4303:
41:
6629:
3613:
5563:
5301:
3181:
3926:
6347:
Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the
3398:
3337:
3817:
3760:
3229:
2826:
2793:
2478:
This property is equivalent to property 3, since the elements of two normal subgroups with trivial intersection necessarily commute, a fact which can be deduced by considering the
2860:
2748:
6528:
486:
461:
424:
1628:
885:
3697:
3670:
3643:
3513:
3486:
3455:
3428:
1540:
1480:
3521:
5473:
5253:
5751:
of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples.
3072:
788:
3828:
888:
6133:
It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group
5726:
This has many of the same properties as the direct product of two groups, and can be characterized algebraically in a similar way.
346:
6843:
6820:
6790:
6739:
6718:
3342:
3284:
296:
3765:
3708:
781:
291:
6688:
4943:
behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.
5226:. This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.
707:
6782:
3195:
774:
5397:
are indecomposable, centerless groups, then the automorphism group is relatively straightforward, being Aut(
1278:
2798:
2765:
5734:
It is also possible to take the direct product of an infinite number of groups. For an infinite sequence
391:
205:
6510:
6436:
5826:
2839:
2727:
1370:
is the group of all vectors in the first quadrant under the operation of component-wise multiplication
123:
6026:
5430:
4177:
3458:
2318:
2132:
6141:
5446:
It is possible to take the direct product of more than two groups at once. Given a finite sequence
2377:
Together, these three properties completely determine the algebraic structure of the direct product
4073:
589:
323:
200:
469:
444:
407:
6862:
3263:
887:. Direct sums play an important role in the classification of abelian groups: according to the
6727:
1607:
864:
739:
529:
4874:
2901:
1936:
1357:
613:
6800:
6766:
6749:
5215:
3675:
3648:
3621:
3491:
3464:
3433:
3406:
2927:
2912:
1848:
553:
541:
159:
93:
8:
6408:
5359:
4294:
814:
128:
23:
3041:
through the center of the cube. A similar fact holds true for the symmetry group of an
6624:{\displaystyle G\times _{Q}H=\{\,(g,h)\in G\times H:\varphi (g)=\chi (h)\,\}{\text{.}}}
6424:
6352:
5986:
5972:, which consists of all elements that have only finitely many non-identity components.
4958:
4616:
4193:
4148:
4134:
4068:
3608:{\displaystyle G\times H=\langle S_{G}\cup S_{H}\mid R_{G}\cup R_{H}\cup R_{P}\rangle }
1525:
1465:
113:
85:
3012:
is the internal direct product of the subgroup of rotations and the two-element group
6839:
6816:
6786:
6735:
6714:
6684:
6364:
5558:{\displaystyle \prod _{i=1}^{n}G_{i}\;=\;G_{1}\times G_{2}\times \cdots \times G_{n}}
5296:{\displaystyle {\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}}
4578:
1594:
843:
839:
518:
361:
255:
684:
6108:
is obtained by relaxing the third condition, so that only one of the two subgroups
5968:. Instead, these subgroups generate a subgroup of the direct product known as the
5358:
are homomorphisms. Such a matrix must have the property that every element in the
3038:
2953:
2523:
2019:
1803:
1598:
1263:
1099:
990:
669:
661:
653:
645:
637:
625:
565:
505:
495:
337:
279:
154:
6835:
6812:
6796:
6762:
6745:
6118:
is required to be normal. The resulting product still consists of ordered pairs
6079:
4680:
4356:
4104:
2998:
2465:
1155:
753:
746:
732:
689:
577:
500:
330:
244:
184:
64:
1071:
The resulting algebraic object satisfies the axioms for a group. Specifically:
5755:
5426:
4489:
3060:
3005:
2876:
2872:
847:
760:
696:
386:
366:
303:
268:
189:
179:
164:
149:
103:
80:
2287:
have the following three important properties: (Saying again that we identify
1727:
The direct product is commutative and associative up to isomorphism. That is,
6856:
6710:
6702:
6039:
5200:
4492:
of the direct products is isomorphic to the direct product of the quotients:
4444:
2942:
2364:
2331:
1799:
1089:
854:
679:
601:
435:
308:
174:
6166:
5230:
4130:
2886:
2062:
1452:
952:
892:
806:
534:
233:
222:
169:
144:
139:
98:
69:
32:
6660:
5219:
4786:
3176:{\displaystyle D_{4n}=\langle r,s\mid r^{2n}=s^{2}=1,sr=r^{-1}s\rangle .}
3042:
2448:
In some contexts, the third property above is replaced by the following:
1917:
1245:
802:
3921:{\displaystyle G\times H=\langle a,b\mid a^{3}=1,b^{5}=1,ab=ba\rangle .}
6774:
4940:
4302:
2479:
858:
701:
429:
6785:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
6348:
522:
5947:
is not generated by the elements of the isomorphic subgroups {
2227:
1249:
891:, every finite abelian group can be expressed as the direct sum of
59:
4355:
This is a special case of the universal property for products in
401:
315:
6130:, but with a slightly more complicated rule for multiplication.
2713:
is the internal direct product of the two-element subgroups {1,
2967:
is the internal direct product of the special orthogonal group
2172:
40:
6197:, is similar to the direct product, except that the subgroups
5592:
6761:(2nd ed.), Lexington, Mass.: Xerox College Publishing,
6734:(3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc.,
5928:
Unlike a finite direct product, the infinite direct product
4539:
Note that it is not true in general that every subgroup of
3009:
6051:
can be expressed uniquely as the product of an element of
2343:
can be expressed uniquely as the product of an element of
838:. This operation is the group-theoretic analogue of the
6021:
as long as it satisfies the following three conditions:
4751:
is simply the
Cartesian product of a conjugacy class in
3266:
for the direct product in terms of the presentations of
2260:, respectively, then we can think of the direct product
5433:, and holds more generally for finite direct products.
5262:
3645:
is a set of relations specifying that each element of
3393:{\displaystyle \ \ H=\langle S_{H}\mid R_{H}\rangle ,}
6531:
5476:
5256:
5090:
It is not true in general that every automorphism of
4238:
be the projection homomorphisms. Then for any group
3831:
3768:
3711:
3678:
3651:
3624:
3524:
3494:
3467:
3436:
3409:
3345:
3332:{\displaystyle G=\langle S_{G}\mid R_{G}\rangle \ \ }
3287:
3198:
3075:
2842:
2801:
2768:
2730:
1610:
1528:
1468:
867:
857:, the direct product is sometimes referred to as the
472:
447:
410:
4606:
which is not the direct product of two subgroups of
3812:{\displaystyle \ \ H=\langle b\mid b^{5}=1\rangle }
3755:{\displaystyle G=\langle a\mid a^{3}=1\rangle \ \ }
2411:is necessarily isomorphic to the direct product of
6623:
5557:
5295:
4946:
4615:The subgroups of direct products are described by
3920:
3811:
3754:
3691:
3664:
3637:
3607:
3507:
3480:
3449:
3422:
3392:
3331:
3223:
3175:
2862:is the internal direct product of these subgroups.
2854:
2820:
2787:
2742:
1622:
1534:
1474:
879:
480:
455:
418:
6854:
5130:is any group, then there exists an automorphism
5075:has a subgroup isomorphic to the direct product
5188:For another example, the automorphism group of
3192:is the internal direct product of the subgroup
6427:, every subdirect product is a fiber product.
1360:under multiplication. Then the direct product
6683:(7 ed.). Cengage Learning. p. 157.
4801:is simply the product of the centralizers of
4638:
1806:of the direct product, up to isomorphism. If
941:The underlying set is the Cartesian product,
782:
6613:
6551:
5380:commutes with every element in the image of
5368:commutes with every element in the image of
3912:
3844:
3806:
3781:
3743:
3718:
3602:
3537:
3384:
3358:
3320:
3294:
3218:
3199:
3167:
3092:
2849:
2843:
2815:
2802:
2782:
2769:
2737:
2731:
889:fundamental theorem of finite abelian groups
828:and constructs a new group, usually denoted
5729:
4741:. It follows that each conjugacy class in
4567:is any non-trivial group, then the product
846:and is one of several important notions of
6250: and
6219:are not required to commute. That is, if
5512:
5508:
4024: and
789:
775:
6612:
6554:
5441:
474:
449:
412:
6829:
6756:
6726:
2405:that satisfy the properties above, then
2203: and
2065:whose orders are relatively prime, then
1720:
6678:
6423:under the projection homomorphisms. By
6009:is isomorphic to the direct product of
4411:. For example, the isomorphic copy of
3224:{\displaystyle \langle r^{2},s\rangle }
6855:
5980:
4192:can be characterized by the following
2926:is the internal direct product of the
2894:of unit complex numbers and the group
2885:is the internal direct product of the
2140:
1916:This follows from the formula for the
347:Classification of finite simple groups
6701:
6358:
4278:, there exists a unique homomorphism
4171:
4166:
2821:{\displaystyle \langle a^{m}\rangle }
2788:{\displaystyle \langle a^{n}\rangle }
2242:, and the second being isomorphic to
6806:
6773:
6663:, then this is a subdirect product.
