25:
4684:
3315:
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is isomorphic to the integers (with the addition operation). From an algebraic point of view, this means that the set of all integers (with the addition operation) is the "only" infinite cyclic group.
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1600:
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3025:
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For all abelian groups there is at least the automorphism that replaces the group elements by their inverses. However, in groups where all elements are equal to their inverses this is the
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2010:
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222:
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141:
between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called
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Whether such a notation is possible without confusion or ambiguity depends on context. For example, the equals sign is not very suitable when the groups are both
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340:
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3800:
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one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to
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42:
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has order 168, as can be found as follows. All 7 non-identity elements play the same role, so we can choose which plays the role of
623:
Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes
108:
1617:
89:
4679:{\displaystyle \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}=\operatorname {Dih} _{2}\oplus \mathbb {Z} _{2}}
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75:
35:
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1134:
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4727:
Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which element corresponds to
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of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group
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2018:
1776:
1714:
1611:
1491:
626:
4180:
4075:
1806:
57:
4809:
3755:
2657:. The first few numbers are 0, 1, 1, 1 and 2 meaning that 4 is the lowest order with more than one group.
1229:
group homomorphism (the inverse function of a bijective group homomorphism is also a group homomorphism).
130:
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1981:
1947:
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3310:{\displaystyle \varphi (x^{a}\cdot x^{b})=\varphi (x^{a+b})=a+b=\varphi (x^{a})+_{n}\varphi (x^{b}),}
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2434:
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of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to
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of non-zero complex numbers with multiplication as the operation is isomorphic to the group
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693:
122:
4153:
3899:
2719:
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2340:
2209:
2175:
2099:
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1903:
1867:
1523:
507:
231:
160:
4855:
4300:
4039:
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1937:
1222:
are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.
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Intuitively, group theorists view two isomorphic groups as follows: For every element
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1940:, but the proof does not indicate how to construct a concrete isomorphism. Examples:
568:
are isomorphic if there exists an isomorphism from one to the other. This is written
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146:
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by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because
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24:
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elements. The lines connecting three points correspond to the group operation:
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There is one more automorphism with this property: multiplying all elements of
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5037:
749:
293:
149:, isomorphic groups have the same properties and need not be distinguished.
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by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of
684:
342:
Spelled out, this means that a group isomorphism is a bijective function
1457:
1237:
In this section some notable examples of isomorphic groups are listed.
2963:{\displaystyle \langle x\rangle =\left\{e,x,\ldots ,x^{n-1}\right\}.}
4407:
while lower powers do not give 1. Thus this automorphism generates
3102:{\displaystyle \varphi :G\to \mathbb {Z} _{n}=\{0,1,\ldots ,n-1\},}
3845:
that is only related to the group structure can be translated via
3586:{\displaystyle f(u^{-1})=f(u)^{-1}\quad {\text{ for all }}u\in G,}
3702:{\displaystyle f(u^{n})=f(u)^{n}\quad {\text{ for all }}u\in G,}
2654:
1380:{\displaystyle (\mathbb {R} ,+)\cong (\mathbb {R} ^{+},\times )}
5084:
Discovering Group Theory: A Transition to
Advanced Mathematics
4806:
we can choose from 4, which determines the rest. Thus we have
1094:(operates with other elements of the group in the same way as
18:
1936:
Some groups can be proven to be isomorphic, relying on the
1657:{\displaystyle \mathbb {Z} _{2}=\mathbb {Z} /2\mathbb {Z} }
1225:
An isomorphism of groups may equivalently be defined as an
4596:
so apart from the identity we can only interchange these.
1704:{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}.}
4564:
because only each of the two elements 1 and 5 generate
5116:. Cambridge: Cambridge University Press. p. 142.
5012:
For abelian groups, all non-trivial automorphisms are
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From the definition, it follows that any isomorphism
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All cyclic groups of a given order are isomorphic to
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The number of distinct groups (up to isomorphism) of
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195:
163:
1595:{\displaystyle \mathbb {R} /\mathbb {Z} \cong S^{1}}
4042:is always a conjugacy class (the same or another).
49:. Unsourced material may be challenged and removed.
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5055:(2nd ed.). New York: John Wiley & Sons.
3367:{\displaystyle G\cong (\mathbb {Z} _{n},+_{n}).}
3020:{\displaystyle G\cong (\mathbb {Z} _{n},+_{n}).}
4844:automorphisms. They correspond to those of the
5150:Journal of the Australian Mathematical Society
5077:
5075:
5023:group, and possibly also outer automorphisms.
8:
4848:, of which the 7 points correspond to the 7
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1317:{\displaystyle (\mathbb {R} ^{+},\times )}
687:of the same group. See also the examples.
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4400:{\displaystyle 3^{6}\equiv 1{\pmod {7}},}
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2709:{\displaystyle (\mathbb {Z} _{n},+_{n}),}
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2049:{\displaystyle (\mathbb {C} ^{*},\cdot )}
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2027:
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2020:
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1949:
1905:
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1808:
1796:{\displaystyle \operatorname {Dih} _{2n}}
1784:
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1734:{\displaystyle \operatorname {Dih} _{2},}
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1513:{\displaystyle \mathbb {R} /\mathbb {Z} }
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109:Learn how and when to remove this message
4197:{\displaystyle \operatorname {Dih} _{3}}
4100:{\displaystyle \operatorname {Aut} (G),}
1823:{\displaystyle \operatorname {Dih} _{n}}
5082:Barnard, Tony & Neil, Hugh (2017).
