820:
3235:
7620:
3279:
52:
5496:
1636:
to give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group to its group structure. From this observation, classifying finite groups becomes a game of finding which combinations/constructions of groups of
6518:
of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in Seress, and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the
5158:
3297:
is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an
1615:
The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group
3642:
Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely. Most of the examples use Sylow's theorem to prove that a group of a particular order is not
3518:
4432:
The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves is the subject of many papers, including
Waterhouse, Scharlau, Casadio and Zappa, Gow, and to some extent Meo.
5491:{\displaystyle |\Omega |={p^{k}m \choose p^{k}}=\prod _{j=0}^{p^{k}-1}{\frac {p^{k}m-j}{p^{k}-j}}=m\prod _{j=1}^{p^{k}-1}{\frac {p^{k-\nu _{p}(j)}m-j/p^{\nu _{p}(j)}}{p^{k-\nu _{p}(j)}-j/p^{\nu _{p}(j)}}}}
4448:-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of Wielandt. In the following, we use
3223:
3156:
6499:
itself. The algorithmic version of this (and many improvements) is described in textbook form in Butler, including the algorithm described in Cannon. These versions are still used in the
1718:
4080:
2395:
1920:
3053:
5942:
1637:
smaller order can be applied to construct a group. For example, a typical application of these theorems is in the classification of finite groups of some fixed cardinality, e.g.
1206:
2461:
3271:
odd, 2 = 2 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are
497:
472:
435:
3079:
2168:
1768:
1853:
4608:
4498:
5899:
4472:
2888:
2811:
2255:
2229:
2126:
1794:
1671:
3773:
2965:
2915:
2775:
2608:
2581:
2500:
2328:
2290:
2004:
1453:
1385:
4546:
2997:
1064:
7623:
2935:
2851:
2831:
2748:
2728:
2708:
2688:
2668:
2648:
2628:
2554:
2534:
1634:
1600:
1580:
1560:
1536:
1516:
1496:
1475:
1425:
1405:
1358:
1338:
1318:
1297:
1273:
1253:
1233:
1168:
1148:
1128:
1108:
1088:
1030:
1010:
990:
967:
932:
1681:
Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of
799:
3450:
6941:
4409:
6633:
3782:≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be
4405:
4192:
shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as
905:
357:
5668:
4845:
4719:
4685:
7677:
4424:'s strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.
1212:
307:
3903:
then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means
3647:. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a
3169:
904:
contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the
2016:
792:
302:
7568:
7262:
7242:
6667:
863:
841:
3106:
834:
3275:, and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.
2890:). However, there are groups that have proper, non-trivial normal subgroups but no normal Sylow subgroups, such as
1684:
4025:
3852:, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30
1957:
718:
17:
2351:
7611:
7309:
6520:
1866:
785:
3026:
3610:-subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow
5904:
7606:
6500:
1173:
402:
216:
2401:
7560:
6569:
6422:
4622:
4437:
889:
134:
4886:
4764:
elements, providing the desired subgroup. This is the maximal possible size of a stabilizer subgroup
7601:
3800:
must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are
7364:
4334:)-conjugate. Burnside's fusion theorem can be used to give a more powerful factorization called a
1067:
828:
600:
334:
211:
99:
480:
455:
418:
3809:
3525:
6998:
4089:. One may easily prove this theorem by Sylow's third theorem. Indeed, observe that the number
3058:
2131:
1929:
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in
1727:
7359:
6533:
4393:
4189:
2041:, then there exists an element (and thus a cyclic subgroup generated by this element) of order
1815:
845:
750:
540:
4587:
4477:
4404:
has on the structure of the entire group. This control is exploited at several stages of the
7461:
Kantor, William M.; Taylor, Donald E. (1988). "Polynomial-time versions of Sylow's theorem".
7114:
6578:
6573:
5866:
4451:
3837:
2860:
2783:
2012:
1931:
624:
2234:
2204:
2105:
1773:
1640:
7578:
7539:
7492:
7445:
7389:
7325:
7272:
7206:
7148:
7098:
7067:
7045:
7015:
6978:
6924:
6871:
4417:
3758:
3289:
reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.
2943:
2893:
2753:
2586:
2559:
2478:
2306:
2268:
1982:
1431:
1363:
564:
552:
170:
104:
7586:
7547:
7500:
7453:
7397:
7333:
7288:
7222:
7164:
7106:
7053:
6994:
6932:
6879:
6599:
8:
6848:
Casadio, Giuseppina; Zappa, Guido (1990). "History of the Sylow theorem and its proofs".
5119:
4521:
4016:
2972:
2335:
1040:
877:
139:
34:
6942:"A formal proof of Sylow's theorem. An experiment in abstract algebra with Isabelle HOL"
1046:
7276:
7210:
7172:
7152:
6982:
6622:
6603:
4335:
4004:
3299:
2920:
2836:
2816:
2733:
2713:
2693:
2673:
2653:
2633:
2613:
2539:
2519:
1619:
1585:
1565:
1545:
1521:
1501:
1481:
1460:
1410:
1390:
1343:
1323:
1303:
1282:
1258:
1238:
1218:
1153:
1133:
1113:
1093:
1073:
1015:
995:
975:
952:
917:
124:
96:
7028:
7010:
7682:
7650:
7634:
7631:
7564:
7527:
7523:
7480:
7476:
7433:
7428:
7409:
7377:
7373:
7313:
7258:
7214:
7194:
7156:
7136:
7086:
7081:
7062:
7033:
6966:
6912:
6859:
6663:
6607:
6507:
4383:
4000:
1721:
529:
372:
266:
7343:"Polynomial-time algorithms for finding elements of prime order and Sylow subgroups"
6421:
The problem of finding a Sylow subgroup of a given group is an important problem in
3007:
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow
695:
7653:
7582:
7543:
7519:
7496:
7472:
7449:
7423:
7393:
7369:
7342:
7329:
7284:
7250:
7218:
7186:
7160:
7128:
7102:
7076:
7049:
7023:
6990:
6986:
6958:
6928:
6902:
6875:
6595:
6587:
6543:
4401:
970:
897:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
7299:
7280:
7574:
7535:
7488:
7441:
7385:
7321:
7268:
7246:
7202:
7144:
7094:
7041:
6974:
6920:
6867:
6515:
5774:
3783:
3648:
2940:
The third bullet point of the third theorem has as an immediate consequence that
2854:
2083:
1539:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
3992:= 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is
7405:
3849:
3250:
3234:
771:
707:
397:
377:
279:
200:
190:
175:
160:
114:
91:
27:
Theorems that help decompose a finite group based on prime factors of its order
6962:
3245:
all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.
7671:
7619:
7531:
7484:
7437:
7381:
7317:
7198:
7140:
7090:
7037:
6970:
6916:
6907:
6888:
6863:
6538:
4421:
3345:
690:
612:
446:
319:
185:
7296:
Cannon, John J. (1971). "Computing local structure of large finite groups".
7508:
Kantor, William M. (1990). "Finding Sylow normalizers in polynomial time".
4413:
3833:
3829:
3813:
3644:
1950:
1276:
912:
901:
545:
244:
233:
180:
155:
150:
109:
80:
43:
7254:
3575:
is a subgroup where all its elements have orders which are powers of
5778:
3845:
3513:{\displaystyle {\begin{bmatrix}x^{im}&0\\0&x^{jm}\end{bmatrix}}}
3249:
A simple illustration of Sylow subgroups and the Sylow theorems are the
1922:. These properties can be exploited to further analyze the structure of
7190:
7132:
6662:. with contribution by Victor J. Katz. Pearson Education. p. 322.
6591:
6121:
4243:)-conjugate. The proof is a simple application of Sylow's theorem: If
2503:
712:
440:
6850:
3278:
7658:
7639:
5505:
remains in any of the factors inside the product on the right. Hence
5097:, one can show the existence of ω of the former type by showing that
533:
6510:, it has been proven, in Kantor and Kantor and Taylor, that a Sylow
893:
70:
3899:
would have a normal subgroup of order 3, and could not be simple.
6548:
5599:
denote the set of points of Ω that are fixed under the action of
3844:
states that if the order of a group is the product of one or two
3801:
2967:
1968:
1033:
885:
412:
326:
7414:
51:
2011:
The following weaker version of theorem 1 was first proved by
7119:
6475:. In other words, a polycyclic generating system of a Sylow
5825:
3817:
3804:, the intersection of these two subgroups is trivial, and so
6748:
4392:
Less trivial applications of the Sylow theorems include the
2917:. Groups that are of prime-power order have no proper Sylow
2780:
An important consequence of
Theorem 2 is that the condition
6893:
3935:
also has 24 distinct elements of order 5. But the order of
5675:
by assumption. The result follows immediately by writing
4408:, and for instance defines the case divisions used in the
3828:
A more complex example involves the order of the smallest
3567:
and all its powers have an order which is a power of
7629:
7175:(1959). "Ein Beweis für die Existenz der Sylowgruppen".
6808:
5152:
places), and can also be shown by a simple computation:
3939:
is only 30, so a simple group of order 30 cannot exist.
3816:
of order 15. Thus, there is only one group of order 15 (
7648:
6700:
5840:
act on Ω by left multiplication. Applying the Lemma to
4436:
One proof of the Sylow theorems exploits the notion of
3684:
Group of order 30, groups of order 20, groups of order
6949:
3459:
3019:-power order) that is maximal for inclusion among all
1724:
to each other and have the largest possible order: if
1215:. Lagrange's theorem states that for any finite group
6676:
5907:
5869:
5161:
4590:
4524:
4480:
4454:
4028:
3761:
3738:= 15 is such a number using the Sylow theorems: Let
3453:
3172:
3109:
3061:
3029:
3002:
2975:
2946:
2923:
2896:
2863:
2839:
2819:
2786:
2756:
2736:
2716:
2696:
2676:
2656:
2636:
2616:
2589:
2562:
2542:
2522:
2481:
2404:
2354:
2309:
2271:
2237:
2207:
2134:
2108:
1985:
1869:
1818:
1776:
1730:
1687:
1643:
1622:
1588:
1568:
1548:
1524:
1504:
1484:
1463:
1434:
1413:
1393:
1366:
1346:
1326:
1306:
1285:
1261:
1241:
1221:
1176:
1156:
1136:
1116:
1096:
1076:
1049:
1018:
998:
978:
955:
920:
483:
458:
421:
2630:-subgroup, and so is conjugate to every other Sylow
1235:
the order (number of elements) of every subgroup of
6784:
6772:
6736:
6688:
6298:denote the set of fixed points of this action. Let
4350:is contained in the center of its normalizer, then
3306:, half the minimal rotation in the dihedral group.