5374:, and every element in the image of
5146:that switches the two factors, i.e.
4119:(or vice versa). In the case where
2131:This fact is closely related to the
6830:Robinson, Derek John Scott (1996),
4620:
3931:
3238:) and the two-element subgroup {1,
2941:and the subgroup consisting of all
13:
6811:(3rd ed.), Berlin, New York:
5436:
3958:. Specifically, define functions
3936:As mentioned above, the subgroups
2266:as containing the original groups
1262:is the group of all two-component
14:
6874:
6430:
5975:
4887:is the product of the centers of
2855:{\displaystyle \langle a\rangle }
2743:{\displaystyle \langle a\rangle }
2171:, and consider the following two
1920:of the cartesian product of sets.
6832:A course in the theory of groups
6757:Herstein, Israel Nathan (1975),
6160:
4549:is the product of a subgroup of
4301:
3247:
2429:is sometimes referred to as the
2236:, the first being isomorphic to
2075:is cyclic as well. That is, if
2034:is the product of the orders of
1861:is the product of the orders of
1142:is the identity element of
39:
6513:, is the following subgroup of
5100:has the above form. (That is,
4947:Automorphisms and endomorphisms
4309:Specifically, the homomorphism
4157:and the composition factors of
2828:are cyclic subgroups of orders
813:is an operation that takes two
6672:
6609:
6603:
6594:
6588:
6567:
6555:
5112:is often a proper subgroup of
4151:of the composition factors of
3672:commutes with each element of
3278:. Specifically, suppose that
708:Infinite dimensional Lie group
1:
6783:Graduate Texts in Mathematics
6681:Contemporary Abstract Algebra
6666:
4293:making the following diagram
3515:are defining relations. Then
2979:and the two-element subgroup
2509:
2248:. If we identify these with
898:
5884:The product of two elements
5413:are not isomorphic, and Aut(
4979:, then the product function
4362:
3030:is the identity element and
2762:are relatively prime. Then
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
7:
6679:Gallian, Joseph A. (2010).
5657:is defined component-wise:
4619:. Other subgroups include
3252:The algebraic structure of
2750:be a cyclic group of order
2087:are relatively prime, then
1812:denotes the trivial group,
1252:. Then the direct product
1232:
1127:is the identity element of
1003:is defined component-wise:
206:List of group theory topics
10:
6879:
6437:Pullback (category theory)
6434:
6362:
6164:
5984:
5891:is defined componentwise:
5827:infinite Cartesian product
4639:Conjugacy and centralizers
4391:, then the direct product
4175:
2226:Both of these are in fact
1923:The order of each element
1098:The direct product has an
5754:More generally, given an
5218:with integer entries and
4766:Along the same lines, if
4757:and a conjugacy class in
4178:Product (category theory)
3231:(which is isomorphic to D
2671:
2636:
2601:
2568:
2559:
2552:
2545:
2540:
2537:
2133:Chinese remainder theorem
1623:{\displaystyle G\times H}
880:{\displaystyle G\oplus H}
5825:are the elements of the
5730:Infinite direct products
4478:is a normal subgroup of
4313:is given by the formula
2871:be the group of nonzero
1583:Then the direct product
1078:The binary operation on
324:Elementary abelian group
201:Glossary of group theory
6728:Herstein, Israel Nathan
6487:be homomorphisms. The
6057:and an element of
5861:with the property that
5429:. This is part of the
2431:internal direct product
2393:group having subgroups
2349:and an element of
1455:with two elements each:
1277:under the operation of
6625:
6282:are presentations for
5801:is defined as follows:
5568:is defined as follows:
5559:
5497:
5442:Finite direct products
5324:is an endomorphism of
5312:is an endomorphism of
5297:
5053:is an automorphism of
4973:is an automorphism of
4244:and any homomorphisms
4133:, it follows that the
3922:
3813:
3756:
3693:
3666:
3639:
3609:
3509:
3482:
3451:
3424:
3394:
3333:
3262:can be used to give a
3225:
3177:
2856:
2822:
2789:
2744:
2423:. In this situation,
2367:with every element of
1624:
1536:
1476:
936:is defined as follows:
881:
740:Linear algebraic group
482:
457:
420:
6809:Undergraduate Algebra
6626:
5560:
5477:
5431:Krull–Schmidt theorem
5298:
3923:
3814:
3757:
3694:
3692:{\displaystyle S_{H}}
3667:
3665:{\displaystyle S_{G}}
3640:
3638:{\displaystyle R_{P}}
3610:
3510:
3508:{\displaystyle R_{H}}
3483:
3481:{\displaystyle R_{G}}
3452:
3450:{\displaystyle S_{H}}
3425:
3423:{\displaystyle S_{G}}
3395:
3334:
3226:
3178:
2904:under multiplication.
2902:positive real numbers
2857:
2823:
2790:
2745:
2049:As a consequence, if
1937:least common multiple
1721:Elementary properties
1625:
1537:
1477:
1358:positive real numbers
882:
483:
458:
421:
6834:, Berlin, New York:
6807:Lang, Serge (2005),
6529:
6139:is referred to as a
5991:Recall that a group
5474:
5254:
5243:can be written as a
5124:.) For example, if
4078:, whose kernels are
3829:
3766:
3709:
3676:
3649:
3622:
3522:
3492:
3465:
3434:
3407:
3343:
3285:
3196:
3073:
2928:special linear group
2913:general linear group
2840:
2836:, respectively, and
2799:
2766:
2728:
2022:, then the order of
1851:of a direct product
1608:
1526:
1466:
865:
470:
445:
408:
16:Mathematical concept
6459:be groups, and let
6397:is any subgroup of
5981:Semidirect products
5970:infinite direct sum
5211:, the group of all
5063:. It follows that
4561:. For example, if
4555:with a subgroup of
4182:The direct product
4135:composition factors
2534:
2281:These subgroups of
2141:Algebraic structure
1630:
1542:
1482:
114:Group homomorphisms
24:Algebraic structure
6621:
6509:, also known as a
6359:Subdirect products
6353:category of groups
6187:, usually denoted
6142:Zappa–Szép product
6094:semidirect product
5987:Semidirect product
5838:; i.e., functions
5555:
5293:
5287:
5229:In general, every
4194:universal property
4172:Universal property
4167:Further properties
4147:are precisely the
3918:
3809:
3752:
3689:
3662:
3635:
3605:
3505:
3478:
3447:
3420:
3390:
3329:
3221:
3173:
2852:
2818:
2785:
2740:
2528:
1994:In particular, if
1620:
1604:
1532:
1522:
1472:
1462:
1216:is the inverse of
1198:is the inverse of
877:
853:In the context of
805:, specifically in
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
6845:978-0-387-94461-6
6822:978-0-387-22025-3
6792:978-0-387-95385-4
6759:Topics in algebra
6741:978-0-13-374562-7
6720:978-0-89871-510-1
6619:
6383:subdirect product
6365:Subdirect product
6045:Every element of
5639:The operation on
5425:, wr denotes the
4735:are conjugate in
4711:are conjugate in
4579:diagonal subgroup
4488:. Moreover, the
4468:are normal, then
4401:is a subgroup of
4385:is a subgroup of
4373:is a subgroup of
3774:
3771:
3751:
3748:
3351:
3348:
3328:
3325:
3052:be odd, and let D
2911:is odd, then the
2706:
2705:
2433:of its subgroups
2358:Every element of
2337:Every element of
2313:, respectively.)