5071:
5007:general linear group over finite fields
1803:is isomorphic to the direct product of
5146:"A Consequence of the Axiom of Choice"
5019:Non-abelian groups have a non-trivial
2012:of all complex numbers under addition.
899:{\displaystyle f(u)\odot f(v)=f(u*v).}
614:{\displaystyle (G,*)\cong (H,\odot ).}
495:{\displaystyle f(u*v)=f(u)\odot f(v).}
5086:. Boca Ratan: CRC Press. p. 94.
4837:{\displaystyle 7\times 6\times 4=168}
3932:of the group. Thus it is a bijection
3782:is an isomorphism between two groups
7:
654:Sometimes one can even simply write
47:adding citations to reliable sources
4386:
3825:then everything that is true about
4290:{\displaystyle \mathbb {Z} _{p-1}}
3865:into a true ditto statement about
3147:{\displaystyle \varphi (x^{a})=a.}
1182:This implies, in particular, that
14:
5040: – One-to-one correspondence
4589:{\displaystyle \mathbb {Z} _{6},}
4557:{\displaystyle \mathbb {Z} _{2},}
4493:{\displaystyle \mathbb {Z} _{6},}
4432:{\displaystyle \mathbb {Z} _{6}.}
4025:{\displaystyle f(u)*f(v)=f(u*v).}
3413:will map the identity element of
1855:{\displaystyle \mathbb {Z} _{2}.}
1460:(with addition) is a subgroup of
4525:{\displaystyle \mathbb {Z} _{6}}
4461:{\displaystyle \mathbb {Z} _{7}}
4348:{\displaystyle \mathbb {Z} _{7}}
4229:{\displaystyle \mathbb {Z} _{p}}
2005:{\displaystyle (\mathbb {C} ,+)}
1971:{\displaystyle (\mathbb {R} ,+)}
1279:, is isomorphic to the group of
1272:{\displaystyle (\mathbb {R} ,+)}
23:
4379:
4177:(which itself is isomorphic to
3681:
3565:
3500:{\displaystyle f(e_{G})=e_{H},}
1664:, and can therefore be written
34:needs additional citations for
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4775:
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3011:
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2700:
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2510:
2498:
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2467:
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2147:
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2115:
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1333:
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1:
4500:in that order or conversely.
3751:is also a group isomorphism.
3744:{\displaystyle f^{-1}:H\to G}
1478:{\displaystyle \mathbb {R} ,}
1074:"behaves in the same way" as
4326:multiplying all elements of
3896:An isomorphism from a group
1449:{\displaystyle \mathbb {Z} }
16:Bijective group homomorphism
4038:under an automorphism of a
3433:to the identity element of
2397:{\displaystyle (H,\odot ),}
1978:is isomorphic to the group
1748:Generalizing this, for all
1520:is isomorphic to the group
217:{\displaystyle (H,\odot ),}
5212:
4599:The automorphism group of
4503:The automorphism group of
4107:itself forms a group, the
2616:{\displaystyle (H,\odot )}
2548:{\displaystyle (H,\odot )}
2263:{\displaystyle (H,\odot )}
2153:{\displaystyle (H,\odot )}
1555:1 (under multiplication):
1422:{\displaystyle f(x)=e^{x}}
831:{\displaystyle (H,\odot )}
690:Conversely, given a group
561:{\displaystyle (H,\odot )}
285:{\displaystyle (H,\odot )}
5163:10.1017/S1446788700031505
5110:The Fascination of Groups
5033:Group isomorphism problem
3758:"being isomorphic" is an
3709:and that the inverse map
960:then the bijection is an
779:{\displaystyle f:G\to H,}
645:{\displaystyle G\cong H.}
145:. From the standpoint of
5051:Herstein, I. N. (1975).
4761:{\displaystyle (1,1,0).}
4720:{\displaystyle (1,0,0).}
3957:{\displaystyle f:G\to G}
3406:{\displaystyle f:G\to H}
3167:{\displaystyle \varphi }
1014:there exists an element
953:{\displaystyle \odot =*}
367:{\displaystyle f:G\to H}
4799:{\displaystyle (0,0,1)}
3928:to itself is called an
2426:{\displaystyle a\in G,}
2337:is an isomorphism from
2096:of an isomorphism from
153:Definition and notation
5107:Budden, F. J. (1972).
4999:
4998:{\displaystyle b+c=a.}
4964:
4963:{\displaystyle a+c=b,}
4930:
4929:{\displaystyle a+b=c,}
4895:
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4800:
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2791:be a cyclic group and
2785:
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2077:
2050:
2006:
1972:
1926:
1890:
1856:
1824:
1797:
1768:
1735:
1705:
1658:
1596:
1541:
1514:
1479:
1450:
1423:
1381:
1318:
1273:
1216:
1196:
1176:
1153:
1128:
1108:
1088:
1068:
1048:
1028:
1008:
985:
954:
928:
900:
832:
800:
780:
742:
719:
718:{\displaystyle (G,*),}
677:
646:
615:
562:
530:
496:
428:
408:
388:
368:
336:
313:
286:
254:
218:
183:
5133:– via VDOC.PUB.
