820:
3095:
7480:
3139:
52:
5356:
1633:
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
6378:
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
5018:
3157:
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
1612:
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
3502:
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
3378:
4292:
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.
5351:{\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)}}}}
4308:-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
3032:
6359:
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
1715:
3940:
3083:
2386:
1917:
2929:
5802:
1634:
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.
1203:
2452:
3131:
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:
2955:
2165:
1765:
1850:
4468:
4358:
5759:
4332:
2802:
2252:
2226:
2123:
1791:
1668:
3633:
2873:
2766:
2599:
2572:
2491:
2325:
2287:
2001:
1450:
1382:
4406:
1061:
7483:
2842:
2822:
2739:
2719:
2699:
2679:
2659:
2639:
2619:
2545:
2525:
1631:
1597:
1577:
1557:
1533:
1513:
1493:
1472:
1422:
1402:
1355:
1335:
1315:
1294:
1270:
1250:
1230:
1165:
1145:
1125:
1105:
1085:
1027:
1007:
987:
967:
932:
1678:
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:
3310:
6801:
4269:
6493:
3642:≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be
4265:
4052:
shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as
905:
357:
5528:
4705:
4579:
4545:
7537:
4284:'s strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.
1209:
307:
3763:
then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means
3507:. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a
904:
contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the
2013:
792:
302:
7428:
7122:
7102:
6527:
863:
841:
2982:
834:
3135:, and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.
2848:. However, there are groups that have proper, non-trivial normal subgroups but no normal Sylow subgroups, such as
1681:
3885:
3712:, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30
1954:
718:
17:
3045:
2348:
7471:
7169:
6380:
1863:
785:
2902:
3470:-subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow
5764:
7466:
6360:
1170:
402:
216:
2392:
7420:
6429:
6282:
4482:
4297:
889:
134:
4746:
4624:
elements, providing the desired subgroup. This is the maximal possible size of a stabilizer subgroup
7461:
3660:
must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are
7224:
4194:)-conjugate. Burnside's fusion theorem can be used to give a more powerful factorization called a
1064:
828:
600:
334:
211:
99:
480:
455:
418:
3669:
3385:
6858:
3949:. One may easily prove this theorem by Sylow's third theorem. Indeed, observe that the number
2934:
2128:
1926:
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in
1724:
7219:
6393:
4253:
4049:
2038:, then there exists an element (and thus a cyclic subgroup generated by this element) of order
1812:
845:
750:
540:
4447:
4337:
4264:
has on the structure of the entire group. This control is exploited at several stages of the
7321:
Kantor, William M.; Taylor, Donald E. (1988). "Polynomial-time versions of Sylow's theorem".
6974:
6438:
6433:
5726:
4311:
3697:
2774:
2009:
1928:
624:
2231:
2201:
2102:
1770:
1637:
7438:
7399:
7352:
7305:
7249:
7185:
7132:
7066:
7008:
6958:
6927:
6905:
6875:
6838:
6784:
6731:
4277:
3618:
3149:
reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.
2851:
2744:
2577:
2550:
2469:
2303:
2265:
1979:
1428:
1360:
564:
552:
170:
104:
7446:
7407:
7360:
7313:
7257:
7193:
7148:
7082:
7024:
6966:
6913:
6854:
6792:
6739:
6459:
8:
6708:
Casadio, Giuseppina; Zappa, Guido (1990). "History of the Sylow theorem and its proofs".
4979:
4381:
3876:
2332:
1037:
877:
139:
34:
6802:"A formal proof of Sylow's theorem. An experiment in abstract algebra with Isabelle HOL"
1043:
7136:
7070:
7032:
7012:
6842:
6482:
6463:
4195:
3864:
3159:
2827:
2807:
2724:
2704:
2684:
2664:
2644:
2624:
2604:
2530:
2510:
1616:
1582:
1562:
1542:
1518:
1498:
1478:
1457:
1407:
1387:
1340:
1320:
1300:
1279:
1255:
1235:
1215:
1150:
1130:
1110:
1090:
1070:
1012:
992:
972:
952:
917:
124:
96:
6888:
6870:
7542:
7510:
7494:
7491:
7424:
7387:
7383:
7340:
7336:
7293:
7288:
7269:
7237:
7233:
7173:
7118:
7074:
7054:
7016:
6996:
6946:
6941:
6922:
6893:
6826:
6772:
6719:
6523:
6467:
6367:
4243:
3860:
1718:
529:
372:
266:
7203:"Polynomial-time algorithms for finding elements of prime order and Sylow subgroups"
6281:
The problem of finding a Sylow subgroup of a given group is an important problem in
2883:
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow
695:
7513:
7442:
7403:
7379:
7356:
7332:
7309:
7283:
7253:
7229:
7202:
7189:
7144:
7110:
7078:
7046:
7020:
6988:
6962:
6936:
6909:
6883:
6850:
6846:
6818:
6788:
6762:
6735:
6455:
6447:
6403:
4261:
897:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
7159:
7140:
7434:
7395:
7348:
7301:
7245:
7181:
7128:
7106:
7062:
7004:
6954:
6901:
6834:
6780:
6727:
6375:
5634:
3643:
3508:
2845:
2080:
1536:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
3852:= 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is
7265:
3709:
3110:
3094:
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
6822:
3105:
all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.
7531:
7479:
7391:
7344:
7297:
7241:
7177:
7058:
7000:
6950:
6897:
6830:
6776:
6767:
6748:
6723:
6398:
4281:
3205:
690:
612:
446:
319:
185:
7156:
Cannon, John J. (1971). "Computing local structure of large finite groups".
7368:
Kantor, William M. (1990). "Finding Sylow normalizers in polynomial time".
4273:
3693:
3689:
3673:
3504:
1947:
1273:
912:
901:
545:
244:
233:
180:
155:
150:
109:
80:
43:
7114:
3435:
is a subgroup where all its elements have orders which are powers of
5638:
3705:
3373:{\displaystyle {\begin{bmatrix}x^{im}&0\\0&x^{jm}\end{bmatrix}}}
3109:
A simple illustration of Sylow subgroups and the Sylow theorems are the
1919:. These properties can be exploited to further analyze the structure of
7050:
6992:
6522:. with contribution by Victor J. Katz. Pearson Education. p. 322.
6451:
5981:
4103:)-conjugate. The proof is a simple application of Sylow's theorem: If
2494:
712:
440:
6710:
3138:
7518:
7499:
5365:
remains in any of the factors inside the product on the right. Hence
4957:, one can show the existence of ω of the former type by showing that
533:
6370:, it has been proven, in Kantor and Kantor and Taylor, that a Sylow
893:
70:
3759:
would have a normal subgroup of order 3, and could not be simple.
6408:
5459:
denote the set of points of Ω that are fixed under the action of
3704:
states that if the order of a group is the product of one or two
3661:
1965:
1030:
885:
412:
326:
2771:
A very important consequence of
Theorem 2 is that the condition
7274:
51:
2008:
The following weaker version of theorem 1 was first proved by
6979:
6335:. In other words, a polycyclic generating system of a Sylow
5685:
3677:
3664:, the intersection of these two subgroups is trivial, and so
3066:
2369:
6608:
4252:
Less trivial applications of the Sylow theorems include the
6753:
3795:
also has 24 distinct elements of order 5. But the order of
5535:
by assumption. The result follows immediately by writing
4268:, and for instance defines the case divisions used in the
3688:
A more complex example involves the order of the smallest
3427:
and all its powers have an order which is a power of
7489:
7035:(1959). "Ein Beweis für die Existenz der Sylowgruppen".
6668:
5012:
places), and can also be shown by a simple computation:
3799:
is only 30, so a simple group of order 30 cannot exist.
3676:
of order 15. Thus, there is only one group of order 15 (
7508:
6560:
5700:
act on Ω by left multiplication. Applying the Lemma to
4296:
One proof of the Sylow theorems exploits the notion of
3544:
Group of order 30, groups of order 20, groups of order
6809:
3319:
2895:-power order) that is maximal for inclusion among all
1721:
to each other and have the largest possible order: if
1212:. Lagrange's theorem states that for any finite group
6536:
5767:
5729:
5021:
4450:
4384:
4340:
4314:
3888:
3621:
3598:= 15 is such a number using the Sylow theorems: Let
3313:
3048:
2985:
2937:
2905:
2878:
2854:
2830:
2810:
2777:
2747:
2727:
2707:
2687:
2667:
2647:
2627:
2607:
2580:
2553:
2533:
2513:
2472:
2395:
2351:
2306:
2268:
2234:
2204:
2131:
2105:
1982:
1866:
1815:
1773:
1727:
1684:
1640:
1619:
1585:
1565:
1545:
1521:
1501:
1481:
1460:
1431:
1410:
1390:
1363:
1343:
1323:
1303:
1282:
1258:
1238:
1218:
1173:
1153:
1133:
1113:
1093:
1073:
1046:
1015:
995:
975:
955:
920:
483:
458:
421:
2621:-subgroup, and so is conjugate to every other Sylow
1232:
the order (number of elements) of every subgroup of
6644:
6632:
6596:
6548:
6158:denote the set of fixed points of this action. Let
4210:is contained in the center of its normalizer, then
3166:, half the minimal rotation in the dihedral group.
892:that give detailed information about the number of
7419:. Cambridge Tracts in Mathematics. Vol. 152.
6680:
6656:
6620:
6134:, and it follows that this number is a divisor of
5796:
5753:
5350:
4462:
4400:
4352:
4326:
3934:
3646:(since it has no distinct conjugates). Similarly,
3627:
3474:-subgroups are conjugate to each other, the Sylow
3372:
3077:
3026:
2949:
2923:
2867:
2836:
2816:
2796:
2760:
2733:
2713:
2693:
2673:
2653:
2633:
2613:
2593:
2566:
2539:
2519:
2485:
2446:
2380:
2319:
2281:
2246:
2220:
2159:
2117:
1995:
1911:
1844:
1785:
1759:
1709:
1662:
1625:
1591:
1571:
1551:
1527:
1507:
1487:
1466:
1444:
1416:
1396:
1376:
1349:
1329:
1309:
1288:
1264:
1244:
1224:
1197:
1159:
1139:
1119:
1099:
1079:
1055:
1021:
1001:
981:
961:
926:
491:
466:
429:
6871:"The mathematical life of Cauchy's group theorem"
6800:Kammüller, Florian; Paulson, Lawrence C. (1999).
5073:
5041:
3292:. Thus by Theorem 1, the order of the Sylow
3162:, which can be represented by rotation through π/
2507:The Sylow theorems imply that for a prime number
7529:
7484:Abstract Algebra/Group Theory/The Sylow Theorems
6799:
6511:
6347:(including the identity) and taking elements of
1539:to each other. Furthermore, the number of Sylow
1208:The Sylow theorems assert a partial converse to
6977:(1980). "The early proofs of Sylow's theorem".
