703:n
3700:D
3692:n
3689:D
3680:2
3678:D
3673:1
3671:D
3663:2
3661:D
3656:1
3654:D
3643:4
3641:K
3632:2
3630:D
3619:2
3617:Z
3608:1
3606:D
3581:n
3578:D
3561:.
3553:1
3550:=
3545:n
3541:)
3537:t
3534:s
3531:(
3528:,
3525:1
3522:=
3517:2
3513:t
3509:,
3506:1
3503:=
3498:2
3494:s
3487:t
3484:,
3481:s
3473:=
3468:n
3463:D
3436:n
3431:D
3409:r
3406:s
3400:t
3380:s
3358:n
3353:D
3331:r
3328:s
3322:s
3319:=
3316:r
3296:1
3293:=
3288:2
3284:s
3256:.
3248:1
3245:=
3240:2
3236:)
3232:r
3229:s
3226:(
3223:=
3218:2
3214:s
3210:=
3205:n
3201:r
3194:s
3191:,
3188:r
3180:=
3163:1
3156:r
3152:=
3147:1
3140:s
3136:r
3133:s
3130:,
3127:2
3124:=
3121:)
3118:s
3115:(
3106:,
3103:n
3100:=
3097:)
3094:r
3091:(
3079:s
3076:,
3073:r
3065:=
3056:n
3051:D
3027:n
3024:D
3013:x
3008:n
3006:/
2998:k
2992:n
2990:/
2986:2
2977:k
2974:r
2955:.
2950:)
2942:n
2938:k
2932:2
2913:n
2909:k
2903:2
2885:n
2881:k
2875:2
2859:n
2855:k
2849:2
2834:(
2829:=
2820:k
2815:s
2793:)
2785:n
2781:k
2775:2
2759:n
2755:k
2749:2
2731:n
2727:k
2721:2
2702:n
2698:k
2692:2
2677:(
2672:=
2663:k
2658:r
2638:n
2616:.
2612:)
2604:0
2599:1
2589:1
2581:0
2573:(
2569:=
2564:3
2559:s
2552:,
2548:)
2540:1
2535:0
2528:0
2523:1
2512:(
2508:=
2503:2
2498:s
2491:,
2487:)
2479:0
2474:1
2467:1
2462:0
2454:(
2450:=
2445:1
2440:s
2433:,
2429:)
2421:1
2413:0
2406:0
2401:1
2393:(
2389:=
2384:0
2379:s
2370:,
2366:)
2358:0
2353:1
2343:1
2338:0
2330:(
2326:=
2321:3
2316:r
2309:,
2305:)
2297:1
2289:0
2282:0
2277:1
2266:(
2262:=
2257:2
2252:r
2245:,
2241:)
2233:0
2228:1
2221:1
2213:0
2205:(
2201:=
2196:1
2191:r
2184:,
2180:)
2172:1
2167:0
2160:0
2155:1
2147:(
2143:=
2138:0
2133:r
2113:4
2111:D
2091:n
2058:n
2037:.
2032:j
2026:i
2021:r
2016:=
2011:j
2006:s
1998:i
1993:s
1987:,
1982:j
1976:i
1971:s
1966:=
1961:j
1956:r
1948:i
1943:s
1937:,
1932:j
1929:+
1926:i
1921:s
1916:=
1911:j
1906:s
1898:i
1893:r
1887:,
1882:j
1879:+
1876:i
1871:r
1866:=
1861:j
1856:r
1848:i
1843:r
1826:n
1821:0
1815:n
1810:0
1805:n
1789:2
1785:1
1780:1
1776:1
1774:s
1772:2
1770:s
1761:0
1759:r
1755:1
1753:r
1749:2
1747:r
1743:0
1741:s
1737:1
1735:s
1731:2
1729:s
1725:2
1723:s
1717:2
1715:r
1711:0
1709:r
1705:1
1703:r
1699:2
1697:s
1693:0
1691:s
1687:1
1685:s
1681:1
1679:s
1673:1
1671:r
1667:2
1665:r
1661:0
1659:r
1655:1
1653:s
1649:2
1647:s
1643:0
1641:s
1637:0
1635:s
1629:1
1627:s
1623:0
1621:s
1617:2
1615:s
1611:1
1609:r
1605:0
1603:r
1599:2
1597:r
1593:2
1591:r
1585:0
1583:s
1579:2
1577:s
1573:1
1571:s
1567:0
1565:r
1561:2
1559:r
1555:1
1553:r
1549:1
1547:r
1541:2
1539:s
1535:1
1533:s
1529:0
1527:s
1523:2
1521:r
1517:1
1515:r
1511:0
1509:r
1505:0
1503:r
1497:2
1495:s
1491:1
1489:s
1485:0
1483:s
1479:2
1477:r
1473:1
1471:r
1467:0
1465:r
1455:2
1451:1
1447:0
1443:2
1439:1
1435:0
1426:3
1424:D
1404:2
1400:1
1396:0
1351:8
1346:D
1317:n
1314:2
1294:n
1274:2
1270:/
1266:n
1246:2
1242:/
1238:n
1218:n
1198:n
1176:n
1171:D
1141:3
1135:n
1112:n
1089:n
1069:n
1066:2
1046:n
1006:n
1003:D
997:n
995:2
993:D
984:n
982:2
975:n
967:n
958:n
955:D
894:.
888:6
863:e
856:t
849:v
745:8
743:E
737:7
735:E
729:6
727:E
721:4
719:F
713:2
711:G
705:)
703:n
693:)
691:n
681:)
679:n
669:)
667:n
657:)
655:n
645:)
643:n
633:)
631:n
621:)
619:n
561:)
548:Z
536:)
523:Z
499:)
486:Z
477:(
390:p
355:Q
347:n
344:D
334:n
331:A
323:n
320:S
312:n
309:Z
83:)
77:(
72:)
68:(
54:.
20:)
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