892:that give detailed information about the number of
7559:. Cambridge Tracts in Mathematics. Vol. 152.
6820:
6796:
6760:
6274:, and it follows that this number is a divisor of
5936:
5893:
5490:
4602:
4540:
4492:
4466:
4074:
3786:(since it has no distinct conjugates). Similarly,
3767:
3614:-subgroups are conjugate to each other, the Sylow
3512:
3218:{\displaystyle n_{p}\equiv 1\ (\mathrm {mod} \ p)}
3217:
3150:
3073:
3047:
2991:
2959:
2929:
2909:
2882:
2845:
2825:
2805:
2769:
2742:
2722:
2702:
2682:
2662:
2642:
2622:
2602:
2575:
2548:
2528:
2494:
2455:
2389:
2322:
2284:
2249:
2223:
2162:
2120:
1998:
1914:
1847:
1788:
1762:
1712:
1665:
1628:
1594:
1574:
1554:
1530:
1510:
1490:
1469:
1447:
1419:
1399:
1379:
1352:
1332:
1312:
1291:
1267:
1247:
1227:
1200:
1162:
1142:
1122:
1102:
1082:
1058:
1024:
1004:
984:
961:
926:
491:
466:
429:
7011:"The mathematical life of Cauchy's group theorem"
6940:Kammüller, Florian; Paulson, Lawrence C. (1999).
5213:
5181:
3432:. Thus by Theorem 1, the order of the Sylow
3302:, which can be represented by rotation through π/
2516:The Sylow theorems imply that for a prime number
7669:
7624:Abstract Algebra/Group Theory/The Sylow Theorems
6939:
6651:
6487:(including the identity) and taking elements of
1542:to each other. Furthermore, the number of Sylow
1211:The Sylow theorems assert a partial converse to
7117:(1980). "The early proofs of Sylow's theorem".
6614:
5536:It may be noted that conversely every subgroup
3888:= 10, since neither 4 nor 7 divides 10, and if
2650:-subgroup. Due to the maximality condition, if
6724:
6712:
5114:(if none existed, that valuation would exceed
4427:
3907:has at least 20 distinct elements of order 3.
3823:
3151:{\displaystyle n_{p}=|\operatorname {Cl} (K)|}
2813:is equivalent to the condition that the Sylow
7510:
7463:
7350:
7239:Fundamental Algorithms for Permutation Groups
7177:
6514:-subgroup and its normalizer can be found in
4003:over 5 elements. It has order 60, and has 24
3699:distinct primes are some of the applications.
3309:Another example are the Sylow p-subgroups of
1090:) that is not a proper subgroup of any other
876:In mathematics, specifically in the field of
793:
7460:
7305:
6847:
6814:
6706:
6574:"Théorèmes sur les groupes de substitutions"
6491:-power order contained in the normalizer of
6479:-subgroup can be found by starting from any
4175: − 1)! ≡ −1 (mod
3015:-subgroup (that is, every element in it has
3055:denote the set of conjugates of a subgroup
1713:{\displaystyle \operatorname {Syl} _{p}(G)}
7113:
6682:
5591:-group, let Ω be a finite set acted on by
4744:The proof will show the existence of some
4075:{\displaystyle (p-1)!\equiv -1{\pmod {p}}}
1275:. The Sylow theorems state that for every
800:
786:
7427:
7363:
7080:
7027:
6906:
4610:, and let Ω denote the set of subsets of
3749:be the number of Sylow 3-subgroups. Then
3713:has more than one Sylow 5-subgroup, then
1518:-subgroups of a group (for a given prime
864:Learn how and when to remove this message
485:
460:
423:
7298:Computers in Algebra and Number Theory (
7245:. Vol. 559. Berlin, New York City:
7171:
7060:
6742:
6694:
6657:
6562:
6294:act on Ω by conjugation, and again let Ω
6216:. For this group action, the stabilizer
3812:of groups of order 3 and 5, that is the
3277:
3233:
2390:{\displaystyle n_{p}\equiv 1{\pmod {p}}}
1562:-subgroups of a group for a given prime
888:named after the Norwegian mathematician
827:This article includes a list of general
7404:
7340:
6790:
6778:
4978:, the ones we are looking for, one has
3637:
3011:-subgroup in an infinite group to be a
1915:{\displaystyle {\text{gcd}}(|G:P|,p)=1}
14:
7670:
7554:
7507:
7295:
7236:
6826:
6802:
6766:
6754:
6623:"Classification of groups of order 60"
6620:
6195:-subgroup. By Theorem 2, the orbit of
4406:classification of finite simple groups
3721:
3048:{\displaystyle \operatorname {Cl} (K)}
2583:. Conversely, if a subgroup has order
906:classification of finite simple groups
358:Classification of finite simple groups
7649:
7630:
6568:
6206:, so by the orbit-stabilizer theorem
4895:, which counts the number of factors
4163: − 2)! ≡ 1 (mod
1070:of every group element is a power of
7410:"Sylow's theorem in polynomial time"
6639:from the original on 28 October 2020
6428:One proof of the existence of Sylow
5937:{\displaystyle p\nmid |\Omega _{0}|}
4500:for the negation of this statement.
4444:acts on itself or on the set of its
4440:in various creative ways. The group
4396:, which studies the control a Sylow
4100:-subgroups in the symmetric group
3705:(Groups of order 60): If the order |
2189:be a prime factor with multiplicity
1427:. Moreover, every subgroup of order
813:
7008:
6886:
6730:
6718:
6471:is also such that is divisible by
4232:-conjugate if and only if they are
4064:
4010:
3742:be a group of order 15 = 3 · 5 and
3730:are such that every group of order
3447:, is the set of diagonal matrices
2379:
1201:{\displaystyle {\text{Syl}}_{p}(G)}
24:
7300:Proc. SIAM-AMS Sympos. Appl. Math.
6889:"Sylow's proof of Sylow's theorem"
6660:A First Course In Abstract Algebra
5920:
5185:
5167:
4474:as notation for "a divides b" and
4267:is contained in the normalizer of
3202:
3199:
3196:
3003:Sylow theorems for infinite groups
2456:{\displaystyle n_{p}=|G:N_{G}(P)|}
833:it lacks sufficient corresponding
25:
7694:
7594:
7243:Lecture Notes in Computer Science
4625:on Ω by left multiplication: for
4410:Alperin–Brauer–Gorenstein theorem
4184:
3881:must equal 1 (mod 3). Therefore,
3553:, its primitive roots have order
7618:
7063:"Die Entdeckung der Sylow-Sätze"
4126:times the number of p-cycles in
2556:-subgroup is of the same order,
818:
50:
7230:
6444:and the index is divisible by
6432:-subgroups is constructive: if
5797:, then there exists an element
5746:, then there exists an element
4057:
4007:of order 5, and 20 of order 3.
3874:must divide 10 ( = 2 · 5), and
2511:
2372:
2193:of the order of a finite group
2102:, then there exists an element
1963:of the order of a finite group
1300:of the order of a finite group
7308:. Vol. 4. Providence RI:
6172:Let Ω be the set of all Sylow
6072:denote the order of any Sylow
5930:
5915:
5888:
5876:
5641:will lie in an orbit of order
5480:
5474:
5443:
5437:
5409:
5403:
5369:
5363:
5171:
5163:
4773:, since for any fixed element
4548:is divisible by a prime power
4534:
4526:
4342:is a finite group whose Sylow
4068:
4058:
4041:
4029:
3953:must divide 6 ( = 2 · 3) and
3924:must divide 6 ( = 2 · 3), and
3734:is cyclic. One can show that
3338: ≡ 1 (mod
3212:
3192:
3144:
3140:
3134:
3124:
3042:
3036:
3023:-subgroups in the group. Let
2985:
2977:
2449:
2445:
2439:
2419:
2383:
2373:
1903:
1893:
1879:
1875:
1828:
1820:
1740:
1732:
1707:
1701:
1653:
1645:
1195:
1189:
1039:(meaning its cardinality is a
719:Infinite dimensional Lie group
13:
1:
7029:10.1016/S0315-0860(03)00003-X
6836:
6521:Magma computer algebra system
6416:
6184:act on Ω by conjugation. Let
5777:to each other (and therefore
4200:is a finite group with Sylow
3333:are primes ≥ 3 and
1610:
1150:-subgroups for a given prime
7678:Theorems about finite groups
7557:Permutation Group Algorithms
7524:10.1016/0196-6774(90)90009-4
7477:10.1016/0196-6774(88)90002-8
7429:10.1016/0022-0000(85)90052-2
7374:10.1016/0196-6774(85)90029-X
7341:Kantor, William M. (1985a).
7082:10.1016/0315-0860(88)90048-1
5148:from it involves a carry in
4956:. This means that for those
4416:whose Sylow 2-subgroup is a
3158:is finite, then every Sylow
1676:
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
7:
7607:Encyclopedia of Mathematics
7061:Scharlau, Winfried (1988).
6526:
6501:GAP computer algebra system
6091:denote the number of Sylow
5765:. In particular, all Sylow
4428:Proof of the Sylow theorems
4283:, but in the normalizer of
3824:Small groups are not simple
3229:
2300:. Then the following hold:
2086:to each other. That is, if
1720:, all members are actually
1605:
217:List of group theory topics
10:
7699:
7561:Cambridge University Press
6658:Fraleigh, John B. (2004).
6423:computational group theory
5671:), which is a multiple of
5118:). This is an instance of
4885:, and therefore using the
4279:are conjugate not only in
3162:-subgroup is conjugate to
3074:{\displaystyle K\subset G}
2857:(Theorem 3 can often show
2163:{\displaystyle g^{-1}Hg=K}
1763:{\displaystyle |G|=p^{n}m}
1407:that divides the order of
6841:
6393:} so that, by the Lemma,
5824:Let Ω be the set of left
5696:over all distinct orbits
5144:digits zero, subtracting
5090:over all distinct orbits
4887:additive p-adic valuation
4251:, then the normalizer of
4194:Burnside's fusion theorem
3978:| = 60 = 2 · 3 · 5, then
3960:must equal 1 (mod 7), so
3946:| = 42 = 2 · 3 · 7. Here
3931:must equal 1 (mod 5). So
1848:{\displaystyle |P|=p^{n}}
6908:10.33232/BIMS.0033.55.63
6815:Kantor & Taylor 1988
6707:Casadio & Zappa 1990
6555:
5564:, namely any one of the
5533:, completing the proof.
