4769:
4391:
2323:
3895:
1574:
4251:
2195:
4080:
3996:
2522:
4141:
1946:
4577:
3515:), but in this case the whole subgroup is a normal subgroup. All members of a particular coset carry out the same permutation of these cosets, but in this case they represent only the 3-element
4192:
2184:
749:
504:
1130:
455:
3183:
is infinite, then all such sets are therefore infinite.) The set of these sets forms a group isomorphic to a subset of the group of permutations, so the number of these sets must divide
789:
4239:
2113:
1324:
2148:
1473:
4471:
1019:
4736:
1699:
2367:
2614:
2426:
361:
4432:
326:
589:
390:
412:
244:
1399:
1205:
2469:
1828:
1751:
1501:
2075:
1777:
3811:
3760:
3647:
2690:
2555:
2042:
840:
696:
287:
160:
96:
4688:
of two distinct index 2 subgroups (which are necessarily normal) gives the third point on the projective line containing these subgroups, and a group must contain
4682:
4612:
1428:
1357:
1234:
1163:
4650:
899:
871:
653:
623:
3920:
yields a third. This is a simple corollary of the above discussion (namely the projectivization of the vector space structure of the elementary abelian group
3679:
128:
4386:{\displaystyle \mathbf {P} (\operatorname {Hom} (G,\mathbf {Z} /p)):=(\operatorname {Hom} (G,\mathbf {Z} /p))\setminus \{0\})/(\mathbf {Z} /p)^{\times }}
1516:
2318:{\displaystyle \{(x,y)\mid x{\text{ is even}}\},\quad \{(x,y)\mid y{\text{ is even}}\},\quad {\text{and}}\quad \{(x,y)\mid x+y{\text{ is even}}\}}
4023:
4006:
does not act on this geometry, nor does it reflect any of the non-abelian structure (in both cases because the quotient is abelian).
3926:
2477:
4833:
4805:
4095:
3553:
There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:
1878:
3391:
4812:
4786:
4009:
However, it is an elementary result, which can be seen concretely as follows: the set of normal subgroups of a given index
592:
4486:
4149:
4874:
4852:
4819:
2159:
705:
460:
3512:
1048:
417:
754:
4801:
4201:
4790:
2086:
1262:
2121:
1446:
4437:
3543:
943:
4912:
4691:
3316:
is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S
1638:
1587:
2343:
2560:
2372:
335:
4407:
303:
3581:
2081:
1706:
524:
370:
395:
184:
4826:
4779:
3913:
2624:
1362:
1168:
3547:
2435:
3890:{\displaystyle \mathbf {E} ^{p}(G)\supseteq \mathbf {A} ^{p}(G)\supseteq \mathbf {O} ^{p}(G).}
1797:
1720:
1486:
4931:
3501:
2050:
1756:
4685:
3917:
3738:
3625:
2659:
2533:
2020:
1610:
809:
665:
364:
256:
175:
133:
65:
4901:
4658:
4588:
3395:
1404:
1333:
1210:
1139:
8:
4629:
3912:
An elementary observation is that one cannot have exactly 2 subgroups of index 2, as the
3422:
1622:
876:
848:
628:
598:
3355:
and also the left coset, so the two are identical.) More generally, a subgroup of index
4883:
3508:
3278:!, but must satisfy other criteria as well. Since the normal subgroup is a subgroup of
2154:
4738:
index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.
3652:
101:
3516:
3426:
2015:
1591:
4014:
2116:
1569:{\displaystyle |G:\operatorname {ker} \;\varphi |=|\operatorname {im} \;\varphi |.}
1480:
250:
166:
is the disjoint union of the left cosets and because each left coset has the same
4623:
3478:
3299:
2752:
2709:
2693:
2045:
1715:
799:
699:
4869:
3901:
843:
4925:
2705:
1999:
414:, namely the set of even integers and the set of odd integers, so the index
4144:
2628:
2472:
2337:
1995:
1862:
1476:
518:
24:
4916:
3405:
is normal, and other properties of subgroups of prime index are given in (
4752:
3532:
3401:
An alternative proof of the result that a subgroup of index lowest prime
1994:
Note that a subgroup of lowest prime index may not exist, such as in any
1780:
1606:
167:
20:
3433:
subgroup (in fact it has three such) of order 8, and thus of index 3 in
3171:. Since the number of possible permutations of cosets is finite, namely
4905:
4887:
2528:
1839:
3343:. (We can arrive at this fact also by noting that all the elements of
4747:
4075:{\displaystyle \mathbf {P} (\operatorname {Hom} (G,\mathbf {Z} /p)).}
1953:
4768:
3139:(that is, for any coset), it shows that multiplying on the right by
2728:
which is realized for the trivial subgroup, or in fact any subgroup
329:
32:
3504:
also has 24 members and a subgroup of index 3 (this time it is a D
3536:
3274:
The index of the normal subgroup not only has to be a divisor of
1714:
As a special case of the orbit-stabilizer theorem, the number of
4194:
A non-trivial such map has as kernel a normal subgroup of index
3331:
of index 2 is a normal subgroup, because the normal subgroup of
3991:{\displaystyle G/\mathbf {E} ^{p}(G)\cong (\mathbf {Z} /p)^{k}}
2786:
can be taken to be the kernel of the natural homomorphism from
3167:
is finite or infinte. Now assume that it is the finite number
2517:{\displaystyle \mathbb {Z} ^{n}\to \mathbb {Z} /p\mathbb {Z} }
43:
4245:) does not change the kernel; thus one obtains a map from
3449:. Multiplying on the right any element of a right coset of
4476:
As a consequence, the number of normal subgroups of index
2794:. Let us explain this in more detail, using right cosets:
2790:
to the permutation group of the left (or right) cosets of
3383:
having no other prime factors. For example, the subgroup
3327:= 2 this gives the rather obvious result that a subgroup
4618:. Further, given two distinct normal subgroups of index
3526:
4136:{\displaystyle \operatorname {Hom} (G,\mathbf {Z} /p),}
3175:!, then there can only be a finite number of sets like
3143:
makes the same permutation of cosets as multiplying by
1941:{\displaystyle |G:\operatorname {Core} (H)|\leq |G:H|!}
3163:
What we have said so far applies whether the index of
4694:
4661:
4632:
4591:
4489:
4440:
4410:
4254:
4204:
4152:
4098:
4026:
3929:
3814:
3741:
3655:
3628:
2662:
2563:
2536:
2480:
2438:
2375:
2346:
2198:
2162:
2124:
2089:
2053:
2023:
1881:
1800:
1759:
1723:
1641:
1519:
1489:
1449:
1407:
1365:
1336:
1265:
1213:
1171:
1142:
1051:
946:
879:
851:
812:
757:
708:
668:
631:
601:
527:
463:
420:
398:
373:
338:
306:
259:
187:
136:
104:
68:
3390:
of the non-abelian group of order 21 is normal (see
4793:. Unsourced material may be challenged and removed.
