48:
2497:
1715:
the analogy becomes strained. There is a different family of examples that more closely mimics the dihedral groups of order 2, but that requires a bit more setup. Let ζ denote a primitive
2136:, though the sheer number of families of such groups grows so quickly that further classifications along these lines are judged difficult for the human mind to comprehend. For example,
3085:
2873:
2614:
493:
468:
431:
2989:
2817:
2757:
2660:
3032:
3409:
3523:
2881:
2211:
3593:
795:
2214:
consisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations.
3388:— An exhaustive catalog of the 340 non-abelian groups of order dividing 64 with detailed tables of defining relations, constants, and
2132:≤ 4 were classified early in the history of group theory, and modern work has extended these classifications to groups whose order divides
2694:
353:
3575:
3557:
3535:
3469:
1044:
303:
2379:
2892:, and others classified those simple groups whose Sylow 2-subgroups were abelian, dihedral, semidihedral, or quaternion.
2428:
2557:
1063:
1040:
788:
298:
3325:
2206:
are due to A. Shalev and independently to C. R. Leedham-Green, both in 1994. They admit a classification of finite
1239:
by conjugation. Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite
2499:, and these are dominated by the classes that are two-step nilpotent. Because of this rapid growth, there is a
714:
3271:
Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2002), "A millennium project: constructing small groups",
3357:
2885:
2137:
781:
1291:
3041:
2872:
classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and
398:
212:
3392:
presentations of each group in the notation the text defines. "Of enduring value to those interested in
3317:
130:
3223:
2851:
20:
2666:-subgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and any
2855:
2847:
1419:
1387:
1379:
835:
596:
330:
207:
95:
2567:
476:
451:
414:
3461:
2954:
2863:
2820:
2783:
2519:
tends to infinity. For instance, of the 49 910 529 484 different groups of order at most 2000,
746:
536:
2726:
3453:
3404:
2500:
1106:
620:
2635:
3507:
3479:
3438:
3389:
3384:
3350:
3292:
3010:
2152:
2057:
1875:
1427:
1295:
1227:-group intersects the center non-trivially as may be proved by considering the elements of
985:
560:
548:
166:
100:
2819:
These groups are related (for different primes), possess important properties such as the
8:
3570:, de Gruyter Expositions in Mathematics 56, vol. 3, Berlin: Walter de Gruyter GmbH,
3552:, de Gruyter Expositions in Mathematics 47, vol. 2, Berlin: Walter de Gruyter GmbH,
3530:, de Gruyter Expositions in Mathematics 46, vol. 1, Berlin: Walter de Gruyter GmbH,
3106:
2775:
2182:
859:
839:
831:
135:
30:
2701:-groups arise both as subgroups and as quotient groups. As subgroups, for a given prime
3442:
3296:
2859:
2173:
1748:
1508:
1423:
1367:
120:
92:
3624:
3597:
3571:
3553:
3531:
3465:
3446:
3426:
3372:
3334:
3321:
2889:
2877:
2831:
Much of the structure of a finite group is carried in the structure of its so-called
2508:
1407:
1048:
525:
368:
262:
3300:
2392: = 2, both the semi-direct products mentioned above are isomorphic to the
691:
3495:
3418:
3307:
3280:
3235:
2901:
2401:
2273:
2222:
2199:
1724:
1504:
1468:
1435:
1248:
913:
874:
676:
668:
660:
652:
644:
632:
572:
512:
502:
344:
286:
161:
3239:
2693:-groups are fundamental tools in understanding the structure of groups and in the
1511:. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of
3503:
3475:
3434:
3380:
3346:
3311:
3288:
3183:
2911:
2850:
of a finite group exert control over the group that was used in the proof of the
2177:
2078:
are those upper triangular matrices with 1s one the diagonal and 0s on the first
1660:
1580:
1255:-group is the subgroup of the center consisting of the central elements of order
1217:
1210:
1131:
959:
760:
753:
739:
696:
584:
507:
337:
251:
191:
71:
974:
3545:
3499:
3342:
2943:-group so has non-trivial center, so given a non-trivial element of the center
2869:
2679:
2393:
2194:. The coclass conjectures were proven in the 1980s using techniques related to
1528:
1489:
1150:
1102:
1001:
925:
767:
703:
393:
373:
310:
275:
196:
186:
171:
156:
110:
87:
3422:
3284:
2906:
3618:
3600:
3430:
3393:
1512:
1485:
1059:
884:
686:
608:
442:
315:
181:
2504:
1458:
1005:
978:
905:
817:
813:
541:
240:
229:
176:
151:
146:
105:
76:
39:
1503:
The dihedral groups are both very similar to and very dissimilar from the
3400:
2823:, and allow one to determine many aspects of the structure of the group.
2195:
2191:
2148:
1732:
1454:
941:
809:
2236:
The trivial group is the only group of order one, and the cyclic group C
2836:
1124:
708:
436:
3605:
2109:
1627:, and its lower central series, upper central series, lower exponent-
529:
3460:, London Mathematical Society Monographs. New Series, vol. 27,
1635:
central series are equal. It is generated by its elements of order
1496:
of order 8 is a non-abelian 2-group. However, every group of order
929:
66:
1430:, so very well understood. The map from the automorphism group of
2511:
of 2-groups among isomorphism classes of groups of order at most
2168:
408:
3376:
2718:
47:
3339:
Finite simple groups (Proc. Instructional Conf., Oxford, 1969)
2370:. The first one can be described in other terms as group UT(3,
1671:
are always regular groups, it is also a minimal such example.
1298:. Generalizing the earlier comments about the socle, a finite
877:. The orders of different elements may be different powers of
2159:-groups into families based on large quotient and subgroups.