4846:orbit-stabilizer theorem
4603:{\displaystyle p\nmid u}
4552:has a subgroup of order
4493:{\displaystyle a\nmid b}
3974:On the other hand, for |
3557:− 1, which implies that
335:Elementary abelian group
212:Glossary of group theory
6963:10.1023/A:1006269330992
5894:{\displaystyle p\nmid }
4467:{\displaystyle a\mid b}
4271:). By Sylow's theorem
3810:internal direct product
3726:Some non-prime numbers
2883:{\displaystyle n_{p}=1}
2806:{\displaystyle n_{p}=1}
2292:be the number of Sylow
2197:, so that the order of
1967:, there exists a Sylow
1582:is congruent to 1 (mod
1387:, the highest power of
1320:, there exists a Sylow
1130:. The set of all Sylow
848:more precise citations.
7302:, New York City, 1970)
7120:Arch. Hist. Exact Sci.
7115:Waterhouse, William C.
6894:Irish Math. Soc. Bull.
6851:Boll. Storia Sci. Mat.
6448:, then the normalizer
6308:and observe that then
5944:, hence in particular
5938:
5895:
5492:
5338:
5255:
5000:, while for any other
4604:
4542:
4494:
4468:
4394:focal subgroup theorem
4354:has a normal subgroup
4139: − 2)!
4076:
3769:
3514:
3290:
3246:
3219:
3152:
3075:
3049:
2993:
2961:
2931:
2911:
2884:
2847:
2827:
2807:
2771:
2744:
2724:
2704:
2684:
2664:
2644:
2624:
2604:
2577:
2550:
2530:
2496:
2457:
2391:
2324:
2286:
2251:
2250:{\displaystyle n>0}
2225:
2224:{\displaystyle p^{n}m}
2164:
2122:
2121:{\displaystyle g\in G}
2037:dividing the order of
2000:
1916:
1849:
1790:
1789:{\displaystyle n>0}
1764:
1714:
1667:
1666:{\displaystyle |G|=60}
1630:
1596:
1576:
1556:
1532:
1512:
1492:
1471:
1449:
1421:
1401:
1381:
1354:
1334:
1314:
1293:
1269:
1249:
1229:
1202:
1164:
1144:
1124:
1104:
1084:
1060:
1026:
1012:, i.e., a subgroup of
1006:
986:
963:
928:
751:Linear algebraic group
493:
468:
431:
7555:Seress, Ákos (2003).
7415:J. Comput. Syst. Sci.
7255:10.1007/3-540-54955-2
6621:Gracia–Saz, Alfonso.
6406:| = 1 (mod
6362:) in particular, and
5955:so there exists some
5939:
5896:
5493:
5305:
5222:
4605:
4543:
4495:
4469:
4141:. On the other hand,
4077:
3770:
3768:{\displaystyle \mid }
3709:| = 60 and
3537:. Since the order of
3515:
3281:
3237:
3220:
3153:
3076:
3050:
2994:
2962:
2960:{\displaystyle n_{p}}
2932:
2912:
2910:{\displaystyle S_{4}}
2885:
2848:
2828:
2808:
2772:
2770:{\displaystyle p^{n}}
2745:
2725:
2705:
2685:
2665:
2645:
2625:
2610:, then it is a Sylow
2605:
2603:{\displaystyle p^{n}}
2578:
2576:{\displaystyle p^{n}}
2551:
2531:
2497:
2495:{\displaystyle N_{G}}
2458:
2392:
2325:
2323:{\displaystyle n_{p}}
2287:
2285:{\displaystyle n_{p}}
2252:
2226:
2165:
2123:
2066:Given a finite group
2029:Given a finite group
2013:Augustin-Louis Cauchy
2001:
1999:{\displaystyle p^{n}}
1932:Mathematische Annalen
1917:
1850:
1791:
1765:
1715:
1668:
1631:
1597:
1577:
1557:
1533:
1513:
1493:
1472:
1450:
1448:{\displaystyle p^{n}}
1422:
1402:
1382:
1380:{\displaystyle p^{n}}
1355:
1335:
1315:
1294:
1270:
1255:divides the order of
1250:
1230:
1203:
1170:is sometimes written
1165:
1145:
1125:
1105:
1085:
1066:or equivalently, the
1061:
1027:
1007:
987:
964:
929:
494:
469:
432:
7312:. pp. 161–176.
6385:. It follows that Ω
6256:, the normalizer of
5905:
5867:
5857:| = (mod
5159:
5136:ends with precisely
5126:notation the number
4588:
4522:
4478:
4452:
4418:quasi-dihedral group
4358:of order coprime to
4121: − 1
4026:
3967:= 1. So, as before,
3895:= 1 then, as above,
3759:
3638:Example applications
3451:
3170:
3107:
3059:
3027:
2973:
2944:
2921:
2894:
2861:
2837:
2817:
2784:
2754:
2734:
2714:
2694:
2674:
2654:
2634:
2614:
2587:
2560:
2540:
2520:
2479:
2402:
2352:
2307:
2269:
2235:
2205:
2132:
2106:
1983:
1867:
1816:
1774:
1728:
1685:
1641:
1620:
1586:
1566:
1546:
1522:
1502:
1482:
1461:
1432:
1411:
1391:
1364:
1344:
1324:
1304:
1283:
1259:
1239:
1219:
1174:
1154:
1134:
1114:
1094:
1074:
1047:
1016:
996:
976:
953:
918:
884:are a collection of
481:
456:
419:
7237:Butler, G. (1991).
6950:J. Automat. Reason.
6534:Frattini's argument
6066: —
5720: —
5581: —
5544:gives rise to sets
4686:stabilizer subgroup
4541:{\displaystyle |G|}
4512: —
4412:classifying finite
4190:Frattini's argument
4005:cyclic permutations
3971:can not be simple.
3793:must divide 3, and
3722:Cyclic group orders
3634:) are all abelian.
3089: —
2992:{\displaystyle |G|}
2750:-subgroup of order
2730:is a subgroup of a
2183: —
2070:and a prime number
2064: —
2033:and a prime number
2027: —
1947: —
1804:, then every Sylow
878:finite group theory
125:Group homomorphisms
35:Algebraic structure
7651:Weisstein, Eric W.
7635:"Sylow p-Subgroup"
7632:Weisstein, Eric W.
7406:Kantor, William M.
7191:10.1007/BF01240818
7133:10.1007/BF00327877
6592:10.1007/BF01442913
6508:permutation groups
6343:). By Theorem 2,
6170:
6080:of a finite group
6060:
6009:. Furthermore, if
5934:
5891:
5844:on Ω, we see that
5822:
5714:
5628:
5579:
5488:
4787:, the right coset
4668:. For a given set
4600:
4562:
4538:
4506:
4490:
4464:
4336:semidirect product
4255:contains not only
4072:
3765:
3510:
3504:
3443:One such subgroup
3300:outer automorphism
3291:
3247:
3215:
3148:
3087:
3071:
3045:
2989:
2957:
2927:
2907:
2880:
2843:
2823:
2803:
2767:
2740:
2720:
2700:
2680:
2660:
2640:
2620:
2600:
2573:
2546:
2526:
2492:
2453:
2387:
2320:
2282:
2247:
2221:
2201:can be written as
2177:
2160:
2118:
2058:
2025:
2015:, and is known as
1996:
1941:
1912:
1845:
1786:
1760:
1710:
1663:
1626:
1592:
1572:
1552:
1528:
1508:
1488:
1467:
1445:
1417:
1397:
1377:
1350:
1330:
1310:
1289:
1265:
1245:
1225:
1213:Lagrange's theorem
1198:
1160:
1140:
1120:
1100:
1080:
1059:{\displaystyle p,}
1056:
1022:
1002:
982:
959:
924:
890:Peter Ludwig Sylow
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
6887:Gow, Rod (1994).
6351:are conjugate in
6168:
6058:
5901:by definition so
5820:
5712:
5703:and reducing mod
5626:
5616:| (mod
5577:
5486:
5297:
5211:
4560:
4504:
4420:. These rely on
4400:-subgroup of the
4001:alternating group
3583:choices for both
3208:
3191:
3085:
2930:{\displaystyle p}
2846:{\displaystyle G}
2826:{\displaystyle p}
2743:{\displaystyle p}
2723:{\displaystyle H}
2703:{\displaystyle G}
2683:{\displaystyle p}
2663:{\displaystyle H}
2643:{\displaystyle p}
2623:{\displaystyle p}
2549:{\displaystyle p}
2529:{\displaystyle p}
2175:
2056:
2023:
1939:
1873:
1629:{\displaystyle G}
1595:{\displaystyle p}
1575:{\displaystyle p}
1555:{\displaystyle p}
1531:{\displaystyle p}
1511:{\displaystyle p}
1491:{\displaystyle G}
1470:{\displaystyle p}
1420:{\displaystyle G}
1400:{\displaystyle p}
1353:{\displaystyle G}
1333:{\displaystyle p}
1313:{\displaystyle G}
1292:{\displaystyle p}
1268:{\displaystyle G}
1248:{\displaystyle G}
1228:{\displaystyle G}
1181:
1163:{\displaystyle p}
1143:{\displaystyle p}
1123:{\displaystyle G}
1103:{\displaystyle p}
1083:{\displaystyle p}
1025:{\displaystyle G}
1005:{\displaystyle G}
985:{\displaystyle p}
962:{\displaystyle G}
927:{\displaystyle p}
874:
873:
866:
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
7690:
7664:
7663:
7654:"Sylow Theorems"
7645:
7644:
7622:
7615:
7602:"Sylow theorems"
7590:
7551:
7512:
7504:
7465:
7457:
7431:
7416:
7401:
7367:
7352:
7347:
7337:
7307:
7301:
7292:
7226:
7179:
7173:Wielandt, Helmut
7168:
7121:
7110:
7084:
7057:
7031:
7009:Meo, M. (2004).