1939:of the orders of
1718:
1717:
1578:
1577:
1535:{\displaystyle H}
1518:
1517:
1475:{\displaystyle G}
1404:) = (
1315:) = (
861:, and is denoted
840:Cartesian product
799:
798:
374:
373:
256:Alternating group
213:
212:
6870:
6848:
6825:
6803:
6769:
6752:
6732:Abstract algebra
6723:
6695:
6694:
6676:
6658:
6644:
6630:
6628:
6627:
6622:
6620:
6617:
6544:
6543:
6522:
6508:
6502:
6496:
6486:
6472:
6458:
6452:
6446:
6422:
6416:
6406:
6396:
6390:
6380:
6374:
6342:
6323:
6306:
6293:
6287:
6277:
6265:
6255:
6249:
6237:
6227:
6218:
6208:
6202:
6196:
6186:
6180:
6156:
6150:
6138:
6129:
6117:
6107:
6101:
6087:
6077:
6071:
6062:
6056:
6050:
6037:
6020:
6014:
6008:
6002:
5996:
5967:
5955:
5946:
5920:
5904:
5890:
5880:
5874:
5860:
5837:
5824:
5806:The elements of
5800:
5778:
5766:
5750:
5718:
5690:
5656:
5635:
5629:
5613:
5590:
5573:The elements of
5564:
5562:
5561:
5556:
5554:
5553:
5535:
5534:
5522:
5521:
5507:
5506:
5496:
5491:
5463:
5385:
5379:
5373:
5367:
5357:
5343:
5329:
5323:
5317:
5311:
5302:
5300:
5299:
5294:
5292:
5291:
5246:
5242:
5225:
5214:
5210:
5197:
5183:
5145:
5135:
5129:
5123:
5111:
5099:
5086:
5074:
5062:
5049:
5004:
4978:
4972:
4966:
4956:
4935:
4916:
4898:
4892:
4886:
4868:
4839:
4812:
4806:
4800:
4784:
4762:
4756:
4750:
4740:
4734:
4725:
4716:
4710:
4701:
4692:
4678:
4660:
4634:
4628:
4611:
4602:
4576:
4566:
4560:
4554:
4548:
4534:
4516:
4487:
4477:
4467:
4461:
4452:
4442:
4441:
4436:
4435:
4426:
4416:
4410:
4400:
4390:
4384:
4378:
4372:
4350:
4322:) =
4312:
4305:
4292:
4277:
4260:
4243:
4237:
4216:
4191:
4162:
4156:
4146:
4128:
4118:
4112:
4102:
4093:It follows that
4090:, respectively.
4089:
4083:
4066:
4057:
4044:
4023:
3999:
3978:
3957:
3947:
3941:
3932:Normal structure
3927:
3925:
3924:
3919:
3887:
3886:
3868:
3867:
3818:
3816:
3815:
3810:
3799:
3798:
3772:
3769:
3761:
3759:
3758:
3753:
3749:
3746:
3736:
3735:
3698:
3696:
3695:
3690:
3688:
3687:
3671:
3669:
3668:
3663:
3661:
3660:
3644:
3642:
3641:
3636:
3634:
3633:
3614:
3612:
3611:
3606:
3601:
3600:
3588:
3587:
3575:
3574:
3562:
3561:
3549:
3548:
3514:
3512:
3511:
3506:
3504:
3503:
3487:
3485:
3484:
3479:
3477:
3476:
3456:
3454:
3453:
3448:
3446:
3445:
3429:
3427:
3426:
3421:
3419:
3418:
3399:
3397:
3396:
3391:
3383:
3382:
3370:
3369:
3349:
3346:
3338:
3336:
3335:
3330:
3326:
3323:
3319:
3318:
3306:
3305:
3277:
3271:
3261:
3230:
3228:
3227:
3222:
3211:
3210:
3182:
3180:
3179:
3174:
3163:
3162:
3132:
3131:
3119:
3118:
3088:
3087:
3039:point reflection
3036:
3029:
3023:
2996:
2990:
2978:
2966:
2954:orthogonal group
2948:Similarly, when
2940:
2925:
2899:
2893:
2884:
2870:
2861:
2859:
2858:
2853:
2827:
2825:
2824:
2819:
2814:
2813:
2794:
2792:
2791:
2786:
2781:
2780:
2749:
2747:
2746:
2741:
2712:
2702:
2697:
2690:
2683:
2676:
2667:
2660:
2655:
2648:
2641:
2632:
2625:
2618:
2613:
2606:
2597:
2590:
2583:
2576:
2571:
2564:
2557:
2550:
2543:
2535:
2533:
2527:
2524:Klein four-group
2521:
2505:
2499:
2493:
2487:
2473:
2463:
2457:
2444:
2438:
2428:
2422:
2416:
2410:
2404:
2398:
2388:
2382:
2372:
2363:
2354:
2348:
2342:
2329:
2312:
2306:
2300:
2293:
2286:
2277:
2271:
2265:
2259:
2253:
2247:
2241:
2235:
2221:
2202:
2180:
2170:
2156:
2150:
2127:
2116:
2086:
2080:
2074:
2060:
2054:
2045:
2039:
2033:
2020:relatively prime
2017:
2016:
2010:
2005:
2004:
1998:
1990:
1989:
1983:
1979:
1973:
1969:
1957:
1950:
1944:
1934:
1912:
1911:
1905:
1902:
1899:
1893:
1889:
1879:
1872:
1866:
1860:
1843:
1837:
1829:
1819:
1811:
1804:identity element
1794:
1788:
1782:
1776:
1763:
1746:
1738:
1631:
1629:
1627:
1626:
1621:
1603:
1599:Klein four-group
1592:
1543:
1541:
1539:
1538:
1533:
1521:
1483:
1481:
1479:
1478:
1473:
1461:
1450:
1444:
1433:
1369:
1356:be the group of
1355:
1344:
1276:
1261:
1244:be the group of
1243:
1227:
1221:
1215:
1209:
1203:
1197:
1191:
1179:
1169:
1147:
1141:
1132:
1126:
1117:
1100:identity element
1087:
1064:
1002:
991:binary operation
985:
975:
965:
950:
935:
922:
919:(with operation
918:
912:
909:(with operation
908:
886:
884:
883:
878:
850:in mathematics.