6474:
5396:It may be noted that conversely every subgroup
3748:= 10, since neither 4 nor 7 divides 10, and if
2641:-subgroup. Due to the maximality condition, if
6584:
6572:
4974:(if none existed, that valuation would exceed
4287:
3767:has at least 20 distinct elements of order 3.
3683:
3027:{\displaystyle n_{p}=|\operatorname {Cl} (K)|}
2804:is equivalent to the condition that the Sylow
7370:
7323:
7210:
7099:Fundamental Algorithms for Permutation Groups
7037:
6374:-subgroup and its normalizer can be found in
3863:over 5 elements. It has order 60, and has 24
3559:distinct primes are some of the applications.
3169:Another example are the Sylow p-subgroups of
1087:) that is not a proper subgroup of any other
876:In mathematics, specifically in the field of
793:
7320:
7165:
6707:
6674:
6566:
6434:"Théorèmes sur les groupes de substitutions"
6351:-power order contained in the normalizer of
6339:-subgroup can be found by starting from any
4035: − 1)! ≡ −1 (mod
2891:-subgroup (that is, every element in it has
2931:denote the set of conjugates of a subgroup
1710:{\displaystyle \operatorname {Syl} _{p}(G)}
6973:
6542:
5451:-group, let Ω be a finite set acted on by
4604:The proof will show the existence of some
3935:{\displaystyle (p-1)!\equiv -1{\pmod {p}}}
1272:. The Sylow theorems state that for every
800:
786:
7287:
7223:
6940:
6887:
6766:
4470:, and let Ω denote the set of subsets of
3609:be the number of Sylow 3-subgroups. Then
3573:has more than one Sylow 5-subgroup, then
1515:-subgroups of a group (for a given prime
864:Learn how and when to remove this message
485:
460:
423:
7158:Computers in Algebra and Number Theory (
7105:. Vol. 559. Berlin, New York City:
7031:
6920:
6602:
6554:
6517:
6422:
6154:act on Ω by conjugation, and again let Ω
6076:. For this group action, the stabilizer
3672:of groups of order 3 and 5, that is the
3137:
3093:
3078:{\displaystyle n_{p}\equiv 1{\bmod {p}}}
2381:{\displaystyle n_{p}\equiv 1{\bmod {p}}}
1559:-subgroups of a group for a given prime
888:named after the Norwegian mathematician
827:This article includes a list of general
7264:
7200:
6650:
6638:
4838:, the ones we are looking for, one has
3497:
2887:-subgroup in an infinite group to be a
1912:{\displaystyle {\text{gcd}}(|G:P|,p)=1}
14:
7530:
7414:
7367:
7155:
7096:
6686:
6662:
6626:
6614:
6483:"Classification of groups of order 60"
6480:
6055:-subgroup. By Theorem 2, the orbit of
4266:classification of finite simple groups
3581:
2924:{\displaystyle \operatorname {Cl} (K)}
2574:. Conversely, if a subgroup has order
906:classification of finite simple groups
358:Classification of finite simple groups
7509:
7490:
6428:
6066:, so by the orbit-stabilizer theorem
4755:, which counts the number of factors
4023: − 2)! ≡ 1 (mod
1067:of every group element is a power of
7270:"Sylow's theorem in polynomial time"
6499:from the original on 28 October 2020
6288:One proof of the existence of Sylow
5797:{\displaystyle p\nmid |\Omega _{0}|}
4360:for the negation of this statement.
4304:acts on itself or on the set of its
4300:in various creative ways. The group
4256:, which studies the control a Sylow
3960:-subgroups in the symmetric group
3565:(Groups of order 60): If the order |
2186:be a prime factor with multiplicity
1424:. Moreover, every subgroup of order
813:
6868:
6746:
6590:
6578:
6331:is also such that is divisible by
4092:-conjugate if and only if they are
3924:
3870:
3602:be a group of order 15 = 3 · 5 and
3590:are such that every group of order
3307:, is the set of diagonal matrices
1198:{\displaystyle {\text{Syl}}_{p}(G)}
24:
7160:Proc. SIAM-AMS Sympos. Appl. Math.
6749:"Sylow's proof of Sylow's theorem"
6520:A First Course In Abstract Algebra
5780:
5045:
5027:
4334:as notation for "a divides b" and
4127:is contained in the normalizer of
2879:Sylow theorems for infinite groups
2447:{\displaystyle n_{p}=|G:N_{G}(P)|}
833:it lacks sufficient corresponding
25:
7554:
7454:
7103:Lecture Notes in Computer Science
4485:on Ω by left multiplication: for
4270:Alperin–Brauer–Gorenstein theorem
4044:
3741:must equal 1 (mod 3). Therefore,
3413:, its primitive roots have order
7478:
6923:"Die Entdeckung der Sylow-Sätze"
3986:times the number of p-cycles in
2547:-subgroup is of the same order,
818:
50:
7090:
6304:and the index is divisible by
6292:-subgroups is constructive: if
5657:, then there exists an element
5606:, then there exists an element
3917:
3867:of order 5, and 20 of order 3.
3734:must divide 10 ( = 2 · 5), and
2502:
2190:of the order of a finite group
2099:, then there exists an element
1960:of the order of a finite group
1297:of the order of a finite group
7168:. Vol. 4. Providence RI:
6032:Let Ω be the set of all Sylow
5932:denote the order of any Sylow
5790:
5775:
5748:
5736:
5501:will lie in an orbit of order
5340:
5334:
5303:
5297:
5269:
5263:
5229:
5223:
5031:
5023:
4633:, since for any fixed element
4408:is divisible by a prime power
4394:
4386:
4202:is a finite group whose Sylow
3928:
3918:
3901:
3889:
3813:must divide 6 ( = 2 · 3) and
3784:must divide 6 ( = 2 · 3), and
3594:is cyclic. One can show that
3198: ≡ 1 (mod
3020:
3016:
3010:
3000:
2918:
2912:
2899:-subgroups in the group. Let
2440:
2436:
2430:
2410:
1900:
1890:
1876:
1872:
1825:
1817:
1737:
1729:
1704:
1698:
1650:
1642:
1192:
1186:
1036:(meaning its cardinality is a
719:Infinite dimensional Lie group
13:
1:
6889:10.1016/S0315-0860(03)00003-X
6696:
6381:Magma computer algebra system
6276:
6044:act on Ω by conjugation. Let
5637:to each other (and therefore
4060:is a finite group with Sylow
3193:are primes ≥ 3 and
1607:
1147:-subgroups for a given prime
7538:Theorems about finite groups
7417:Permutation Group Algorithms
7384:10.1016/0196-6774(90)90009-4
7337:10.1016/0196-6774(88)90002-8
7289:10.1016/0022-0000(85)90052-2
7234:10.1016/0196-6774(85)90029-X
7201:Kantor, William M. (1985a).
6942:10.1016/0315-0860(88)90048-1
5008:from it involves a carry in
4816:. This means that for those
4276:whose Sylow 2-subgroup is a
3034:is finite, then every Sylow
1673:
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
7:
7467:Encyclopedia of Mathematics
6921:Scharlau, Winfried (1988).
6386:
6361:GAP computer algebra system
5951:denote the number of Sylow
5625:. In particular, all Sylow
4288:Proof of the Sylow theorems
4143:, but in the normalizer of
3684:Small groups are not simple
3089:
2297:. Then the following hold:
2083:to each other. That is, if
1717:, all members are actually
1602:
217:List of group theory topics
10:
7559:
7421:Cambridge University Press
6518:Fraleigh, John B. (2004).
6283:computational group theory
5531:), which is a multiple of
4978:). This is an instance of
4745:, and therefore using the
4139:are conjugate not only in
3038:-subgroup is conjugate to
2950:{\displaystyle K\subset G}
2160:{\displaystyle g^{-1}Hg=K}
1760:{\displaystyle |G|=p^{n}m}
1404:that divides the order of
6701:
6253:} so that, by the Lemma,
5684:Let Ω be the set of left
5556:over all distinct orbits
5004:digits zero, subtracting
4950:over all distinct orbits
4747:additive p-adic valuation
4111:, then the normalizer of
4054:Burnside's fusion theorem
3838:| = 60 = 2 · 3 · 5, then
3820:must equal 1 (mod 7), so
3806:| = 42 = 2 · 3 · 7. Here
3791:must equal 1 (mod 5). So
1845:{\displaystyle |P|=p^{n}}
6768:10.33232/BIMS.0033.55.63
6675:Kantor & Taylor 1988
6567:Casadio & Zappa 1990
6415:
5424:, namely any one of the
5393:, completing the proof.
4706:orbit-stabilizer theorem
4463:{\displaystyle p\nmid u}
4412:has a subgroup of order
4353:{\displaystyle a\nmid b}
3834:On the other hand, for |
3417:− 1, which implies that
335:Elementary abelian group
212:Glossary of group theory
6823:10.1023/A:1006269330992
5754:{\displaystyle p\nmid }
4327:{\displaystyle a\mid b}
4131:). By Sylow's theorem
3670:internal direct product
3586:Some non-prime numbers
2797:{\displaystyle n_{p}=1}
2289:be the number of Sylow
2194:, so that the order of
1964:, there exists a Sylow
1579:is congruent to 1 (mod
1384:, the highest power of
1317:, there exists a Sylow
1127:. The set of all Sylow
848:more precise citations.
7162:, New York City, 1970)
6980:Arch. Hist. Exact Sci.
6975:Waterhouse, William C.
6754:Irish Math. Soc. Bull.
6711:Boll. Storia Sci. Mat.
6308:, then the normalizer
6168:and observe that then
5804:, hence in particular
5798:
5755:
5352:
5198:
5115:
4860:, while for any other
4464:
4402:
4354:
4328:
4254:focal subgroup theorem
4214:has a normal subgroup
3999: − 2)!
3936:
3629:
3374:
3150:
3106:
3079:
3028:
2951:
2925:
2869:
2838:
2818:
2798:
2762:
2735:
2715:
2695:
2675:
2655:
2635:
2615:
2595:
2568:
2541:
2521:
2487:
2448:
2382:
2321:
2283:
2248:
2247:{\displaystyle n>0}
2222:
2221:{\displaystyle p^{n}m}
2161:
2119:
2118:{\displaystyle g\in G}
2034:dividing the order of
1997:
1913:
1846:
1787:
1786:{\displaystyle n>0}
1761:
1711:
1664:
1663:{\displaystyle |G|=60}
1627:
1593:
1573:
1553:
1529:
1509:
1489:
1468:
1446:
1418:
1398:
1378:
1351:
1331:
1311:
1290:
1266:
1246:
1226:
1199:
1161:
1141:
1121:
1101:
1081:
1057:
1023:
1009:, i.e., a subgroup of
1003:
983:
963:
928:
751:Linear algebraic group
493:
468:
431:
7415:Seress, Ákos (2003).