7005:
7003:
6997:. Archived from
6951:
6946:
6936:
6910:
6895:
6883:
6852:
6830:
6824:
6818:
6812:
6806:
6800:
6794:
6788:
6782:
6776:
6770:
6764:
6758:
6752:
6746:
6740:
6734:
6728:
6722:
6716:
6710:
6704:
6698:
6692:
6686:
6680:
6674:
6673:
6655:
6649:
6648:
6646:
6644:
6638:
6630:math.toronto.edu
6627:
6618:
6612:
6611:
6566:
6544:Maximal subgroup
6411:
6405:
6398:
6327:
6317:
6307:
6286:
6281:
6273:
6255:
6219:
6215:
6190:
6183:
6179:
6163:
6149:
6144:
6136:
6108:
6067:
6064:
6041:
6039:
6033:
6027:
6017:-subgroup, then
5986:
5964:
5954:
5952:
5943:
5941:
5940:
5935:
5933:
5928:
5927:
5918:
5900:
5898:
5897:
5892:
5862:
5856:
5852:
5835:
5804:
5796:
5772:
5753:
5745:
5733:
5721:
5718:
5706:
5702:
5699:
5695:
5693:
5682:
5680:
5674:
5659:
5657:
5648:
5640:
5636:
5621:
5615:
5608:
5602:
5594:
5590:
5586:
5582:
5568:distinct cosets
5550:
5532:
5516:
5501:and no power of
5497:
5495:
5494:
5489:
5487:
5485:
5484:
5483:
5473:
5472:
5458:
5447:
5446:
5436:
5435:
5414:
5413:
5412:
5402:
5401:
5387:
5373:
5372:
5362:
5361:
5340:
5337:
5330:
5329:
5319:
5298:
5296:
5289:
5288:
5278:
5268:
5267:
5257:
5254:
5247:
5246:
5236:
5218:
5217:
5216:
5210:
5209:
5200:
5196:
5195:
5184:
5174:
5166:
5135:
5133:
5120:Kummer's theorem
5113:
5108:
5096:
5093:
5089:
5087:
5076:
5074:
5068:
5062:
5043:
5038:
5025:
5020:
5003:
4999:
4994:
4977:
4972:
4959:
4955:
4946:
4933:
4917:
4884:
4877:
4875:
4869:
4860:
4840:
4835:
4829:
4828:
4815:
4802:
4799:is contained in
4798:
4794:
4790:
4786:
4771:
4767:
4758:
4754:
4750:
4740:
4710:
4674:
4667:
4641:
4634:
4621:
4613:
4609:
4607:
4606:
4601:
4583:
4571:
4547:
4545:
4544:
4539:
4537:
4529:
4513:
4510:
4499:
4497:
4496:
4491:
4473:
4471:
4470:
4465:
4443:
4402:derived subgroup
4378:= {1}, that is,
4299:that normalizes
4208:and two subsets
4180:
4168:
4156:
4140:
4125:
4123:
4122:
4116:
4113:
4085:for every prime
4081:
4079:
4078:
4073:
4071:
4017:Wilson's theorem
4011:Wilson's theorem
3863:is simple, and |
3855:
3774:
3772:
3771:
3766:
3674: <
3658:Groups of order
3601:
3562:
3552:
3519:
3517:
3516:
3511:
3509:
3508:
3501:
3500:
3474:
3473:
3431:
3405:
3391:
3386: + 1)(
3370: − 1)(
3344:, which are all
3343:
3293:By contrast, if
3224:
3222:
3221:
3216:
3206:
3205:
3189:
3182:
3181:
3165:
3161:
3157:
3155:
3154:
3149:
3147:
3127:
3119:
3118:
3102:
3098:
3094:
3090:
3080:
3078:
3077:
3072:
3054:
3052:
3051:
3046:
3022:
3018:
3010:
2998:
2996:
2995:
2990:
2988:
2980:
2966:
2964:
2963:
2958:
2956:
2955:
2936:
2934:
2933:
2928:
2916:
2914:
2913:
2908:
2906:
2905:
2889:
2887:
2886:
2881:
2873:
2872:
2852:
2850:
2849:
2844:
2832:
2830:
2829:
2824:
2812:
2810:
2809:
2804:
2796:
2795:
2776:
2774:
2773:
2768:
2766:
2765:
2749:
2747:
2746:
2741:
2729:
2727:
2726:
2721:
2709:
2707:
2706:
2701:
2689:
2687:
2686:
2681:
2669:
2667:
2666:
2661:
2649:
2647:
2646:
2641:
2629:
2627:
2626:
2621:
2609:
2607:
2606:
2601:
2599:
2598:
2582:
2580:
2579:
2574:
2572:
2571:
2555:
2553:
2552:
2547:
2535:
2533:
2532:
2527:
2501:
2499:
2498:
2493:
2491:
2490:
2474:
2470:
2466:
2462:
2460:
2459:
2454:
2452:
2438:
2437:
2422:
2414:
2413:
2396:
2394:
2393:
2388:
2386:
2364:
2363:
2345:
2341:
2333:
2329:
2327:
2326:
2321:
2319:
2318:
2299:
2295:
2291:
2289:
2288:
2283:
2281:
2280:
2264:
2261:does not divide
2260:
2256:
2254:
2253:
2248:
2230:
2228:
2227:
2222:
2217:
2216:
2200:
2196:
2192:
2188:
2184:
2181:
2169:
2167:
2166:
2161:
2147:
2146:
2127:
2125:
2124:
2119:
2101:
2097:
2093:
2089:
2081:
2077:
2073:
2069:
2065:
2062:
2049:
2044:
2040:
2036:
2032:
2028:
2017:Cauchy's theorem
2005:
2003:
2002:
1997:
1995:
1994:
1977:
1971:
1966:
1962:
1955:
1948:
1945:
1925:
1921:
1919:
1918:
1913:
1896:
1882:
1874:
1871:
1862:
1858:
1854:
1852:
1851:
1846:
1844:
1843:
1831:
1823:
1811:
1807:
1803:
1800:does not divide
1799:
1795:
1793:
1792:
1787:
1769:
1767:
1766:
1761:
1756:
1755:
1743:
1735:
1719:
1717:
1716:
1711:
1697:
1696:
1672:
1670:
1669:
1664:
1656:
1648:
1635:
1633:
1632:
1627:
1601:
1599:
1598:
1593:
1581:
1579:
1578:
1573:
1561:
1559:
1558:
1553:
1537:
1535:
1534:
1529:
1517:
1515:
1514:
1509:
1498:, and the Sylow
1497:
1495:
1494:
1489:
1476:
1474:
1473:
1468:
1454:
1452:
1451:
1446:
1444:
1443:
1426:
1424:
1423:
1418:
1406:
1404:
1403:
1398:
1386:
1384:
1383:
1378:
1376:
1375:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1319:
1317:
1316:
1311:
1298:
1296:
1295:
1290:
1274:
1272:
1271:
1266:
1254:
1252:
1251:
1246:
1234:
1232:
1231:
1226:
1207:
1205:
1204:
1199:
1188:
1187:
1182:
1179:
1169:
1167:
1166:
1161:
1149:
1147:
1146:
1141:
1129:
1127:
1126:
1121:
1109:
1107:
1106:
1101:
1089:
1087:
1086:
1081:
1065:
1063:
1062:
1057:
1031:
1029:
1028:
1023:
1011:
1009:
1008:
1003:
991:
989:
988:
983:
968:
966:
965:
960:
933:
931:
930:
925:
869:
862:
858:
855:
849:
844:this article by
835:inline citations
822:
821:
814:
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
238:
65:
64:
54:
41:
30:
29:
21:
7698:
7697:
7693:
7692:
7691:
7689:
7688:
7687:
7668:
7667:
7600:
7597:
7571:
7345:
7265:
7247:Springer-Verlag
7233:
7001:
6944:
6844:
6839:
6834:
6833:
6825:
6821:
6813:
6809:
6801:
6797:
6789:
6785:
6777:
6773:
6765:
6761:
6753:
6749:
6741:
6737:
6729:
6725:
6717:
6713:
6705:
6701:
6693:
6689:
6683:Waterhouse 1980
6681:
6677:
6670:
6656:
6652:
6642:
6640:
6636:
6625:
6619:
6615:
6567:
6563:
6558:
6553:
6529:
6516:polynomial time
6457:
6419:
6414:
6404:
6400:
6399:| ≡ |
6396:
6394:
6388:
6371:
6356:
6337:
6319:
6309:
6306:
6299:
6297:
6280:
6277:
6275:
6270:
6265:
6263:
6249:
6234:
6230:
6226:
6224:
6217:
6212:
6207:
6204:
6185:
6181:
6177:
6166:
6156:
6151:
6143:
6140:
6138:
6134:
6129:
6114:
6105:
6100:
6098:
6089:
6083:
6065:
6062:
6055:
6035:
6034:| = |
6029:
6028:| = |
6020:
6018:
5985:
5981:
5978:
5963:
5956:
5951:
5947:
5945:
5929:
5923:
5919:
5914:
5906:
5903:
5902:
5868:
5865:
5864:
5854:
5853:| ≡ |
5851:
5847:
5845:
5833:
5818:
5802:
5794:
5781:), that is, if
5770:
5751:
5743:
5731:
5719:
5716:
5709:
5704:
5700:
5697:
5692:
5689:
5686:
5684:
5678:
5676:
5672:
5665:
5655:
5650:
5647:
5644:
5642:
5638:
5634:
5631:
5624:
5619:
5614:
5610:
5609:| ≡ |
5606:
5604:
5600:
5598:
5592:
5588:
5584:
5580:
5574:
5559:
5558:
5554:
5548:
5545:
5522:
5514:
5511:
5506:
5468:
5464:
5463:
5459:
5454:
5431:
5427:
5420:
5416:
5415:
5397:
5393:
5392:
5388:
5383:
5357:
5353:
5346:
5342:
5341:
5339:
5325:
5321:
5320:
5309:
5284:
5280:
5279:
5263:
5259:
5258:
5256:
5242:
5238:
5237:
5226:
5212:
5205:
5201:
5191:
5187:
5186:
5180:
5179:
5178:
5170:
5162:
5160:
5157:
5156:
5132:
5129:
5127:
5122:(since in base
5106:
5103:
5098:
5094:
5091:
5086:
5083:
5080:
5078:
5072:
5070:
5061:
5060:
5056:
5053:
5050:
5045:
5037:
5036:
5032:
5029:
5027:
5019:
5016:
5013:
5010:
5005:
5001:
4993:
4990:
4987:
4984:
4979:
4971:
4970:
4966:
4963:
4961:
4957:
4945:
4942:
4939:
4932:
4929:
4926:
4923:
4916:
4915:
4911:
4908:
4905:
4900:
4893:
4882:
4879:
4874:
4871:
4870:| = |
4868:
4865:
4862:
4859:
4858:
4854:
4851:
4849:
4839:
4834:
4831:
4830:| ≤ |
4826:
4825:
4824:
4820:
4817:
4816:| = |
4814:
4813:
4809:
4806:
4804:
4800:
4796:
4795:
4792:
4788:
4785:
4781:
4777:
4774:
4772:
4769:
4765:
4759:
4756:
4752:
4748:
4745:
4738:
4734:
4730:
4726:
4722:
4717:
4714:
4708:
4704:
4700:
4696:
4692:
4688:
4683:
4682:
4678:
4672:
4669:
4665:
4661:
4657:
4654:
4650:
4646:
4643:
4639:
4636:
4633:
4629:
4626:
4619:
4611:
4589:
4586:
4585:
4570:
4567:
4565:
4558:
4533:
4525:
4523:
4520:
4519:
4517:
4514:A finite group
4511:
4508:
4479:
4476:
4475:
4453:
4450:
4449:
4441:
4430:
4328:
4237:
4196:states that if
4187:
4170:
4158:
4150:
4142:
4134:
4131:
4117:
4114:
4111:
4110:
4108:
4105:
4094:
4056:
4027:
4024:
4023:
4013:
3998:
3991:
3984:
3966:
3959:
3952:
3942:Next, suppose |
3930:
3923:
3916:
3894:
3887:
3880:
3873:
3853:
3826:
3799:
3792:
3781:
3760:
3757:
3756:
3755:
3748:
3724:
3649:normal subgroup
3640:
3633:
3624:
3592:
3558:
3547:
3545:
3536:
3503:
3502:
3493:
3489:
3487:
3481:
3480:
3475:
3466:
3462:
3455:
3454:
3452:
3449:
3448:
3422:
3413:
3407:
3406:, the order of
3393:
3390: − 1)
3365:
3363:
3354:
3348:. The order of
3334:
3324:
3315:
3288:
3266:
3244:
3232:
3227:
3195:
3177:
3173:
3171:
3168:
3167:
3163:
3159:
3143:
3123:
3114:
3110:
3108:
3105:
3104:
3100:
3096:
3092:
3088:
3060:
3057:
3056:
3028:
3025:
3024:
3020:
3016:
3008:
3005:
2984:
2976:
2974:
2971:
2970:
2951:
2947:
2945:
2942:
2941:
2922:
2919:
2918:
2901:
2897:
2895:
2892:
2891:
2868:
2864:
2862:
2859:
2858:
2855:normal subgroup
2838:
2835:
2834:
2818:
2815:
2814:
2791:
2787:
2785:
2782:
2781:
2761:
2757:
2755:
2752:
2751:
2735:
2732:
2731:
2715:
2712:
2711:
2695:
2692:
2691:
2675:
2672:
2671:
2655:
2652:
2651:
2635:
2632:
2631:
2615:
2612:
2611:
2594:
2590:
2588:
2585:
2584:
2567:
2563:
2561:
2558:
2557:
2541:
2538:
2537:
2521:
2518:
2517:
2514:
2509:
2486:
2482:
2480:
2477:
2476:
2472:
2468:
2464:
2448:
2433:
2429:
2418:
2409:
2405:
2403:
2400:
2399:
2371:
2359:
2355:
2353:
2350:
2349:
2343:
2339:
2334:, which is the
2331:
2314:
2310:
2308:
2305:
2304:
2297:
2293:
2276:
2272:
2270:
2267:
2266:
2262:
2258:
2236:
2233:
2232:
2212:
2208:
2206:
2203:
2202:
2198:
2194:
2190:
2186:
2182:
2179:
2172:
2139:
2135:
2133:
2130:
2129:
2107:
2104:
2103:
2099:
2095:
2091:
2087:
2079:
2075:
2071:
2067:
2063:
2060:
2053:
2047:
2042:
2038:
2034:
2030:
2026:
2009:
1990:
1986:
1984:
1981:
1980:
1975:
1969:
1964:
1960:
1953:
1946:
1943:
1923:
1892:
1878:
1870:
1868:
1865:
1864:
1860:
1856:
1839:
1835:
1827:
1819:
1817:
1814:
1813:
1809:
1805:
1801:
1797:
1775:
1772:
1771:
1751:
1747:
1739:
1731:
1729:
1726:
1725:
1692:
1688:
1686:
1683:
1682:
1679:
1652:
1644:
1642:
1639:
1638:
1621:
1618:
1617:
1613:
1608:
1587:
1584:
1583:
1567:
1564:
1563:
1547:
1544:
1543:
1523:
1520:
1519:
1503:
1500:
1499:
1483:
1480:
1479:
1462:
1459:
1458:
1439:
1435:
1433:
1430:
1429:
1412:
1409:
1408:
1392:
1389:
1388:
1371:
1367:
1365:
1362:
1361:
1345:
1342:
1341:
1325:
1322:
1321:
1305:
1302:
1301:
1284:
1281:
1280:
1260:
1257:
1256:
1240:
1237:
1236:
1220:
1217:
1216:
1183:
1178:
1177:
1175:
1172:
1171:
1155:
1152:
1151:
1135:
1132:
1131:
1115:
1112:
1111:
1095:
1092:
1091:
1075:
1072:
1071:
1048:
1045:
1044:
1017:
1014:
1013:
997:
994:
993:
977:
974:
973:
954:
951:
950:
947:-Sylow subgroup
919:
916:
915:
870:
859:
853:
850:
840:Please help to
839:
823:
819:
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
484:
482:
479:
478:
459:
457:
454:
453:
422:
420:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
7696:
7686:
7685:
7680:
7666:
7665:
7646:
7627:
7616:
7596:
7595:External links
7593:
7592:
7591:
7569:
7552:
7518:(4): 523–563.
7505:
7458:
7422:(3): 359–394.
7402:
7365:10.1.1.74.3690
7358:(4): 478–514.
7338:
7306:SIAM-AMS Proc.
7293:
7263:
7232:
7229:
7228:
7227:
7185:(1): 401–402.
7169:
7127:(3): 279–290.
7111:
7068:Historia Math.
7058:
7022:(2): 196–221.
7016:Historia Math.
7006:
7004:on 2006-01-03.
6957:(3): 235–264.
6937:
6884:
6854:(in Italian).
6843:
6840:
6838:
6835:
6832:
6831:
6819:
6807:
6795:
6783:
6771:
6759:
6747:
6735:
6723:
6711:
6699:
6687:
6675:
6668:
6650:
6613:
6586:(4): 584–594.
6560:
6559:
6557:
6554:
6552:
6551:
6546:
6541:
6536:
6530:
6528:
6525:
6455:
6418:
6415:
6402:
6386:
6369:
6354:
6335:
6304:
6295:
6278:
6268:
6261:
6247:
6232:
6228:
6220:
6210:
6202:
6176:-subgroups of
6167:
6158:≡ 1 (mod
6154:
6141:
6132:
6112:
6103:
6096:
6095:-subgroups of
6087:
6081:
6056:
5998:and therefore
5983:
5979:
5961:
5949:
5932:
5926:
5922:
5917:
5913:
5910:
5890:
5887:
5884:
5881:
5878:
5875:
5872:
5849:
5819:
5793:-subgroups of
5769:-subgroups of
5710:
5690:
5687:
5683:as the sum of
5663:
5653:
5645:
5632:
5625:
5617:
5612:
5596:
5575:
5556:
5555:
5552:
5546:
5520:
5509:
5499:
5498:
5482:
5479:
5476:
5471:
5467:
5462:
5457:
5453:
5450:
5445:
5442:
5439:
5434:
5430:
5426:
5423:
5419:
5411:
5408:
5405:
5400:
5396:
5391:
5386:
5382:
5379:
5376:
5371:
5368:
5365:
5360:
5356:
5352:
5349:
5345:
5336:
5333:
5328:
5324:
5318:
5315:
5312:
5308:
5304:
5301:
5295:
5292:
5287:
5283:
5277:
5274:
5271:
5266:
5262:
5253:
5250:
5245:
5241:
5235:
5232:
5229:
5225:
5221:
5215:
5208:
5204:
5199:
5194:
5190:
5183:
5177:
5173:
5169:
5165:
5130:
5101:
5084:
5081:
5077:is the sum of
5058:
5057:
5054:
5048:
5034:
5033:
5030:
5017:
5014:
5008:
4991:
4988:
4982:
4968:
4967:
4964:
4943:
4937:
4930:
4927:
4921:
4913:
4912:
4909:
4903:
4891:
4880:
4872:
4866:
4863:
4856:
4855:
4852:
4837:
4832:
4822:
4821:
4818:
4811:
4810:
4807:
4791:
4783:
4779:
4775:
4768:
4755:
4746:
4736:
4732:
4728:
4724:
4715:
4712:
4706:
4702:
4698:
4694:
4690:
4680:
4679:
4676:
4670:
4663:
4659:
4655:
4652:
4648:
4644:
4637:
4631:
4627:
4599:
4596:
4593:
4568:
4559:
4536:
4532:
4528:
4515:
4502:
4489:
4486:
4483:
4463:
4460:
4457:
4429:
4426:
4326:
4235:
4216:normalized by
4186:
4185:Fusion results
4183:
4151:≡ 1 (mod
4146:
4129:
4103:
4092:
4083:
4082:
4070:
4067:
4063:
4060:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4012:
4009:
3996:
3989:
3982:
3964:
3957:
3950:
3928:
3921:
3914:
3892:
3885:
3878:
3871:
3825:
3822:
3820:isomorphism).