1703:) provides an analogue for the dihedral group for all primes
1205:, creating an infinite descent. As a corollary, every finite
2190:-groups of fixed coclass as perturbations of finitely many
1477:
are both 2-groups of order 4, but they are not isomorphic.
2766:-subgroup), and various others. As quotients, the largest
2400:
of order 8. The other non-abelian group of order 8 is the
1278:, and so it too has a non-trivial center. The preimage in
1093:, and the result follows from the Correspondence Theorem.
16:
Group in which the order of every element is a power of p
3337:(1971), "Global and local properties of finite groups",
2678:-subgroup. This and other properties are proved in the
1515:, that is those groups of order 2 and nilpotency class
2685:
1047:
for groups. A proof sketch is as follows: because the
3270:
3044:
3013:
2957:
2786:
2729:
2638:
2570:
2431:
2421:
The number of isomorphism classes of groups of order
2140:
and James K. Senior classified groups of order 2 for
1695:) is the dihedral group of order 8, so in some sense
479:
454:
417:
2862:help describe the structure of groups as acting on
2492:{\displaystyle p^{{\frac {2}{27}}n^{3}+O(n^{8/3})}}
2374:) of unitriangular matrices over finite field with
2105:is the least integer at least as large as the base
1434:into this general linear group has been studied by
1116:This forms the basis for many inductive methods in
3452:
3170:
3131:
3079:
3026:
3007:(since the subgroup they generate must have order
2983:
2811:
2751:
2654:
2608:
2491:
1979:, the set of invertible linear transformations of
1862:) are irregular groups of maximal class and order
487:
462:
425:
3038:is central, so the group is abelian, and in fact
2340: ≠ 2, one is a semi-direct product of C
3616:
3591:
3566:Berkovich, Yakov; Janko, Zvonimir (2011-06-16),
3273:International Journal of Algebra and Computation
2717:-subgroup not unique but all conjugate) and the
2532:, or just over 99%, are 2-groups of order 1024.
1878:are another fundamental family of examples. Let
1382:is a proper quotient of the group, every finite
1089:. We may now apply the inductive hypothesis to
965:The remainder of this article deals with finite
3410:Journal für die reine und angewandte Mathematik
2540:Every finite group whose order is divisible by
2162:An entirely different method classifies finite
1374:-groups are well studied. Just as every finite
969:-groups. For an example of an infinite abelian
3565:
3543:
3486:Sims, Charles (1965), "Enumerating p-groups",
3356:
3144:
1869:
1674:
1453:-groups of the same order are not necessarily
1438:, who showed that the kernel of this map is a
2358:, and the other is a semi-direct product of C
1719:th root of unity in the complex numbers, let
789:
3458:The structure of groups of prime power order
1651:-group of maximal class, since it has order
1378:-group has a non-trivial center so that the
1101:One of the first standard results using the
2544:contains a subgroup which is a non-trivial
1522:
1335:. If a normal subgroup is not contained in
916:(the number of its elements) is a power of
3405:"The classification of prime-power groups"
3333:
3253:
2147:Rather than classify the groups by order,
1591:-subgroups of the general linear group GL(
1539:-groups. Denote the cyclic group of order
796:
782:
3522:
3176:
2333:. There are also two non-abelian groups.
1842:) is the dihedral group of order 2. When
1623: − 1). It has nilpotency class
481:
456:
419:
3306:
3118:
2293:There are three abelian groups of order
2246:. There are exactly two groups of order
2172:, that is, the difference between their
1770:acts as multiplication by ζ. The powers
3224:"On the number of groups of order 1024"
3221:
3215:
2991:), or it generates a subgroup of order
2548:-group, namely a cyclic group of order
1113:-group cannot be the trivial subgroup.
3617:
2695:classification of finite simple groups
1918:to be the vector space generated by {
1173:, but then there is a smaller example
1004:since by definition every element has
354:Classification of finite simple groups
3592:
2564:-group of maximal possible order: if
2503:conjecture asserting that almost all
1096:
3485:
3399:
3209:
3157:
3080:{\displaystyle G=C_{p}\times C_{p}.}
2947:this either generates the group (so
2858:of elementary abelian groups called
1711: = 2. However, for higher
1039:. This follows by induction, using
977:, and for an example of an infinite
2770:-group quotient is the quotient of
2686:Application to structure of a group
1631:central series, and upper exponent-
1318:, and any normal subgroup of order
1306:contains normal subgroups of order
13:
3515:
14:
3636:
3585:
2552:generated by an element of order
2535:
2268:. For example, the cyclic group C
2119:
1599:) are direct products of various
873:, and not fewer, is equal to the
3313:Theory of groups of finite order
2826:
2186:described the set of all finite
1361:
1243:-group is central and has order
46:
3246:
3190:. Stack Exchange. 24 March 2012
2935:To prove that a group of order
2416:
1882:be a vector space of dimension
1547:(1), and the wreath product of
1535:are very important examples of
1027:has a normal subgroup of order
3202:
3171:Leedham-Green & McKay 2002
3163:
3150:
3137:
3132:Leedham-Green & McKay 2002
3124:
3111:
3100:
2939:is abelian, note that it is a
2929:
2803:
2797:
2746:
2740:
2580:
2572:
2507:are 2-groups: the fraction of
2484:
2463:
2290:are both 2-groups of order 4.
1830:-group of maximal class. When
1782:), and the example groups are
715:Infinite dimensional Lie group
1:
3263:
3240:10.1080/00927872.2021.2006680
3222:Burrell, David (2021-12-08).