4572:{\displaystyle (p^{k+1}-1)/(p-1)=1+p+\cdots +p^{k}}
3904:and the transfer homomorphism, as discussed there.
2954:which perform a given permutation on the cosets of
4730:
4676:
4644:
4606:
4571:
4465:
4426:
4400:subgroups. Conversely, a normal subgroup of index
4385:
4233:
4186:
4135:
4074:
3990:
3889:
3754:
3673:
3641:
3371:is finite) is necessarily normal, as the index of
2684:
2608:
2549:
2516:
2463:
2420:
2361:
2317:
2178:
2142:
2107:
2069:
2036:
1940:
1822:
1771:
1745:
1693:
1568:
1495:
1467:
1422:
1393:
1351:
1318:
1228:
1199:
1157:
1124:
1013:
893:
865:
834:
783:
743:
690:
647:
617:
583:
498:
449:
406:
384:
355:
320:
281:
238:
154:
122:
90:
2656:is said to be infinite. In this case, the index
1983:is normal, as the index of its core must also be
1690:
54:, or equivalently, the number of right cosets of
4923:
4187:{\displaystyle \mathbf {F} _{p}=\mathbf {Z} /p.}
3649:normal subgroup that contains the derived group
3542:and have interesting structure, as described at
3900:These groups have important connections to the
3697:-group (not necessarily elementary) onto which
2179:{\displaystyle \mathbb {Z} \oplus \mathbb {Z} }
3711:) is the intersection of all normal subgroups
3602:) is the intersection of all normal subgroups
3489:perform the same permutation of the cosets of
1401:is finite, then equality holds if and only if
1207:is finite, then equality holds if and only if
744:{\displaystyle |\mathbb {Z} :2\mathbb {Z} |=2}
499:{\displaystyle |\mathbb {Z} :n\mathbb {Z} |=n}
253:if some of them are infinite). Thus the index
3801:As these are weaker conditions on the groups
3363:is the smallest prime factor of the order of
2801:that leave all cosets the same form a group.
1971:is 2, or for a finite group the lowest prime
1125:{\displaystyle |G:H\cap K|\leq |G:H|\,|G:K|,}
702:that may be finite or infinite. For example,
4614:corresponds to no normal subgroups of index
4346:
4340:
3778:-group (not necessarily abelian) onto which
3485:. All elements from any particular coset of
2312:
2277:
2264:
2235:
2228:
2199:
450:{\displaystyle |\mathbb {Z} :2\mathbb {Z} |}
4085:In detail, the space of homomorphisms from
3477:, corresponding to the six elements of the
784:{\displaystyle |\mathbb {R} :\mathbb {Z} |}
4473:which shows that this map is a bijection.
4234:{\displaystyle (\mathbf {Z} /p)^{\times }}
3298:must also correspond to a subgroup of the
2732:of infinite cardinality less than that of
1998:of non-prime order, or more generally any
1554:
1534:
4853:Learn how and when to remove this message
4198:and multiplying the map by an element of
3187:!. Furthermore, it must be a multiple of
2510:
2497:
2483:
2349:
2172:
2164:
1099:
988:
772:
764:
726:
715:
481:
470:
438:
427:
400:
378:
349:
314:
3584:, and is the largest elementary abelian
3535:index are kernels of surjective maps to
2631:of index 2, which is necessarily normal.
4913:Subgroup of least prime index is normal
4434:up to a choice of "which coset maps to
3095:, as desired. Now we show that for any
2751:(finite or infinite) always contains a
4924:
3907:
3441:. This dihedral group has a 4-member D
3195:contains the same number of cosets of
2186:has three subgroups of index 2, namely
2108:{\displaystyle \operatorname {SO} (n)}
1319:{\displaystyle |H:H\cap K|\leq |G:K|,}
4917:Groupprops, The Group Properties Wiki
4902:Normality of subgroups of prime index
3527:Normal subgroups of prime power index
2762:), also of finite index. In fact, if
2143:{\displaystyle \operatorname {O} (n)}
1468:{\displaystyle \varphi \colon G\to H}
4791:adding citations to reliable sources
4762:
4466:{\displaystyle 1\in \mathbf {Z} /p,}
3457:gives a member of the same coset of
3018:. Assume that multiplying the coset
2716:has a countable number of cosets in
2644:has an infinite number of cosets in
1991:equals its core, i.e., it is normal.