3797:
3790:
3779:
3764:
3753:
3746:
3723:
3720:
3719:
3718:
3703:
3700:
3682:
3679:
3656:
3639:
3636:
3629:
3622:
3618:-subgroups of
3551: − 1
3541:
3532:
3526:primitive root
3507:
3499:
3496:
3492:
3488:
3486:
3483:
3482:
3479:
3476:
3472:
3469:
3465:
3461:
3460:
3458:
3436:-subgroups is
3418:
3411:
3404: + 1
3359:
3352:
3320:
3313:
3286:
3261:
3251:dihedral group
3242:
3231:
3228:
3214:
3211:
3204:
3201:
3198:
3194:
3188:
3185:
3180:
3176:
3146:
3142:
3139:
3136:
3133:
3130:
3126:
3122:
3117:
3113:
3083:
3070:
3067:
3064:
3044:
3041:
3038:
3035:
3032:
3004:
3001:
2987:
2983:
2979:
2954:
2950:
2926:
2904:
2900:
2879:
2876:
2871:
2867:
2842:
2822:
2802:
2799:
2794:
2790:
2764:
2760:
2739:
2719:
2699:
2679:
2659:
2639:
2619:
2597:
2593:
2570:
2566:
2545:
2525:
2513:
2510:
2508:
2507:
2489:
2485:
2451:
2447:
2444:
2441:
2436:
2432:
2428:
2425:
2421:
2417:
2412:
2408:
2397:
2385:
2382:
2378:
2375:
2370:
2367:
2362:
2358:
2347:
2317:
2313:
2296:-subgroups of
2279:
2275:
2246:
2243:
2240:
2220:
2215:
2211:
2173:
2159:
2156:
2153:
2150:
2145:
2142:
2138:
2117:
2114:
2111:
2098:-subgroups of
2078:-subgroups of
2054:
2021:
1993:
1989:
1937:
1911:
1908:
1905:
1902:
1899:
1895:
1891:
1888:
1885:
1881:
1877:
1842:
1838:
1834:
1830:
1826:
1822:
1785:
1782:
1779:
1759:
1754:
1750:
1746:
1742:
1738:
1734:
1709:
1706:
1703:
1700:
1695:
1691:
1678:
1675:
1662:
1659:
1655:
1651:
1647:
1625:
1612:
1609:
1607:
1604:
1591:
1571:
1551:
1527:
1507:
1487:
1466:
1442:
1438:
1416:
1396:
1374:
1370:
1349:
1329:
1309:
1288:
1264:
1244:
1224:
1197:
1194:
1191:
1186:
1159:
1139:
1119:
1099:
1079:
1055:
1052:
1021:
1001:
981:
958:
923:
882:Sylow theorems
872:
871:
826:
824:
817:
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
18:Sylow subgroup
9:
6:
4:
3:
2:
7695:
7684:
7681:
7679:
7676:
7675:
7673:
7661:
7660:
7655:
7652:
7647:
7642:
7641:
7636:
7633:
7628:
7625:
7621:
7617:
7613:
7609:
7608:
7603:
7599:
7598:
7588:
7584:
7580:
7576:
7572:
7570:9780521661034
7566:
7562:
7558:
7553:
7549:
7545:
7541:
7537:
7533:
7529:
7525:
7521:
7517:
7513:
7511:J. Algorithms
7506:
7502:
7498:
7494:
7490:
7486:
7482:
7478:
7474:
7470:
7466:
7464:J. Algorithms
7459:
7455:
7451:
7447:
7443:
7439:
7435:
7430:
7425:
7421:
7417:
7411:
7407:
7403:
7399:
7395:
7391:
7387:
7383:
7379:
7375:
7371:
7366:
7361:
7357:
7353:
7351:J. Algorithms
7344:
7339:
7335:
7331:
7327:
7323:
7319:
7315:
7311:
7303:
7294:
7290:
7286:
7282:
7278:
7274:
7270:
7266:
7264:9783540549550
7260:
7256:
7252:
7248:
7244:
7240:
7235:
7234:
7224:
7220:
7216:
7212:
7208:
7204:
7200:
7196:
7192:
7188:
7184:
7181:(in German).
7180:
7174:
7170:
7166:
7162:
7158:
7154:
7150:
7146:
7142:
7138:
7134:
7130:
7126:
7122:
7116:
7112:
7108:
7104:
7100:
7096:
7092:
7088:
7083:
7078:
7074:
7071:(in German).
7070:
7069:
7064:
7059:
7055:
7051:
7047:
7043:
7039:
7035:
7030:
7025:
7021:
7018:
7017:
7012:
7007:
7000:
6996:
6992:
6988:
6984:
6980:
6976:
6972:
6968:
6964:
6960:
6956:
6952:
6943:
6938:
6934:
6930:
6926:
6922:
6918:
6914:
6909:
6904:
6901:(33): 55–63.
6900:
6896:
6890:
6885:
6881:
6877:
6873:
6869:
6865:
6861:
6857:
6853:
6846:
6845:
6828:
6823:
6816:
6811:
6804:
6799:
6792:
6787:
6780:
6775:
6768:
6763:
6757:, Chapter 16.
6756:
6751:
6744:
6743:Wielandt 1959
6739:
6732:
6727:
6720:
6715:
6708:
6703:
6696:
6695:Scharlau 1988
6691:
6684:
6679:
6671:
6669:9788178089973
6665:
6661:
6654:
6635:
6631:
6624:
6617:
6609:
6605:
6601:
6597:
6593:
6589:
6585:
6582:(in French).
6581:
6580:
6575:
6571:
6565:
6561:
6550:
6547:
6545:
6542:
6540:
6539:Hall subgroup
6537:
6535:
6532:
6531:
6524:
6522:
6517:
6513:
6509:
6504:
6502:
6498:
6494:
6490:
6486:
6482:
6478:
6474:
6470:
6466:
6462:
6458:
6451:
6447:
6443:
6440:-subgroup of
6439:
6435:
6431:
6426:
6424:
6413:
6409:
6392:
6384:
6380:
6376:
6372:
6366:is normal in
6365:
6361:
6357:
6350:
6346:
6342:
6338:
6331:
6326:
6322:
6316:
6312:
6302:
6293:
6288:
6285:
6271:
6259:
6253:
6246:
6242:
6238:
6223:
6213:
6205:
6198:
6194:
6188:
6175:
6165:
6161:
6157:
6148:
6135:
6127:
6123:
6119:
6115:
6106:
6094:
6090:
6079:
6075:
6071:
6054:
6052:
6048:
6045:
6038:
6032:
6026:
6023:
6016:
6012:
6008:
6004:
6001:
5997:
5993:
5990:
5976:
5972:
5968:
5959:
5924:
5911:
5908:
5885:
5882:
5879:
5873:
5870:
5860:
5843:
5839:
5831:
5827:
5817:
5815:
5811:
5808:
5800:
5792:
5788:
5784:
5780:
5776:
5768:
5764:
5760:
5757:
5749:
5742:-subgroup of
5741:
5737:
5730:-subgroup of
5729:
5725:
5708:
5670:
5666:
5656:
5649:|/|
5637:not fixed by
5623:
5573:
5571:
5567:
5563:
5543:
5539:
5534:
5531:
5527:
5523:
5512:
5504:
5477:
5469:
5465:
5460:
5455:
5451:
5448:
5440:
5432:
5428:
5424:
5421:
5417:
5406:
5398:
5394:
5389:
5384:
5380:
5377:
5374:
5366:
5358:
5354:
5350:
5347:
5343:
5334:
5331:
5326:
5322:
5316:
5313:
5310:
5306:
5302:
5299:
5293:
5290:
5285:
5281:
5275:
5272:
5269:
5264:
5260:
5251:
5248:
5243:
5239:
5233:
5230:
5227:
5223:
5219:
5206:
5202:
5197:
5192:
5188:
5175:
5155:
5154:
5153:
5151:
5147:
5143:
5139:
5125:
5121:
5117:
5112:
5104:
5066:
5063:|) <
5051:
5042:
5028:0 < |
5024:
5021:|) >
5011:
4998:
4985:
4976:
4954:
4950:
4940:
4924:
4906:
4898:
4894:
4888:
4861:| |
4847:
4842:
4803:; therefore,
4763:
4742:
4721:
4687:
4624:
4617:
4597:
4594:
4591:
4582:
4579:
4575:
4557:
4555:
4551:
4530:
4501:
4487:
4484:
4481:
4461:
4458:
4455:
4447:
4439:
4434:
4425:
4423:
4422:J. L. Alperin
4419:
4415:
4414:simple groups
4411:
4407:
4403:
4399:
4395:
4390:
4388:
4386:
4381:
4377:
4373:
4369:
4365:
4361:
4357:
4353:
4349:
4345:
4341:
4337:
4333:
4329:
4322:
4318:
4314:
4310:
4306:
4302:
4298:
4294:
4290:
4286:
4282:
4278:
4274:
4270:
4266:
4262:
4258:
4254:
4250:
4246:
4242:
4238:
4231:
4227:
4223:
4219:
4215:
4211:
4207:
4203:
4199:
4195:
4191:
4182:
4178:
4174:
4166:
4162:
4154:
4149:
4145:
4138:
4132:
4120:
4106:
4099:
4095:
4088:
4065:
4061:
4053:
4050:
4047:
4044:
4038:
4035:
4032:
4022:
4021:
4020:
4018:
4008:
4006:
4002:
3995:
3988:
3981:
3977:
3972:
3970:
3963:
3956:
3949:
3945:
3940:
3938:
3934:
3927:
3920:
3913:
3908:
3906:
3902:
3898:
3891:
3884:
3877:
3870:
3867:| = 30, then
3866:
3862:
3857:
3854:(= 2 · 3 · 5)
3851:
3848:, then it is
3847:
3843:
3841:
3835:
3831:
3821:
3819:
3815:
3811:
3807:
3803:
3796:
3789:
3785:
3778:
3762:
3752:
3745:
3741:
3737:
3733:
3729:
3716:
3712:
3708:
3704:
3701:
3698:
3694:
3690:
3687:
3683:
3680:
3677:
3673:
3669:
3665:
3661:
3657:
3654:
3653:
3652:
3650:
3646:
3635:
3632:
3628:
3621:
3617:
3613:
3609:
3605:
3602:. This means
3600:
3596:
3590:
3586:
3582:
3578:
3574:
3570:
3566:
3561:
3556:
3550:
3544:
3540:
3535:
3531:
3527:
3523:
3505:
3497:
3494:
3490:
3484:
3477:
3470:
3467:
3463:
3456:
3446:
3441:
3439:
3435:
3429:
3426:
3421:
3417:
3410:
3403:
3400:
3397: =
3396:
3389:
3385:
3381:
3377:
3374: −
3373:
3369:
3362:
3358:
3351:
3347:
3341:
3337:
3332:
3328:
3323:
3319:
3312:
3307:
3305:
3301:
3296:
3285:
3280:
3276:
3274:
3270:
3265:
3260:
3256:
3252:
3241:
3236:
3226:
3209:
3186:
3183:
3178:
3174:
3137:
3131:
3128:
3120:
3115:
3111:
3099:-subgroup of
3082:
3068:
3065:
3062:
3039:
3033:
3030:
3014:
3000:
2981:
2969:
2952:
2948:
2938:
2924:
2902:
2898:
2877:
2874:
2869:
2865:
2856:
2840:
2833:-subgroup of
2820:
2800:
2797:
2792:
2788:
2778:
2762:
2758:
2737:
2717:
2697:
2690:-subgroup of
2677:
2657:
2637:
2617:
2595:
2591:
2568:
2564:
2543:
2523:
2505:
2487:
2483:
2471:-subgroup of
2467:is any Sylow
2442:
2434:
2430:
2426:
2423:
2415:
2410:
2406:
2398:
2380:
2376:
2368:
2365:
2360:
2356:
2348:
2342:-subgroup in
2338:of the Sylow
2337:
2315:
2311:
2303:
2302:
2301:
2277:
2273:
2244:
2241:
2238:
2218:
2213:
2209:
2171:
2157:
2154:
2151:
2148:
2143:
2140:
2136:
2115:
2112:
2109:
2085:
2052:
2050:
2020:
2018:
2014:
2008:
2006:
1991:
1987:
1973:
1959:
1952:
1936:
1934:
1933:
1927:
1909:
1906:
1900:
1897:
1889:
1886:
1883:
1840:
1836:
1832:
1824:
1783:
1780:
1777:
1757:
1752:
1748:
1744:
1736:
1723:
1704:
1698:
1693:
1689:
1674:
1660:
1657:
1649:
1623:
1603:
1589:
1569:
1549:
1541:
1525:
1505:
1485:
1478:-subgroup of
1477:
1464:
1455:
1440:
1436:
1414:
1394:
1372:
1368:
1347:
1340:-subgroup of
1327:
1307:
1299:
1286:
1278:
1262:
1242:
1222:
1214:
1209:
1192:
1184:
1157:
1137:
1117:
1110:-subgroup of
1097:
1077:
1069:
1053:
1050:
1042:
1038:
1036:
1019:
999:
992:-subgroup of
979:
972:
956:
949:) of a group
948:
946:
941:
939:
921:
914:
909:
907:
903:
900:that a given
899:
895:
891:
887:
883:
879:
868:
865:
857:
854:November 2018
847:
843:
837:
836:
830:
825:
816:
815:
812:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
32:
31:
19:
7657:
7638:
7626:at Wikibooks
7605:
7556:
7515:
7509:
7468:
7462:
7419:
7413:
7355:
7349:
7297:
7238:
7182:
7176:
7124:
7118:
7075:(1): 40–52.