2411:
2082:−1 superdiagonals. The group
1344:, then its intersection with
991:
928:guarantee the existence of a
3093:
2917:
2848:elementary abelian subgroups
2609:{\displaystyle |G|=n=p^{k}m}
1216:In another direction, every
846:. That is, for each element
488:{\displaystyle \mathbb {Z} }
463:{\displaystyle \mathbb {Z} }
426:{\displaystyle \mathbb {Z} }
7:
3568:Groups of Prime Power Order
3550:Groups of Prime Power Order
3528:Groups of Prime Power Order
3360:; Senior, James K. (1964),
2984:{\displaystyle G=C_{p^{2}}}
2895:
2515:is thought to tend to 1 as
2242:is the only group of order
2228:
2151:proposed using a notion of
1870:Unitriangular matrix groups
1739:be a cyclic group of order
1675:Generalized dihedral groups
1445:
1394:induces an automorphism on
1294:and these groups begin the
213:List of group theory topics
10:
3641:
3318:Cambridge University Press
3145:Hall Jr. & Senior 1964
3003:not in its orbit generate
2951:is cyclic, hence abelian:
2378:elements, also called the
2202:. The final proofs of the
1866:, but are not isomorphic.
1531:of cyclic groups of order
1062:(see below), according to
947:that divides the order of
912:-group if and only if its
18:
3423:10.1515/crll.1940.182.130
3285:10.1142/S0218196702001115
3228:Communications in Algebra
3034:) but they commute since
2812:{\displaystyle O^{p}(G).}
2560:. In fact, it contains a
1727:generated by it, and let
1643:. The second such group,
1386:-group has a non-trivial
1085:is necessarily normal in
865:such that the product of
21:n-group (category theory)
3500:10.1112/plms/s3-15.1.151
2922:
2864:symplectic vector spaces
2752:{\displaystyle O_{p}(G)}
2674:is contained in a Sylow
2250:, both abelian, namely C
2069:. In terms of matrices,
2056:), and the terms of its
1774:are normal subgroups of
1743:generated by an element
1667:. Since groups of order
1523:Iterated wreath products
1420:elementary abelian group
1390:. Every automorphism of
1388:outer automorphism group
1380:inner automorphism group
1109:of a non-trivial finite
890:-groups are also called
331:Elementary abelian group
208:Glossary of group theory
19:Not to be confused with
3488:Proc. London Math. Soc.
3462:Oxford University Press
3456:; McKay, Susan (2002),
2212:directed coclass graphs
2022:is a vector space over
2005:form a subgroup of Aut(
1874:The Sylow subgroups of
920:. Given a finite group
3184:"Every group of order
3081:
3028:
2985:
2821:focal subgroup theorem
2813:
2753:
2656:
2655:{\displaystyle p^{k},}
2610:
2493:
2155:which gathered finite
1735:generated by 1−ζ. Let
1639:, but its exponent is
1567: + 1). Then
1045:Correspondence Theorem
838:of every element is a
747:Linear algebraic group
489:
464:
427:
3396:" (from the preface).
3371:, London: Macmillan,
3362:The Groups of Order 2
3082:
3029:
3027:{\displaystyle p^{2}}
2986:
2852:Feit–Thompson theorem
2814:
2754:
2657:
2611:
2494:
2380:Heisenberg group mod
2217:Every group of order
1876:general linear groups
1822:and nilpotency class
1655:and nilpotency class
1231:which are fixed when
490:
465:
428:
3454:Leedham-Green, C. R.
3042:
3011:
2955:
2784:
2759:(the unique largest
2727:
2636:
2568:
2429:
2153:isoclinism of groups
2124:The groups of order
2058:lower central series
1428:general linear group
1322:is contained in the
1296:upper central series
1181:whose normalizer in
1077:. Being central in
1019:is a group of order
986:Tarski monster group
477:
452:
415:
2860:extraspecial groups
2509:isomorphism classes
2183:coclass conjectures
2093:, nilpotency class
1725:cyclotomic integers
1683: = 2 and
1509:semidihedral groups
1457:; for example, the
1414:. The quotient G/Φ(
1270:-group, then so is
860:nonnegative integer
121:Group homomorphisms
31:Algebraic structure
3598:Weisstein, Eric W.
3544:Berkovich, Yakov;
3358:Hall Jr., Marshall
3335:Glauberman, George
3077:
3024:
2981:
2856:central extensions
2809:
2779:-residual subgroup
2749:
2705:one has the Sylow
2652:
2606:
2489:
2174:composition length
1749:semidirect product
1424:automorphism group
1354:has size at least
1302:-group with order
1149:, because for any
1145:properly contains
1123:For instance, the
1097:Non-trivial center
597:Special orthogonal
485:
460:
423:
304:Lagrange's theorem
3577:978-3-1102-0717-0
3559:978-3-1102-0419-3
3537:978-3-1102-0418-6
3471:978-0-19-853548-5
3345:, pp. 1–64,
3308:Burnside, William
2999:and some element
2890:George Glauberman
2878:Daniel Gorenstein
2445:
2200:powerful p-groups
2166:-groups by their
2138:Marshall Hall Jr.
2044:-subgroup of Aut(
1619: − 1)/(
1579:-subgroup of the
1505:quaternion groups
1408:Frattini subgroup
1282:of the center of
1169:, and so also in
858:, there exists a
806:
805:
381:
380:
263:Alternating group
220:
219:
3632:
3611:
3610:
3580:
3562:
3540:
3524:Berkovich, Yakov
3510:
3482:
3449:
3417:(182): 130–141,
3387:
3353:
3330:
3303:
3257:
3250:
3244:
3243:
3234:(6): 2408–2410.