837:
827:
821:
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
485:
484:
479:
477:
462:
460:
459:
454:
452:
425:
423:
422:
417:
415:
338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
228:
227:
54:
53:
43:
30:
19:
18:
6878:
6877:
6873:
6872:
6871:
6869:
6868:
6867:
6853:
6852:
6846:
6836:Springer-Verlag
6823:
6813:Springer-Verlag
6793:
6742:
6721:
6698:
6691:
6677:
6673:
6669:
6646:
6632:
6616:
6539:
6535:
6530:
6527:
6526:
6514:
6504:
6498:
6492:
6474:
6460:
6454:
6448:
6442:
6439:
6433:
6425:Goursat's lemma
6418:
6412:
6398:
6392:
6386:
6376:
6370:
6367:
6361:
6339:
6332:
6327:
6326:
6321:
6314:
6308:
6298:
6289:
6283:
6274:
6269:
6268:
6263:
6257:
6251:
6246:
6241:
6240:
6235:
6229:
6223:
6210:
6204:
6198:
6188:
6182:
6176:
6169:
6163:
6152:
6146:
6134:
6119:
6109:
6103:
6097:
6083:
6073:
6067:
6058:
6052:
6046:
6029:
6016:
6010:
6004:
5998:
5997:with subgroups
5992:
5989:
5983:
5978:
5966:
5957:
5953:
5948:
5944:
5939:
5929:
5926:
5906:
5894:
5885:
5876:
5872:
5862:
5858:
5853:
5839:
5835:
5830:
5822:
5817:
5807:
5798:
5793:
5783:
5779:of groups, the
5777:
5768:
5764:
5759:
5748:
5741:
5735:
5732:
5724:
5715:
5711:
5705:
5699:
5692:
5687:
5681:
5673:
5667:
5660:
5655:
5646:
5640:
5631:
5627:
5620:
5615:
5611:
5602:
5595:
5589:
5580:
5574:
5549:
5545:
5530:
5526:
5517:
5513:
5502:
5498:
5492:
5481:
5475:
5472:
5471:
5464:of groups, the
5462:
5453:
5447:
5444:
5439:
5437:Generalizations
5381:
5375:
5369:
5363:
5345:
5331:
5325:
5319:
5313:
5307:
5286:
5285:
5280:
5274:
5273:
5268:
5258:
5257:
5255:
5252:
5251:
5244:
5234:
5223:
5212:
5199:
5189:
5181:
5174:
5167:
5160:
5150:
5137:
5131:
5125:
5113:
5101:
5091:
5076:
5064:
5054:
5048:
5029:
5009:
4980:
4974:
4968:
4962:
4952:
4949:
4918:
4903:
4894:
4888:
4878:
4873:Similarly, the
4862:
4849:
4841:
4829:
4817:
4808:
4802:
4790:
4767:
4758:
4752:
4742:
4736:
4733:
4727:
4724:
4718:
4712:
4709:
4703:
4700:
4694:
4693:if and only if
4684:
4676:
4669:
4662:
4658:
4651:
4644:
4641:
4630:
4624:
4617:Goursat's lemma
4607:
4584:
4568:
4562:
4556:
4550:
4540:
4514:
4496:
4479:
4469:
4463:
4457:
4448:
4439:
4438:
4433:
4428:
4427:is the product
4418:
4412:
4402:
4392:
4386:
4380:
4374:
4368:
4365:
4357:category theory
4349:
4341:
4331:
4325:
4317:
4310:
4279:
4268:
4262:
4251:
4245:
4239:
4223:
4218:
4202:
4197:
4183:
4180:
4174:
4169:
4158:
4152:
4138:
4120:
4114:
4108:
4094:
4085:
4079:
4064:
4059:
4055:
4050:
4030:
4025:
4009:
4004:
3985:
3980:
3964:
3959:
3949:
3943:
3937:
3934:
3882:
3878:
3863:
3859:
3830:
3827:
3826:
3794:
3790:
3767:
3764:
3763:
3731:
3727:
3710:
3707:
3706:
3702:For example if
3683:
3679:
3677:
3674:
3673:
3656:
3652:
3650:
3647:
3646:
3629:
3625:
3623:
3620:
3619:
3596:
3592:
3583:
3579:
3570:
3566:
3557:
3553:
3544:
3540:
3523:
3520:
3519:
3499:
3495:
3493:
3490:
3489:
3472:
3468:
3466:
3463:
3462:
3459:generating sets
3457:are (disjoint)
3441:
3437:
3435:
3432:
3431:
3414:
3410:
3408:
3405:
3404:
3378:
3374:
3365:
3361:
3344:
3341:
3340:
3314:
3310:
3301:
3297:
3286:
3283:
3282:
3273:
3267:
3253:
3250:
3245:
3237:
3206:
3202:
3197:
3194:
3193:
3191:
3155:
3151:
3127:
3123:
3111:
3107:
3080:
3076:
3074:
3071:
3070:
3058:
3031:
3025:
3013:
2999:identity matrix
2992:
2980:
2968:
2956:
2943:scalar matrices
2930:
2915:
2895:
2889:
2880:
2873:complex numbers
2866:
2841:
2838:
2837:
2809:
2805:
2800:
2797:
2796:
2776:
2772:
2767:
2764:
2763:
2729:
2726:
2725:
2708:
2700:
2693:
2686:
2679:
2672:
2663:
2658:
2651:
2644:
2637:
2628:
2621:
2616:
2609:
2602:
2593:
2586:
2579:
2574:
2569:
2560:
2553:
2546:
2541:
2529:
2517:
2512:
2501:
2495:
2489:
2483:
2469:
2459:
2453:
2452:3′. Both
2440:
2434:
2424:
2418:
2412:
2406:
2400:
2394:
2384:
2383:. That is, if
2378:
2368:
2359:
2350:
2344:
2338:
2321:
2308:
2302:
2295:
2288:
2282:
2273:
2267:
2261:
2255:
2249:
2243:
2237:
2231:
2204:
2185:
2176:
2158:
2157:be groups, let
2152:
2146:
2143:
2138:
2114:
2090:
2082:
2076:
2066:
2056:
2050:
2041:
2035:
2023:
2014:
2008:
2007:
2002:
1996:
1995:
1987:
1981:
1977:
1971:
1967:
1955:
1954:
1946:
1940:
1924:
1909:
1903:
1900:
1897:
1891:
1887:
1877:
1876:
1868:
1862:
1852:
1839:
1838:for any groups
1827:
1817:
1813:
1807:
1790:
1784:
1778:
1777:for any groups
1761:
1748:
1736:
1728:
1723:
1609:
1606:
1605:
1584:
1581:
1527:
1524:
1523:
1467:
1464:
1463:
1446:
1440:
1431:
1424:
1417:
1410:
1403:
1396:
1389:
1382:
1375:
1361:
1351:
1342:
1335:
1328:
1321:
1314:
1307:
1300:
1293:
1286:
1279:vector addition
1266:
1253:
1239:
1235:
1223:
1217:
1211:
1205:
1199:
1193:
1181:
1171:
1159:
1143:
1140:
1134:
1128:
1125:
1119:
1115:
1109:
1103:
1079:
1069:
1062:
1055:
1048:
1041:
1034:
1027:
1020:
1013:
1006:
994:
977:
967:
955:
951:. That is, the
942:
927:
920:
914:
910:
904:
901:
866:
863:
862:
829:
823:
817:
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
723:
722:
673:
665:
657:
649:
641:
614:Special unitary
525:
511:
510:
492:
491:
473:
471:
468:
467:
448:
446:
443:
442:
411:
409:
406:
405:
397:
396:
387:Discrete groups
376:
375:
331:Frobenius group
276:
263:
252:
245:Symmetric group
241:
225:
215:
214:
65:Normal subgroup
51:
31:
22:
17:
12:
11:
5:
6876:
6866:
6865:
6863:Group products
6851:
6850:
6844:
6827:
6821:
6804:
6791:
6771:
6754:
6740:
6724:
6719:
6703:Artin, Michael
6697:
6696:
6689:
6670:
6668:
6665:
6615:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6553:
6550:
6547:
6542:
6538:
6534:
6435:Main article:
6432:
6431:Fiber products
6429:
6381:are groups, a
6363:Main article:
6360:
6357:
6345:
6344:
6337:
6330:
6324:
6319:
6312:
6280:
6279:
6272:
6266:
6261:
6244:
6238:
6233:
6165:Main article:
6162:
6159:
6090:
6089:
6064:
6043:
5985:Main article:
5982:
5979:
5977:
5976:Other products
5974:
5958:
5951:
5942:
5931:
5925:
5924:
5923:
5922:
5882:
5875:for each
5870:
5856:
5845:
5833:
5820:
5809:
5803:
5796:
5785:
5781:direct product
5769:
5762:
5756:indexed family
5746:
5739:
5731:
5728:
5723:
5722:
5721:
5720:
5713:
5709:
5703:
5697:
5685:
5679:
5671:
5665:
5651:
5644:
5637:
5625:
5618:
5607:
5600:
5585:
5578:
5570:
5566:
5565:
5552:
5548:
5544:
5541:
5538:
5533:
5529:
5525:
5520:
5516:
5511:
5505:
5501:
5495:
5490:
5487:
5484:
5480:
5466:direct product
5458:
5451:
5443:
5440:
5438:
5435:
5427:wreath product
5304:
5303:
5290:
5284:
5281:
5279:
5276:
5275:
5272:
5269:
5267:
5264:
5263:
5261:
5186:
5185:
5179:
5172:
5165:
5158:
5051:
5050:
5046:
5027:
4948:
4945:
4938:
4937:
4917: =
4871:
4870:
4858:
4845:
4840: =
4821:
4731:
4722:
4707:
4698:
4674:
4667:
4656:
4649:
4640:
4637:
4621:fiber products
4604:
4603:
4537:
4536:
4364:
4361:
4353:
4352:
4347:
4337:
4327:
4323:
4307:
4306:
4264:
4247:
4221:
4200:
4176:Main article:
4173:
4170:
4168:
4165:
4062:
4053:
4047:
4046:
4028:
4007:
3983:
3962:
3948:are normal in
3933:
3930:
3929:
3928:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3885:
3881:
3877:
3874:
3871:
3866:
3862:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3820:
3819:
3808:
3805:
3802:
3797:
3793:
3789:
3786:
3783:
3780:
3777:
3745:
3742:
3739:
3734:
3730:
3726:
3723:
3720:
3717:
3714:
3686:
3682:
3659:
3655:
3632:
3628:
3616:
3615:
3604:
3599:
3595:
3591:
3586:
3582:
3578:
3573:
3569:
3565:
3560:
3556:
3552:
3547:
3543:
3539:
3536:
3533:
3530:
3527:
3502:
3498:
3475:
3471:
3444:
3440:
3417:
3413:
3401:
3400:
3389:
3386:
3381:
3377:
3373:
3368:
3364:
3360:
3357:
3354:
3322:
3317:
3313:
3309:
3304:
3300:
3296:
3293:
3290:
3249:
3246:
3244:
3243:
3232:
3220:
3217:
3214:
3209:
3205:
3201:
3186:
3184:
3183:
3172:
3169:
3166:
3161:
3158:
3154:
3150:
3147:
3144:
3141:
3138:
3135:
3130:
3126:
3122:
3117:
3114:
3110:
3106:
3103:
3100:
3097:
3094:
3091:
3086:
3083:
3079:
3061:dihedral group
3053:
3046:
3006:symmetry group
3002:
2946:
2905:
2877:multiplication
2863:
2851:
2848:
2845:
2817:
2812:
2808:
2804:
2784:
2779:
2775:
2771:
2739:
2736:
2733:
2722:
2704:
2703:
2698:
2691:
2684:
2677:
2669:
2668:
2661:
2656:
2649:
2642:
2634:
2633:
2626:
2619:
2614:
2607:
2599:
2598:
2591:
2584:
2577:
2572:
2566:
2565:
2558:
2551:
2544:
2539:
2513:
2511:
2508:
2476:
2475:
2375:
2374:
2356:
2335:
2278:as subgroups.