1956:function; this is discussed further
1794:Similarly, the number of conjugates
4867:
3564:) is the intersection of all index
3406:
1869:satisfies the following inequality:
1507:is equal to the order of the image:
1014:{\displaystyle |G:K|=|G:H|\,|H:K|.}
13:
4731:{\displaystyle 0,1,3,7,15,\ldots }
2125:
363:be the subgroup consisting of the
14:
4943:
4895:
4875:The American Mathematical Monthly
4337:
3544:Focal subgroup theorem: Subgroups
2635:
1957:
1694:{\displaystyle |Gx|=|G:G_{x}|.\!}
289:measures the "relative sizes" of
178:of the two groups by the formula
4767:
4448:
4412:
4404:determines a non-trivial map to
4361:
4319:
4279:
4256:
4209:
4169:
4155:
4115:
4051:
4028:
3966:
3940:
3865:
3841:
3817:
3513:point groups in three dimensions
3392:List of small non-abelian groups
2362:{\displaystyle \mathbb {Z} ^{n}}
1963:As a corollary, if the index of
521:, the formula may be written as
16:Mathematics group theory concept
4778:needs additional citations for
4089:to the (cyclic) group of order
4017:, namely the projective space
3308:, the group of permutations of
2738:
2609:{\displaystyle (p^{n}-1)/(p-1)}
2421:{\displaystyle (p^{n}-1)/(p-1)}
2276:
2270:
2234:
356:{\displaystyle H=2\mathbb {Z} }
328:be the group of integers under
4535:
4523:
4515:
4490:
4427:{\displaystyle \mathbf {Z} /p}
4374:
4357:
4349:
4334:
4331:
4309:
4300:
4294:
4291:
4269:
4260:
4222:
4205:
4127:
4105:
4066:
4063:
4041:
4032:
3979:
3962:
3956:
3950:
3881:
3875:
3857:
3851:
3833:
3827:
3805:one obtains the containments
3668:
3656:
3496:On the other hand, the group T
3351:constitute the right coset of
3339:and therefore be identical to
2678:
2664:
2603:
2591:
2583:
2564:
2493:
2458:
2439:
2415:
2403:
2395:
2376:
2292:
2280:
2250:
2238:
2214:
2202:
2137:
2131:
2102:
2096:
1931:
1917:
1909:
1905:
1899:
1883:
1683:
1662:
1654:
1643:
1559:
1547:
1539:
1521:
1459:
1387:
1367:
1309:
1295:
1287:
1267:
1193:
1173:
1115:
1101:
1095:
1081:
1073:
1053:
1004:
990:
984:
970:
962:
948:
873:, since the underlying set of
828:
814:
777:
759:
731:
710:
684:
670:
641:
633:
611:
603:
577:
569:
559:
551:
543:
529:
486:
465:
443:
422:
321:{\displaystyle G=\mathbb {Z} }
275:
261:
232:
224:
219:
205:
197:
189:
174:, the index is related to the
149:
137:
117:
105:
84:
70:
1:
4870:"On Subgroups of Prime Index"
4758:
3412:
3247:. Since this is true for any
3135:. Since this is true for any
2696:. For example, the index of
1838:is equal to the index of the
1779:is equal to the index of the
1621:is equal to the index of the
912:
842:is equal to the order of the
584:{\displaystyle |G:H|=|G|/|H|}
385:{\displaystyle 2\mathbb {Z} }
249:(interpret the quantities as
3727:is a (possibly non-abelian)
3445:subgroup, which we may call
3119:. This is because the coset
3026:gives elements of the coset
3022:on the right by elements of
407:{\displaystyle \mathbb {Z} }
239:{\displaystyle |G|=|G:H||H|}
7:
4741:
4143:is a vector space over the
3312:objects. So for example if
2747:of finite index in a group
2008:
1394:{\displaystyle |H:H\cap K|}
1200:{\displaystyle |G:H\cap K|}
10:
4948:
3473:. There are six cosets of
3425:has 24 elements. It has a
3255:must be a member of A, so
2981:First let us show that if
2950:be the set of elements of
1975:that divides the order of
4868:Lam, T. Y. (March 2004),
3693:) is the largest abelian
2992:, then any other element
2720:. Note that the index of
2464:{\displaystyle (p^{n}-1)}
1952:where ! denotes the
506:for any positive integer
3582:elementary abelian group
3396:Frobenius group#Examples
2774:will be some divisor of
2724:is at most the order of
2082:special orthogonal group
1823:{\displaystyle gHg^{-1}}
1746:{\displaystyle gxg^{-1}}
1707:orbit-stabilizer theorem
1496:{\displaystyle \varphi }
1479:, then the index of the
901:is the set of cosets of
4241:(a non-zero number mod
3790:) is also known as the
3290:times its index inside
3199:. Finally, if for some
2942:Let us call this group
2712:, depending on whether
2625:infinite dihedral group
2432:, corresponding to the
457:is 2. More generally,
62:. The index is denoted
42:is the number of left
4732:
4678:
4646:
4608:
4573:
4467:
4428:
4387:
4235:
4188:
4137:
4076:
3992:
3891:
3756:
3675:
3643:
3548:focal subgroup theorem
3437:, which we shall call
3379:! and thus must equal
3271:is a normal subgroup.
3239:(by the definition of
3191:because each coset of
3115:will be an element of
2686:
2610:
2551:
2518:
2465:
2422:
2363:
2319:
2180:
2144:
2109:
2071:
2070:{\displaystyle S_{n},}
2038:
1942:
1824:
1773:
1772:{\displaystyle x\in G}
1747:
1695:
1570:
1497:
1469:
1424:
1395:
1353:
1320:
1230:
1201:
1159:
1126:
1015:
895:
867:
836:
785:
745:
692:
649:
619:
585:
500:
451:
408:
386:
357:
322:
283:
240:
156:
124:
92:
4802:"Index of a subgroup"
4733:
4679:
4647:
4609:
4574:
4468:
4429:
4388:
4236:
4189:
4138:
4077:
3993:
3892:
3757:
3755:{\displaystyle p^{k}}
3676:
3644:
3642:{\displaystyle p^{k}}
3502:pyritohedral symmetry
3335:must have index 2 in
2687:
2685:{\displaystyle |G:H|}
2611:
2552:
2550:{\displaystyle F_{n}}
2519:
2466:
2423:
2364:
2320:
2181:
2150:, and thus is normal.
2145:
2110:
2072:
2039:
2037:{\displaystyle A_{n}}
1943:
1825:
1774:
1748:
1705:This is known as the
1696:
1571:
1498:
1470:
1425:
1396:
1354:
1321:
1231:
1202:
1160:
1127:
1016:
896:
868:
837:
835:{\displaystyle |G:N|}
786:
746:
693:
691:{\displaystyle |G:H|}
650:
620:
586:
501:
452:
409:
387:
358:
323:
284:
282:{\displaystyle |G:H|}
241:
157:
155:{\displaystyle (G:H)}
125:
93:
91:{\displaystyle |G:H|}
4787:improve this article
4692:
4686:symmetric difference
4677:{\displaystyle p=2,}
4659:
4630:
4607:{\displaystyle k=-1}
4589:
4487:
4438:
4408:
4252:
4202:
4150:
4096:
4024:
3927:
3918:symmetric difference
3812:
3739:
3653:
3626:
3531:Normal subgroups of
3073:, or in other words
2962:is a right coset of
2778:! and a multiple of
2770:, then the index of
2660:
2648:, then the index of
2561:
2534:
2478:
2436:
2373:
2344:
2196:
2160:
2122:
2087:
2051:
2021:
1879:
1798:
1757:
1721:
1639:
1517:
1487:
1447:
1423:{\displaystyle HK=G}
1405:
1363:
1352:{\displaystyle HK=G}
1334:
1263:
1229:{\displaystyle HK=G}
1211:
1169:
1158:{\displaystyle HK=G}
1140:
1049:
944:
877:
849:
810:
755:
706:
666:
629:
599:
525:
461:
418:
396:
371:
336:
304:
257:
185:
134:
102:
66:
4645:{\displaystyle p+1}
3908:Geometric structure
3423:octahedral symmetry
2616:subgroups of index
2428:subgroups of index
2332:More generally, if
2115:has index 2 in the
2077:and thus is normal.