3219:
3213:
3206:
3200:
3199:
3197:
3195:
3180:
3174:
3167:
3161:
3154:
3148:
3141:
3135:
3128:
3122:
3115:
3109:
3104:
3087:
3086:
3084:
3083:
3078:
3073:
3072:
3060:
3059:
3033:
3031:
3030:
3025:
3023:
3022:
2990:
2988:
2987:
2982:
2980:
2979:
2978:
2977:
2933:
2902:Elementary group
2839:of non-identity
2818:
2816:
2815:
2810:
2796:
2795:
2758:
2756:
2755:
2750:
2739:
2738:
2661:
2659:
2658:
2653:
2648:
2647:
2620:does not divide
2615:
2613:
2612:
2607:
2602:
2601:
2583:
2575:
2558:Cauchy's theorem
2531:
2530:
2527:
2524:
2498:
2496:
2495:
2490:
2488:
2487:
2483:
2482:
2478:
2456:
2455:
2446:
2438:
2402:quaternion group
2274:Klein four-group
2204:coclass theorems
2180:. The so-called
2178:nilpotency class
1983:which take each
1834: = 2,
1687: = 2,
1607:). It has order
1469:Klein four-group
1165:is contained in
1064:Cauchy's theorem
1041:Cauchy's theorem
875:identity element
798:
791:
784:
740:Algebraic groups
513:Hyperbolic group
503:Arithmetic group
494:
492:
491:
486:
484:
469:
467:
466:
461:
459:
432:
430:
429:
424:
422:
345:Schur multiplier
299:Cauchy's theorem
287:Quaternion group
235:
234:
61:
60:
50:
37:
26:
25:
3640:
3639:
3635:
3634:
3633:
3631:
3630:
3629:
3615:
3614:
3588:
3583:
3578:
3560:
3546:Janko, Zvonimir
3538:
3518:
3516:Further reading
3513:
3472:
3328:
3266:
3261:
3260:
3254:Glauberman 1971
3251:
3247:
3220:
3216:
3207:
3203:
3193:
3191:
3182:
3181:
3177:
3168:
3164:
3155:
3151:
3142:
3138:
3129:
3125:
3116:
3112:
3105:
3101:
3096:
3091:
3090:
3068:
3064:
3055:
3051:
3043:
3040:
3039:
3018:
3014:
3012:
3009:
3008:
2973:
2969:
2968:
2964:
2956:
2953:
2952:
2934:
2930:
2925:
2920:
2912:Regular p-group
2898:
2833:local subgroups
2829:
2791:
2787:
2785:
2782:
2781:
2734:
2730:
2728:
2725:
2724:
2688:
2662:called a Sylow
2643:
2639:
2637:
2634:
2633:
2628:has a subgroup
2597:
2593:
2579:
2571:
2569:
2566:
2565:
2538:
2528:
2525:
2522:
2520:
2474:
2470:
2466:
2451:
2447:
2437:
2436:
2432:
2430:
2427:
2426:
2419:
2414:
2407:
2399:
2369:
2363:
2357:
2351:
2345:
2332:
2326:
2320:
2314:
2308:
2302:
2289:
2285:
2281:
2271:
2267:
2261:
2255:
2241:
2234:
2122:
2097:, and exponent
2088:
2077:
2068:
2039:
2017:
2004:
1991:
1971:. For each 1 ≤
1962:
1945:
1936:
1926:
1917:
1908:
1899:
1892:
1872:
1723:be the ring of
1677:
1659:, but is not a
1647:(2), is also a
1581:symmetric group
1575:) is the Sylow
1529:wreath products
1525:
1495:
1476:
1466:
1448:
1364:
1353:
1343:
1334:
1218:normal subgroup
1132:proper subgroup
1099:
1069:has a subgroup
994:
812:, specifically
802:
773:
772:
761:Abelian variety
754:Reductive group
742:
732:
731:
730:
729:
680:
672:
664:
656:
648:
621:Special unitary
532:
518:
517:
499:
498:
480:
478:
475:
474:
455:
453:
450:
449:
418:
416:
413:
412:
404:
403:
394:Discrete groups
383:
382:
338:Frobenius group
283:
270:
259:
252:Symmetric group
248:
232:
222:
221:
72:Normal subgroup
58:
38:
29:
24:
17:
12:
11:
5:
3638:
3628:
3627:
3613:
3612:
3587:
3586:External links
3584:
3582:
3581:
3576:
3563:
3558:
3541:
3536:
3519:
3517:
3514:
3512:
3511:
3483:
3470:
3450:
3397:
3354:
3343:Academic Press
3341:, Boston, MA:
3331:
3326:
3304:
3279:(5): 623–644,
3267:
3265:
3262:
3259:
3258:
3245:
3214:
3201:
3188:is metabelian"
3175:
3162:
3149:
3136:
3134:, p. 214)
3123:
3110:
3098:
3097:
3095:
3092:
3089:
3088:
3076:
3071:
3067:
3063:
3058:
3054:
3050:
3047:
3021:
3017:
2976:
2972:
2967:
2963:
2960:
2927:
2926:
2924:
2921:
2919:
2916:
2915:
2914:
2909:
2904:
2897:
2894:
2870:Richard Brauer
2828:
2825:
2808:
2805:
2802:
2799:
2794:
2790:
2748:
2745:
2742:
2737:
2733:
2687:
2684:
2680:Sylow theorems
2651:
2646:
2642:
2605:
2600:
2596:
2592:
2589:
2586:
2582:
2578:
2574:
2556:obtained from
2537:
2536:Within a group
2534:
2486:
2481:
2477:
2473:
2469:
2465:
2462:
2459:
2454:
2450:
2444:
2441:
2435:
2418:
2415:
2413:
2410:
2405:
2397:
2394:dihedral group
2365:
2359:
2353:
2347:
2346: × C
2341:
2328:
2327: × C
2322:
2321: × C
2316:
2310:
2309: × C
2304:
2298:
2287:
2286: × C
2283:
2279:
2269:
2263:
2262: × C
2257:
2251:
2237:
2233:
2227:
2121:
2120:Classification
2118:
2086:
2073:
2064:
2037:
2013:
1996:
1987:
1958:
1941:
1931:
1922:
1913:
1904:
1897:
1890:
1871:
1868:
1794:) =
1676:
1673:
1615: = (
1524:
1521:
1493:
1490:dihedral group
1474:
1464:
1447:
1444:
1363:
1360:
1348:
1339:
1330:
1290:is called the
1247:. Indeed, the
1151:counterexample
1103:class equation
1098:
1095:
1031:for every 1 ≤
993:
990:
926:Sylow theorems
804:
803:
801:
800:
793:
786:
778:
775:
774:
771:
770:
768:Elliptic curve
764:
763:
757:
756:
750:
749:
743:
738:
737:
734:
733:
728:
727:
724:
721:
717:
713:
712:
711:
706:
704:Diffeomorphism
700:
699:
694:
689:
683:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
641:
640:
629:
628:
617:
616:
605:
604:
593:
592:
581:
580:
569:
568:
561:Special linear
557:
556:
549:General linear
545:
544:
539:
533:
524:
523:
520:
519:
516:
515:
510:
505:
497:
496:
483:
471:
458:
445:
443:Modular groups
441:
440:
439:
434:
421:
405:
402:
401:
396:
390:
389:
388:
385:
384:
379:
378:
377:
376:
371:
366:
363:
357:
356:
350:
349:
348:
347:
341:
340:
334:
333:
328:
319:
318:
316:Hall's theorem
313:
311:Sylow theorems
307:
306:
301:
293:
292:
291:
290:
284:
279:
276:Dihedral group
272:
271:
266:
260:
255:
249:
244:
233:
228:
227:
224:
223:
218:
217:
216:
215:
210:
202:
201:
200:
199:
194:
189:
184:
179:
174:
169:
167:multiplicative
164:
159:
154:
149:
141:
140:
139:
138:
133:
125:
124:
116:
115:
114:
113:
111:Wreath product
108:
103:
98:
96:direct product
90:
88:Quotient group
82:
81:
80:
79:
74:
69:
59:
56:
55:
52:
51:
43:
42:
15:
9:
6:
4:
3:
2:
3637:
3626:
3623:
3622:
3620:
3608:
3607:
3602:
3599:
3595:
3594:Rowland, Todd
3590:
3589:
3579:
3573:
3569:
3564:
3561:
3555:
3551:
3547:
3542:
3539:
3533:
3529:
3525:
3521:
3520:
3509:
3505:
3501:
3497:
3493:
3489:
3484:
3481:
3477:
3473:
3467:
3463:
3459:
3455:
3451:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3416:
3412:
3411:
3406:
3402:
3398:
3395:
3394:finite groups
3391:
3386:
3382:
3378:
3374:
3370:
3366:
3363:
3359:
3355:
3352:
3348:
3344:
3340:
3336:
3332:
3329:
3327:9781440035456
3323:
3319:
3315:
3314:
3309:
3305:
3302:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3269:
3268:
3255:
3249:
3241:
3237:
3233:
3229:
3225:
3218:
3211:
3205:
3189:
3187:
3179:
3172:
3166:
3159:
3153:
3146:
3140:
3133:
3127:
3120:
3119:Burnside 1897
3114:
3108:
3103:
3099:
3074:
3069:
3065:
3061:
3056:
3052:
3048:
3045:
3037:
3019:
3015:
3006:
3002:
2998:
2994:
2974:
2970:
2965:
2961:
2958:
2950:
2946:
2942:
2938:
2932:
2928:
2913:
2910:
2908:
2905:
2903:
2900:
2899:
2893:
2891:
2887:
2886:Michio Suzuki
2883:
2882:Helmut Bender
2879:
2875:
2871:
2867:
2865:
2861:
2857:
2853:
2849:
2844:
2842:
2838:
2834:
2827:Local control
2824:
2822:
2806:
2800:
2792:
2788:
2780:
2778:
2773:
2769:
2765:
2762:
2743:
2735:
2731:
2723:
2721:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2683:
2681:
2677:
2673:
2670:-subgroup of
2669:
2665:
2649:
2644:
2640:
2631:
2627:
2623:
2619:
2603:
2598:
2594:
2590:
2587:
2584:
2576:
2563:
2559:
2555:
2551:
2547:
2543:
2533:
2518:
2514:
2510:
2506:
2505:finite groups
2502:
2479:
2475:
2471:
2467:
2460:
2457:
2452:
2448:
2442:
2439:
2433:
2424:
2409:
2403:
2395:
2391:
2386:
2384:
2383:
2377:
2373:
2368:
2362:
2356:
2350:
2344:
2339:
2334:
2331:
2325:
2319:
2313:
2307:
2301:
2296:
2291:
2278:
2275:
2266:
2260:
2254:
2249:
2245:
2240:
2232:
2226:
2224:
2220:
2215:
2213:
2209:
2205:
2201:
2197:
2193:
2189:
2185:
2184:
2179:
2175:
2171:
2170:
2165:
2160:
2158:
2154:
2150:
2145:
2144:≤ 6 in 1964.