7275:J. Comput. Syst. Sci.
7115:10.1007/3-540-54955-2
6481:Gracia–Saz, Alfonso.
6266:| = 1 (mod
6222:) in particular, and
5815:so there exists some
5799:
5756:
5353:
5165:
5082:
4465:
4403:
4355:
4329:
4001:. On the other hand,
3937:
3630:
3628:{\displaystyle \mid }
3569:| = 60 and
3397:. Since the order of
3375:
3141:
3097:
3080:
3029:
2952:
2926:
2870:
2868:{\displaystyle S_{4}}
2839:
2819:
2799:
2763:
2761:{\displaystyle p^{n}}
2736:
2716:
2696:
2676:
2656:
2636:
2616:
2601:, then it is a Sylow
2596:
2594:{\displaystyle p^{n}}
2569:
2567:{\displaystyle p^{n}}
2542:
2522:
2488:
2486:{\displaystyle N_{G}}
2449:
2383:
2322:
2320:{\displaystyle n_{p}}
2284:
2282:{\displaystyle n_{p}}
2249:
2223:
2162:
2120:
2063:Given a finite group
2026:Given a finite group
2010:Augustin-Louis Cauchy
1998:
1996:{\displaystyle p^{n}}
1929:Mathematische Annalen
1914:
1847:
1788:
1762:
1712:
1665:
1628:
1594:
1574:
1554:
1530:
1510:
1490:
1469:
1447:
1445:{\displaystyle p^{n}}
1419:
1399:
1379:
1377:{\displaystyle p^{n}}
1352:
1332:
1312:
1291:
1267:
1252:divides the order of
1247:
1227:
1200:
1167:is sometimes written
1162:
1142:
1122:
1102:
1082:
1063:or equivalently, the
1058:
1024:
1004:
984:
964:
929:
494:
469:
432:
7172:. pp. 161–176.
6245:. It follows that Ω
6116:, the normalizer of
5765:
5727:
5717:| = (mod
5019:
4996:ends with precisely
4986:notation the number
4448:
4382:
4338:
4312:
4278:quasi-dihedral group
4218:of order coprime to
3981: − 1
3886:
3827:= 1. So, as before,
3755:= 1 then, as above,
3619:
3498:Example applications
3311:
3046:
2983:
2935:
2903:
2852:
2828:
2808:
2775:
2745:
2725:
2705:
2685:
2665:
2645:
2625:
2605:
2578:
2551:
2531:
2511:
2470:
2393:
2349:
2304:
2266:
2232:
2202:
2129:
2103:
1980:
1864:
1813:
1771:
1725:
1682:
1638:
1617:
1583:
1563:
1543:
1519:
1499:
1479:
1458:
1429:
1408:
1388:
1361:
1341:
1321:
1301:
1280:
1256:
1236:
1216:
1171:
1151:
1131:
1111:
1091:
1071:
1044:
1013:
993:
973:
953:
918:
884:are a collection of
481:
456:
419:
7097:Butler, G. (1991).
6810:J. Automat. Reason.
6394:Frattini's argument
5926: —
5580: —
5441: —
5404:gives rise to sets
4546:stabilizer subgroup
4401:{\displaystyle |G|}
4372: —
4272:classifying finite
4050:Frattini's argument
3865:cyclic permutations
3831:can not be simple.
3653:must divide 3, and
3582:Cyclic group orders
3494:) are all abelian.
2965: —
2741:-subgroup of order
2721:is a subgroup of a
2180: —
2067:and a prime number
2061: —
2030:and a prime number
2024: —
1944: —
1801:, then every Sylow
878:finite group theory
125:Group homomorphisms
35:Algebraic structure
7511:Weisstein, Eric W.
7495:"Sylow p-Subgroup"
7492:Weisstein, Eric W.
7266:Kantor, William M.
7051:10.1007/BF01240818
6993:10.1007/BF00327877
6452:10.1007/BF01442913
6368:permutation groups
6203:). By Theorem 2,
6030:
5940:of a finite group
5920:
5869:. Furthermore, if
5794:
5751:
5704:on Ω, we see that
5682:
5574:
5488:
5439:
5348:
4647:, the right coset
4528:. For a given set
4460:
4422:
4398:
4366:
4350:
4324:
4196:semidirect product
4115:contains not only
3932:
3625:
3370:
3364:
3303:One such subgroup
3160:outer automorphism
3151:
3107:
3075:
3024:
2963:
2947:
2921:
2865:
2834:
2814:
2794:
2758:
2731:
2711:
2691:
2671:
2651:
2631:
2611:
2591:
2564:
2537:
2517:
2483:
2444:
2378:
2317:
2279:
2244:
2218:
2198:can be written as
2174:
2157:
2115:
2055:
2022:
2012:, and is known as
1993:
1938:
1909:
1842:
1783:
1757:
1707:
1660:
1623:
1589:
1569:
1549:
1525:
1505:
1485:
1464:
1442:
1414:
1394:
1374:
1347:
1327:
1307:
1286:
1262:
1242:
1222:
1210:Lagrange's theorem
1195:
1157:
1137:
1117:
1097:
1077:
1056:{\displaystyle p,}
1053:
1019:
999:
979:
959:
924:
890:Peter Ludwig Sylow
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
6747:Gow, Rod (1994).
6211:are conjugate in
6028:
5918:
5761:by definition so
5680:
5572:
5563:and reducing mod
5486:
5476:| (mod
5437:
5346:
5157:
5071:
4420:
4364:
4280:. These rely on
4260:-subgroup of the
3861:alternating group
3443:choices for both
2961:
2837:{\displaystyle G}
2817:{\displaystyle p}
2734:{\displaystyle p}
2714:{\displaystyle H}
2694:{\displaystyle G}
2674:{\displaystyle p}
2654:{\displaystyle H}
2634:{\displaystyle p}
2614:{\displaystyle p}
2540:{\displaystyle p}
2520:{\displaystyle p}
2172:
2053:
2020:
1936:
1870:
1626:{\displaystyle G}
1592:{\displaystyle p}
1572:{\displaystyle p}
1552:{\displaystyle p}
1528:{\displaystyle p}
1508:{\displaystyle p}
1488:{\displaystyle G}
1467:{\displaystyle p}
1417:{\displaystyle G}
1397:{\displaystyle p}
1350:{\displaystyle G}
1330:{\displaystyle p}
1310:{\displaystyle G}
1289:{\displaystyle p}
1265:{\displaystyle G}
1245:{\displaystyle G}
1225:{\displaystyle G}
1178:
1160:{\displaystyle p}
1140:{\displaystyle p}
1120:{\displaystyle G}
1100:{\displaystyle p}
1080:{\displaystyle p}
1022:{\displaystyle G}
1002:{\displaystyle G}
982:{\displaystyle p}
962:{\displaystyle G}
927:{\displaystyle p}
874:
873:
866:
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
7550:
7524:
7523:
7514:"Sylow Theorems"
7505:
7504:
7482:
7475:
7462:"Sylow theorems"
7450:
7411:
7372:
7364:
7325:
7317:
7291:
7276:
7261:
7227:
7212:
7207:
7197:
7167:
7161:
7152:
7086:
7039:
7033:Wielandt, Helmut
7028:
6981:
6970:
6944:
6917:
6891:
6869:Meo, M. (2004).
6865:
6863:
6857:. Archived from
6811:
6806:
6796:
6770:
6755:
6743:
6712:
6690:
6684:
6678:
6672:
6666:
6660:
6654:
6648:
6642:
6636:
6630:
6624:
6618:
6612:
6606:
6600:
6594:
6588:
6582:
6576:
6570:
6564:
6558:
6552:
6546:
6540:
6534:
6533:
6515:
6509:
6508:
6506:
6504:
6498:
6490:math.toronto.edu
6487:
6478:
6472:
6471:
6426:
6404:Maximal subgroup
6271:
6265:
6258:
6187:
6177:
6167:
6146:
6141:
6133:
6115:
6079:
6075:
6050:
6043:
6039:
6023:
6009:
6004:
5996:
5968:
5927:
5924:
5901:
5899:
5893:
5887:
5877:-subgroup, then
5846:
5824:
5814:
5812:
5803:
5801:
5800:
5795:
5793:
5788:
5787:
5778:
5760:
5758:
5757:
5752:
5722:
5716:
5712:
5695:
5664:
5656:
5632:
5613:
5605:
5593:
5581:
5578:
5566:
5562:
5559:
5555:
5553:
5542:
5540:
5534:
5519:
5517:
5508:
5500:
5496:
5481:
5475:
5468:
5462:
5454:
5450:
5446:
5442:
5428:distinct cosets
5410:
5392:
5376:
5361:and no power of
5357:
5355:
5354:
5349:
5347:
5345:
5344:
5343:
5333:
5332:
5318:
5307:
5306:
5296:
5295:
5274:
5273:
5272:
5262:
5261:
5247:
5233:
5232:
5222:
5221:
5200:
5197:
5190:
5189:
5179:
5158:
5156:
5149:
5148:
5138:
5128:
5127:
5117:
5114:
5107:
5106:
5096:
5078:
5077:
5076:
5070:
5069:
5060:
5056:
5055:
5044:
5034:
5026:
4995:
4993:
4980:Kummer's theorem
4973:
4968:
4956:
4953:
4949:
4947:
4936:
4934:
4928:
4922:
4903:
4898:
4885:
4880:
4863:
4859:
4854:
4837:
4832:
4819:
4815:
4806:
4793:
4777:
4744:
4737:
4735:
4729:
4720:
4700:
4695:
4689:
4688:
4675:
4662:
4659:is contained in
4658:
4654:
4650:
4646:
4631:
4627:
4618:
4614:
4610:
4600:
4570:
4534:
4527:
4501:
4494:
4481:
4473:
4469:
4467:
4466:
4461:
4443:
4431:
4407:
4405:
4404:
4399:
4397:
4389:
4373:
4370:
4359:
4357:
4356:
4351:
4333:
4331:
4330:
4325:
4303:
4262:derived subgroup
4238:= {1}, that is,
4159:that normalizes
4068:and two subsets
4040:
4028:
4016:
4000:
3985:
3983:
3982:
3976:
3973:
3945:for every prime
3941:
3939:
3938:
3933:
3931:
3877:Wilson's theorem
3871:Wilson's theorem
3723:is simple, and |
3715:
3634:
3632:
3631:
3626:
3534: <
3518:Groups of order
3461:
3422:
3412:
3379:
3377:
3376:
3371:
3369:
3368:
3361:
3360:
3334:
3333:
3291:
3265:
3251:
3246: + 1)(
3230: − 1)(
3204:, which are all
3203:
3153:By contrast, if
3084:
3082:
3081:
3076:
3074:
3073:
3058:
3057:
3041:
3037:
3033:
3031:
3030:
3025:
3023:
3003:
2995:
2994:
2978:
2974:
2970:
2966:
2956:
2954:
2953:
2948:
2930:
2928:
2927:
2922:
2898:
2894:
2886:
2874:
2872:
2871:
2866:
2864:
2863:
2843:
2841:
2840:
2835:
2823:
2821:
2820:
2815:
2803:
2801:
2800:
2795:
2787:
2786:
2767:
2765:
2764:
2759:
2757:
2756:
2740:
2738:
2737:
2732:
2720:
2718:
2717:
2712:
2700:
2698:
2697:
2692:
2680:
2678:
2677:
2672:
2660:
2658:
2657:
2652:
2640:
2638:
2637:
2632:
2620:
2618:
2617:
2612:
2600:
2598:
2597:
2592:
2590:
2589:
2573:
2571:
2570:
2565:
2563:
2562:
2546:
2544:
2543:
2538:
2526:
2524:
2523:
2518:
2492:
2490:
2489:
2484:
2482:
2481:
2465:
2461:
2457:
2453:
2451:
2450:
2445:
2443:
2429:
2428:
2413:
2405:
2404:
2387:
2385:
2384:
2379:
2377:
2376:
2361:
2360:
2342:
2338:
2330:
2326:
2324:
2323:
2318:
2316:
2315:
2296:
2292:
2288:
2286:
2285:
2280:
2278:
2277:
2261:
2258:does not divide
2257:
2253:
2251:
2250:
2245:
2227:
2225:
2224:
2219:
2214:
2213:
2197:
2193:
2189:
2185:
2181:
2178:
2166:
2164:
2163:
2158:
2144:
2143:
2124:
2122:
2121:
2116:
2098:
2094:
2090:
2086:
2078:
2074:
2070:
2066:
2062:
2059:
2046:
2041:
2037:
2033:
2029:
2025:
2014:Cauchy's theorem
2002:
2000:
1999:
1994:
1992:
1991:
1974:
1968:
1963:
1959:
1952:
1945:
1942:
1922:
1918:
1916:
1915:
1910:
1893:
1879:
1871:
1868:
1859:
1855:
1851:
1849:
1848:
1843:
1841:
1840:
1828:
1820:
1808:
1804:
1800:
1797:does not divide
1796:
1792:
1790:
1789:
1784:
1766:
1764:
1763:
1758:
1753:
1752:
1740:
1732:
1716:
1714:
1713:
1708:
1694:
1693:
1669:
1667:
1666:
1661:
1653:
1645:
1632:
1630:
1629:
1624:
1598:
1596:
1595:
1590:
1578:
1576:
1575:
1570:
1558:
1556:
1555:
1550:
1534:
1532:
1531:
1526:
1514:
1512:
1511:
1506:
1495:, and the Sylow
1494:
1492:
1491:
1486:
1473:
1471:
1470:
1465:
1451:
1449:
1448:
1443:
1441:
1440:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1373:
1372:
1356:
1354:
1353:
1348:
1336:
1334:
1333:
1328:
1316:
1314:
1313:
1308:
1295:
1293:
1292:
1287:
1271:
1269:
1268:
1263:
1251:
1249:
1248:
1243:
1231:
1229:
1228:
1223:
1204:
1202:
1201:
1196:
1185:
1184:
1179:
1176:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1062:
1060:
1059:
1054:
1028:
1026:
1025:
1020:
1008:
1006:
1005:
1000:
988:
986:
985:
980:
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:
7558:
7557:
7553:
7552:
7551:
7549:
7548:
7547:
7528:
7527:
7460:
7457:
7431:
7205:
7125:
7107:Springer-Verlag
7093:
6861:
6804:
6704:
6699:
6694:
6693:
6685:
6681:
6673:
6669:
6661:
6657:
6649:
6645:
6637:
6633:
6625:
6621:
6613:
6609:
6601:
6597:
6589:
6585:
6577:
6573:
6565:
6561:
6553:
6549:
6543:Waterhouse 1980
6541:
6537:
6530:
6516:
6512:
6502:
6500:
6496:
6485:
6479:
6475:
6427:
6423:
6418:
6413:
6389:
6376:polynomial time
6317:
6279:
6274:
6264:
6260:
6259:| ≡ |
6256:
6254:
6248:
6231:
6216:
6197:
6179:
6169:
6166:
6159:
6157:
6140:
6137:
6135:
6130:
6125:
6123:
6109:
6094:
6090:
6086:
6084:
6077:
6072:
6067:
6064:
6045:
6041:
6037:
6026:
6016:
6011:
6003:
6000:
5998:
5994:
5989:
5974:
5965:
5960:
5958:
5949:
5943:
5925:
5922:
5915:
5895:
5894:| = |
5889:
5888:| = |
5880:
5878:
5845:
5841:
5838:
5823:
5816:
5811:
5807:
5805:
5789:
5783:
5779:
5774:
5766:
5763:
5762:
5728:
5725:
5724:
5714:
5713:| ≡ |
5711:
5707:
5705:
5693:
5678:
5662:
5654:
5641:), that is, if
5630:
5611:
5603:
5591:
5579:
5576:
5569:
5564:
5560:
5557:
5552:
5549:
5546:
5544:
5538:
5536:
5532:
5525:
5515:
5510:
5507:
5504:
5502:
5498:
5494:
5491:
5484:
5479:
5474:
5470:
5469:| ≡ |
5466:
5464:
5460:
5458:
5452:
5448:
5444:
5440:
5434:
5419:
5418:
5414:
5408:
5405:
5382:
5374:
5371:
5366:
5328:
5324:
5323:
5319:
5314:
5291:
5287:
5280:
5276:
5275:
5257:
5253:
5252:
5248:
5243:
5217:
5213:
5206:
5202:
5201:
5199:
5185:
5181:
5180:
5169:
5144:
5140:
5139:
5123:
5119:
5118:
5116:
5102:
5098:
5097:
5086:
5072:
5065:
5061:
5051:
5047:
5046:
5040:
5039:
5038:
5030:
5022:
5020:
5017:
5016:
4992:
4989:
4987:
4982:(since in base
4966:
4963:
4958:
4954:
4951:
4946:
4943:
4940:
4938:
4932:
4930:
4921:
4920:
4916:
4913:
4910:
4905:
4897:
4896:
4892:
4889:
4887:
4879:
4876:
4873:
4870:
4865:
4861:
4853:
4850:
4847:
4844:
4839:
4831:
4830:
4826:
4823:
4821:
4817:
4805:
4802:
4799:
4792:
4789:
4786:
4783:
4776:
4775:
4771:
4768:
4765:
4760:
4753:
4742:
4739:
4734:
4731:
4730:| = |
4728:
4725:
4722:
4719:
4718:
4714:
4711:
4709:
4699:
4694:
4691:
4690:| ≤ |
4686:
4685:
4684:
4680:
4677:
4676:| = |
4674:
4673:
4669:
4666:
4664:
4660:
4656:
4655:
4652:
4648:
4645:
4641:
4637:
4634:
4632:
4629:
4625:
4619:
4616:
4612:
4608:
4605:
4598:
4594:
4590:
4586:
4582:
4577:
4574:
4568:
4564:
4560:
4556:
4552:
4548:
4543:
4542:
4538:
4532:
4529:
4525:
4521:
4517:
4514:
4510:
4506:
4503:
4499:
4496:
4493:
4489:
4486:
4479:
4471:
4449:
4446:
4445:
4430:
4427:
4425:
4418:
4393:
4385:
4383:
4380:
4379:
4377:
4374:A finite group
4371:
4368:
4339:
4336:
4335:
4313:
4310:
4309:
4301:
4290:
4188:
4097:
4056:states that if
4047:
4030:
4018:
4010:
4002:
3994:
3991:
3977:
3974:
3971:
3970:
3968:
3965:
3954:
3916:
3887:
3884:
3883:
3873:
3858:
3851:
3844:
3826:
3819:
3812:
3802:Next, suppose |
3790:
3783:
3776:
3754:
3747:
3740:
3733:
3713:
3686:
3659:
3652:
3641:
3620:
3617:
3616:
3615:
3608:
3584:
3509:normal subgroup
3500:
3493:
3484:
3452:
3418:
3407:
3405:
3396:
3363:
3362:
3353:
3349:
3347:
3341:
3340:
3335:
3326:
3322:
3315:
3314:
3312:
3309:
3308:
3282:
3273:
3267:
3266:, the order of
3253:
3250: − 1)
3225:
3223:
3214:
3208:. The order of
3194:
3184:
3175:
3148:
3126:
3104:
3092:
3087:
3069:
3065:
3053:
3049:
3047:
3044:
3043:
3039:
3035:
3019:
2999:
2990:
2986:
2984:
2981:
2980:
2976:
2972:
2968:
2964:
2936:
2933:
2932:
2904:
2901:
2900:
2896:
2892:
2884:
2881:
2859:
2855:
2853:
2850:
2849:
2846:normal subgroup
2829:
2826:
2825:
2809:
2806:
2805:
2782:
2778:
2776:
2773:
2772:
2752:
2748:
2746:
2743:
2742:
2726:
2723:
2722:
2706:
2703:
2702:
2686:
2683:
2682:
2666:
2663:
2662:
2646:
2643:
2642:
2626:
2623:
2622:
2606:
2603:
2602:
2585:
2581:
2579:
2576:
2575:
2558:
2554:
2552:
2549:
2548:
2532:
2529:
2528:
2512:
2509:
2508:
2505:
2500:
2477:
2473:
2471:
2468:
2467:
2463:
2459:
2455:
2439:
2424:
2420:
2409:
2400:
2396:
2394:
2391:
2390:
2372:
2368:
2356:
2352:
2350:
2347:
2346:
2340:
2336:
2331:, which is the
2328:
2311:
2307:
2305:
2302:
2301:
2294:
2290:
2273:
2269:
2267:
2264:
2263:
2259:
2255:
2233:
2230:
2229:
2209:
2205:
2203:
2200:
2199:
2195:
2191:
2187:
2183:
2179:
2176:
2169:
2136:
2132:
2130:
2127:
2126:
2104:
2101:
2100:
2096:
2092:
2088:
2084:
2076:
2072:
2068:
2064:
2060:
2057:
2050:
2044:
2039:
2035:
2031:
2027:
2023:
2006:
1987:
1983:
1981:
1978:
1977:
1972:
1966:
1961:
1957:
1950:
1943:
1940:
1920:
1889:
1875:
1867:
1865:
1862:
1861:
1857:
1853:
1836:
1832:
1824:
1816:
1814:
1811:
1810:
1806:
1802:
1798:
1794:
1772:
1769:
1768:
1748:
1744:
1736:
1728:
1726:
1723:
1722:
1689:
1685:
1683:
1680:
1679:
1676:
1649:
1641:
1639:
1636:
1635:
1618:
1615:
1614:
1610:
1605:
1584:
1581:
1580:
1564:
1561:
1560:
1544:
1541:
1540:
1520:
1517:
1516:
1500:
1497:
1496:
1480:
1477:
1476:
1459:
1456:
1455:
1436:
1432:
1430:
1427:
1426:
1409:
1406:
1405:
1389:
1386:
1385:
1368:
1364:
1362:
1359:
1358:
1342:
1339:
1338:
1322:
1319:
1318:
1302:
1299:
1298:
1281:
1278:
1277:
1257:
1254:
1253:
1237:
1234:
1233:
1217:
1214:
1213:
1180:
1175:
1174:
1172:
1169:
1168:
1152:
1149:
1148:
1132:
1129:
1128:
1112:
1109:
1108:
1092:
1089:
1088:
1072:
1069:
1068:
1045:
1042:
1041:
1014:
1011:
1010:
994:
991:
990:
974:
971:
970:
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:
7556:
7546:
7545:
7540:
7526:
7525:
7506:
7487:
7476:
7456:
7455:External links
7453:
7452:
7451:
7429:
7412:
7378:(4): 523–563.