2224:
2223:
2142:
2139:
2137:
2136:
2130:
2129:
2047:
1993:
1992:
1921:
1915:
1914:
1845:
1796:
1724:
1722:
1719:
1716:
1715:
1712:
1709:
1706:
1703:
1699:
1698:
1695:
1692:
1689:
1686:
1682:
1681:
1678:
1675:
1672:
1669:
1665:
1664:
1661:
1658:
1655:
1652:
1648:
1647:
1644:
1641:
1638:
1635:
1619:
1616:
1613:
1580:
1579:
1576:
1575:
1572:
1569:
1565:
1564:
1561:
1558:
1554:
1553:
1550:
1547:
1531:
1519:
1516:
1515:
1512:
1509:
1505:
1504:
1501:
1498:
1494:
1493:
1490:
1487:
1471:
1458:
1457:
1456:
1436:
1435:
1429:
1422:
1415:
1408:
1401:
1394:
1387:
1380:
1372:
1371:
1347:
1346:
1340:
1333:
1326:
1319:
1312:
1305:
1298:
1291:
1283:
1282:
1234:
1231:
1230:
1229:
1158:of an element
1152:
1149:
1136:
1121:
1111:
1105:
1096:
1093:
1076:
1068:
1067:
1066:
1065:
1060:
1053:
1046:
1039:
1032:
1025:
1018:
1011:
987:
938:
925:direct product
900:
897:
876:
873:
870:
855:abelian groups
848:direct product
811:direct product
797:
796:
794:
793:
786:
779:
771:
768:
767:
764:
763:
761:Elliptic curve
757:
756:
750:
749:
743:
742:
736:
731:
730:
727:
726:
721:
720:
717:
714:
710:
706:
705:
704:
699:
697:Diffeomorphism
693:
692:
687:
682:
676:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
633:
622:
621:
610:
609:
598:
597:
586:
585:
574:
573:
562:
561:
554:Special linear
550:
549:
542:General linear
538:
537:
532:
526:
517:
516:
513:
512:
509:
508:
503:
498:
490:
489:
476:
464:
451:
438:
436:Modular groups
434:
433:
432:
427:
414:
398:
395:
394:
389:
383:
382:
381:
378:
377:
372:
371:
370:
369:
364:
359:
356:
350:
349:
343:
342:
341:
340:
334:
333:
327:
326:
321:
312:
311:
309:Hall's theorem
306:
304:Sylow theorems
300:
299:
294:
286:
285:
284:
283:
277:
272:
269:Dihedral group
265:
264:
259:
253:
248:
242:
237:
226:
221:
220:
217:
216:
211:
210:
209:
208:
203:
195:
194:
193:
192:
187:
182:
177:
172:
167:
162:
160:multiplicative
157:
152:
147:
142:
134:
133:
132:
131:
126:
118:
117:
109:
108:
107:
106:
104:Wreath product
101:
96:
91:
89:direct product
83:
81:Quotient group
75:
74:
73:
72:
67:
62:
52:
49:
48:
45:
44:
36:
35:
15:
9:
6:
4:
3:
2:
6875:
6864:
6861:
6860:
6858:
6847:
6841:
6837:
6833:
6828:
6824:
6818:
6814:
6810:
6805:
6802:
6798:
6794:
6788:
6784:
6780:
6776:
6772:
6768:
6764:
6760:
6755:
6751:
6747:
6743:
6737:
6733:
6729:
6725:
6722:
6716:
6712:
6711:Prentice Hall
6708:
6704:
6700:
6699:
6692:
6690:9780547165097
6686:
6682:
6675:
6671:
6664:
6662:
6657:
6653:
6649:
6643:
6639:
6635:
6606:
6600:
6597:
6591:
6585:
6582:
6579:
6576:
6573:
6570:
6564:
6561:
6558:
6548:
6545:
6540:
6536:
6532:
6524:
6521:
6517:
6512:
6507:
6501:
6495:
6490:
6489:fiber product
6485:
6481:
6477:
6471:
6467:
6463:
6457:
6451:
6445:
6438:
6428:
6426:
6421:
6415:
6410:
6405:
6401:
6395:
6389:
6384:
6379:
6373:
6366:
6356:
6354:
6350:
6340:
6333:
6322:
6315:
6305:
6301:
6297:
6296:
6295:
6292:
6286:
6275:
6264:
6254:
6247:
6236:
6226:
6222:
6221:
6220:
6217:
6213:
6207:
6201:
6195:
6191:
6185:
6179:
6174:
6168:
6161:Free products
6158:
6155:
6149:
6144:
6143:
6137:
6131:
6127:
6123:
6116:
6112:
6106:
6100:
6095:
6086:
6081:
6076:
6070:
6065:
6061:
6055:
6049:
6044:
6041:
6036:
6032:
6028:
6024:
6023:
6022:
6019:
6013:
6007:
6001:
5995:
5988:
5973:
5971:
5965:
5961:
5954:
5945:
5938:
5934:
5918:
5914:
5910:
5902:
5898:
5893:
5892:
5889:
5883:
5879:
5873:
5866:
5859:
5852:
5848:
5843:
5836:
5828:
5823:
5816:
5812:
5805:
5804:
5802:
5799:
5792:
5788:
5782:
5776:
5772:
5765:
5757:
5752:
5745:
5738:
5727:
5716:
5702:
5696:
5688:
5678:
5674:
5664:
5659:
5658:
5654:
5650:
5643:
5638:
5634:
5628:
5621:
5610:
5606:
5599:
5594:
5588:
5584:
5577:
5572:
5571:
5569:
5550:
5546:
5542:
5539:
5536:
5531:
5527:
5523:
5518:
5514:
5509:
5503:
5499:
5493:
5488:
5485:
5482:
5478:
5470:
5469:
5468:
5467:
5461:
5457:
5450:
5434:
5432:
5428:
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5392:
5387:
5384:
5378:
5372:
5366:
5361:
5356:
5352:
5348:
5342:
5338:
5334:
5328:
5322:
5316:
5310:
5288:
5282:
5277:
5270:
5265:
5259:
5250:
5249:
5248:
5241:
5237:
5232:
5227:
5221:
5217:
5209:
5207:
5203:
5196:
5192:
5178:
5171:
5164:
5157:
5153:
5149:
5148:
5147:
5144:
5140:
5134:
5128:
5121:
5117:
5109:
5105:
5098:
5094:
5088:
5084:
5080:
5072:
5068:
5061:
5057:
5044:
5040:
5036:
5032:
5025:
5021:
5017:
5013:
5008:
5007:
5006:
5003:
4999:
4995:
4991:
4987:
4983:
4977:
4971:
4965:
4960:
4955:
4944:
4942:
4933:
4929:
4925:
4921:
4914:
4910:
4906:
4902:
4901:
4900:
4897:
4891:
4885:
4881:
4876:
4866:
4861:
4857:
4853:
4848:
4844:
4837:
4833:
4828:
4824:
4820:
4816:
4815:
4814:
4811:
4805:
4798:
4794:
4788:
4783:
4779:
4775:
4771:
4764:
4761:
4755:
4749:
4745:
4739:
4730:
4721:
4715:
4706:
4697:
4691:
4687:
4682:
4673:
4666:
4655:
4648:
4643:Two elements
4636:
4633:
4627:
4622:
4618:
4613:
4610:
4600:
4596:
4592:
4588:
4583:
4582:
4581:
4580:
4575:
4571:
4565:
4559:
4553:
4547:
4543:
4532:
4528:
4524:
4520:
4512:
4508:
4504:
4500:
4495:
4494:
4493:
4491:
4486:
4482:
4476:
4472:
4466:
4460:
4454:
4451:
4446:
4431:
4425:
4421:
4415:
4409:
4405:
4399:
4395:
4389:
4383:
4377:
4371:
4360:
4358:
4345:
4340:
4335:
4330:
4321:
4316:
4315:
4314:
4304:
4300:
4299:
4298:
4296:
4291:
4287:
4283:
4276:
4272:
4267:
4259:
4255:
4250:
4242:
4236:
4232:
4228:
4224:
4215:
4211:
4207:
4203:
4195:
4190:
4186:
4179:
4164:
4161:
4155:
4150:
4145:
4141:
4136:
4132:
4127:
4123:
4117:
4111:
4106:
4101:
4097:
4091:
4088:
4082:
4077:
4076:homomorphisms
4075:
4070:
4069:homomorphisms
4065:
4056:
4043:
4039:
4035:
4031:
4022:
4018:
4014:
4010:
4003:
4002:
4001:
3998:
3994:
3990:
3986:
3977:
3973:
3969:
3965:
3956:
3952:
3946:
3940:
3915:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3883:
3879:
3875:
3872:
3869:
3864:
3860:
3856:
3853:
3850:
3847:
3841:
3838:
3835:
3832:
3825:
3824:
3823:
3803:
3800:
3795:
3791:
3787:
3784:
3778:
3775:
3740:
3737:
3732:
3728:
3724:
3721:
3715:
3712:
3705:
3704:
3703:
3700:
3684:
3680:
3657:
3653:
3630:
3626:
3597:
3593:
3589:
3584:
3580:
3576:
3571:
3567:
3563:
3558:
3554:
3550:
3545:
3541:
3534:
3531:
3528:
3525:
3518:
3517:
3516:
3500:
3496:
3473:
3469:
3460:
3442:
3438:
3415:
3411:
3387:
3379:
3375:
3371:
3366:
3362:
3355:
3352:
3315:
3311:
3307:
3302:
3298:
3291:
3288:
3281:
3280:
3279:
3276:
3270:
3265:
3260:
3256:
3248:Presentations
3241:
3236:
3215:
3212:
3207:
3203:
3190:
3170:
3164:
3159:
3156:
3152:
3148:
3145:
3142:
3139:
3136:
3133:
3128:
3124:
3120:
3115:
3112:
3108:
3104:
3101:
3098:
3095:
3089:
3084:
3081:
3077:
3069:
3068:
3066:
3062:
3057:
3051:
3047:
3044:
3040:
3035:
3028:
3021:
3017:
3011:
3007:
3003:
3000:
2995:
2988:
2984:
2976:
2972:
2964:
2960:
2955:
2951:
2947:
2944:
2938:
2934:
2929:
2923:
2919:
2914:
2910:
2906:
2903:
2898:
2892:
2888:
2883:
2878:
2874:
2869:
2864:
2846:
2835:
2831:
2810:
2806:
2777:
2773:
2761:
2757:
2753:
2734:
2723:
2720:
2716:
2711:
2699:
2696:
2692:
2689:
2685:
2682:
2678:
2675:
2670:
2666:
2662:
2657:
2654:
2650:
2647:
2643:
2640:
2635:
2631:
2627:
2624:
2620:
2615:
2612:
2608:
2605:
2600:
2596:
2592:
2589:
2585:
2582:
2578:
2573:
2567:
2563:
2556:
2549:
2536:
2532:
2525:
2520:
2515:
2514:
2507:
2504:
2498:
2492:
2486:
2481:
2472:
2467:
2462:
2456:
2451:
2450:
2449:
2446:
2443:
2437:
2432:
2427:
2421:
2415:
2409:
2403:
2397:
2392:
2387:
2381:
2371:
2366:
2362:
2357:
2353:
2347:
2341:
2336:
2333:
2328:
2324:
2320:
2316:
2315:
2314:
2311:
2305:
2298:
2291:
2285:
2279:
2276:
2270:
2264:
2258:
2252:
2246:
2240:
2234:
2229:
2219:
2215:
2211:
2207:
2200:
2196:
2192:
2188:
2184:
2183:
2182:
2179:
2174:
2169:
2165:
2161:
2155:
2149:
2134:
2126:
2123:
2119:
2112:
2109:
2105:
2101:
2098:
2094:
2089:
2088:
2085:
2079:
2073:
2069:
2064:
2063:cyclic groups
2059:
2053:
2048:
2044:
2038:
2031:
2027:
2021:
2013:
2001:
1986:
1976:
1965:
1961:
1953:
1952:
1949:
1943:
1938:
1932:
1928:
1922:
1919:
1908:
1896:
1886:
1882:
1875:
1874:
1871:
1865:
1859:
1855:
1850:
1846:
1842:
1836:
1832:
1826:
1822:
1816:
1810:
1805:
1801:
1800:trivial group
1797:
1793:
1787:
1781:
1774:
1770:
1766:
1760:
1756:
1752:
1745:
1741:
1735:
1731:
1726:
1725:
1713:
1710:
1707:
1704:
1701:
1700:
1696:
1693:
1690:
1687:
1684:
1683:
1679:
1676:
1673:
1670:
1667:
1666:
1662:
1659:
1656:
1653:
1650:
1649:
1645:
1642:
1639:
1636:
1633:
1632:
1617:
1614:
1611:
1602:
1600:
1596:
1591:
1587:
1573:
1570:
1567:
1566:
1562:
1559:
1556:
1555:
1551:
1548:
1545:
1544:
1529:
1520:
1513:
1510:
1507:
1506:
1502:
1499:
1496:
1495:
1491:
1488:
1485:
1484:
1469:
1460:
1459:
1454:
1453:cyclic groups
1449:
1443:
1438:
1437:
1428:
1421:
1414:
1407:
1400:
1393:
1386:
1379:
1374:
1373:
1368:
1364:
1359:
1354:
1349:
1348:
1339:
1332:
1325:
1318:
1311:
1304:
1297:
1290:
1285:
1284:
1280:
1274:
1270:
1265:
1260:
1256:
1251:
1247:
1242:
1237:
1236:
1226:
1220:
1214:
1208:
1202:
1196:
1189:
1185:
1178:
1174:
1167:
1163:
1157:
1153:
1150:
1146:
1139:
1131:
1124:
1114:
1108:
1101:
1097:
1094:
1091:
1086:
1082:
1077:
1075:Associativity
1074:
1073:
1072:
1059:
1052:
1045:
1038:
1031:
1024:
1017:
1010:
1005:
1004:
1001:
997:
992:
988:
984:
980:
974:
970:
963:
959:
954:
953:ordered pairs
949:
945:
940:
939:
937:
934:
930:
926:
917:
907:
903:Given groups
896:
894:
893:cyclic groups
890:
874:
871:
868:
860:
856:
851:
849:
845:
841:
836:
832:
826:
820:
816:
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
345:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
111:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
21:
20:
6831:
6808:
6778:
6758:
6731:
6706:
6680:
6674:
6661:epimorphisms
6655:
6651:
6647:
6641:
6637:
6633:
6525:
6519:
6515:
6505:
6499:
6493:
6488:
6483:
6479:
6475:
6469:
6465:
6461:
6455:
6449:
6443:
6440:
6419:
6413:
6409:surjectively
6403:
6399:
6393:
6387:
6382:
6377:
6371:
6368:
6346:
6335:
6328:
6317:
6310:
6303:
6299:
6290:
6284:
6281:
6270:
6259:
6252:
6242:
6231:
6224:
6215:
6211:
6205:
6199:
6193:
6189:
6183:
6177:
6173:free product
6172:
6170:
6167:Free product
6153:
6147:
6140:
6135:
6132:
6125:
6121:
6114:
6110:
6104:
6098:
6093:
6091:
6084:
6074:
6068:
6059:
6053:
6047:
6034:
6030:
6027:intersection
6017:
6011:
6005:
5999:
5993:
5990:
5969:
5963:
5959:
5949:
5940:
5936:
5932:
5927:
5916:
5912:
5908:
5900:
5896:
5887:
5877:
5868:
5864:
5854:
5850:
5846:
5841:
5831:
5829:of the sets
5818:
5814:
5810:
5794:
5790:
5786:
5780:
5774:
5770:
5760:
5753:
5743:
5736:
5733:
5725:
5707:
5700:
5694:
5683:
5676:
5669:
5662:
5652:
5648:
5641:
5632:
5623:
5616:
5608:
5604:
5597:
5586:
5582:
5575:
5567:
5465:
5459:
5455:
5448:
5445:
5422:
5418:
5414:
5410:
5406:
5402:
5398:
5394:
5390:
5388:
5382:
5376:
5370:
5364:
5354:
5350:
5346:
5340:
5336:
5332:
5326:
5320:
5314:
5308:
5305:
5239:
5235:
5231:endomorphism
5228:
5205:
5201:
5194:
5190:
5187:
5176:
5169:
5162:
5155:
5151:
5142:
5138:
5132:
5126:
5119:
5115:
5107:
5103:
5096:
5092:
5089:
5082:
5078:
5070:
5066:
5059:
5055:
5052:
5042:
5038:
5034:
5030:
5023:
5019:
5015:
5011:
5001:
4997:
4993:
4989:
4985:
4981:
4975:
4969:
4963:
4959:automorphism
4953:
4950:
4939:
4931:
4927:
4923:
4919:
4912:
4908:
4904:
4895:
4889:
4883:
4879:
4872:
4864:
4859:
4855:
4851:
4846:
4842:
4835:
4831:
4826:
4822:
4818:
4809:
4803:
4796:
4792:
4781:
4777:
4773:
4769:
4765:
4759:
4753:
4747:
4743:
4737:
4728:
4719:
4713:
4704:
4695:
4689:
4685:
4671:
4664:
4653:
4646:
4642:
4631:
4625:
4614:
4608:
4605:
4598:
4594:
4590:
4586:
4573:
4569:
4563:
4557:
4551:
4545:
4541:
4538:
4530:
4526:
4522:
4518:
4510:
4506:
4502:
4498:
4484:
4480:
4474:
4470:
4464:
4458:
4455:
4449:
4447:subgroup of
4429:
4423:
4419:
4413:
4407:
4403:
4397:
4393:
4387:
4381:
4375:
4369:
4366:
4354:
4343:
4338:
4333:
4328:
4319:
4308:
4289:
4285:
4281:
4274:
4270:
4265:
4257:
4253:
4248:
4240:
4234:
4230:
4226:
4219:
4213:
4209:
4205:
4198:
4188:
4184:
4181:
4159:
4153:
4143:
4139:
4131:finite group
4125:
4121:
4115:
4109:
4099:
4095:
4092:
4086:
4080:
4072:
4060:
4051:
4048:
4041:
4037:
4033:
4026:
4020:
4016:
4012:
4005:
3996:
3992:
3988:
3981:
3975:
3971:
3967:
3960:
3954:
3950:
3944:
3938:
3935:
3821:
3701:
3617:
3402:
3274:
3268:
3264:presentation
3258:
3254:
3251:
3239:
3234:
3188:
3064:
3055:
3049:
3033:
3026:
3019:
3015:
2997:denotes the
2993:
2986:
2982:
2974:
2970:
2962:
2958:
2949:
2936:
2932:
2921:
2917:
2908:
2896:
2890:
2887:circle group
2881:
2867:
2833:
2829:
2759:
2755:
2751:
2718:
2714:
2709:
2694:
2687:
2680:
2673:
2664:
2652:
2645:
2638:
2629:
2622:
2610:
2603:
2594:
2587:
2580:
2561:
2554:
2547:
2530:
2518:
2502:
2496:
2490:
2484:
2477:
2470:
2460:
2454:
2447:
2441:
2435:
2430:
2425:
2419:
2413:
2407:
2401:
2395:
2390:
2385:
2379:
2376:
2369:
2360:
2351:
2345:
2339:
2326:
2322:
2319:intersection
2309:
2303:
2296:
2289:
2283:
2280:
2274:
2268:
2262:
2256:
2250:
2244:
2238:
2232:
2225:
2217:
2213:
2209:
2205:
2198:
2194:
2193:, 1) :
2190:
2186:
2177:
2167:
2163:
2159:
2153:
2147:
2144:
2124:
2121:
2117:
2110:
2107:
2103:
2099:
2096:
2092:
2083:
2077:
2071:
2067:
2057:
2051:
2042:
2036:
2029:
2025:
2011:
1999:
1984:
1974:
1963:
1959:
1947:
1941:
1930:
1926:
1906:
1894:
1884:
1880:
1869:
1863:
1857:
1853:
1840:
1834:
1830:
1824:
1820:
1814:
1808:
1791:
1785:
1779:
1772:
1768:
1764:
1758:
1754:
1750:
1743:
1739:
1733:
1729:
1589:
1585:
1582:
1447:
1441:
1426:
1419:
1412:
1405:
1398:
1391:
1384:
1377:
1366:
1362:
1352:
1337:
1330:
1323:
1316:
1309:
1302:
1295:
1288:
1272:
1268:
1258:
1254:
1246:real numbers
1240:
1224:
1218:
1212:
1206:
1200:
1194:
1187:
1183:
1180:is the pair
1176:
1172:
1165:
1161:
1144:
1137:
1129:
1122:
1112:
1106:
1084:
1080:
1070:
1057:
1050:
1043:
1036:
1029:
1022:
1015:
1008:
999:
995:
982:
978:
972:
968:
961:
957:
947:
943:
932:
928:
924:
915:
905:
902:
852:
834:
830:
824:
818:
810:
807:group theory
800:
629:
617:
605:
593:
581:
569:
557:
545:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
88:
70:Group action
33:Group theory
28:Group theory
27:
6775:Lang, Serge
6407:which maps
5220:determinant
5005:defined by
4941:Normalizers
4787:centralizer
4071:, known as
3043:icosahedron
2952:is odd the
1918:cardinality
1090:associative
803:mathematics
519:Topological
358:alternating
6667:References
5417:) wr 2 if
4074:projection
3063:of order 4
2717:} and {1,
2480:commutator
2208:′ = { (1,
1595:isomorphic
899:Definition
859:direct sum
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
6601:χ
6586:φ
6577:×
6571:∈
6537:×
6349:coproduct
5630:for each
5543:×
5540:⋯
5537:×
5524:×
5479:∏
5283:δ
5278:γ
5271:β
5266:α
4681:conjugate
4593:) :
4363:Subgroups
4105:extension
3913:⟩
3857:∣
3845:⟨
3836:×
3807:⟩
3788:∣
3782:⟨
3744:⟩
3725:∣
3719:⟨
3603:⟩
3590:∪
3577:∪
3564:∣
3551:∪
3538:⟨
3529:×
3385:⟩
3372:∣
3359:⟨
3321:⟩
3308:∣
3295:⟨
3219:⟩
3200:⟨
3168:⟩
3157:−
3105:∣
3093:⟨
2850:⟩
2844:⟨
2816:⟩
2803:⟨
2783:⟩
2770:⟨
2738:⟩
2732:⟨
2228:subgroups
2212:) :
1867:and
1615:×
1102:, namely
872:⊕
690:Conformal
578:Euclidean
185:nilpotent
6857:Category
6777:(2002),
6730:(1996),
6705:(1991),
6511:pullback
5706:′, ...,
5682:′, ...,
5614:, where
5401:) × Aut(
5216:matrices
5106:) × Aut(
5081:) × Aut(
4490:quotient
4437:, where
2879:. Then
2754:, where
2510:Examples
2365:commutes
2175:of
1418:,
1250:addition
1233:Examples
1222:in
1192:, where
1151:Inverses
1118:, where
1095:Identity
966:, where
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
6801:1878556
6779:Algebra
6767:0356988
6750:1375019
6707:Algebra
6351:in the
6294:, then
6040:trivial
5956: }
5767: }
5758:{
5668:, ...,
5603:, ...,
5454:, ...,
5247:matrix
4585:Δ = { (
4445:trivial
4443:is the
4295:commute
4196:. Let
3059:be the
3037:is the
2522:be the
2482:of any
2332:trivial
2189:′ = { (
2173:subsets
1935:is the
1901:
1802:is the
1597:to the
1264:vectors
1156:inverse
923:), the
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
6842:
6819:
6799:
6789:
6765:
6748:
6738:
6717:
6687:
6453:, and
6080:normal
5647:× ⋯ ×
5593:tuples
5581:× ⋯ ×
5330:, and
5306:where
4957:is an
4875:center
4785:, the
4577:has a
4103:is an
3773:
3770:
3750:
3747:
3618:where
3403:where
3350:
3347:
3327:
3324:
3185:Then D
3024:where
2991:where
2875:under
2466:normal
1789:, and
1714:(1,1)
1702:(a,b)
1697:(a,1)
1685:(1,b)
1680:(1,b)
1668:(a,1)
1663:(a,b)
1651:(1,1)
1646:(a,b)
1643:(1,b)
1640:(a,1)
1637:(1,1)
1248:under
1210:, and
913:) and
815:groups
809:, the
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
6503:over
6411:onto
6066:Both
5895:(ƒ •
5749:, ...