2143:
2139:
2135:
2131:
2127:
2117:
2115:
2111:
2108:
2104:
2100:
2096:
2092:
2085:
2081:
2076:
2072:
2067:
2063:
2060:are just the
2059:
2055:
2051:
2047:
2043:
2036:
2032:
2029:
2025:
2021:
2016:
2012:
2008:
2003:
1999:
1995:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1961:
1957:
1954:, and define
1953:
1949:
1944:
1940:
1934:
1930:
1925:
1921:
1916:
1912:
1909:} and define
1907:
1903:
1896:
1889:
1886:with basis {
1885:
1881:
1877:
1867:
1865:
1861:
1857:
1853:
1849:
1846:is odd, both
1845:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1781:
1777:
1773:
1769:
1765:
1761:
1757:
1753:
1750:
1746:
1742:
1738:
1734:
1730:
1726:
1722:
1718:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1672:
1670:
1666:
1664:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1530:
1527:The iterated
1520:
1518:
1514:
1513:maximal class
1510:
1506:
1501:
1499:
1491:
1487:
1483:
1478:
1473:
1470:
1463:
1460:
1456:
1452:
1443:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1362:Automorphisms
1359:
1357:
1351:
1347:
1342:
1338:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1292:second center
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1260:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1219:
1214:
1212:
1208:
1204:
1200:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1161:, the center
1160:
1156:
1152:
1148:
1144:
1140:
1136:
1133:
1129:
1126:
1121:
1119:
1114:
1112:
1108:
1104:
1094:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1065:
1061:
1057:
1053:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1015:is prime and
1014:
1009:
1007:
1003:
999:
989:
987:
983:
980:
976:
972:
968:
963:
961:
957:
954:Every finite
952:
950:
946:
943:
939:
935:
931:
927:
923:
919:
915:
911:
907:
902:
900:
896:
894:
889:
886:
882:
880:
876:
872:
868:
864:
861:
857:
853:
849:
845:
841:
837:
834:in which the
833:
829:
827:
822:
819:
815:
811:
799:
794:
792:
787:
785:
780:
779:
777:
776:
769:
766:
765:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
736:
735:
725:
722:
719:
718:
716:
710:
707:
705:
702:
701:
698:
695:
693:
690:
688:
685:
684:
681:
675:
673:
667:
665:
659:
657:
651:
649:
643:
642:
638:
634:
631:
630:
626:
622:
619:
618:
614:
610:
607:
606:
602:
598:
595:
594:
590:
586:
583:
582:
578:
574:
571:
570:
566:
562:
559:
558:
554:
550:
547:
546:
543:
540:
538:
535:
534:
531:
527:
522:
521:
514:
511:
509:
506:
504:
501:
500:
472:
447:
446:
444:
438:
435:
410:
407:
406:
400:
397:
395:
392:
391:
387:
386:
375:
372:
370:
367:
364:
361:
360:
359:
358:
355:
352:
351:
346:
343:
342:
339:
336:
335:
332:
329:
327:
325:
321:
320:
317:
314:
312:
309:
308:
305:
302:
300:
297:
296:
295:
294:
288:
285:
282:
277:
274:
273:
269:
264:
261:
258:
253:
250:
247:
242:
239:
238:
237:
236:
231:
230:Finite groups
226:
225:
214:
211:
209:
206:
205:
204:
203:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
144:
143:
142:
137:
134:
132:
129:
128:
127:
126:
123:
122:
118:
117:
112:
109:
107:
104:
102:
99:
97:
94:
91:
89:
86:
85:
84:
83:
78:
75:
73:
70:
68:
65:
64:
63:
62:
57:Basic notions
54:
53:
49:
45:
44:
41:
36:
32:
28:
27:
22:
3604:
3567:
3549:
3527:
3491:
3490:, Series 3,
3487:
3457:
3414:
3408:
3401:Hall, Philip
3368:
3364:
3361:
3338:
3312:
3276:
3272:
3248:
3231:
3227:
3217:
3204:
3192:. Retrieved
3185:
3178:
3165:
3152:
3139:
3126:
3113:
3102:
3035:
3004:
3000:
2996:
2992:
2948:
2944:
2940:
2936:
2931:
2868:
2845:
2843:-subgroups.
2840:
2832:
2830:
2776:
2771:
2767:
2763:
2760:
2719:
2714:
2710:
2706:
2702:
2698:
2690:
2689:
2675:
2671:
2667:
2663:
2629:
2625:
2621:
2617:
2561:
2553:
2549:
2545:
2541:
2539:
2516:
2512:
2422:
2420:
2417:Among groups
2389:
2387:
2381:
2375:
2371:
2366:
2360:
2354:
2348:
2342:
2337:
2335:
2329:
2323:
2317:
2311:
2305:
2299:
2294:
2292:
2276:
2264:
2258:
2252:
2247:
2243:
2238:
2235:
2230:
2218:
2216:
2207:
2203:
2196:Lie algebras
2192:pro-p groups
2187:
2181:
2167:
2163:
2161:
2156:
2146:
2141:
2133:
2129:
2125:
2123:
2113:
2106:
2102:
2098:
2094:
2090:
2083:
2079:
2074:
2070:
2065:
2061:
2053:
2049:
2045:
2041:
2034:
2030:
2027:
2023:
2019:
2014:
2010:
2006:
2001:
1997:
1993:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1959:
1955:
1951:
1947:
1942:
1938:
1932:
1928:
1923:
1919:
1914:
1910:
1905:
1901:
1894:
1887:
1883:
1879:
1873:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1818:) has order
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1744:
1740:
1736:
1728:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1678:
1668:
1662:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1592:
1588:
1584:
1576:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1526:
1516:
1502:
1500:is abelian.