7365:
7318:
7282:(3): 359–394.
7262:
7225:10.1.1.74.3690
7218:(4): 478–514.
7198:
7166:SIAM-AMS Proc.
7153:
7123:
7092:
7089:
7088:
7087:
7045:(1): 401–402.
7029:
6987:(3): 279–290.
6971:
6928:Historia Math.
6918:
6882:(2): 196–221.
6876:Historia Math.
6866:
6864:on 2006-01-03.
6817:(3): 235–264.
6797:
6744:
6714:(in Italian).
6703:
6700:
6698:
6695:
6692:
6691:
6679:
6667:
6655:
6643:
6631:
6619:
6607:
6595:
6583:
6571:
6559:
6547:
6535:
6528:
6510:
6473:
6446:(4): 584–594.
6420:
6419:
6417:
6414:
6412:
6411:
6406:
6401:
6396:
6390:
6388:
6385:
6315:
6278:
6275:
6262:
6246:
6229:
6214:
6195:
6164:
6155:
6138:
6128:
6121:
6107:
6092:
6088:
6080:
6070:
6062:
6036:-subgroups of
6027:
6018:≡ 1 (mod
6014:
6001:
5992:
5972:
5963:
5956:
5955:-subgroups of
5947:
5941:
5916:
5858:and therefore
5843:
5839:
5821:
5809:
5792:
5786:
5782:
5777:
5773:
5770:
5750:
5747:
5744:
5741:
5738:
5735:
5732:
5709:
5679:
5653:-subgroups of
5629:-subgroups of
5570:
5550:
5547:
5543:as the sum of
5523:
5513:
5505:
5492:
5485:
5477:
5472:
5456:
5435:
5416:
5415:
5412:
5406:
5380:
5369:
5359:
5358:
5342:
5339:
5336:
5331:
5327:
5322:
5317:
5313:
5310:
5305:
5302:
5299:
5294:
5290:
5286:
5283:
5279:
5271:
5268:
5265:
5260:
5256:
5251:
5246:
5242:
5239:
5236:
5231:
5228:
5225:
5220:
5216:
5212:
5209:
5205:
5196:
5193:
5188:
5184:
5178:
5175:
5172:
5168:
5164:
5161:
5155:
5152:
5147:
5143:
5137:
5134:
5131:
5126:
5122:
5113:
5110:
5105:
5101:
5095:
5092:
5089:
5085:
5081:
5075:
5068:
5064:
5059:
5054:
5050:
5043:
5037:
5033:
5029:
5025:
4990:
4961:
4944:
4941:
4937:is the sum of
4918:
4917:
4914:
4908:
4894:
4893:
4890:
4877:
4874:
4868:
4851:
4848:
4842:
4828:
4827:
4824:
4803:
4797:
4790:
4787:
4781:
4773:
4772:
4769:
4763:
4751:
4740:
4732:
4726:
4723:
4716:
4715:
4712:
4697:
4692:
4682:
4681:
4678:
4671:
4670:
4667:
4651:
4643:
4639:
4635:
4628:
4615:
4606:
4596:
4592:
4588:
4584:
4575:
4572:
4566:
4562:
4558:
4554:
4550:
4540:
4539:
4536:
4530:
4523:
4519:
4515:
4512:
4508:
4504:
4497:
4491:
4487:
4459:
4456:
4453:
4428:
4419:
4396:
4392:
4388:
4375:
4362:
4349:
4346:
4343:
4323:
4320:
4317:
4289:
4286:
4186:
4095:
4076:normalized by
4046:
4045:Fusion results
4043:
4011:≡ 1 (mod
4006:
3989:
3963:
3952:
3943:
3942:
3930:
3927:
3923:
3920:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3872:
3869:
3856:
3849:
3842:
3824:
3817:
3810:
3788:
3781:
3774:
3752:
3745:
3738:
3731:
3685:
3682:
3680:isomorphism).
3657:
3650:
3639:
3624:
3613:
3606:
3583:
3580:
3579:
3578:
3563:
3560:
3542:
3539:
3516:
3499:
3496:
3489:
3482:
3478:-subgroups of
3411: − 1
3401:
3392:
3386:primitive root
3367:
3359:
3356:
3352:
3348:
3346:
3343:
3342:
3339:
3336:
3332:
3329:
3325:
3321:
3320:
3318:
3296:-subgroups is
3278:
3271:
3264: + 1
3219:
3212:
3180:
3173:
3146:
3121:
3111:dihedral group
3102:
3091:
3088:
3072:
3068:
3064:
3061:
3056:
3052:
3022:
3018:
3015:
3012:
3009:
3006:
3002:
2998:
2993:
2989:
2959:
2946:
2943:
2940:
2920:
2917:
2914:
2911:
2908:
2880:
2877:
2862:
2858:
2833:
2813:
2793:
2790:
2785:
2781:
2755:
2751:
2730:
2710:
2690:
2670:
2650:
2630:
2610:
2588:
2584:
2561:
2557:
2536:
2516:
2504:
2501:
2499:
2498:
2480:
2476:
2442:
2438:
2435:
2432:
2427:
2423:
2419:
2416:
2412:
2408:
2403:
2399:
2388:
2375:
2371:
2367:
2364:
2359:
2355:
2344:
2314:
2310:
2293:-subgroups of
2276:
2272:
2243:
2240:
2237:
2217:
2212:
2208:
2170:
2156:
2153:
2150:
2147:
2142:
2139:
2135:
2114:
2111:
2108:
2095:-subgroups of
2075:-subgroups of
2051:
2018:
1990:
1986:
1934:
1908:
1905:
1902:
1899:
1896:
1892:
1888:
1885:
1882:
1878:
1874:
1839:
1835:
1831:
1827:
1823:
1819:
1782:
1779:
1776:
1756:
1751:
1747:
1743:
1739:
1735:
1731:
1706:
1703:
1700:
1697:
1692:
1688:
1675:
1672:
1659:
1656:
1652:
1648:
1644:
1622:
1609:
1606:
1604:
1601:
1588:
1568:
1548:
1524:
1504:
1484:
1463:
1439:
1435:
1413:
1393:
1371:
1367:
1346:
1326:
1306:
1285:
1261:
1241:
1221:
1194:
1191:
1188:
1183:
1156:
1136:
1116:
1096:
1076:
1052:
1049:
1018:
998:
978:
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:
7555:
7544:
7541:
7539:
7536:
7535:
7533:
7521:
7520:
7515:
7512:
7507:
7502:
7501:
7496:
7493:
7488:
7485:
7481:
7477:
7473:
7469:
7468:
7463:
7459:
7458:
7448:
7444:
7440:
7436:
7432:
7430:9780521661034
7426:
7422:
7418:
7413:
7409:
7405:
7401:
7397:
7393:
7389:
7385:
7381:
7377:
7373:
7371:J. Algorithms
7366:
7362:
7358:
7354:
7350:
7346:
7342:
7338:
7334:
7330:
7326:
7324:J. Algorithms
7319:
7315:
7311:
7307:
7303:
7299:
7295:
7290:
7285:
7281:
7277:
7271:
7267:
7263:
7259:
7255:
7251:
7247:
7243:
7239:
7235:
7231:
7226:
7221:
7217:
7213:
7211:J. Algorithms
7204:
7199:
7195:
7191:
7187:
7183:
7179:
7175:
7171:
7163:
7154:
7150:
7146:
7142:
7138:
7134:
7130:
7126:
7124:9783540549550
7120:
7116:
7112:
7108:
7104:
7100:
7095:
7094:
7084:
7080:
7076:
7072:
7068:
7064:
7060:
7056:
7052:
7048:
7044:
7041:(in German).
7040:
7034:
7030:
7026:
7022:
7018:
7014:
7010:
7006:
7002:
6998:
6994:
6990:
6986:
6982:
6976:
6972:
6968:
6964:
6960:
6956:
6952:
6948:
6943:
6938:
6934:
6931:(in German).
6930:
6929:
6924:
6919:
6915:
6911:
6907:
6903:
6899:
6895:
6890:
6885:
6881:
6878:
6877:
6872:
6867:
6860:
6856:
6852:
6848:
6844:
6840:
6836:
6832:
6828:
6824:
6820:
6816:
6812:
6803:
6798:
6794:
6790:
6786:
6782:
6778:
6774:
6769:
6764:
6761:(33): 55–63.
6760:
6756:
6750:
6745:
6741:
6737:
6733:
6729:
6725:
6721:
6717:
6713:
6706:
6705:
6688:
6683:
6676:
6671:
6664:
6659:
6652:
6647:
6640:
6635:
6628:
6623:
6617:, Chapter 16.
6616:
6611:
6604:
6603:Wielandt 1959
6599:
6592:
6587:
6580:
6575:
6568:
6563:
6556:
6555:Scharlau 1988
6551:
6544:
6539:
6531:
6529:9788178089973
6525:
6521:
6514:
6495:
6491:
6484:
6477:
6469:
6465:
6461:
6457:
6453:
6449:
6445:
6442:(in French).