7072:
7066:
7019:
7014:
6999:the original
6954:
6948:
6898:
6892:
6858:(1): 29–75.
6855:
6849:
6822:
6810:
6798:
6791:Kantor 1985b
6786:
6779:Kantor 1985a
6774:
6762:
6750:
6738:
6726:
6714:
6702:
6690:
6678:
6659:
6653:
6641:. Retrieved
6629:
6616:
6583:
6577:
6564:
6511:
6505:
6496:
6492:
6488:
6484:
6480:
6476:
6472:
6468:
6464:
6460:
6453:
6449:
6445:
6441:
6437:
6433:
6429:
6427:
6420:
6407:
6390:
6382:
6378:
6374:
6367:
6363:
6359:
6352:
6348:
6344:
6340:
6333:
6329:
6324:
6320:
6314:
6310:
6300:
6291:
6289:
6283:
6266:
6257:
6251:
6244:
6240:
6236:
6225:is given by
6221:
6208:
6200:
6196:
6192:
6186:
6173:
6171:
6159:
6152:
6146:
6130:
6125:
6117:
6110:
6101:
6099:. Then (a)
6092:
6085:
6077:
6073:
6069:
6057:
6050:
6046:
6043:
6036:
6030:
6024:
6021:
6014:
6010:
6006:
6002:
5999:
5995:
5991:
5988:
5974:
5970:
5966:
5965:. With this
5957:
5858:
5841:
5837:
5829:
5823:
5813:
5809:
5806:
5798:
5790:
5786:
5782:
5766:
5762:
5758:
5755:
5747:
5739:
5735:
5727:
5723:
5711:
5667:denotes the
5661:
5651:
5630:Any element
5629:
5587:be a finite
5576:
5569:
5565:
5561:
5541:
5537:
5535:
5529:
5525:
5518:
5507:
5502:
5500:
5149:
5145:
5141:
5137:
5123:
5115:
5110:
5099:
5064:
5046:
5040:
5039:| <
5022:
5006:
4996:
4980:
4974:
4952:
4948:
4935:
4919:
4901:
4896:
4889:
4843:
4761:
4743:
4615:
4580:
4577:
4573:
4563:
4553:
4549:
4518:whose order
4503:
4445:
4438:group action
4435:
4431:
4397:
4391:
4384:
4379:
4375:
4371:
4367:
4363:
4359:
4355:
4351:
4347:
4343:
4339:
4331:
4324:
4320:
4316:
4312:
4308:
4304:
4300:
4296:
4292:
4288:
4284:
4280:
4276:
4272:
4268:
4264:
4260:
4256:
4252:
4248:
4244:
4240:
4233:
4229:
4225:
4221:
4217:
4213:
4209:
4205:
4201:
4197:
4193:
4188:
4176:
4172:
4164:
4160:
4152:
4147:
4143:
4136:
4127:
4118:
4101:
4097:
4090:
4086:
4084:
4019:states that
4014:
3993:
3986:
3979:
3975:
3973:
3968:
3961:
3954:
3947:
3943:
3941:
3936:
3932:
3925:
3918:
3911:
3909:
3904:
3900:
3896:
3889:
3882:
3875:
3868:
3864:
3860:
3858:
3846:prime powers
3839:
3832:that is not
3830:simple group
3827:
3814:cyclic group
3808:must be the
3805:
3794:
3787:
3776:
3750:
3743:
3739:
3735:
3731:
3727:
3725:
3714:
3710:
3706:
3696:
3692:
3688:
3685:
3675:
3671:
3670:primes with
3667:
3663:
3659:
3641:
3630:
3626:
3619:
3615:
3611:
3607:
3603:
3598:
3594:
3588:
3584:
3580:
3579:. There are
3576:
3572:
3568:
3564:
3559:
3554:
3548:
3542:
3538:
3533:
3529:
3521:
3444:
3442:
3437:
3433:
3427:
3424:
3419:
3415:
3408:
3401:
3398:
3394:
3387:
3383:
3379:
3375:
3371:
3367:
3360:
3356:
3349:
3339:
3335:
3330:
3326:
3321:
3317:
3310:
3308:
3303:
3294:
3292:
3283:
3272:
3268:
3263:
3258:
3254:
3248:
3239:
3084:
3012:
3006:
2939:
2937:-subgroups.
2779:
2536:every Sylow
2515:
2512:Consequences
2502:denotes the
2174:
2074:, all Sylow
2055:
2046:
2022:
2010:
1979:
1958:multiplicity
1951:prime factor
1938:
1930:
1928:
1680:
1614:
1457:
1428:
1279:
1277:prime factor
1210:
1034:
944:
943:
937:
935:
913:prime number
910:
902:finite group
881:
875:
860:
851:
832:
811:
640:
628:
616:
604:
592:
580:
568:
556:
327:
314:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
7471:(1): 1–17.
7178:Arch. Math.
6827:Seress 2003
6803:Kantor 1990
6767:Cannon 1971
6755:Butler 1991
6495:but not in
6377:), so then
6191:be a Sylow
6013:is a Sylow
5738:is a Sylow
5595:, and let Ω
4303:, and then
4291:normalizes
4096:of Sylow's
3917:= 6, since
3838:Burnside's
3606:is a Sylow
3095:is a Sylow
1978:, of order
1863:-group and
1855:. That is,
1456:is a Sylow
942:(sometimes
846:introducing
530:Topological
369:alternating
7672:Categories
7587:1028.20002
7548:0731.20005
7501:0642.20019
7454:0573.20022
7398:0604.20001
7334:0253.20027
7289:0785.20001
7231:Algorithms
7223:0092.02403
7165:0436.01006
7107:0637.01006
7054:1065.01009
6995:0943.68149
6933:0829.01011
6880:0721.01008
6837:References
6600:04.0056.02
6579:Math. Ann.
6483:-subgroup
6417:Algorithms
6150:, and (c)
6122:normalizer
6076:-subgroup
5969:, we have
5953:| ≠ 0
5789:are Sylow
5779:isomorphic
5754:such that
5669:stabilizer
5551:for which
5517:|) =
5109:|) =
4995:|) =
4947:|) =
4934:|) =
4918:|) +
4899:, one has
4751:for which
4584:such that
4387:-nilpotent
4346:-subgroup
4315:, so that
4204:-subgroup
3717:is simple.
2504:normalizer
2094:are Sylow
1949:For every
1812:has order
1808:-subgroup
1722:isomorphic
1611:Motivation
1032:that is a
829:references
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
7659:MathWorld
7640:MathWorld
7612:EMS Press
7532:0196-6774
7485:0196-6774
7438:1090-2724
7408:(1985b).
7382:0196-6774
7360:CiteSeerX
7318:0160-7634
7215:119816392
7199:0003-9268
7157:123685226
7141:0003-9519
7091:0315-0860
7038:0315-0860
6971:0168-7433
6917:0791-5578
6864:0392-4432
6608:121928336
6570:Sylow, L.
6199:has size
6120:) is the
5921:Ω
5912:∤
5874:∤
5775:conjugate
5540:of order
5466:ν
5449:−
5429:ν
5425:−
5395:ν
5378:−
5355:ν
5351:−
5332:−
5307:∏
5291:−
5273:−
5249:−
5224:∏
5168:Ω
4973:| =
4878:for each
4836:| =
4595:∤
4572:| =
4485:∤
4459:∣
4295:for some
4287:. Hence
4259:but also
4157:. Hence,
4051:−
4048:≡
4036:−
3985:= 10 and
3910:As well,
3763:∣
3702:Example-3
3681:Example-2
3655:Example-1
3597:| =
3591:, making
3423:) =
3325:), where
3184:≡
3132:
3066:⊂
3034:
2366:≡
2141:−
2113:∈
2084:conjugate
2024:Corollary
1972:-subgroup
1699:
1677:Statement
1540:conjugate
1360:of order
940:-subgroup
896:of fixed
894:subgroups
701:Conformal
589:Euclidean
196:nilpotent
7683:P-groups
6731:Meo 2004
6719:Gow 1994
6634:Archived
6572:(1872).
6527:See also
6328:so that
6318:for all
6290:Now let
6276:= |
6264:. Thus,
6180:and let
6137:divides
6042:so that
5977:for all
5836:and let
5603:. Then
5069:. Since
5044:implies
5004:one has
4848:we have
4718:for its
4684:for its
4675:, write
4614:of size
4015:Part of
3850:solvable
3392:. Since
3230:Examples
2463:, where
2330:divides
2231:, where
1606:Theorems
886:theorems
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
7614:, 2001
7579:1970241
7540:1079450
7493:0925595
7446:0805654
7390:0813589
7326:0367027
7273:1225579
7207:0147529
7149:0575718
7099:0931678
7046:2055642
6987:1449341
6979:1721912
6925:1313412
6872:1096350
6549:p-group
6282:|/
6235:|
6145:|/
6128:), (b)
6109:(where
6059:Theorem
5863:. Now
5713:Theorem
5660:(where
5513:(|
5105:(|
5052:(|
5012:(|
4986:(|
4941:(|
4925:(|
4907:(|
4844:By the
4731:|
4697:|
4658:|
4505:Theorem
4263:(since
4220:, then
4124:
4109:
3842:theorem
3802:coprime
3524:is any
3430:′
3346:abelian
3253:of the
3086:Theorem
2968:divides
2710:, then
2670:is any
2176:Theorem
2057:Theorem
1940:Theorem
971:maximal
842:improve
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
7585:
7577:
7567:
7546:
7538:
7530:
7499:
7491:
7483:
7452:
7444:
7436:
7396:
7388:
7380:
7362:
7332:
7324:
7316:
7287:
7281:395110
7279:
7271:
7261:
7221:
7213:
7205:
7197:
7163:
7155:
7147:
7139:
7105:
7097:
7089:
7052:
7044:
7036:
6993:
6985:
6977:
6969:
6931:
6923:
6915:
6878:
6870:
6862:
6842:Proofs
6666:
6606:
6598:
6395:|
6139:|
6084:. Let
6061:
6040:|
6019:|
5946:|
5846:|
5826:cosets
5715:
5694:|
5685:|
5681:|
5677:|
5658:|
5643:|
5605:|
5134:|
5128:|
5088:|
5079:|
5075:|
5071:|
4962:|
4876:|
4850:|
4805:|
4741:in Ω.