2044:has index 2 in the
1861:, the index of the
894:{\displaystyle G/N}
866:{\displaystyle G/N}
648:{\displaystyle |G|}
618:{\displaystyle |H|}
4728:
4674:
4642:
4604:
4569:
4463:
4424:
4383:
4231:
4184:
4133:
4072:
3988:
3887:
3795:-residual subgroup
3762:normal subgroup):
3752:
3671:
3639:
3588:-group onto which
3568:normal subgroups;
3546:and elaborated at
3509:prismatic symmetry
2682:
2606:
2547:
2514:
2461:
2418:
2359:
2315:
2176:
2155:free abelian group
2140:
2105:
2067:
2034:
1938:
1820:
1769:
1743:
1691:
1566:
1493:
1465:
1420:
1391:
1349:
1316:
1226:
1197:
1155:
1122:
1011:
891:
863:
832:
781:
741:
688:
645:
615:
593:Lagrange's theorem
581:
496:
447:
404:
392:has two cosets in
382:
353:
318:
279:
236:
152:
120:
88:
4863:
4862:
4855:
4837:
3774:) is the largest
3523:symmetric group.
3519:in the 6-member S
3517:alternating group
3453:by an element of
3160:
3159:
2939:
2938:
2310:
2274:
2262:
2226:
2016:alternating group
1857:is a subgroup of
1330:with equality if
1249:are subgroups of
1241:Equivalently, if
1136:with equality if
1035:are subgroups of
930:is a subgroup of
922:is a subgroup of
591:, and it implies
300:For example, let
4939:
4890:
4858:
4851:
4847:
4844:
4838:
4836:
4795:
4771:
4763:
4737:
4735:
4734:
4729:
4683:
4681:
4680:
4675:
4652:such subgroups.
4651:
4649:
4648:
4643:
4613:
4611:
4610:
4605:
4578:
4576:
4575:
4570:
4568:
4567:
4522:
4508:
4507:
4472:
4470:
4469:
4464:
4456:
4451:
4433:
4431:
4430:
4425:
4420:
4415:
4396:to normal index
4392:
4390:
4389:
4384:
4382:
4381:
4369:
4364:
4356:
4327:
4322:
4287:
4282:
4259:
4240:
4238:
4237:
4232:
4230:
4229:
4217:
4212:
4193:
4191:
4190:
4185:
4177:
4172:
4164:
4163:
4158:
4142:
4140:
4139:
4134:
4123:
4118:
4081:
4079:
4078:
4073:
4059:
4054:
4031:
4015:projective space
3997:
3995:
3994:
3989:
3987:
3986:
3974:
3969:
3949:
3948:
3943:
3937:
3896:
3894:
3893:
3888:
3874:
3873:
3868:
3850:
3849:
3844:
3826:
3825:
3820:
3761:
3759:
3758:
3753:
3751:
3750:
3680:
3678:
3677:
3674:{\displaystyle }
3672:
3648:
3646:
3645:
3640:
3638:
3637:
3347:that are not in
3147:, and therefore
2970:
2969:
2805:
2804:
2797:The elements of
2691:
2689:
2688:
2683:
2681:
2667:
2615:
2613:
2612:
2607:
2590:
2576:
2575:
2556:
2554:
2553:
2548:
2546:
2545:
2523:
2521:
2520:
2515:
2513:
2505:
2500:
2492:
2491:
2486:
2470:
2468:
2467:
2462:
2451:
2450:
2427:
2425:
2424:
2419:
2402:
2388:
2387:
2368:
2366:
2365:
2360:
2358:
2357:
2352:
2324:
2322:
2321:
2316:
2311:
2308:
2275:
2272:
2263:
2260:
2227:
2224:
2185:
2183:
2182:
2177:
2175:
2167:
2149:
2147:
2146:
2141:
2117:orthogonal group
2114:
2112:
2111:
2106:
2076:
2074:
2073:
2068:
2063:
2062:
2043:
2041:
2040:
2035:
2033:
2032:
1947:
1945:
1944:
1939:
1934:
1920:
1912:
1886:
1829:
1827:
1826:
1821:
1819:
1818:
1778:
1776:
1775:
1770:
1752:
1750:
1749:
1744:
1742:
1741:
1700:
1698:
1697:
1692:
1686:
1681:
1680:
1665:
1657:
1646:
1575:
1573:
1572:
1567:
1562:
1550:
1542:
1524:
1502:
1500:
1499:
1494:
1474:
1472:
1471:
1466:
1429:
1427:
1426:
1421:
1400:
1398:
1397:
1392:
1390:
1370:
1358:
1356:
1355:
1350:
1325:
1323:
1322:
1317:
1312:
1298:
1290:
1270:
1235:
1233:
1232:
1227:
1206:
1204:
1203:
1198:
1196:
1176:
1164:
1162:
1161:
1156:
1131:
1129:
1128:
1123:
1118:
1104:
1098:
1084:
1076:
1056:
1020:
1018:
1017:
1012:
1007:
993:
987:
973:
965:
951:
900:
898:
897:
892:
887:
872:
870:
869:
864:
859:
841:
839:
838:
833:
831:
817:
790:
788:
787:
782:
780:
775:
767:
762:
750:
748:
747:
742:
734:
729:
718:
713:
697:
695:
694:
689:
687:
673:
654:
652:
651:
646:
644:
636:
624:
622:
621:
616:
614:
606:
590:
588:
587:
582:
580:
572:
567:
562:
554:
546:
532:
505:
503:
502:
497:
489:
484:
473:
468:
456:
454:
453:
448:
446:
441:
430:
425:
413:
411:
410:
405:
403:
391:
389:
388:
383:
381:
362:
360:
359:
354:
352:
327:
325:
324:
319:
317:
288:
286:
285:
280:
278:
264:
251:cardinal numbers
245:
243:
242:
237:
235:
227:
222:
208:
200:
192:
161:
159:
158:
153:
129:
127:
126:
123:{\displaystyle }
121:
97:
95:
94:
89:
87:
73:
4947:
4946:
4942:
4941:
4940:
4938:
4937:
4936:
4922:
4921:
4898:
4893:
4859:
4848:
4842:
4839:
4796:
4794:
4784:
4772:
4761:
4744:
4693:
4690:
4689:
4660:
4657:
4656:
4631:
4628:
4627:
4624:projective line
4590:
4587:
4586:
4563:
4559:
4518:
4497:
4493:
4488:
4485:
4484:
4452:
4447:
4439:
4436:
4435:
4416:
4411:
4409:
4406:
4405:
4377:
4373:
4365:
4360:
4352:
4323:
4318:
4283:
4278:
4255:
4253:
4250:
4249:
4225:
4221:
4213:
4208:
4203:
4200:
4199:
4173:
4168:
4159:
4154:
4153:
4151:
4148:
4147:
4119:
4114:
4097:
4094:
4093:
4055:
4050:
4027:
4025:
4022:
4021:
3982:
3978:
3970:
3965:
3944:
3939:
3938:
3933:
3928:
3925:
3924:
3910:
3902:Sylow subgroups
3869:
3864:
3863:
3845:
3840:
3839:
3821:
3816:
3815:
3813:
3810:
3809:
3746:
3742:
3740:
3737:
3736:
3654:
3651:
3650:
3633:
3629:
3627:
3624:
3623:
3529:
3522:
3507:
3499:
3484:
3479:symmetric group
3444:
3432:
3415:
3389:
3323:In the case of
3319:
3307:
3300:symmetric group
3294:. Its index in
3282:, its index in
3219:, then for any
3161:
3123:is the same as
3086:
3079:
3064:
3058:
3047:
3036:
3009:
2998:
2987:
2975:
2940:
2922:
2912:
2905:
2898:
2880:
2870:
2810:
2753:normal subgroup
2741:
2694:cardinal number
2677:
2663:
2661:
2658:
2657:
2638:
2629:cyclic subgroup
2586:
2571:
2567:
2562:
2559:
2558:
2541:
2537:
2535:
2532:
2531:
2527:Similarly, the
2509:
2501:
2496:
2487:
2482:
2481:
2479:
2476:
2475:
2446:
2442:
2437:
2434:
2433:
2398:
2383:
2379:
2374:
2371:
2370:
2353:
2348:
2347:
2345:
2342:
2341:
2307:
2271:
2259:
2223:
2197:
2194:
2193:
2171:
2163:
2161:
2158:
2157:
2123:
2120:
2119:
2088:
2085:
2084:
2058:
2054:
2052:
2049:
2048:
2046:symmetric group
2028:
2024:
2022:
2019:
2018:
2011:
1930:
1916:
1908:
1882:
1880:
1877:
1876:
1811:
1807:
1799:
1796:
1795:
1758:
1755:
1754:
1734:
1730:
1722:
1719:
1718:
1682:
1676:
1672:
1661:
1653:
1642:
1640:
1637:
1636:
1558:
1546:
1538:
1520:
1518:
1515:
1514:
1488:
1485:
1484:
1448:
1445:
1444:
1443:are groups and
1406:
1403:
1402:
1386:
1366:
1364:
1361:
1360:
1335:
1332:
1331:
1308:
1294:
1286:
1266:
1264:
1261:
1260:
1212:
1209:
1208:
1192:
1172:
1170:
1167:
1166:
1141:
1138:
1137:
1114:
1100:
1094:
1080:
1072:
1052:
1050:
1047:
1046:
1003:
989:
983:
969:
961:
947:
945:
942:
941:
915:
883:
878:
875:
874:
855:
850:
847:
846:
827:
813:
811:
808:
807:
800:normal subgroup
776:
771:
763:
758:
756:
753:
752:
730:
725:
714:
709:
707:
704:
703:
700:cardinal number
683:
669:
667:
664:
663:
640:
632:
630:
627:
626:
610:
602:
600:
597:
596:
576:
568:
563:
558:
550:
542:
528:
526:
523:
522:
485:
480:
469:
464:
462:
459:
458:
442:
437:
426:
421:
419:
416:
415:
399:
397:
394:
393:
377:
372:
369:
368:
348:
337:
334:
333:
313:
305:
302:
301:
274:
260:
258:
255:
254:
231:
223:
218:
204:
196:
188:
186:
183:
182:
135:
132:
131:
103:
100:
99:
83:
69:
67:
64:
63:
23:, specifically
17:
12:
11:
5:
4945:
4935:
4934:
4920:
4919:
4909:
4897:
4896:External links
4894:
4892:
4891:
4882:(3): 256–258,
4864:
4861:
4860:
4775:
4773:
4766:
4760:
4757:
4756:
4755:
4750:
4743:
4740:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4673:
4670:
4667:
4664:
4641:
4638:
4635:
4626:consisting of
4622:one obtains a
4603:
4600:
4597:
4594:
4580:
4579:
4566:
4562:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4521:
4517:
4514:
4511:
4506:
4503:
4500:
4496:
4492:
4462:
4459:
4455:
4450:
4446:
4443:
4423:
4419:
4414:
4394:
4393:
4380:
4376:
4372:
4368:
4363:
4359:
4355:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4326:
4321:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4286:
4281:
4277:
4274:
4271:
4268:
4265:
4262:
4258:
4228:
4224:
4220:
4216:
4211:
4207:
4183:
4180:
4176:
4171:
4167:
4162:
4157:
4132:
4129:
4126:
4122:
4117:
4113:
4110:
4107:
4104:
4101:
4083:
4082:
4071:
4068:
4065:
4062:
4058:
4053:
4049:
4046:
4043:
4040:
4037:
4034:
4030:
4000:
3999:
3985:
3981:
3977:
3973:
3968:
3964:
3961:
3958:
3955:
3952:
3947:
3942:
3936:
3932:
3909:
3906:
3898:
3897:
3886:
3883:
3880:
3877:
3872:
3867:
3862:
3859:
3856:
3853:
3848:
3843:
3838:
3835:
3832:
3829:
3824:
3819:
3799:
3798:
3749:
3745:
3731:-group (i.e.,
3702:
3670:
3667:
3664:
3661:
3658:
3636:
3632:
3618:-group (i.e.,
3614:is an abelian
3593:
3528:
3525:
3520:
3505:
3497:
3482:
3442:
3430:
3414:
3411:
3387:
3317:
3303:
3267:and therefore
3158:
3157:
3084:
3077:
3062:
3056:
3045:
3034:
3007:
2996:
2985:
2977:
2976:
2973:
2968:
2937:
2936:
2920:
2910:
2903:
2896:
2878:
2868:
2812:
2811:
2808:
2803:
2740:
2737:
2692:is actually a
2680:
2676:
2673:
2670:
2666:
2637:
2636:Infinite index
2634:
2633:
2632:
2621:
2605:
2602:
2599:
2596:
2593:
2589:
2585:
2582:
2579:
2574:
2570:
2566:
2544:
2540:
2525:
2512:
2508:
2504:
2499:
2495:
2490:
2485:
2460:
2457:
2454:
2449:
2445:
2441:
2417:
2414:
2411:
2408:
2405:
2401:
2397:
2394:
2391:
2386:
2382:
2378:
2356:
2351:
2329:
2328:
2327:
2326:
2314:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2269:
2266:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2233:
2230:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2188:
2187:
2174:
2170:
2166:
2151:
2139:
2136:
2133:
2130:
2127:
2104:
2101:
2098:
2095:
2092:
2078:
2066:
2061:
2057:
2031:
2027:
2010:
2007:
2006:
2005:
2004:
2003:
1992:
1950:
1949:
1948:
1937:
1933:
1929:
1926:
1923:
1919:
1915:
1911:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1885:
1871:
1870:
1851:
1830:of a subgroup
1817:
1814:
1810:
1806:
1803:
1792:
1768:
1765:
1762:
1753:of an element
1740:
1737:
1733:
1729:
1726:
1711:
1710:
1703:
1702:
1701:
1689:
1685:
1679:
1675:
1671:
1668:
1664:
1660:
1656:
1652:
1649:
1645:
1631:
1630:
1579:
1578:
1577:
1576:
1565:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1530:
1527:
1523:
1509:
1508:
1492:
1464:
1461:
1458:
1455:
1452:
1432:
1431:
1419:
1416:
1413:
1410:
1389:
1385:
1382:
1379:
1376:
1373:
1369:
1348:
1345:
1342:
1339:
1328:
1327:
1326:
1315:
1311:
1307:
1304:
1301:
1297:
1293:
1289:
1285:
1282:
1279:
1276:
1273:
1269:
1255:
1254:
1238:
1237:
1225:
1222:
1219:
1216:
1195:
1191:
1188:
1185:
1182:
1179:
1175:
1154:
1151:
1148:
1145:
1134:
1133:
1132:
1121:
1117:
1113:
1110:
1107:
1103:
1097:
1093:
1090:
1087:
1083:
1079:
1075:
1071:
1068:
1065:
1062:
1059:
1055:
1041:
1040:
1024:
1023:
1022:
1021:
1010:
1006:
1002:
999:
996:
992:
986:
982:
979:
976:
972:
968:
964:
960:
957:
954:
950:
936:
935:
914:
911:
890:
886:
882:
862:
858:
854:
844:quotient group
830:
826:
823:
820:
816:
779:
774:
770:
766:
761:
740:
737:
733:
728:
724:
721:
717:
712:
686:
682:
679:
676:
672:
643:
639:
635:
613:
609:
605:
579:
575:
571:
566:
561:
557:
553:
549:
545:
541:
538:
535:
531:
495:
492:
488:
483:
479:
476:
472:
467:
445:
440:
436:
433:
429:
424:
402:
380:
376:
351:
347:
344:
341:
316:
312:
309:
277:
273:
270:
267:
263:
247:
246:
234:
230:
226:
221:
217:
214:
211:
207:
203:
199:
195:
191:
151:
148:
145:
142:
139:
119:
116:
113:
110:
107:
86:
82:
79:
76:
72:
15:
9:
6:
4:
3:
2:
4944:
4933:
4930:
4929:
4927:
4918:
4914:
4910:
4907:
4903:
4900:
4899:
4889:
4885:
4881:
4877:
4876:
4871:
4866:
4865:
4857:
4854:
4846:
4835:
4832:
4828:
4825:
4821:
4818:
4814:
4811:
4807:
4804: –
4803:
4799:
4798:Find sources:
4792:
4788:
4782:
4781:
4776:This article
4774:
4770:
4765:
4764:
4754:
4751:
4749:
4746:
4745:
4739:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4687:
4671:
4668:
4665:
4662:
4653:
4639:
4636:
4633:
4625:
4621:
4617:
4601:
4598:
4595:
4592:
4585:
4564:
4560:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4532:
4529:
4526:
4519:
4512:
4509:
4504:
4501:
4498:
4494:
4483:
4482:
4481:
4479:
4474:
4460:
4457:
4453:
4444:
4441:
4421:
4417:
4403:
4399:
4378:
4370:
4366:
4353:
4343:
4328:
4324:
4315:
4312:
4306:
4303:
4297:
4288:
4284:
4275:
4272:
4266:
4263:
4248:
4247:
4246:
4244:
4226:
4218:
4214:
4197:
4181:
4178:
4174:
4165:
4160:
4146:
4130:
4124:
4120:
4111:
4108:
4102:
4099:
4092:
4088:
4069:
4060:
4056:
4047:
4044:
4038:
4035:
4020:
4019:
4018:
4016:
4012:
4007:
4005:
4002:and further,
3983:
3975:
3971:
3959:
3953:
3945:
3934:
3930:
3923:
3922:
3921:
3919:
3915:
3905:
3903:
3884:
3878:
3870:
3860:
3854:
3846:
3836:
3830:
3822:
3808:
3807:
3806:
3804:
3796:
3794:
3789:
3785:
3781:
3777:
3773:
3769:
3765:
3747:
3743:
3734:
3730:
3726:
3722:
3718:
3714:
3710:
3706:
3703:
3700:
3696:
3692:
3688:
3684:
3665:
3662:
3659:
3634:
3630:
3621:
3617:
3613:
3609:
3605:
3601:
3597:
3594:
3591:
3587:
3583:
3579:
3575:
3571:
3567:
3563:
3559:
3556:
3555:
3554:
3551:
3549:
3545:
3541:
3539:
3534:
3524:
3518:
3514:
3510:
3503:
3494:
3492:
3488:
3480:
3476:
3472:
3469:is normal in
3468:
3464:
3460:
3456:
3452:
3448:
3440:
3436:
3428:
3424:
3420:
3410:
3408:
3404:
3399:
3397:
3393:
3386:
3382:
3378:
3374:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3342:
3338:
3334:
3330:
3326:
3321:
3315:
3311:
3306:
3301:
3297:
3293:
3289:
3285:
3281:
3277:
3272:
3270:
3266:
3262:
3259:implies that
3258:
3254:
3250:
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3218:
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3156:
3154:
3150:
3146:
3142:
3138:
3134:
3130:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3098:
3094:
3090:
3083:
3076:
3072:
3068:
3061:
3055:
3051:
3044:
3040:
3033:
3029:
3025:
3021:
3017:
3013:
3006:
3002:
2995:
2991:
2984:
2979:
2978:
2972:
2971:
2967:
2965:
2961:
2957:
2953:
2949:
2945:
2935:
2933:
2929:
2925:
2919:
2915:
2909:
2902:
2895:
2891:
2887:
2883:
2877:
2873:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2835:
2832:and likewise
2831:
2827:
2823:
2819:
2814:
2813:
2807:
2806:
2802:
2800:
2795:
2793:
2789:
2785:
2781:
2777:
2773:
2769:
2765:
2761:
2757:
2754:
2750:
2746:
2736:
2735:
2731:
2727:
2723:
2719:
2715:
2711:
2707:
2703:
2699:
2695:
2674:
2671:
2668:
2655:
2651:
2647:
2643:
2630:
2626:
2622:
2619:
2600:
2597:
2594:
2587:
2580:
2577:
2572:
2568:
2542:
2538:
2530:
2526:
2506:
2502:
2488:
2474:
2473:homomorphisms
2455:
2452:
2447:
2443:
2431:
2412:
2409:
2406:
2399:
2392:
2389:
2384:
2380:
2354:
2339:
2335:
2331:
2330:
2309: is even
2304:
2301:
2298:
2295:
2289:
2286:
2283:
2267:
2261: is even
2256:
2253:
2247:
2244:
2241:
2231:
2225: is even
2220:
2217:
2211:
2208:
2205:
2192:
2191:
2190:
2189:
2168:
2156:
2152:
2134:
2128:
2118:
2099:
2093:
2090:
2083:
2079:
2064:
2059:
2055:
2047:
2029:
2025:
2017:
2013:
2012:
2001:
2000:perfect group
1997:
1993:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1962:
1961:
1959:
1955:
1951:
1935:
1927:
1924:
1921:
1913:
1902:
1896:
1893:
1890:
1887:
1875:
1874:
1873:
1872:
1868:
1864:
1860:
1856:
1852:
1849:
1845:
1841:
1837:
1833:
1815:
1812:
1808:
1804:
1801:
1793:
1790:
1786:
1782:
1766:
1763:
1760:
1738:
1735:
1731:
1727:
1724:
1717:
1713:
1712:
1708:
1704:
1687:
1677:
1673:
1669:
1666:
1658:
1650:
1647:
1635:
1634:
1633:
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1604:
1601: ∈
1600:
1596:
1593:
1589:
1585:
1581:
1580:
1563:
1555:
1551:
1543:
1535:
1531:
1528:
1525:
1513:
1512:
1511:
1510:
1506:
1490:
1482:
1478:
1462:
1456:
1453:
1450:
1442:
1438:
1434:
1433:
1417:
1414:
1411:
1408:
1383:
1380:
1377:
1374:
1371:
1346:
1343:
1340:
1337:
1329:
1313:
1305:
1302:
1299:
1291:
1283:
1280:
1277:
1274:
1271:
1259:
1258:
1257:
1256:
1252:
1248:
1244:
1240:
1239:
1223:
1220:
1217:
1214:
1189:
1186:
1183:
1180:
1177:
1152:
1149:
1146:
1143:
1135:
1119:
1111:
1108:
1105:
1091:
1088:
1085:
1077:
1069:
1066:
1063:
1060:
1057:
1045:
1044:
1043:
1042:
1038:
1034:
1030:
1026:
1025:
1008:
1000:
997:
994:
980:
977:
974:
966:
958:
955:
952:
940:
939:
938:
937:
933:
929:
925:
921:
917:
916:
910:
908:
904:
888:
884:
880:
860:
856:
852:
845:
824:
821:
818:
805:
801:
797:
792:
791:is infinite.
768:
738:
735:
722:
719:
701:
698:is a nonzero
680:
677:
674:
662:is infinite,
661:
656:
637:
607:
594:
573:
564:
555:
547:
539:
536:
533:
520:
516:
511:
509:
493:
490:
477:
474:
434:
431:
374:
366:
365:even integers
345:
342:
339:
331:
310:
307:
298:
296:
292:
271:
268:
265:
252:
228:
215:
212:
209:
201:
193:
181:
180:
179:
177:
173:
169:
165:
146:
143:
140:
114:
111:
108:
80:
77:
74:
61:
57:
53:
49:
45:
41:
37:
34:
30:
26:
22:
4932:Group theory
4879:
4873:
4849:
4843:January 2010
4840:
4830:
4823:
4816:
4809:
4797:
4785:Please help
4780:verification
4777:
4654:
4619:
4615:
4583:
4581:
4477:
4475:
4401:
4397:
4395:
4242:
4195:
4145:finite field
4090:
4086:
4084:
4010:
4008:
4003:
4001:
3911:
3899:
3802:
3800:
3792:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3735:is an index
3732:
3728:
3724:
3720:
3716:
3712:
3708:
3704:
3698:
3694:
3690:
3686:
3682:
3622:is an index
3619:
3615:
3611:
3607:
3603:
3599:
3595:
3589:
3585:
3577:
3573:
3569:
3565:
3561:
3557:
3552:
3537:
3530:
3495:
3490:
3486:
3474:
3470:
3466:
3462:
3458:
3454:
3450:
3446:
3438:
3434:
3418:
3416:
3402:
3400:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3328:
3324:
3322:
3313:
3309:
3304:
3295:
3291:
3287:
3283:
3279:
3275:
3273:
3268:
3264:
3260:
3256:
3252:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3208:
3204:
3200:
3196:
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3162:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3081:
3074:
3070:
3066:
3059:
3053:
3049:
3042:
3038:
3031:
3027:
3023:
3019:
3015:
3011:
3004:
3000:
2993:
2989:
2982:
2980:
2963:
2959:
2955:
2951:
2947:
2943:
2941:
2931:
2927:
2923:
2917:
2913:
2907:
2900:
2893:
2889:
2885:
2881:
2875:
2871:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2821:
2817:
2815:
2798:
2796:
2791:
2787:
2783:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2748:
2744:
2742:
2739:Finite index
2733:
2729:
2725:
2721:
2717:
2713:
2701:
2697:
2653:
2649:
2645:
2641:
2639:
2617:
2429:
2333:
1996:simple group
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1866:
1858:
1854:
1847:
1843:
1835:
1831:
1788:
1784:
1626:
1618:
1614:
1605:. Then the
1602:
1598:
1594:
1583:
1504:
1477:homomorphism
1440:
1436:
1250:
1246:
1242:
1036:
1032:
1028:
931:
927:
923:
919:
906:
902:
803:
795:
793:
659:
657:
514:
512:
507:
299:
294:
290:
248:
171:
163:
59:
55:
51:
47:
39:
35:
28:
25:group theory
18:
4753:Codimension
3533:prime power
3511:group, see
3227:, but also
3225:G dca = dxc
2743:A subgroup
2710:uncountable
2471:nontrivial
1863:normal core
1781:centralizer
1607:cardinality
1586:be a group
38:in a group
21:mathematics
4906:PlanetMath
4813:newspapers
4759:References
3914:complement
3782:surjects.