5000:
4965:
4931:
4896:
4876:
4839:
4801:
4763:
4722:
4681:
4591:
4559:
4527:
4495:
4463:
4434:
4402:
4350:
4321:
4292:
4257:
4231:
4199:
4172:
4170:{\displaystyle S_{3}}
4146:. For that group all
4131:
4102:
4067:
4027:
3959:
3923:
3921:{\displaystyle (G,*)}
3883:
3860:
3840:
3820:
3797:
3777:
3746:
3704:
3628:
3608:
3588:
3502:
3451:
3428:
3408:
3369:
3312:
3169:
3149:
3104:
3022:
2965:
2889:
2869:
2849:
2829:
2806:
2786:
2764:
2738:
2736:{\displaystyle +_{n}}
2711:
2644:
2618:
2582:
2580:{\displaystyle (G,*)}
2555:are isomorphic, then
2550:
2518:
2516:{\displaystyle (G,*)}
2484:
2482:{\displaystyle f(a).}
2452:
2428:
2399:
2364:
2362:{\displaystyle (G,*)}
2332:
2309:
2285:
2270:are isomorphic, then
2265:
2233:
2231:{\displaystyle (G,*)}
2199:
2197:{\displaystyle (G,*)}
2155:
2123:
2121:{\displaystyle (G,*)}
2078:
2076:{\displaystyle S^{1}}
2051:
2007:
1973:
1927:
1925:{\displaystyle (G,*)}
1898:infinite cyclic group
1891:
1889:{\displaystyle (G,*)}
1857:
1825:
1798:
1769:
1736:
1706:
1659:
1610:is isomorphic to the
1597:
1542:
1540:{\displaystyle S^{1}}
1515:
1480:
1451:
1424:
1382:
1319:
1283:under multiplication
1281:positive real numbers
1274:
1217:
1197:
1177:
1154:
1129:
1109:
1089:
1069:
1049:
1029:
1009:
986:
955:
929:
901:
833:
801:
781:
743:
720:
678:
647:
616:
563:
531:
529:{\displaystyle (G,*)}
497:
429:
409:
389:
369:
337:
314:
287:
255:
253:{\displaystyle (G,*)}
219:
184:
182:{\displaystyle (G,*)}
4974:
4939:
4905:
4885:
4874:{\displaystyle a,b,}
4856:
4810:
4772:
4731:
4690:
4603:
4568:
4536:
4507:
4472:
4443:
4411:
4359:
4330:
4319:{\displaystyle n=7,}
4301:
4266:
4243:
4211:
4181:
4154:
4140:trivial automorphism
4117:
4076:
4053:
3968:
3936:
3900:
3869:
3849:
3829:
3806:
3786:
3766:
3760:equivalence relation
3713:
3637:
3617:
3597:
3593:and more generally,
3515:
3459:
3437:
3417:
3385:
3321:
3178:
3158:
3113:
3033:
2974:
2898:
2878:
2858:
2838:
2815:
2795:
2775:
2750:
2720:
2669:
2633:
2623:is locally finite.
2595:
2589:locally finite group
2559:
2527:
2495:
2461:
2457:equals the order of
2441:
2408:
2373:
2341:
2321:
2298:
2274:
2242:
2210:
2176:
2132:
2100:
2060:
2019:
1982:
1948:
1904:
1868:
1834:
1807:
1777:
1755:
1715:
1711:Another notation is
1668:
1618:
1561:
1524:
1492:
1464:
1438:
1391:
1387:via the isomorphism
1330:
1287:
1249:
1206:
1186:
1163:
1140:
1118:
1114:). For instance, if
1098:
1078:
1058:
1038:
1018:
995:
975:
938:
912:
842:
810:
790:
755:
729:
694:
676:{\displaystyle G=H.}
658:
627:
572:
540:
508:
438:
418:
398:
378:
346:
323:
303:
264:
232:
193:
161:
43:improve this article
5014:outer automorphisms
3684: for all
3568: for all
3174:is bijective. Then
927:{\displaystyle H=G}
58:"Group isomorphism"
5021:inner automorphism
4995:
4960:
4926:
4901:on one line means
4891:
4871:
4834:
4796:
4758:
4717:
4676:
4586:
4554:
4522:
4490:
4458:
4429:
4397:
4345:
4316:
4287:
4255:{\displaystyle p,}
4252:
4226:
4194:
4167:
4129:{\displaystyle G.}
4126:
4110:automorphism group
4097:
4065:{\displaystyle G,}
4062:
4022:
3954:
3918:
3881:{\displaystyle H,}
3878:
3855:
3835:
3818:{\displaystyle H,}
3815:
3792:
3772:
3741:
3699:
3623:
3603:
3583:
3497:
3449:{\displaystyle H,}
3446:
3423:
3403:
3364:
3317:which proves that
3307:
3164:
3144:
3099:
3017:
2970:We will show that
2960:
2884:
2864:
2854:be a generator of
2844:
2827:{\displaystyle G.}
2824:
2801:
2781:
2762:{\displaystyle n.}
2759:
2733:
2706:
2639:
2613:
2577:
2545:
2513:
2479:
2447:
2423:
2394:
2359:
2327:
2304:
2280:
2260:
2228:
2194:
2150:
2118:
2073:
2046:
2002:
1968:
1922:
1886:
1852:
1820:
1793:
1767:{\displaystyle n,}
1764:
1731:
1701:
1654:
1592:
1537:
1510:
1475:
1446:
1419:
1377:
1314:
1269:
1212:
1192:
1175:{\displaystyle h.}
1172:
1152:{\displaystyle G,}
1149:
1124:
1104:
1084:
1064:
1044:
1024:
1007:{\displaystyle G,}
1004:
981:
950:
924:
896:
828:
796:
776:
741:{\displaystyle H,}
738:
715:
673:
642:
611:
558:
526:
492:
424:
404:
384:
374:such that for all
364:
335:{\displaystyle H.}
332:
309:
297:group homomorphism
282:
250:
214:
179:
5053:Topics in Algebra
4894:{\displaystyle c}
4532:is isomorphic to
4297:For example, for
3858:{\displaystyle f}
3838:{\displaystyle G}
3795:{\displaystyle G}
3775:{\displaystyle f}
3685:
3626:{\displaystyle n}
3606:{\displaystyle n}
3569:
3507:that it will map
3426:{\displaystyle G}
2894:is then equal to
2887:{\displaystyle G}
2867:{\displaystyle G}
2847:{\displaystyle x}
2804:{\displaystyle n}
2784:{\displaystyle G}
2743:denotes addition
2642:{\displaystyle n}
2450:{\displaystyle a}
2330:{\displaystyle f}
2307:{\displaystyle H}
2283:{\displaystyle G}
1614:of two copies of
1241:The group of all
1215:{\displaystyle H}
1195:{\displaystyle G}
1127:{\displaystyle g}
1107:{\displaystyle g}
1087:{\displaystyle g}
1067:{\displaystyle h}
1047:{\displaystyle H}
1027:{\displaystyle h}
984:{\displaystyle g}
799:{\displaystyle H}
427:{\displaystyle G}
407:{\displaystyle v}
387:{\displaystyle u}
312:{\displaystyle G}
226:group isomorphism
157:Given two groups
127:group isomorphism
119:
118:
111:
93:
5203:
5175:
5174:
5172:
5170:
5165:
5141:
5135:
5134:
5132:
5130:
5115:
5104:
5098:
5097:
5079:
5066:
5004:
5002:
5001:
4996:
4969:
4967:
4966:
4961:
4935:
4933:
4932:
4927:
4900:
4898:
4897:
4892:
4880:
4878:
4877:
4872:
4851:
4843:
4841:
4840:
4835:
4805:
4803:
4802:
4797:
4767:
4765:
4764:
4759:
4726:
4724:
4723:
4718:
4685:
4683:
4682:
4677:
4675:
4674:
4669:
4660:
4659:
4647:
4646:
4641:
4632:
4631:
4626:
4617:
4616:
4611:
4595:
4593:
4592:
4587:
4582:
4581:
4576:
4563:
4561:
4560:
4555:
4550:
4549:
4544:
4531:
4529:
4528:
4523:
4521:
4520:
4515:
4499:
4497:
4496:
4491:
4486:
4485:
4480:
4467:
4465:
4464:
4459:
4457:
4456:
4451:
4438:
4436:
4435:
4430:
4425:
4424:
4419:
4406:
4404:
4403:
4398:
4393:
4371:
4370:
4354:
4352:
4351:
4346:
4344:
4343:
4338:
4325:
4323:
4322:
4317:
4296:
4294:
4293:
4288:
4286:
4285:
4274:
4261:
4259:
4258:
4253:
4235:
4233:
4232:
4227:
4225:
4224:
4219:
4203:
4201:
4200:
4195:
4193:
4192:
4176:
4174:
4173:
4168:
4166:
4165:
4144:Klein four-group
4135:
4133:
4132:
4127:
4106:
4104:
4103:
4098:
4071:
4069:
4068:
4063:
4031:
4029:
4028:
4023:
3963:
3961:
3960:
3955:
3927:
3925:
3924:
3919:
3888:and vice versa.
3887:
3885:
3884:
3879:
3864:
3862:
3861:
3856:
3844:
3842:
3841:
3836:
3824:
3822:
3821:
3816:
3801:
3799:
3798:
3793:
3781:
3779:
3778:
3773:
3750:
3748:
3747:
3742:
3728:
3727:
3708:
3706:
3705:
3700:
3686:
3683:
3680:
3679:
3655:
3654:
3632:
3630:
3629:
3624:
3612:
3610:
3609:
3604:
3592:
3590:
3589:
3584:
3570:
3567:
3564:
3563:
3536:
3535:
3506:
3504:
3503:
3498:
3493:
3492:
3477:
3476:
3455:
3453:
3452:
3447:
3432:
3430:
3429:
3424:
3412:
3410:
3409:
3404:
3373:
3371:
3370:
3365:
3357:
3356:
3344:
3343:
3338:
3316:
3314:
3313:
3308:
3300:
3299:
3284:
3283:
3271:
3270:
3237:
3236:
3209:
3208:
3196:
3195:
3173:
3171:
3170:
3165:
3153:
3151:
3150:
3145:
3131:
3130:
3108:
3106:
3105:
3100:
3059:
3058:
3053:
3026:
3024:
3023:
3018:
3010:
3009:
2997:
2996:
2991:
2969:
2967:
2966:
2961:
2956:
2952:
2951:
2950:
2893:
2891:
2890:
2885:
2873:
2871:
2870:
2865:
2853:
2851:
2850:
2845:
2833:
2831:
2830:
2825:
2811:be the order of
2810:
2808:
2807:
2802:
2790:
2788:
2787:
2782:
2768:
2766:
2765:
2760:
2742:
2740:
2739:
2734:
2732:
2731:
2715:
2713:
2712:
2707:
2699:
2698:
2686:
2685:
2680:
2648:
2646:
2645:
2640:
2622:
2620:
2619:
2614:
2586:
2584:
2583:
2578:
2554:
2552:
2551:
2546:
2522:
2520:
2519:
2514:
2488:
2486:
2485:
2480:
2456:
2454:
2453:
2448:
2432:
2430:
2429:
2424:
2403:
2401:
2400:
2395:
2368:
2366:
2365:
2360:
2336:
2334:
2333:
2328:
2313:
2311:
2310:
2305:
2289:
2287:
2286:
2281:
2269:
2267:
2266:
2261:
2237:
2235:
2234:
2229:
2203:
2201:
2200:
2195:
2159:
2157:
2156:
2151:
2127:
2125:
2124:
2119:
2083:mentioned above.