6441:
6440:
6435:
6431:
6425:
6421:
6410:
6407:
6405:
6402:
6400:
6399:Hall subgroup
6397:
6395:
6392:
6391:
6384:
6382:
6377:
6373:
6369:
6364:
6362:
6358:
6354:
6350:
6346:
6342:
6338:
6334:
6330:
6326:
6322:
6318:
6311:
6307:
6303:
6300:-subgroup of
6299:
6295:
6291:
6286:
6284:
6273:
6269:
6252:
6244:
6240:
6236:
6232:
6226:is normal in
6225:
6221:
6217:
6210:
6206:
6202:
6198:
6191:
6186:
6182:
6176:
6172:
6162:
6153:
6148:
6145:
6131:
6119:
6113:
6106:
6102:
6098:
6083:
6073:
6065:
6058:
6054:
6048:
6035:
6025:
6021:
6017:
6008:
5995:
5987:
5983:
5979:
5975:
5966:
5954:
5950:
5939:
5935:
5931:
5914:
5912:
5908:
5905:
5898:
5892:
5886:
5883:
5876:
5872:
5868:
5864:
5861:
5857:
5853:
5850:
5836:
5832:
5828:
5819:
5784:
5771:
5768:
5745:
5742:
5739:
5733:
5730:
5720:
5703:
5699:
5691:
5687:
5677:
5675:
5671:
5668:
5660:
5652:
5648:
5644:
5640:
5636:
5628:
5624:
5620:
5617:
5609:
5602:-subgroup of
5601:
5597:
5590:-subgroup of
5589:
5585:
5568:
5530:
5526:
5516:
5509:|/|
5497:not fixed by
5483:
5433:
5431:
5427:
5423:
5403:
5399:
5394:
5391:
5387:
5383:
5372:
5364:
5337:
5329:
5325:
5320:
5315:
5311:
5308:
5300:
5292:
5288:
5284:
5281:
5277:
5266:
5258:
5254:
5249:
5244:
5240:
5237:
5234:
5226:
5218:
5214:
5210:
5207:
5203:
5194:
5191:
5186:
5182:
5176:
5173:
5170:
5166:
5162:
5159:
5153:
5150:
5145:
5141:
5135:
5132:
5129:
5124:
5120:
5111:
5108:
5103:
5099:
5093:
5090:
5087:
5083:
5079:
5066:
5062:
5057:
5052:
5048:
5035:
5015:
5014:
5013:
5011:
5007:
5003:
4999:
4985:
4981:
4977:
4972:
4964:
4926:
4923:|) <
4911:
4902:
4888:0 < |
4884:
4881:|) >
4871:
4858:
4845:
4836:
4814:
4810:
4800:
4784:
4766:
4758:
4754:
4748:
4721:| |
4707:
4702:
4663:; therefore,
4623:
4602:
4581:
4547:
4484:
4477:
4457:
4454:
4451:
4442:
4439:
4435:
4417:
4415:
4411:
4390:
4361:
4347:
4344:
4341:
4321:
4318:
4315:
4307:
4299:
4294:
4285:
4283:
4282:J. L. Alperin
4279:
4275:
4274:simple groups
4271:
4267:
4263:
4259:
4255:
4250:
4248:
4246:
4241:
4237:
4233:
4229:
4225:
4221:
4217:
4213:
4209:
4205:
4201:
4197:
4193:
4189:
4182:
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4126:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4091:
4087:
4083:
4079:
4075:
4071:
4067:
4063:
4059:
4055:
4051:
4042:
4038:
4034:
4026:
4022:
4014:
4009:
4005:
3998:
3992:
3980:
3966:
3959:
3955:
3948:
3925:
3921:
3913:
3910:
3907:
3904:
3898:
3895:
3892:
3882:
3881:
3880:
3878:
3868:
3866:
3862:
3855:
3848:
3841:
3837:
3832:
3830:
3823:
3816:
3809:
3805:
3800:
3798:
3794:
3787:
3780:
3773:
3768:
3766:
3762:
3758:
3751:
3744:
3737:
3730:
3727:| = 30, then
3726:
3722:
3717:
3714:(= 2 · 3 · 5)
3711:
3708:, then it is
3707:
3703:
3701:
3695:
3691:
3681:
3679:
3675:
3671:
3667:
3663:
3656:
3649:
3645:
3638:
3622:
3612:
3605:
3601:
3597:
3593:
3589:
3576:
3572:
3568:
3564:
3561:
3558:
3554:
3550:
3547:
3543:
3540:
3537:
3533:
3529:
3525:
3521:
3517:
3514:
3513:
3512:
3510:
3506:
3495:
3492:
3488:
3481:
3477:
3473:
3469:
3465:
3462:. This means
3460:
3456:
3450:
3446:
3442:
3438:
3434:
3430:
3426:
3421:
3416:
3410:
3404:
3400:
3395:
3391:
3387:
3383:
3365:
3357:
3354:
3350:
3344:
3337:
3330:
3327:
3323:
3316:
3306:
3301:
3299:
3295:
3289:
3286:
3281:
3277:
3270:
3263:
3260:
3257: =
3256:
3249:
3245:
3241:
3237:
3234: −
3233:
3229:
3222:
3218:
3211:
3207:
3201:
3197:
3192:
3188:
3183:
3179:
3172:
3167:
3165:
3161:
3156:
3145:
3140:
3136:
3134:
3130:
3125:
3120:
3116:
3112:
3101:
3096:
3086:
3070:
3062:
3059:
3054:
3050:
3013:
3007:
3004:
2996:
2991:
2987:
2975:-subgroup of
2958:
2944:
2941:
2938:
2915:
2909:
2906:
2890:
2876:
2860:
2856:
2847:
2831:
2824:-subgroup of
2811:
2791:
2788:
2783:
2779:
2769:
2753:
2749:
2728:
2708:
2688:
2681:-subgroup of
2668:
2648:
2628:
2608:
2586:
2582:
2559:
2555:
2534:
2514:
2496:
2478:
2474:
2462:-subgroup of
2458:is any Sylow
2433:
2425:
2421:
2417:
2414:
2406:
2401:
2397:
2389:
2373:
2365:
2362:
2357:
2353:
2345:
2339:-subgroup in
2335:of the Sylow
2334:
2312:
2308:
2300:
2299:
2298:
2274:
2270:
2241:
2238:
2235:
2215:
2210:
2206:
2168:
2154:
2151:
2148:
2145:
2140:
2137:
2133:
2112:
2109:
2106:
2082:
2049:
2047:
2017:
2015:
2011:
2005:
2003:
1988:
1984:
1970:
1956:
1949:
1933:
1931:
1930:
1924:
1906:
1903:
1897:
1894:
1886:
1883:
1880:
1837:
1833:
1829:
1821:
1780:
1777:
1774:
1754:
1749:
1745:
1741:
1733:
1720:
1701:
1695:
1690:
1686:
1671:
1657:
1654:
1646:
1620:
1600:
1586:
1566:
1546:
1538:
1522:
1502:
1482:
1475:-subgroup of
1474:
1461:
1452:
1437:
1433:
1411:
1391:
1369:
1365:
1344:
1337:-subgroup of
1324:
1304:
1296:
1283:
1275:
1259:
1239:
1219:
1211:
1206:
1189:
1181:
1154:
1134:
1114:
1107:-subgroup of
1094:
1074:
1066:
1050:
1047:
1039:
1035:
1033:
1016:
996:
989:-subgroup of
976:
969:is a maximal
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:
7517:
7498:
7486:at Wikibooks
7465:
7416:
7375:
7369:
7328:
7322:
7279:
7273:
7215:
7209:
7157:
7098:
7042:
7036:
6984:
6978:
6935:(1): 40–52.
6932:
6926:
6879:
6874:
6859:the original
6814:
6808:
6758:
6752:
6718:(1): 29–75.
6715:
6709:
6682:
6670:
6658:
6651:Kantor 1985b
6646:
6639:Kantor 1985a
6634:
6622:
6610:
6598:
6586:
6574:
6562:
6550:
6538:
6519:
6513:
6501:. Retrieved
6489:
6476:
6443:
6437:
6424:
6371:
6365:
6356:
6352:
6348:
6344:
6340:
6336:
6332:
6328:
6324:
6320:
6313:
6309:
6305:
6301:
6297:
6293:
6289:
6287:
6280:
6267:
6250:
6242:
6238:
6234:
6227:
6223:
6219:
6212:
6208:
6204:
6200:
6193:
6189:
6184:
6180:
6174:
6170:
6160:
6151:
6149:
6143:
6126:
6117:
6111:
6104:
6100:
6096:
6085:is given by
6081:
6068:
6060:
6056:
6052:
6046:
6033:
6031:
6019:
6012:
6006:
5990:
5985:
5977:
5970:
5961:
5959:. Then (a)
5952:
5945:
5937:
5933:
5929:
5917:
5910:
5906:
5903:
5896:
5890:
5884:
5881:
5874:
5870:
5866:
5862:
5859:
5855:
5851:
5848:
5834:
5830:
5826:
5825:. With this
5817:
5718:
5701:
5697:
5689:
5683:
5673:
5669:
5666:
5658:
5650:
5646:
5642:
5626:
5622:
5618:
5615:
5607:
5599:
5595:
5587:
5583:
5571:
5527:denotes the
5521:
5511:
5490:Any element
5489:
5447:be a finite
5436:
5429:
5425:
5421:
5401:
5397:
5395:
5389:
5385:
5378:
5367:
5362:
5360:
5009:
5005:
5001:
4997:
4983:
4975:
4970:
4959:
4924:
4906:
4900:
4899:| <
4882:
4866:
4856:
4840:
4834:
4812:
4808:
4795:
4779:
4761:
4756:
4749:
4703:
4621:
4603:
4475:
4440:
4437:
4433:
4423:
4413:
4409:
4378:whose order
4363:
4305:
4298:group action
4295:
4291:
4257:
4251:
4244:
4239:
4235:
4231:
4227:
4223:
4219:
4215:
4211:
4207:
4203:
4199:
4191:
4184:
4180:
4176:
4172:
4168:
4164:
4160:
4156:
4152:
4148:
4144:
4140:
4136:
4132:
4128:
4124:
4120:
4116:
4112:
4108:
4104:
4100:
4093:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4048:
4036:
4032:
4024:
4020:
4012:
4007:
4003:
3996:
3987:
3978:
3961:
3957:
3950:
3946:
3944:
3879:states that
3874:
3853:
3846:
3839:
3835:
3833:
3828:
3821:
3814:
3807:
3803:
3801:
3796:
3792:
3785:
3778:
3771:
3769:
3764:
3760:
3756:
3749:
3742:
3735:
3728:
3724:
3720:
3718:
3706:prime powers
3699:
3692:that is not
3690:simple group
3687:
3674:cyclic group
3668:must be the
3665:
3654:
3647:
3636:
3610:
3603:
3599:
3595:
3591:
3587:
3585:
3574:
3570:
3566:
3556:
3552:
3548:
3545:
3535:
3531:
3530:primes with
3527:
3523:
3519:
3501:
3490:
3486:
3479:
3475:
3471:
3467:
3463:
3458:
3454:
3448:
3444:
3440:
3439:. There are
3436:
3432:
3428:
3424:
3419:
3414:
3408:
3402:
3398:
3393:
3389:
3381:
3304:
3302:
3297:
3293:
3287:
3284:
3279:
3275:
3268:
3261:
3258:
3254:
3247:
3243:
3239:
3235:
3231:
3227:
3220:
3216:
3209:
3199:
3195:
3190:
3186:
3181:
3177:
3170:
3168:
3163:
3154:
3152:
3143:
3132:
3128:
3123:
3118:
3114:
3108:
3099:
2960:
2888:
2882:
2770:
2527:every Sylow
2506:
2503:Consequences
2493:denotes the
2171:
2071:, all Sylow
2052:
2043:
2019:
2007:
1976:
1955:multiplicity
1948:prime factor
1935:
1927:
1925:
1677:
1611:
1454:
1425:
1276:
1274:prime factor
1207:
1031:
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:
7331:(1): 1–17.