4566:|
4507:
4169:. So,
4133:, ie.
3999:, the
3834:cyclic
3784:normal
3775:5 and
3645:simple
3571:. So,
3267:. For
3257:-gon,
3207:
3190:
3166:, and
3103:, and
2265:. Let
2178:
2059:
1942:
1796:where
1538:) are
1037:-group
936:Sylow
911:For a
880:, the
831:, but
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
7346:(PDF)
7277:S2CID
7211:S2CID
7153:S2CID
7002:(PDF)
6983:S2CID
6945:(PDF)
6643:8 May
6637:(PDF)
6626:(PDF)
6604:S2CID
6556:Notes
6463:) of
6436:is a
6243:} =
6169:Proof
5987:, so
5821:Proof
5805:with
5726:is a
5627:Proof
5578:Lemma
4960:with
4720:orbit
4561:Proof
4338:: if
3818:up to
3378:) = (
3364:) is
2853:is a
2336:index
2128:with
1956:with
1859:is a
1770:with
1068:order
1041:power
969:is a
898:order
730:Sp(∞)
727:SU(∞)
140:image
7565:ISBN
7528:ISSN
7481:ISSN
7434:ISSN
7378:ISSN
7314:ISSN
7259:ISBN
7195:ISSN
7137:ISSN
7087:ISSN
7034:ISSN
6967:ISSN
6913:ISSN
6899:0033
6860:ISSN
6664:ISBN
6645:2021
6347:and
6068:Let
5785:and
5773:are
5734:and
5583:Let
5528:) =
5026:(as
4760:has
4711:and
4651:= {
4635:and
4623:acts
4564:Let
4370:and
4323:are
4319:and
4275:and
4228:are
4224:and
4212:and
3695:and
3666:and
3587:and
3528:of
3329:and
2475:and
2257:and
2242:>
2185:Let
2090:and
2082:are
1781:>
934:, a
724:O(∞)
713:Loop
532:and
7583:Zbl
7544:Zbl
7520:doi
7497:Zbl
7473:doi
7450:Zbl
7424:doi
7394:Zbl
7370:doi
7330:Zbl
7310:AMS
7285:Zbl
7251:doi
7219:Zbl
7187:doi
7161:Zbl
7129:doi
7103:Zbl
7077:doi
7050:Zbl
7024:doi
6991:Zbl
6959:doi
6929:Zbl
6903:doi
6876:Zbl
6596:JFM
6588:doi
6506:In
6467:in
6389:= {
6315:xQx
6303:∈ Ω
6260:in
6237:gPg
6189:∈ Ω
6124:of
6063:(3)
5992:HgP
5971:hgP
5960:∈ Ω
5832:in
5828:of
5801:in
5750:in
5722:If
5717:(2)
5635:∈ Ω
5549:∈ Ω
4883:∈ Ω
4749:∈ Ω
4673:∈ Ω
4640:∈ Ω
4509:(1)
4382:is
4107:is
4062:mod
3859:If
3840:p q
3651:.
3563:or
3546:is
3282:In
3238:In
3091:If
2377:mod
2180:(3)
2061:(2)
2045:in
1974:of
1944:(1)
1872:gcd
1690:Syl
1602:).
1180:Syl
1043:of
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
7674::
7656:.
7637:.
7610:,
7604:,
7581:.
7575:MR
7573:.
7563:.
7542:.
7536:MR
7534:.
7526:.
7516:11
7514:.
7495:.
7489:MR
7487:.
7479:.
7467:.
7448:.
7442:MR
7440:.
7432:.
7420:30
7418:.
7412:.
7392:.
7386:MR
7384:.
7376:.
7368:.
7354:.
7348:.
7328:.
7322:MR
7320:.
7304:.
7283:.
7275:.
7269:MR
7267:.
7257:.
7249:.
7241:.
7217:.
7209:.
7203:MR
7201:.
7193:.
7183:10
7159:.
7151:.
7145:MR
7143:.
7135:.
7125:21
7123:.
7101:.
7095:MR
7093:.
7085:.
7073:15
7065:.
7048:.
7042:MR
7040:.
7032:.
7020:31
7013:.
6989:.
6981:.
6975:MR
6973:.
6965:.
6955:23
6953:.
6947:.
6927:.
6921:MR
6919:.
6911:.
6897:.
6891:.
6874:.
6868:MR
6866:.
6856:10
6632:.
6628:.
6602:.
6594:.
6576:.
6523:.
6503:.
6452:=
6425:.
6412:.
6381:=
6332:≤
6323:∈
6313:=
6287:.
6272:=
6239:=
6231:∈
6214:=
6164:.
6107:=
6053:.
6049:=
6047:Hg
6025:Hg
6005:≤
6003:Hg
5994:=
5982:∈
5975:gP
5973:=
5967:gP
5958:gP
5816:.
5812:=
5810:Hg
5761:≤
5759:Hg
5707:.
5622:.
5572:.
5570:Hg
5560:=
5140:+
4951:+
4841:.
4782:⊆
4778:∈
4735:∈
4723:{
4705:=
4693:∈
4689:{
4662:∈
4642:,
4630:∈
4618:.
4576:=
4574:pm
4556:.
4389:.
4368:PK
4366:=
4362:,
4311:=
4307:=
4289:gh
4181:.
3856:.
3836:.
3691:,
3662:,
3660:pq
3620:GL
3520:,
3440:.
3409:GL
3382:)(
3350:GL
3311:GL
3287:12
3225:.
3129:Cl
3081:.
3031:Cl
2999:.
2777:.
2170:.
2051:.
2019:.
2007:.
1935:.
1926:.
1673:.
1661:60
1208:.
908:.
615:U(
591:E(
579:O(
37:→
7662:.
7643:.
7589:.
7550:.
7522::
7503:.
7475::
7469:9
7456:.
7426::
7400:.
7372::
7356:6
7336:.
7291:.
7253::
7225:.
7189::
7167:.
7131::
7109:.
7079::
7056:.
7026::
6961::
6935:.
6905::
6882:.
6829:.
6817:.
6805:.
6793:.
6781:.
6769:.
6745:.
6733:.
6721:.
6709:.
6697:.
6685:.
6672:.
6647:.
6610:.
6590::
6584:5
6512:p
6497:H
6493:H
6489:p
6485:H
6481:p
6477:p
6473:p
6469:G
6465:H
6461:H
6459:(
6456:G
6454:N
6450:N
6446:p
6442:G
6438:p
6434:H
6430:p
6410:)
6408:p
6403:0
6401:Ω
6397:Ω
6391:P
6387:0
6383:Q
6379:P
6375:Q
6373:(
6370:G
6368:N
6364:Q
6360:Q
6358:(
6355:G
6353:N
6349:Q
6345:P
6341:Q
6339:(
6336:G
6334:N
6330:P
6325:P
6321:x
6311:Q
6305:0
6301:Q
6296:0
6292:P
6284:q
6279:G
6269:p
6267:n
6262:G
6258:P
6254:)
6252:P
6250:(
6248:G
6245:N
6241:P
6233:G
6229:g
6227:{
6222:P
6218:G
6211:p
6209:n
6203:p
6201:n
6197:P
6193:p
6187:P
6182:G
6178:G
6174:p
6162:)
6160:p
6155:p
6153:n
6147:q
6142:G
6133:p
6131:n
6126:P
6118:P
6116:(
6113:G
6111:N
6104:p
6102:n
6097:G
6093:p
6088:p
6086:n
6082:G
6078:P
6074:p
6070:q
6051:P
6044:g
6037:P
6031:H
6022:g
6015:p
6011:H
6007:P
6000:g
5996:P
5989:g
5984:H
5980:h
5962:0
5950:0
5948:Ω
5931:|
5925:0
5916:|
5909:p
5889:]
5886:P
5883::
5880:G
5877:[
5871:p
5861:)
5859:p
5855:Ω
5850:0
5848:Ω
5842:H
5838:H
5834:G
5830:P
5814:K
5807:g
5803:G
5799:g
5795:G
5791:p
5787:K
5783:H
5771:G
5767:p
5763:P
5756:g
5752:G
5748:g
5744:G
5740:p
5736:P
5732:G
5728:p
5724:H
5705:p
5701:x
5698:H
5691:x
5688:H
5679:Ω
5673:p
5664:x
5662:H
5654:x
5652:H
5646:H
5639:H
5633:x
5620:)
5618:p
5613:0
5611:Ω
5607:Ω
5601:H
5597:0
5593:H
5589:p
5585:H
5566:m
5562:H
5557:ω
5553:G
5547:ω
5542:p
5538:H
5530:r
5526:m
5524:(
5521:p
5519:ν
5515:Ω
5510:p
5508:ν
5503:p
5481:)
5478:j
5475:(
5470:p
5461:p
5456:/
5452:j
5444:)
5441:j
5438:(
5433:p
5422:k
5418:p
5410:)
5407:j
5404:(
5399:p
5390:p
5385:/
5381:j
5375:m
5370:)
5367:j
5364:(
5359:p
5348:k
5344:p
5335:1
5327:k
5323:p
5317:1
5314:=
5311:j
5303:m
5300:=
5294:j
5286:k
5282:p
5276:j
5270:m
5265:k
5261:p
5252:1
5244:k
5240:p
5234:0
5231:=
5228:j
5220:=
5214:)
5207:k
5203:p
5198:m
5193:k
5189:p
5182:(
5176:=
5172:|
5164:|
5150:r
5146:p
5142:r
5138:k
5131:G
5124:p
5116:r
5111:r
5107:Ω
5102:p
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1992:n
1988:p
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