1497:
1481:
1479:
1471:
1461:
1459:cyclic group
1450:
1449:
1439:
1431:
1415:
1411:
1403:
1399:
1395:
1391:
1383:
1375:
1371:
1368:automorphism
1365:
1355:
1349:
1345:
1340:
1336:
1331:
1327:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1261:
1256:
1252:
1251:of a finite
1244:
1240:
1236:
1232:
1228:
1224:
1223:of a finite
1220:
1215:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1146:
1142:
1138:
1137:of a finite
1134:
1127:
1122:
1117:
1115:
1110:
1105:is that the
1100:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1055:
1051:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1010:
1006:finite order
997:
995:
984:-group, see
981:
975:Prüfer group
973:-group, see
970:
966:
964:
955:
953:
948:
944:
937:
933:
921:
917:
909:
906:finite group
903:
898:
892:
891:
887:
883:
878:
870:
866:
862:
855:
851:
847:
843:
825:
824:
820:
818:prime number
814:group theory
807:
636:
624:
612:
600:
588:
576:
564:
552:
323:
322:
280:
267:
256:
245:
241:Cyclic group
119:
106:Free product
77:Group action
40:Group theory
35:Group theory
34:
3494:: 151–166,
2907:Prüfer rank
2874:John Walter
2837:normalizers
2709:-subgroups
2210:-groups in
2149:Philip Hall
2040:is a Sylow
1747:. Form the
1733:prime ideal
1587:). Maximal
1480:Nor need a
1402:), where Φ(
1060:non-trivial
942:prime power
810:mathematics
526:Topological
365:alternating
3369:≤ 6)
3264:References
2854:. Certain
2846:The large
2412:Prevalence
2297:, namely C
2282:which is C
2223:metabelian
2176:and their
2089:has order
2009:) denoted
1946:} for 1 ≤
1826:, so is a
1484:-group be
1455:isomorphic
1370:groups of
1326:th center
1209:-group is
1125:normalizer
1000:-group is
992:Properties
958:-group is
940:for every
897:or simply
869:copies of
816:, given a
633:Symplectic
573:Orthogonal
530:Lie groups
437:Free group
162:continuous
101:Direct sum
3606:MathWorld
3601:"p-Group"
3447:122817195
3431:0075-4102
3210:Sims 1965
3194:7 January
3158:Hall 1940
3094:Citations
3062:×
2918:Footnotes
2713:(largest
2632:of order
2425:grows as
2110:logarithm
1963:= 0 when
1442:-group.
1406:) is the
1310:with 0 ≤
1211:nilpotent
1120:-groups.
1073:of order
960:nilpotent
936:of order
697:Conformal
585:Euclidean
192:nilpotent
3625:P-groups
3619:Category
3548:(2008),
3526:(2008),
3403:(1940),
3377:64016861
3310:(1897),
3301:31716675
2896:See also
2501:folklore
2272:and the
2128:for 0 ≤
1850:(2) and
1661:regular
1507:and the
1467:and the
1446:Examples
1436:Burnside
1422:and its
1418:) is an
1235:acts on
1043:and the
1002:periodic
930:subgroup
895:-primary
692:Poincaré
537:Solenoid
409:Integers
399:Lattices
374:sporadic
369:Lie type
197:solvable
187:dihedral
172:additive
157:infinite
67:Subgroup
3508:0169921
3480:1918951
3439:0003389
3390:lattice
3385:0168631
3351:0352241
3293:1935567
2774:by the
2315:, and C
2169:coclass
2048:) = GL(
2033:, then
1937:, ...,
1900:, ...,
1731:be the
1559:(1) as
1555:) with
1486:abelian
1141:-group
1023:, then
899:primary
885:Abelian
854:-group
687:Lorentz
609:Unitary
508:Lattice
448:PSL(2,
182:abelian
93:(Semi-)
3596:&
3574:
3556:
3534:
3506:
3478:
3468:
3445:
3437:
3429:
3383:
3375:
3349:
3324:
3299:
3291:
2835:, the
2761:normal
2616:where
2364:with C
2352:with C
2229:Up to
2101:where
1766:where
1665:-group
1611:where
1488:; the
1107:center
1049:center
996:Every
979:simple
924:, the
828:-group
542:Circle
473:SL(2,
362:cyclic
326:-group
177:cyclic
152:finite
147:simple
131:kernel
3443:S2CID
3297:S2CID
3107:proof
2923:Notes
2722:-core
2624:then
2256:and C
2018:. If
1967:>
1758:) of
1707:when
1679:When
1426:is a
1266:is a
1249:socle
1153:with
1130:of a
914:order
908:is a
850:of a
840:power
836:order
832:group
830:is a
726:Sp(∞)
723:SU(∞)
136:image
3572:ISBN
3554:ISBN
3532:ISBN
3466:ISBN
3427:ISSN
3415:1940
3373:LCCN
3322:ISBN
3196:2016
2388:For
2336:For
2198:and
1762:and
1583:Sym(
1366:The
823:, a
720:O(∞)
709:Loop
528:and
3496:doi
3419:doi
3281:doi
3236:doi
2995:so
2529:289
2526:367
2523:487
2396:Dih
2303:, C
2221:is
2112:of
1992:to
1838:(2,
1543:as
1492:Dih
1410:of
1398:/Φ(
1262:If
1189:is
1091:G/H
1058:is
1054:of
1011:If
962:.
951:.
932:of
842:of
808:In
635:Sp(
623:SU(
599:SO(
563:SL(
551:GL(
3621::
3603:.
3504:MR
3502:,
3492:15
3476:MR
3474:,
3464:,
3441:,
3435:MR
3433:,
3425:,
3413:,
3407:,
3381:MR
3379:,
3347:MR
3320:,
3316:,
3295:,
3289:MR
3287:,
3277:12
3275:,
3232:50
3230:.
3226:.
3005:G,
2993:p,
2945:g,
2888:,
2884:,
2880:,
2876:,
2866:.
2697:.
2682:.
2622:m,
2521:49
2443:27
2408:.
2385:.
2225:.
2116:.
2052:,
1975:≤
1950:≤
1935:+1
1927:,
1893:,
1806:.
1802:)/
1519:.
1358:.