7038:Arch. Math.
6687:Seress 2003
6663:Kantor 1990
6627:Cannon 1971
6615:Butler 1991
6355:but not in
6237:), so then
6051:be a Sylow
5873:is a Sylow
5598:is a Sylow
5455:, and let Ω
4163:, and then
4151:normalizes
3956:of Sylow's
3777:= 6, since
3698:Burnside's
3466:is a Sylow
2971:is a Sylow
1975:, of order
1860:-group and
1852:. That is,
1453:is a Sylow
942:(sometimes
846:introducing
530:Topological
369:alternating
7532:Categories
7447:1028.20002
7408:0731.20005
7361:0642.20019
7314:0573.20022
7258:0604.20001
7194:0253.20027
7149:0785.20001
7091:Algorithms
7083:0092.02403
7025:0436.01006
6967:0637.01006
6914:1065.01009
6855:0943.68149
6793:0829.01011
6740:0721.01008
6697:References
6460:04.0056.02
6439:Math. Ann.
6343:-subgroup
6277:Algorithms
6010:, and (c)
5982:normalizer
5936:-subgroup
5829:, we have
5813:| ≠ 0
5649:are Sylow
5639:isomorphic
5614:such that
5529:stabilizer
5411:for which
5377:|) =
4969:|) =
4855:|) =
4807:|) =
4794:|) =
4778:|) +
4759:, one has
4611:for which
4444:such that
4247:-nilpotent
4206:-subgroup
4175:, so that
4064:-subgroup
3577:is simple.
2495:normalizer
2091:are Sylow
1946:For every
1809:has order
1805:-subgroup
1719:isomorphic
1608:Motivation
1029:that is a
829:references
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
7519:MathWorld
7500:MathWorld
7472:EMS Press
7392:0196-6774
7345:0196-6774
7298:1090-2724
7268:(1985b).
7242:0196-6774
7220:CiteSeerX
7178:0160-7634
7075:119816392
7059:0003-9268
7017:123685226
7001:0003-9519
6951:0315-0860
6898:0315-0860
6831:0168-7433
6777:0791-5578
6724:0392-4432
6468:121928336
6430:Sylow, L.
6059:has size
5980:) is the
5781:Ω
5772:∤
5734:∤
5635:conjugate
5400:of order
5326:ν
5309:−
5289:ν
5285:−
5255:ν
5238:−
5215:ν
5211:−
5192:−
5167:∏
5151:−
5133:−
5109:−
5084:∏
5028:Ω
4833:| =
4738:for each
4696:| =
4455:∤
4432:| =
4345:∤
4319:∣
4155:for some
4147:. Hence
4119:but also
4017:. Hence,
3911:−
3908:≡
3896:−
3845:= 10 and
3770:As well,
3623:∣
3562:Example-3
3541:Example-2
3515:Example-1
3457:| =
3451:, making
3283:) =
3185:), where
3060:≡
3008:
2942:⊂
2910:
2363:≡
2138:−
2110:∈
2081:conjugate
2021:Corollary
1969:-subgroup
1696:
1674:Statement
1537:conjugate
1357:of order
940:-subgroup
896:of fixed
894:subgroups
701:Conformal
589:Euclidean
196:nilpotent
7543:P-groups
6591:Meo 2004
6579:Gow 1994
6494:Archived
6432:(1872).
6387:See also
6188:so that
6178:for all
6150:Now let
6136:= |
6124:. Thus,
6040:and let
5997:divides
5902:so that
5837:for all
5696:and let
5463:. Then
4929:. Since
4904:implies
4864:one has
4708:we have
4578:for its
4544:for its
4535:, write
4474:of size
3875:Part of
3710:solvable
3252:. Since
3090:Examples
2454:, where
2327:divides
2228:, where
1603: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
7474:, 2001
7439:1970241
7400:1079450
7353:0925595
7306:0805654
7250:0813589
7186:0367027
7133:1225579
7067:0147529
7009:0575718
6959:0931678
6906:2055642
6847:1449341
6839:1721912
6785:1313412
6732:1096350
6409:p-group
6142:|/
6095:|
6005:|/
5988:), (b)
5969:(where
5919:Theorem
5723:. Now
5573:Theorem
5520:(where
5373:(|
4965:(|
4912:(|
4872:(|
4846:(|
4801:(|
4785:(|
4767:(|
4704:By the
4591:|
4557:|
4518:|
4365:Theorem
4123:(since
4080:, then
3984:
3969:
3702:theorem
3662:coprime
3384:is any
3290:′
3206:abelian
3113:of the
2962:Theorem
2701:, then
2661:is any
2173:Theorem
2054:Theorem
1937:Theorem
842:improve
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
7445:
7437:
7427:
7406:
7398:
7390:
7359:
7351:
7343:
7312:
7304:
7296:
7256:
7248:
7240:
7222:
7192:
7184:
7176:
7147:
7141:395110
7139:
7131:
7121:
7081:
7073:
7065:
7057:
7023:
7015:
7007:
6999:
6965:
6957:
6949:
6912:
6904:
6896:
6853:
6845:
6837:
6829:
6791:
6783:
6775:
6738:
6730:
6722:
6702:Proofs
6526:
6466:
6458:
6255:|
5999:|
5944:. Let
5921:
5900:|
5879:|
5806:|
5706:|
5686:cosets
5575:
5554:|
5545:|
5541:|
5537:|
5518:|
5503:|
5465:|
4994:|
4988:|
4948:|
4939:|
4935:|
4931:|
4822:|
4736:|
4710:|
4665:|
4601:in Ω.
4426:|
4367:
4029:. So,
3993:, ie.
3859:, the
3694:cyclic
3644:normal
3635:5 and
3505:simple
3431:. So,
3127:. For
3117:-gon,
3042:, and
2979:, and
2262:. Let
2175:
2056:
1939:
1793:where
1535:) are
1034:-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
7206:(PDF)
7137:S2CID
7071:S2CID
7013:S2CID
6862:(PDF)
6843:S2CID
6805:(PDF)
6503:8 May
6497:(PDF)
6486:(PDF)
6464:S2CID
6416:Notes
6323:) of
6296:is a
6103:} =
6029:Proof
5847:, so
5681:Proof
5665:with
5586:is a
5487:Proof
5438:Lemma
4820:with
4580:orbit
4421:Proof
4198:: if
3678:up to
3238:) = (
3224:) is
2844:is a
2333:index
2125:with
1953:with
1856:is a
1767:with
1065:order
1038:power
898:order
730:Sp(∞)
727:SU(∞)
140:image
7425:ISBN
7388:ISSN
7341:ISSN
7294:ISSN
7238:ISSN
7174:ISSN
7119:ISBN
7055:ISSN
6997:ISSN
6947:ISSN
6894:ISSN
6827:ISSN
6773:ISSN
6759:0033
6720:ISSN
6524:ISBN
6505:2021
6207:and
5928:Let
5645:and
5633:are
5594:and
5443:Let
5388:) =
4886:(as
4620:has
4571:and
4511:= {
4495:and
4483:acts
4424:Let
4230:and
4183:are
4179:and
4135:and
4088:are
4084:and
4072:and
3555:and
3526:and
3447:and
3388:of
3189:and
2466:and
2254:and
2239:>
2182:Let
2087:and
2079:are
1778:>
934:, a
724:O(∞)
713:Loop
532:and
7443:Zbl
7404:Zbl
7380:doi
7357:Zbl
7333:doi
7310:Zbl
7284:doi
7254:Zbl
7230:doi
7190:Zbl
7170:AMS
7145:Zbl
7111:doi
7079:Zbl
7047:doi
7021:Zbl
6989:doi
6963:Zbl
6937:doi
6910:Zbl
6884:doi
6851:Zbl
6819:doi
6789:Zbl
6763:doi
6736:Zbl
6456:JFM
6448:doi
6366:In
6327:in
6249:= {
6175:xQx
6163:∈ Ω
6120:in
6097:gPg
6049:∈ Ω
5984:of
5923:(3)
5852:HgP
5831:hgP
5820:∈ Ω
5692:in
5688:of
5661:in
5610:in
5582:If
5577:(2)
5495:∈ Ω
5409:∈ Ω
4743:∈ Ω
4609:∈ Ω
4533:∈ Ω
4500:∈ Ω
4369:(1)
4242:is
3967:is
3922:mod
3719:If
3700:p q
3511:.
3423:or
3406:is
3142:In
3098:In
3067:mod
2967:If
2370:mod
2177:(3)
2058:(2)
2042:in
1971:of
1941:(1)
1869:gcd
1687:Syl
1599:).
1177:Syl
1040:of
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
7534::
7516:.
7497:.
7470:,
7464:,
7441:.
7435:MR
7433:.
7423:.
7402:.
7396:MR
7394:.
7386:.
7376:11
7374:.
7355:.
7349:MR
7347:.
7339:.
7327:.
7308:.
7302:MR
7300:.
7292:.
7280:30
7278:.
7272:.
7252:.
7246:MR
7244:.
7236:.
7228:.
7214:.
7208:.
7188:.
7182:MR
7180:.
7164:.
7143:.
7135:.
7129:MR
7127:.
7117:.
7109:.
7101:.
7077:.
7069:.
7063:MR
7061:.
7053:.
7043:10
7019:.
7011:.
7005:MR
7003:.
6995:.
6985:21
6983:.
6961:.
6955:MR
6953:.
6945:.
6933:15
6925:.
6908:.
6902:MR
6900:.
6892:.
6880:31
6873:.
6849:.
6841:.
6835:MR
6833:.
6825:.
6815:23
6813:.
6807:.
6787:.
6781:MR
6779:.
6771:.
6757:.
6751:.
6734:.
6728:MR
6726:.
6716:10
6492:.
6488:.
6462:.
6454:.
6436:.
6383:.
6363:.
6312:=
6285:.
6272:.
6241:=
6192:≤
6183:∈
6173:=
6147:.