3719:such that
3606:such that
3421:of chiral
3417:The group
2906:∈ H) then
2782:; indeed,
2766:has index
2529:free group
1840:normalizer
1716:conjugates
1623:stabilizer
1597:, and let
913:Properties
332:, and let
162:. Because
4748:Virtually
4726:…
4599:−
4582:for some
4554:⋯
4530:−
4510:−
4445:∈
4379:×
4338:∖
4307:
4267:
4227:×
4103:
4039:
3960:≅
3916:of their
3861:⊇
3837:⊇
3701:surjects.
3592:surjects.
3231:for some
3229:dca = hdc
3087:for some
3010:for some
2706:countable
2598:−
2578:−
2494:→
2453:−
2410:−
2390:−
2296:∣
2254:∣
2218:∣
2169:⊕
2129:
2094:
1987:and thus
1954:factorial
1914:≤
1897:
1813:−
1764:∈
1736:−
1556:φ
1536:φ
1491:φ
1460:→
1454::
1451:φ
1381:∩
1292:≤
1281:∩
1187:∩
1078:≤
1067:∩
4926:Category
4742:See also
3580:) is an
3463:Hca = Hc
3427:dihedral
3413:Examples
3407:Lam 2004
3375:divides
3286:must be
3215:we have
2884:for all
2009:Examples
625:divides
367:. Then
330:addition
33:subgroup
4888:4145135
4827:scholar
4013:form a
3540:-groups
3257:ca = xc
3245:hd = dx
3217:ca = xc
3052:, then
3003:equals
2958:. Then
2848:, then
2704:may be
1609:of the
1359:. (If
1165:. (If
806:, then
4886:
4829:
4822:
4815:
4808:
4800:
3359:where
3243:), so
3179:. (If
2946:. Let
2892:(with
2627:has a
1617:under
1588:acting
1481:kernel
1253:, then
1039:, then
934:, then
751:, but
519:finite
176:orders
44:cosets
27:, the
4915:" at
4884:JSTOR
4834:JSTOR
4820:books
3127:, so
3030:. If
2974:Proof
2926:, so
2864:. If
2809:Proof
2340:then
2338:prime
1979:then
1958:below
1611:orbit
1590:on a
1475:is a
798:is a
658:When
595:that
513:When
31:of a
29:index
4806:news
4684:the
4655:For
4480:is
3394:and
3367:(if
3207:and
3133:Hcab
3103:and
3041:and
2850:Hcab
2758:(of
2623:The
2557:has
2369:has
2153:The
2080:The
2014:The
1894:Core
1582:Let
1439:and
1245:and
1031:and
926:and
293:and
168:size
4904:at
4880:111
4789:by
4304:Hom
4264:Hom
4100:Hom
4036:Hom
3715:of
3681:):
3500:of
3465:).
3409:).
3398:).
3261:cac
3129:Hcb
3125:Hca
2999:of
2928:Hca
2834:Hcb
2818:Hca
2816:If
2708:or
2700:in
2652:in
2640:If
2336:is
2273:and
1967:in
1865:of
1853:If
1846:in
1842:of
1834:in
1787:in
1783:of
1625:of
1613:of
1592:set
1532:ker
1503:in
1483:of
1435:If
1027:If
918:If
905:in
802:of
794:If
517:is
170:as
130:or
98:or
58:in
50:in
46:of
19:In
4928::
4878:,
4872:,
4720:15
4620:p,
4584:k;
4298::=
4196:p,
4091:p,
3803:K,
3550:.
3506:2h
3493:.
3381:p,
3320:.
3263:∈
3251:,
3235:∈
3223:∈
3211:∈
3203:∈
3155:.
3149:ab
3141:ab
3131:=
3121:Hc
3113:ab
3111:,
3082:ab
3071:Hc
3069:∈
3067:hc
3065:=
3054:cb
3050:hd
3048:=
3043:cb
3037:=
3032:cb
3028:Hd
3020:Hc
3005:ab
2966:.
2934:.
2932:Hc
2930:⊂
2916:=
2914:ca
2899:,
2888:∈
2874:=
2872:ca
2860:∈
2856:∀
2854:Hc
2852:⊂
2844:∈
2840:∀
2838:Hc
2836:⊂
2828:∈
2824:∀
2822:Hc
2820:⊂
2734:G.
2726:G,
2091:SO
1985:p,
1977:G,
1960:.
1552:im
1430:.)
1236:.)
909:.
655:.
510:.
297:.
4911:"
4908:.
4856:)
4850:(
4845:)
4841:(
4831:·
4824:·
4817:·
4810:·
4783:.
4723:,
4717:,
4714:7
4711:,
4708:3
4705:,
4702:1
4699:,
4696:0
4672:,
4669:2
4666:=
4663:p
4640:1
4637:+
4634:p
4616:p
4602:1
4596:=
4593:k
4565:k
4561:p
4557:+
4551:+
4548:p
4545:+
4542:1
4539:=
4536:)
4533:1
4527:p
4524:(
4520:/
4516:)
4513:1
4505:1
4502:+
4499:k
4495:p
4491:(
4478:p
4461:,
4458:p
4454:/
4449:Z
4442:1
4422:p
4418:/
4413:Z
4402:p
4398:p
4375:)
4371:p
4367:/
4362:Z
4358:(
4354:/
4350:)
4347:}
4344:0
4341:{
4335:)
4332:)
4329:p
4325:/
4320:Z
4316:,
4313:G
4310:(
4301:(
4295:)
4292:)
4289:p
4285:/
4280:Z
4276:,
4273:G
4270:(
4261:(
4257:P
4243:p
4223:)
4219:p
4215:/
4210:Z
4206:(
4182:.
4179:p
4175:/
4170:Z
4166:=
4161:p
4156:F
4131:,
4128:)
4125:p
4121:/
4116:Z
4112:,
4109:G
4106:(
4087:G
4070:.
4067:)
4064:)
4061:p
4057:/
4052:Z
4048:,
4045:G
4042:(
4033:(
4029:P
4011:p
4004:G
3998:,
3984:k
3980:)
3976:p
3972:/
3967:Z
3963:(
3957:)
3954:G
3951:(
3946:p
3941:E
3935:/
3931:G
3885:.
3882:)
3879:G
3876:(
3871:p
3866:O
3858:)
3855:G
3852:(
3847:p
3842:A
3834:)
3831:G
3828:(
3823:p
3818:E
3797:.
3793:p
3788:G
3786:(
3784:O
3780:G
3776:p
3772:G
3770:(
3768:O
3766:/
3764:G
3748:k
3744:p
3733:K
3729:p
3725:K
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