1352:+1
1314:≤
1259:.
1213:.
1197:=
1157:=
1081:,
1035:≤
1008:.
988:.
904:A
901:.
881:.
611:U(
587:E(
575:O(
33:→
3609:.
3498::
3421::
3367:n
3365:(
3283::
3256:)
3252:(
3242:.
3238::
3212:)
3208:(
3198:.
3186:p
3173:)
3169:(
3160:)
3156:(
3147:)
3143:(
3130:(
3121:)
3117:(
3075:.
3070:p
3066:C
3057:p
3053:C
3049:=
3046:G
3036:g
3020:2
3016:p
3001:h
2997:g
2975:2
2971:p
2966:C
2962:=
2959:G
2949:G
2941:p
2937:p
2841:p
2807:.
2804:)
2801:G
2798:(
2793:p
2789:O
2777:p
2772:G
2768:p
2764:p
2747:)
2744:G
2741:(
2736:p
2732:O
2720:p
2715:p
2711:P
2707:p
2703:p
2699:p
2691:p
2676:p
2672:G
2668:p
2664:p
2650:,
2645:k
2641:p
2630:P
2626:G
2618:p
2604:m
2599:k
2595:p
2591:=
2588:n
2585:=
2581:|
2577:G
2573:|
2562:p
2554:p
2550:p
2546:p
2542:p
2517:n
2513:n
2485:)
2480:3
2476:/
2472:8
2468:n
2464:(
2461:O
2458:+
2453:3
2449:n
2440:2
2434:p
2423:p
2406:8
2404:Q
2398:4
2390:p
2382:p
2376:p
2372:p
2367:p
2361:p
2355:p
2349:p
2343:p
2338:p
2330:p
2324:p
2318:p
2312:p
2306:p
2300:p
2295:p
2288:2
2284:2
2280:4
2277:V
2270:4
2265:p
2259:p
2253:p
2248:p
2244:p
2239:p
2231:p
2219:p
2208:p
2188:p
2164:p
2157:p
2142:n
2134:p
2130:n
2126:p
2114:n
2107:p
2103:k
2099:p
2095:n
2091:p
2087:1
2084:U
2080:m
2075:m
2071:U
2066:m
2062:U
2054:p
2050:n
2046:V
2042:p
2038:1
2035:U
2031:Z
2028:p
2026:/
2024:Z
2020:V
2015:m
2011:U
2007:V
2002:m
2000:+
1998:i
1994:V
1989:i
1985:V
1981:V
1977:n
1973:m
1969:n
1965:i
1960:i
1956:V
1952:n
1948:i
1943:n
1939:e
1933:i
1929:e
1924:i
1920:e
1915:i
1911:V
1906:n
1902:e
1898:2
1895:e
1891:1
1888:e
1884:n
1880:V
1864:p
1860:p
1858:,
1856:p
1854:(
1852:E
1848:W
1844:p
1840:n
1836:E
1832:p
1828:p
1824:n
1820:p
1816:n
1814:,
1812:p
1810:(
1808:E
1804:P
1800:p
1798:(
1796:E
1792:n
1790:,
1788:p
1786:(
1784:E
1780:p
1778:(
1776:E
1772:P
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1756:p
1754:(
1752:E
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1741:p
1737:G
1729:P
1721:Z
1717:p
1713:n
1709:n
1705:p
1701:n
1699:(
1697:W
1693:n
1691:(
1689:W
1685:n
1681:p
1669:p
1663:p
1657:p
1653:p
1649:p
1645:W
1641:p
1637:p
1633:p
1629:p
1625:p
1621:p
1617:p
1613:k
1609:p
1605:n
1603:(
1601:W
1597:Q
1595:,
1593:n
1589:p
1585:p
1577:p
1573:n
1571:(
1569:W
1565:n
1563:(
1561:W
1557:W
1553:n
1551:(
1549:W
1545:W
1541:p
1537:p
1533:p
1517:n
1498:p
1494:4
1482:p
1475:4
1472:V
1465:4
1462:C
1451:p
1440:p
1432:G
1416:G
1412:G
1404:G
1400:G
1396:G
1392:G
1384:p
1376:p
1372:p
1356:p
1350:i
1346:Z
1341:i
1337:Z
1332:i
1328:Z
1324:i
1320:p
1316:n
1312:i
1308:p
1304:p
1300:p
1288:Z
1286:/
1284:G
1280:G
1276:Z
1274:/
1272:G
1268:p
1264:G
1257:p
1253:p
1245:p
1241:p
1237:N
1233:G
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1225:p
1221:N
1207:p
1203:Z
1201:/
1199:H
1195:Z
1193:/
1191:N
1187:Z
1185:/
1183:G
1179:Z
1177:/
1175:H
1171:H
1167:N
1163:Z
1159:N
1155:H
1147:H
1143:G
1139:p
1135:H
1128:N
1118:p
1111:p
1087:G
1083:H
1079:G
1075:p
1071:H
1067:Z
1056:G
1052:Z
1037:k
1033:m
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1021:p
1017:G
1013:p
998:p
982:p
971:p
967:p
956:p
949:G
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922:G
918:p
910:p
893:p
888:p
879:p
871:g
867:p
863:n
856:G
852:p
848:g
844:p
826:p
821:p
797:e
790:t
783:v
679:8
677:E
671:7
669:E
663:6
661:E
655:4
653:F
647:2
645:G
639:)
637:n
627:)
625:n
615:)
613:n
603:)
601:n
591:)
589:n
579:)
577:n
567:)
565:n
555:)
553:n
495:)
482:Z
470:)
457:Z
433:)
420:Z
411:(
324:p
289:Q
281:n
278:D
268:n
265:A
257:n
254:S
246:n
243:Z
23:.
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