6132:=
6099:=
6091:∈
6074:=
6024:.
5967:=
5913:.
5909:=
5907:Hg
5885:Hg
5865:≤
5863:Hg
5854:=
5842:∈
5835:gP
5833:=
5827:gP
5818:gP
5676:.
5672:=
5670:Hg
5621:≤
5619:Hg
5567:.
5482:.
5432:.
5430:Hg
5420:=
5000:+
4811:+
4701:.
4642:⊆
4638:∈
4595:∈
4583:{
4565:=
4553:∈
4549:{
4522:∈
4502:,
4490:∈
4478:.
4436:=
4434:pm
4416:.
4249:.
4228:PK
4226:=
4222:,
4171:=
4167:=
4149:gh
4041:.
3716:.
3696:.
3551:,
3522:,
3520:pq
3480:GL
3380:,
3300:.
3269:GL
3242:)(
3210:GL
3171:GL
3147:12
3085:.
3005:Cl
2957:.
2907:Cl
2875:.
2768:.
2167:.
2048:.
2016:.
2004:.
1932:.
1923:.
1670:.
1658:60
1205:.
908:.
615:U(
591:E(
579:O(
37:→
7522:.
7503:.
7449:.
7410:.
7382::
7363:.
7335::
7329:9
7316:.
7286::
7260:.
7232::
7216:6
7196:.
7151:.
7113::
7085:.
7049::
7027:.
6991::
6969:.
6939::
6916:.
6886::
6821::
6795:.
6765::
6742:.
6689:.
6677:.
6665:.
6653:.
6641:.
6629:.
6605:.
6593:.
6581:.
6569:.
6557:.
6545:.
6532:.
6507:.
6470:.
6450::
6444:5
6372:p
6357:H
6353:H
6349:p
6345:H
6341:p
6337:p
6333:p
6329:G
6325:H
6321:H
6319:(
6316:G
6314:N
6310:N
6306:p
6302:G
6298:p
6294:H
6290:p
6270:)
6268:p
6263:0
6261:Ω
6257:Ω
6251:P
6247:0
6243:Q
6239:P
6235:Q
6233:(
6230:G
6228:N
6224:Q
6220:Q
6218:(
6215:G
6213:N
6209:Q
6205:P
6201:Q
6199:(
6196:G
6194:N
6190:P
6185:P
6181:x
6171:Q
6165:0
6161:Q
6156:0
6152:P
6144:q
6139:G
6129:p
6127:n
6122:G
6118:P
6114:)
6112:P
6110:(
6108:G
6105:N
6101:P
6093:G
6089:g
6087:{
6082:P
6078:G
6071:p
6069:n
6063:p
6061:n
6057:P
6053:p
6047:P
6042:G
6038:G
6034:p
6022:)
6020:p
6015:p
6013:n
6007:q
6002:G
5993:p
5991:n
5986:P
5978:P
5976:(
5973:G
5971:N
5964:p
5962:n
5957:G
5953:p
5948:p
5946:n
5942:G
5938:P
5934:p
5930:q
5911:P
5904:g
5897:P
5891:H
5882:g
5875:p
5871:H
5867:P
5860:g
5856:P
5849:g
5844:H
5840:h
5822:0
5810:0
5808:Ω
5791:|
5785:0
5776:|
5769:p
5749:]
5746:P
5743::
5740:G
5737:[
5731:p
5721:)
5719:p
5715:Ω
5710:0
5708:Ω
5702:H
5698:H
5694:G
5690:P
5674:K
5667:g
5663:G
5659:g
5655:G
5651:p
5647:K
5643:H
5631:G
5627:p
5623:P
5616:g
5612:G
5608:g
5604:G
5600:p
5596:P
5592:G
5588:p
5584:H
5565:p
5561:x
5558:H
5551:x
5548:H
5539:Ω
5533:p
5524:x
5522:H
5514:x
5512:H
5506:H
5499:H
5493:x
5480:)
5478:p
5473:0
5471:Ω
5467:Ω
5461:H
5457:0
5453:H
5449:p
5445:H
5426:m
5422:H
5417:ω
5413:G
5407:ω
5402:p
5398:H
5390:r
5386:m
5384:(
5381:p
5379:ν
5375:Ω
5370:p
5368:ν
5363:p
5341:)
5338:j
5335:(
5330:p
5321:p
5316:/
5312:j
5304:)
5301:j
5298:(
5293:p
5282:k
5278:p
5270:)
5267:j
5264:(
5259:p
5250:p
5245:/
5241:j
5235:m
5230:)
5227:j
5224:(
5219:p
5208:k
5204:p
5195:1
5187:k
5183:p
5177:1
5174:=
5171:j
5163:m
5160:=
5154:j
5146:k
5142:p
5136:j
5130:m
5125:k
5121:p
5112:1
5104:k
5100:p
5094:0
5091:=
5088:j
5080:=
5074:)
5067:k
5063:p
5058:m
5053:k
5049:p
5042:(
5036:=
5032:|
5024:|
5010:r
5006:p
5002:r
4998:k
4991:G
4984:p
4976:r
4971:r
4967:Ω
4962:p
4960:ν
4955:ω
4952:G
4945:ω
4942:G
4933:Ω
4927:)
4925:k
4919:ω
4915:G
4909:p
4907:ν
4901:p
4895:ω
4891:G
4883:r
4878:ω
4875:G
4869:p
4867:ν
4862:ω
4857:r
4852:ω
4849:G
4843:p
4841:ν
4835:p
4829:ω
4825:G
4818:ω
4813:r
4809:k
4804:G
4798:p
4796:ν
4791:ω
4788:G
4782:p
4780:ν
4774:ω
4770:G
4764:p
4762:ν
4757:p
4752:p
4750:ν
4741:ω
4733:G
4727:ω
4724:G
4717:ω
4713:G
4698:p
4693:ω
4687:α
4683:ω
4679:G
4672:ω
4668:G
4661:ω
4657:α
4653:ω
4649:G
4644:G
4640:ω
4636:α
4630:ω
4626:G
4622:p
4617:ω
4613:G
4607:ω
4599:}
4597:G
4593:g
4589:ω
4587:⋅
4585:g
4576:ω
4573:G
4569:}
4567:ω
4563:ω
4561:⋅
4559:g
4555:G
4551:g
4541:ω
4537:G
4531:ω
4526:}
4524:ω
4520:x
4516:x
4513:g
4509:ω
4507:⋅
4505:g
4498:ω
4492:G
4488:g
4480:G
4476:p
4472:G
4458:u
4452:p
4441:u
4438:p
4429:G
4414:p
4410:p
4395:|
4391:G
4387:|
4376:G
4348:b
4342:a
4322:b
4316:a
4306:p
4302:G
4258:p
4245:p
4240:G
4236:K
4234:∩
4232:P
4224:G
4220:P
4216:K
4212:G
4208:P
4204:p
4200:G
4192:P
4190:(
4187:G
4185:N
4181:B
4177:A
4173:B
4169:B
4165:A
4161:B
4157:h
4153:P
4145:B
4141:G
4137:P
4133:P
4129:A
4125:P
4121:P
4117:P
4113:B
4109:A
4107:=
4105:B
4101:P
4099:(
4096:G
4094:N
4090:G
4086:B
4082:A
4078:P
4074:B
4070:A
4066:P
4062:p
4058:G
4039:)
4037:p
4033:p
4031:(
4027:)
4025:p
4021:p
4019:(
4015:)
4013:p
4008:p
4004:n
3997:p
3995:(
3990:p
3988:S
3979:p
3975:/
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3958:p
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3951:n
3947:p
3929:)
3926:p
3919:(
3914:1
3905:!
3902:)
3899:1
3893:p
3890:(
3857:5
3854:A
3850:5
3847:n
3843:3
3840:n
3836:G
3829:G
3825:7
3822:n
3818:7
3815:n
3811:7
3808:n
3804:G
3797:G
3793:G
3789:5
3786:n
3782:5
3779:n
3775:5
3772:n
3765:G
3761:G
3757:G
3753:3
3750:n
3746:3
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3739:3
3736:n
3732:3
3729:n
3725:G
3721:G
3666:G
3658:5
3655:n
3651:5
3648:n
3640:3
3637:n
3614:3
3611:n
3607:3
3604:n
3600:G
3596:n
3592:n
3588:n
3575:G
3571:G
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3538:.
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3532:p
3528:q
3524:p
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3476:p
3472:p
3468:p
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3420:x
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3390:F
3382:x
3366:]
3358:m
3355:j
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3338:0
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3328:i
3324:x
3317:[
3305:P
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3294:p
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3274:(
3272:2
3262:m
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3215:(
3213:2
3202:)
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3176:(
3174:2
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3119:D
3115:n
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3063:1
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3036:p
3021:|
3017:)
3014:K
3011:(
3001:|
2997:=
2992:p
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2973:p
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2939:K
2919:)
2916:K
2913:(
2897:p
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2889:p
2885:p
2861:4
2857:S
2832:G
2812:p
2792:1
2789:=
2784:p
2780:n
2754:n
2750:p
2729:p
2709:H
2689:G
2669:p
2649:H
2629:p
2609:p
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2560:n
2556:p
2535:p
2515:p
2497:.
2479:G
2475:N
2464:G
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2456:P
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2437:)
2434:P
2431:(
2426:G
2422:N
2418::
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2411:|
2407:=
2402:p
2398:n
2374:p
2366:1
2358:p
2354:n
2343:.
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2313:p
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2275:p
2271:n
2260:m
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2242:0
2236:n
2216:m
2211:n
2207:p
2196:G
2192:G
2188:n
2184:p
2155:K
2152:=
2149:g
2146:H
2141:1
2134:g
2113:G
2107:g
2097:G
2093:p
2089:K
2085:H
2077:G
2073:p
2069:p
2065:G
2045:G
2040:p
2036:G
2032:p
2028:G
1989:n
1985:p
1973:G
1967:p
1962:G
1958:n
1951:p
1921:G
1907:1
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1901:)
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1891:|
1887:P
1884::
1881:G
1877:|
1873:(
1858:p
1854:P
1838:n
1834:p
1830:=
1826:|
1822:P
1818:|
1807:P
1803:p
1799:m
1795:p
1781:0
1775:n
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1750:n
1746:p
1742:=
1738:|
1734:G
1730:|
1705:)
1702:G
1699:(
1691:p
1655:=
1651:|
1647:G
1643:|
1621:G
1587:p
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1523:p
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1462:p
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1392:p
1370:n
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1240:G
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1193:)
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1187:(
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867:)
861:(
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683:8
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675:7
673:E
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617:n
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499:)
486:Z
474:)
461:Z
437:)
424:Z
415:(
328:p
293:Q
285:n
282:D
272:n
269:A
261:n
258:S
250:n
247:Z
20:)
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