5405:) if
5389:When
5360:image
5245:2 × 2
5213:2 × 2
5168:) = (
4525:) × (
4505:) / (
4336:), ƒ
4149:union
4129:is a
4049:Then
3822:then
3008:of a
2707:Then
2301:with
2102:) × (
1970:= lcm
1849:order
1711:(a,1)
1708:(1,b)
1705:(a,b)
1694:(1,1)
1691:(a,b)
1688:(1,b)
1677:(a,b)
1674:(1,1)
1671:(a,1)
1660:(1,b)
1657:(a,1)
1654:(1,1)
1390:) × (
1301:) + (
1035:) = (
1021:) · (
719:Sp(∞)
716:SU(∞)
129:image
6840:ISBN
6817:ISBN
6787:ISBN
6736:ISBN
6715:ISBN
6685:ISBN
6659:are
6645:and
6497:and
6473:and
6441:Let
6417:and
6391:and
6375:and
6288:and
6203:and
6181:and
6171:The
6151:and
6102:and
6078:are
6072:and
6025:The
6015:and
6003:and
5911:) •
5867:) ∈
5591:are
5409:and
5393:and
5344:and
5204:(2,
5114:Aut(
5102:Aut(
5077:Aut(
5065:Aut(
5026:) =
4967:and
4926:) ×
4893:and
4854:) ×
4807:and
4776:) ∈
4726:and
4717:and
4702:and
4679:are
4661:and
4629:and
4462:and
4440:{1}
4434:{1}
4379:and
4261:and
4217:and
4084:and
4067:are
4058:and
4040:) =
4019:) =
3979:and
3942:and
3762:and
3488:and
3461:and
3430:and
3339:and
3272:and
3048:Let
3010:cube
3004:The
2865:Let
2832:and
2795:and
2758:and
2724:Let
2516:Let
2464:are
2458:and
2439:and
2417:and
2399:and
2317:The
2307:and
2294:and
2272:and
2254:and
2151:and
2145:Let
2081:and
2061:are
2055:and
2040:and
2018:are
2006:and
1945:and
1847:The
1798:The
1757:) ×
1747:and
1445:and
1439:Let
1350:Let
1238:Let
1154:The
1133:and
989:The
976:and
844:sets
822:and
713:O(∞)
702:Loop
521:and
6631:If
6491:of
6385:of
6369:If
6209:of
6175:of
6145:of
6096:of
6082:in
6038:is
6033:∩
5886:ƒ,
5844:→ ⋃
5840:ƒ:
5362:of
5233:of
5198:is
5136:of
5037:),
4961:of
4951:If
4877:of
4789:of
4683:in
4623:of
4456:If
4417:in
4367:If
4280:ƒ:
4137:of
4113:by
4107:of
4000:by
2969:SO(
2931:SL(
2916:GL(
2907:If
2900:of
2526::
2500:in
2488:in
2468:in
2391:any
2389:is
2330:is
2230:of
1767:× (
1593:is
1451:be
1204:in
1170:of
1110:, 1
1088:is
993:on
842:of
801:In
628:Sp(
616:SU(
592:SO(
556:SL(
544:GL(
6859::
6838:,
6815:,
6797:MR
6795:,
6781:,
6763:MR
6746:MR
6744:,
6713:,
6709:,
6654:→
6650::
6640:→
6636::
6634:𝜑
6523::
6518:×
6482:→
6478::
6468:→
6464::
6462:𝜑
6447:,
6402:×
6355:.
6334:∪
6316:∪
6309:〈
6307:=
6302:∗
6258:〈
6256:=
6230:〈
6228:=
6214:∗
6192:∗
6157:.
6124:,
6113:,
6092:A
5907:ƒ(
5905:=
5899:)(
5863:ƒ(
5742:,
5717:′)
5691:=
5689:′)
5675:)(
5622:∈
5421:≅
5386:.
5353:→
5349::
5339:→
5335::
5318:,
5238:×
5224:±1
5222:,
5202:GL
5193:×
5175:,
5161:,
5141:×
5118:×
5095:×
5087:.
5069:×
5058:×
5022:,
5018:)(
5014:×
5000:×
4996:→
4992:×
4988::
4984:×
4911:×
4899::
4882:×
4834:,
4813::
4795:,
4780:×
4772:,
4763:.
4746:×
4688:×
4670:,
4652:,
4635:.
4612:.
4601:}
4597:∈
4589:,
4572:×
4544:×
4529:/
4521:/
4513:)
4509:×
4501:×
4483:×
4473:×
4453:.
4432:×
4422:×
4406:×
4396:×
4359:.
4346:)
4318:ƒ(
4297::
4288:×
4284:→
4273:→
4269::
4256:→
4252::
4233:→
4229:×
4225::
4212:→
4208:×
4204::
4187:×
4163:.
4142:×
4124:×
4098:×
4036:,
4015:,
3995:→
3991:×
3987::
3974:→
3970:×
3966::
3953:×
3699:.
3257:×
3242:}.
3067::
3022:},
3018:,
3014:{−
2989:},
2985:,
2981:{−
2973:,
2961:,
2957:O(
2935:,
2920:,
2752:mn
2721:}.
2506:.
2494:,
2445:.
2325:∩
2216:∈
2197:∈
2181::
2166:×
2162:=
2122:mn
2120:/
2113:)
2106:/
2095:/
2070:×
2032:)
2028:,
1988:|)
1980:,
1972:(|
1962:,
1951::
1929:,
1890:=
1883:×
1873::
1856:×
1833:×
1823:×
1783:,
1771:×
1753:×
1742:×
1732:×
1634:*
1601::
1588:×
1574:1
1568:b
1563:b
1557:1
1552:b
1549:1
1546:*
1514:1
1508:a
1503:a
1497:1
1492:a
1489:1
1486:*
1425:×
1411:×
1397:,
1383:,
1365:×
1336:+
1329:,
1322:+
1308:,
1294:,
1271:,
1257:×
1186:,
1175:×
1164:,
1104:(1
1083:×
1056:∆
1049:,
1042:*
1028:,
1014:,
998:×
981:∈
971:∈
960:,
946:×
931:×
906:G
895:.
833:×
604:U(
580:E(
568:O(
26:→
6849:.
6826:.
6770:.
6753:.
6693:.
6656:Q
6652:H
6648:χ
6642:Q
6638:G
6618:.
6614:}
6610:)
6607:h
6604:(
6598:=
6595:)
6592:g
6589:(
6583::
6580:H
6574:G
6568:)
6565:h
6562:,
6559:g
6556:(
6552:{
6549:=
6546:H
6541:Q
6533:G
6520:H
6516:G
6506:Q
6500:H
6494:G
6484:Q
6480:H
6476:χ
6470:Q
6466:G
6456:Q
6450:H
6444:G
6420:H
6414:G
6404:H
6400:G
6394:H
6388:G
6378:H
6372:G
6343:.
6341:〉
6338:H
6336:R
6331:G
6329:R
6325:|
6320:H
6318:S
6313:G
6311:S
6304:H
6300:G
6291:H
6285:G
6278:,
6276:〉
6273:H
6271:R
6267:|
6262:H
6260:S
6253:H
6248:〉
6245:G
6243:R
6239:|
6234:G
6232:S
6225:G
6216:H
6212:G
6206:H
6200:G
6194:H
6190:G
6184:H
6178:G
6154:H
6148:G
6136:P
6128:)
6126:h
6122:g
6120:(
6115:H
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