2082:
2080:
2079:
2074:
2072:
2071:
2055:
2053:
2052:
2047:
2036:
2035:
2030:
2011:
2009:
2008:
2003:
1992:
1977:
1975:
1974:
1969:
1958:
1931:
1929:
1928:
1923:
1895:
1893:
1892:
1887:
1861:
1859:
1858:
1853:
1848:
1847:
1842:
1829:
1827:
1826:
1821:
1819:
1818:
1802:
1800:
1799:
1794:
1792:
1791:
1773:
1771:
1770:
1765:
1741:because it is a
1740:
1738:
1737:
1732:
1727:
1726:
1710:
1708:
1707:
1702:
1697:
1696:
1691:
1682:
1681:
1676:
1663:
1661:
1660:
1655:
1653:
1645:
1640:
1632:
1631:
1626:
1608:Klein four-group
1601:
1599:
1598:
1593:
1591:
1590:
1578:
1573:
1568:
1546:
1544:
1543:
1538:
1536:
1535:
1519:
1517:
1516:
1511:
1509:
1504:
1499:
1484:
1482:
1481:
1476:
1471:
1455:
1453:
1452:
1447:
1445:
1428:
1426:
1425:
1420:
1418:
1417:
1386:
1384:
1383:
1378:
1367:
1366:
1361:
1340:
1323:
1321:
1320:
1315:
1304:
1303:
1298:
1278:
1276:
1275:
1270:
1259:
1245:under addition,
1221:
1219:
1218:
1213:
1201:
1199:
1198:
1193:
1181:
1179:
1178:
1173:
1158:
1156:
1155:
1150:
1133:
1131:
1130:
1125:
1113:
1111:
1110:
1105:
1093:
1091:
1090:
1085:
1073:
1071:
1070:
1065:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1013:
1011:
1010:
1005:
990:
988:
987:
982:
959:
957:
956:
951:
933:
931:
930:
925:
905:
903:
902:
897:
837:
835:
834:
829:
805:
803:
802:
797:
785:
783:
782:
777:
747:
745:
744:
739:
724:
722:
721:
716:
682:
680:
679:
674:
651:
649:
648:
643:
620:
618:
617:
612:
567:
565:
564:
559:
535:
533:
532:
527:
501:
499:
498:
493:
433:
431:
430:
425:
413:
411:
410:
405:
393:
391:
390:
385:
373:
371:
370:
365:
341:
339:
338:
333:
318:
316:
315:
310:
291:
289:
288:
283:
259:
257:
256:
251:
223:
221:
220:
215:
188:
186:
185:
180:
123:abstract algebra
114:
107:
103:
100:
94:
92:
51:
27:
19:
5211:
5210:
5206:
5205:
5204:
5202:
5201:
5200:
5181:
5180:
5179:
5178:
5168:
5166:
5143:
5142:
5138:
5128:
5126:
5124:
5113:
5106:
5105:
5101:
5094:
5081:
5080:
5073:
5063:
5050:
5047:
5029:
4972:
4971:
4937:
4936:
4903:
4902:
4883:
4882:
4854:
4853:
4849:
4808:
4807:
4770:
4769:
4729:
4728:
4688:
4687:
4664:
4651:
4636:
4621:
4606:
4601:
4600:
4571:
4566:
4565:
4539:
4534:
4533:
4510:
4505:
4504:
4475:
4470:
4469:
4446:
4441:
4440:
4414:
4409:
4408:
4362:
4357:
4356:
4333:
4328:
4327:
4299:
4298:
4269:
4264:
4263:
4241:
4240:
4214:
4209:
4208:
4184:
4179:
4178:
4157:
4152:
4151:
4115:
4114:
4074:
4073:
4051:
4050:
4040:conjugacy class
3966:
3965:
3934:
3933:
3898:
3897:
3894:
3867:
3866:
3847:
3846:
3827:
3826:
3804:
3803:
3784:
3783:
3764:
3763:
3716:
3711:
3710:
3671:
3646:
3635:
3634:
3615:
3614:
3595:
3594:
3552:
3524:
3513:
3512:
3484:
3468:
3457:
3456:
3435:
3434:
3415:
3414:
3383:
3382:
3379:
3348:
3333:
3319:
3318:
3291:
3275:
3262:
3222:
3200:
3187:
3176:
3175:
3156:
3155:
3122:
3111:
3110:
3048:
3031:
3030:
3001:
2986:
2972:
2971:
2936:
2917:
2913:
2896:
2895:
2876:
2875:
2856:
2855:
2836:
2835:
2813:
2812:
2793:
2792:
2773:
2772:
2748:
2747:
2723:
2718:
2717:
2690:
2675:
2667:
2666:
2663:
2653:A000001 in the
2631:
2630:
2593:
2592:
2591:if and only if
2557:
2556:
2525:
2524:
2493:
2492:
2459:
2458:
2439:
2438:
2406:
2405:
2371:
2370:
2339:
2338:
2319:
2318:
2296:
2295:
2294:if and only if
2272:
2271:
2240:
2239:
2208:
2207:
2174:
2173:
2167:
2163:
2130:
2129:
2098:
2097:
2090:
2063:
2058:
2057:
2025:
2017:
2016:
1980:
1979:
1946:
1945:
1938:axiom of choice
1902:
1901:
1866:
1865:
1837:
1832:
1831:
1810:
1805:
1804:
1780:
1775:
1774:
1753:
1752:
1718:
1713:
1712:
1686:
1671:
1666:
1665:
1621:
1616:
1615:
1582:
1559:
1558:
1549:complex numbers
1527:
1522:
1521:
1490:
1489:
1462:
1461:
1436:
1435:
1409:
1389:
1388:
1356:
1328:
1327:
1293:
1285:
1284:
1247:
1246:
1235:
1204:
1203:
1184:
1183:
1161:
1160:
1138:
1137:
1116:
1115:
1096:
1095:
1076:
1075:
1056:
1055:
1036:
1035:
1016:
1015:
993:
992:
973:
972:
936:
935:
910:
909:
840:
839:
808:
807:
788:
787:
753:
752:
727:
726:
692:
691:
656:
655:
625:
624:
570:
569:
538:
537:
506:
505:
504:The two groups
436:
435:
416:
415:
396:
395:
376:
375:
344:
343:
321:
320:
301:
300:
262:
261:
230:
229:
191:
190:
159:
158:
155:
137:that sets up a
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
5209:
5207:
5199:
5198:
5193:
5183:
5182:
5177:
5176:
5156:(3): 306–308.
5136:
5122:
5099:
5092:
5070:
5069:
5068:
5067:
5061:
5046:
5043:
5042:
5041:
5035:
5028:
5025:
4994:
4991:
4988:
4985:
4982:
4979:
4959:
4956:
4953:
4950:
4947:
4944:
4925:
4922:
4919:
4916:
4913:
4910:
4890:
4870:
4867:
4864:
4861:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4673:
4668:
4663:
4658:
4654:
4650:
4645:
4640:
4635:
4630:
4625:
4620:
4615:
4610:
4585:
4580:
4575:
4553:
4548:
4543:
4519:
4514:
4489:
4484:
4479:
4455:
4450:
4428:
4423:
4418:
4396:
4392:
4389:
4385:
4382:
4377:
4374:
4369:
4365:
4342:
4337:
4315:
4312:
4309:
4306:
4284:
4281:
4278:
4273:
4251:
4248:
4223:
4218:
4191:
4187:
4164:
4160:
4142:, e.g. in the
4125:
4122:
4096:
4093:
4090:
4087:
4084:
4081:
4061:
4058:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3953:
3950:
3947:
3944:
3941:
3917:
3914:
3911:
3908:
3905:
3893:
3890:
3877:
3874:
3854:
3834:
3814:
3811:
3791:
3771:
3740:
3737:
3734:
3731:
3726:
3723:
3719:
3698:
3695:
3692:
3689:
3678:
3674:
3670:
3667:
3664:
3661:
3658:
3653:
3649:
3645:
3642:
3622:
3602:
3582:
3579:
3576:
3573:
3562:
3559:
3555:
3551:
3548:
3545:
3542:
3539:
3534:
3531:
3527:
3523:
3520:
3496:
3491:
3487:
3483:
3480:
3475:
3471:
3467:
3464:
3445:
3442:
3422:
3402:
3399:
3396:
3393:
3390:
3378:
3375:
3363:
3360:
3355:
3351:
3347:
3342:
3337:
3332:
3329:
3326:
3306:
3303:
3298:
3294:
3290:
3287:
3282:
3278:
3274:
3269:
3265:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3235:
3232:
3229:
3225:
3221:
3218:
3215:
3212:
3207:
3203:
3199:
3194:
3190:
3186:
3183:
3163:
3143:
3140:
3137:
3134:
3129:
3125:
3121:
3118:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3057:
3052:
3047:
3044:
3041:
3038:
3016:
3013:
3008:
3004:
3000:
2995:
2990:
2985:
2982:
2979:
2959:
2955:
2949:
2946:
2943:
2939:
2935:
2932:
2929:
2926:
2923:
2920:
2916:
2912:
2909:
2906:
2903:
2883:
2863:
2843:
2823:
2820:
2800:
2780:
2758:
2755:
2730:
2726:
2705:
2702:
2697:
2693:
2689:
2684:
2679:
2674:
2662:
2659:
2638:
2612:
2609:
2606:
2603:
2600:
2576:
2573:
2570:
2567:
2564:
2544:
2541:
2538:
2535:
2532:
2512:
2509:
2506:
2503:
2500:
2478:
2475:
2472:
2469:
2466:
2446:
2422:
2419:
2416:
2413:
2393:
2390:
2387:
2384:
2381:
2378:
2358:
2355:
2352:
2349:
2346:
2326:
2303:
2279:
2259:
2256:
2253:
2250:
2247:
2227:
2224:
2221:
2218:
2215:
2193:
2190:
2187:
2184:
2181:
2165:
2161:
2149:
2146:
2143:
2140:
2137:
2117:
2114:
2111:
2108:
2105:
2089:
2086:
2085:
2084:
2070:
2066:
2045:
2042:
2039:
2034:
2029:
2024:
2013:
2001:
1998:
1995:
1991:
1987:
1967:
1964:
1961:
1957:
1953:
1934:
1933:
1921:
1918:
1915:
1912:
1909:
1885:
1882:
1879:
1876:
1873:
1862:
1851:
1846:
1841:
1817:
1813:
1790:
1787:
1783:
1763:
1760:
1746:
1743:dihedral group
1730:
1725:
1721:
1700:
1695:
1690:
1685:
1680:
1675:
1652:
1648:
1644:
1639:
1635:
1630:
1625:
1612:direct product
1604:
1603:
1602:
1589:
1585:
1581:
1577:
1572:
1567:
1553:absolute value
1534:
1530:
1508:
1503:
1498:
1474:
1470:
1444:
1432:
1431:
1430:
1416:
1412:
1408:
1405:
1402:
1399:
1396:
1376:
1373:
1370:
1365:
1360:
1355:
1352:
1349:
1346:
1343:
1339:
1335:
1313:
1310:
1307:
1302:
1297:
1292:
1268:
1265:
1262:
1258:
1254:
1234:
1231:
1211:
1191:
1171:
1168:
1148:
1145:
1123:
1103:
1083:
1063:
1043:
1023:
1003:
1000:
980:
949:
946:
943:
923:
920:
917:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
827:
824:
821:
818:
815:
795:
775:
772:
769:
766:
763:
760:
737:
734:
714:
711:
708:
705:
702:
699:
672:
669:
666:
663:
641:
638:
635:
632:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
557:
554:
551:
548:
545:
525:
522:
519:
516:
513:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
434:it holds that
423:
403:
383:
363:
360:
357:
354:
351:
331:
328:
308:
281:
278:
275:
272:
269:
249:
246:
243:
240:
237:
213:
210:
207:
204:
201:
198:
178:
175:
172:
169:
166:
154:
151:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
5208:
5197:
5194:
5192:
5189:
5188:
5186:
5164:
5159:
5155:
5151:
5147:
5140:
5137:
5125:
5119:
5112:
5111:
5103:
5100:
5095:
5093:9781138030169
5089:
5085:
5078:
5076:
5072:
5064:
5058:
5054:
5049:
5048:
5044:
5039:
5036:
5034:
5031:
5030:
5026:
5024:
5022:
5017:
5015:
5010:
5008:
4992:
4989:
4986:
4983:
4980:
4977:
4957:
4954:
4951:
4948:
4945:
4942:
4923:
4920:
4917:
4914:
4911:
4908:
4888:
4868:
4865:
4862:
4859:
4847:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4790:
4787:
4784:
4781:
4778:
4755:
4749:
4746:
4743:
4740:
4737:
4714:
4708:
4705:
4702:
4699:
4696:
4671:
4661:
4656:
4652:
4648:
4643:
4633:
4628:
4618:
4613:
4597:
4583:
4578:
4551:
4546:
4517:
4501:
4487:
4482:
4453:
4426:
4421:
4394:
4387:
4383:
4375:
4372:
4367:
4363:
4340:
4313:
4310:
4307:
4304:
4282:
4279:
4276:
4249:
4246:
4239:
4221:
4205:
4189:
4185:
4162:
4158:
4149:
4145:
4141:
4136:
4123:
4120:
4112:
4111:
4094:
4088:
4082:
4079:
4059:
4056:
4048:
4043:
4041:
4037:
4032:
4019:
4013:
4010:
4007:
4001:
3998:
3992:
3986:
3983:
3977:
3971:
3951:
3945:
3942:
3939:
3931:
3912:
3909:
3906:
3892:Automorphisms
3891:
3889:
3875:
3872:
3852:
3832:
3812:
3809:
3789:
3769:
3761:
3757:
3752:
3738:
3732:
3729:
3724:
3721:
3717:
3696:
3693:
3690:
3687:
3676:
3668:
3662:
3659:
3651:
3647:
3640:
3620:
3613:th powers to
3600:
3580:
3577:
3574:
3571:
3560:
3557:
3549:
3543:
3540:
3532:
3529:
3525:
3518:
3511:to inverses,
3510:
3494:
3489:
3485:
3481:
3473:
3469:
3462:
3443:
3440:
3420:
3400:
3394:
3391:
3388:
3376:
3374:
3361:
3353:
3349:
3345:
3340:
3327:
3324:
3304:
3296:
3292:
3285:
3280:
3276:
3267:
3263:
3256:
3253:
3250:
3247:
3244:
3241:
3233:
3230:
3227:
3223:
3216:
3213:
3205:
3201:
3197:
3192:
3188:
3181:
3161:
3141:
3138:
3135:
3127:
3123:
3116:
3096:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3060:
3055:
3042:
3039:
3036:
3027:
3014:
3006:
3002:
2998:
2993:
2980:
2977:
2957:
2953:
2947:
2944:
2941:
2937:
2933:
2930:
2927:
2924:
2921:
2918:
2914:
2910:
2904:
2881:
2861:
2841:
2821:
2818:
2798:
2778:
2769:
2756:
2753:
2746:
2728:
2724:
2703:
2695:
2691:
2687:
2682:
2661:Cyclic groups
2660:
2658:
2656:
2652:
2636:
2629:
2624:
2607:
2604:
2601:
2590:
2571:
2568:
2565:
2539:
2536:
2533:
2507:
2504:
2501:
2489:
2476:
2470:
2464:
2444:
2436:
2420:
2417:
2414:
2411:
2404:then for any
2391:
2385:
2382:
2379:
2353:
2350:
2347:
2324:
2315:
2301:
2293:
2277:
2254:
2251:
2248:
2222:
2219:
2216:
2204:
2188:
2185:
2182:
2172:of the group
2171:
2144:
2141:
2138:
2112:
2109:
2106:
2095:
2087:
2068:
2064:
2040:
2037:
2032:
2014:
1996:
1993:
1962:
1959:
1943:
1942:
1941:
1939:
1916:
1913:
1910:
1899:
1880:
1877:
1874:
1863:
1849:
1844:
1815:
1811:
1788:
1785:
1781:
1761:
1758:
1751:
1747:
1744:
1728:
1723:
1719:
1698:
1693:
1683:
1678:
1646:
1642:
1633:
1628:
1613:
1609:
1605:
1587:
1583:
1579:
1570:
1557:
1556:
1554:
1550:
1532:
1528:
1501:
1488:
1472:
1459:
1433:
1414:
1410:
1406:
1400:
1394:
1371:
1368:
1363:
1350:
1344:
1341:
1326:
1325:
1308:
1305:
1300:
1282:
1263:
1260:
1244:
1240:
1239:
1238:
1232:
1230:
1228:
1223:
1209:
1189:
1169:
1166:
1159:then so does
1146:
1143:
1136:
1121:
1101:
1081:
1061:
1041:
1021:
1001:
998:
978:
969:
967:
963:
947:
944:
941:
921:
918:
915:
906:
893:
887:
884:
881:
875:
872:
866:
860:
857:
851:
845:
822:
819:
816:
793:
773:
770:
764:
761:
758:
751:
735:
732:
712:
706:
703:
700:
688:
686:
670:
667:
664:
661:
652:
639:
636:
633:
630:
621:
608:
602:
599:
596:
590:
584:
581:
578:
552:
549:
546:
520:
517:
514:
502:
489:
483:
477:
474:
468:
462:
459:
453:
450:
447:
441:
421:
401:
381:
361:
355:
352:
349:
329:
326:
306:
298:
295:
276:
273:
270:
244:
241:
238:
227:
211:
205:
202:
199:
173:
170:
167:
152:
150:
148:
144:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
5191:Group theory
5169:21 September
5167:. Retrieved
5153:
5149:
5144:Ash (1973).
5139:
5127:. Retrieved
5109:
5102:
5083:
5052:
5018:
5011:
4850:non-identity
4598:
4502:
4238:prime number
4206:
4148:permutations
4137:
4108:
4044:
4033:
3930:automorphism
3895:
3753:
3380:
3377:Consequences
3028:
2770:
2664:
2649:is given by
2625:
2490:
2316:
2314:is abelian.
2205:
2160:is always {e
2091:
1935:
1487:factor group
1243:real numbers
1236:
1224:
970:
965:
962:automorphism
907:
838:by defining
786:we can make
689:
653:
622:
503:
225:
156:
147:group theory
142:
133:between two
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
4072:denoted by
4047:composition
3633:th powers,
991:of a group
5185:Categories
5129:12 October
5123:0521080169
5062:0471010901
5045:References
4846:Fano plane
3964:such that
2164:}, where e
2088:Properties
2015:The group
1944:The group
1434:The group
1227:invertible
1054:such that
143:isomorphic
69:newspapers
5196:Morphisms
5038:Bijection
5005:See also
4823:×
4817:×
4662:⊕
4634:⊕
4619:⊕
4373:≡
4280:−
4083:
4011:∗
3984:∗
3949:→
3913:∗
3736:→
3722:−
3691:∈
3575:∈
3558:−
3530:−
3398:→
3328:≅
3286:φ
3257:φ
3217:φ
3198:⋅
3182:φ
3162:φ
3154:Clearly,
3117:φ
3088:−
3079:…
3046:→
3037:φ
2981:≅
2945:−
2931:…
2908:⟩
2902:⟨
2608:⊙
2572:∗
2540:⊙
2508:∗
2415:∈
2386:⊙
2354:∗
2255:⊙
2223:∗
2189:∗
2145:⊙
2113:∗
2041:⋅
2033:∗
1917:∗
1881:∗
1684:×
1580:≅
1372:×
1351:≅
1309:×
1135:generates
948:∗
942:⊙
885:∗
858:⊙
823:⊙
768:→
750:bijection
707:∗
685:subgroups
634:≅
603:⊙
591:≅
585:∗
553:⊙
521:∗
475:⊙
451:∗
359:→
294:bijective
277:⊙
245:∗
206:⊙
174:∗
139:bijection
99:June 2015
5027:See also
3756:relation
3509:inverses
3109:so that
3029:Define
2834:Letting
2651:sequence
2170:identity
1485:and the
1458:integers
1233:Examples
806:a group
131:function
2292:abelian
2168:is the
1900:, then
83:scholar
5120:
5090:
5059:
4236:for a
2745:modulo
2716:where
2094:kernel
1896:is an
748:and a
725:a set
135:groups
85:
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5114:(PDF)
4036:image
3762:. If
2628:order
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2435:order
299:from
292:is a
228:from
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90:JSTOR
76:books
5171:2013
5131:2022
5118:ISBN
5088:ISBN
5057:ISBN
4970:and
4881:and
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4045:The
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3802:and
3754:The
2771:Let
2655:OEIS
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2433:the
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1606:The
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934:and
536:and
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189:and
125:, a
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