# Groups 2 Groups Janko Vol 2

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<p>de Gruyter Expositions in Mathematics 47EditorsV. P. Maslov, Academy of Sciences, MoscowW. D. Neumann, Columbia University, New YorkR. O. Wells, Jr., International University, BremenGroups of Prime Power OrderVolume 2byYakov Berkovich and Zvonimir JankoWalter de Gruyter Berlin New YorkAuthorsYakov Berkovich Zvonimir JankoJerusalem str. 53, apt. 15 Mathematisches InstitutAfula 18251 Ruprecht-Karls-Universitt HeidelbergIsrael Im Neuenheimer Feld 288E-Mail: berkov@math.haifa.ac.il 69120 HeidelbergE-Mail: janko@mathi.uni-heidelberg.deMathematics Subject Classification 2000: 20-02, 20D15, 20E07Key words: Finite p-group theory, minimal nonabelian subgroups, metacyclic subgroups, extraspe-cial subgroups, equally partitioned groups, p-groups with given maximal subgroups, 2-groups withfew cyclic subgroups of given order, Wards theorem on quaternion-free groups, 2-groups withsmall centralizers of an involution, Blackburns theorem on minimal nonmetacyclic groups. Printed on acid-free paper which falls within the guidelinesof the ANSI to ensure permanence and durability.ISSN 0938-6572ISBN 978-3-11-020419-3Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.Copyright 2008 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.All rights reserved, including those of translation into foreign languages. No part of this book maybe reproduced or transmitted in any form or by any means, electronic or mechanical, includingphotocopy, recording, or any information storage or retrieval system, without permission in writingfrom the publisher.Typeset using the authors TeX files: Kay Dimler, Mncheberg.Printing and binding: Hubert & Co. GmbH & Co. KG, Gttingen.Cover design: Thomas Bonnie, Hamburg.ContentsList of denitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . viiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv:46 Degrees of irreducible characters of Suzuki ]-groups . . . . . . . . . . 1:47 On the number of metacyclic epimorphic images of nite ]-groups . . . 14:48 On 2-groups with small centralizer of an involution, I . . . . . . . . . . 19:49 On 2-groups with small centralizer of an involution, II . . . . . . . . . . 28:50 Jankos theorem on 2-groups without normal elementary abelian subgroupsof order 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43:51 2-groups with self centralizing subgroup isomorphic to ES . . . . . . . . 52:52 2-groups with D2-subgroup of small order . . . . . . . . . . . . . . . . 75:53 2-groups G with c2(G) = 4 . . . . . . . . . . . . . . . . . . . . . . . . 96:54 2-groups G with cn(G) = 4, n > 2 . . . . . . . . . . . . . . . . . . . . 109:55 2-groups G with small subgroup '. G [ o(.) = 2n) . . . . . . . . . . 122:56 Theorem of Ward on quaternion-free 2-groups . . . . . . . . . . . . . . 134:57 Nonabelian 2-groups all of whose minimal nonabelian subgroups are iso-morphic and have exponent 4 . . . . . . . . . . . . . . . . . . . . . . . 140:58 Non-Dedekindian ]-groups all of whose nonnormal subgroups of the sameorder are conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147:59 ]-groups with few nonnormal subgroups . . . . . . . . . . . . . . . . . 150:60 The structure of the Burnside group of order 212. . . . . . . . . . . . . 151:61 Groups of exponent 4 generated by three involutions . . . . . . . . . . . 163:62 Groups with large normal closures of nonnormal cyclic subgroups . . . . 169:63 Groups all of whose cyclic subgroups of composite orders are normal . . 172vi Groups of prime power order:64 ]-groups generated by elements of given order . . . . . . . . . . . . . . 179:65 A2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188:66 A new proof of Blackburns theorem on minimal nonmetacyclic 2-groups 197:67 Determination of U2-groups . . . . . . . . . . . . . . . . . . . . . . . . 202:68 Characterization of groups of prime exponent . . . . . . . . . . . . . . . 206:69 Elementary proofs of some Blackburns theorems . . . . . . . . . . . . 209:70 Non-2-generator ]-groups all of whose maximal subgroups are 2-generator 214:71 Determination of A2-groups . . . . . . . . . . . . . . . . . . . . . . . . 233:72 An-groups, n > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248:73 Classication of modular ]-groups . . . . . . . . . . . . . . . . . . . . 257:74 ]-groups with a cyclic subgroup of index ]2. . . . . . . . . . . . . . . 274:75 Elements of order _ 4 in ]-groups . . . . . . . . . . . . . . . . . . . . 277:76 ]-groups with few A1-subgroups . . . . . . . . . . . . . . . . . . . . . 282:77 2-groups with a self-centralizing abelian subgroup of type (4. 2) . . . . . 316:78 Minimal nonmodular ]-groups . . . . . . . . . . . . . . . . . . . . . . 323:79 Nonmodular quaternion-free 2-groups . . . . . . . . . . . . . . . . . . 334:80 Minimal non-quaternion-free 2-groups . . . . . . . . . . . . . . . . . . 356:81 Maximal abelian subgroups in 2-groups . . . . . . . . . . . . . . . . . . 361:82 A classication of 2-groups with exactly three involutions . . . . . . . . 368:83 ]-groups G with D2(G) or D+2(G) extraspecial . . . . . . . . . . . . . 396:84 2-groups whose nonmetacyclic subgroups are generated by involutions . 399:85 2-groups with a nonabelian Frattini subgroup of order 16 . . . . . . . . 402:86 ]-groups G with metacyclic D+2(G) . . . . . . . . . . . . . . . . . . . 406:87 2-groups with exactly one nonmetacyclic maximal subgroup . . . . . . . 412:88 Hall chains in normal subgroups of ]-groups . . . . . . . . . . . . . . . 437:89 2-groups with exactly six cyclic subgroups of order 4 . . . . . . . . . . 454Contents vii:90 Nonabelian 2-groups all of whose minimal nonabelian subgroups are of or-der 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463:91 Maximal abelian subgroups of ]-groups . . . . . . . . . . . . . . . . . 467:92 On minimal nonabelian subgroups of ]-groups . . . . . . . . . . . . . . 474AppendixA.16 Some central products . . . . . . . . . . . . . . . . . . . . . . . . . . . 485A.17 Alternate proofs of characterization theorems of Miller and Janko on 2-groups, and some related results . . . . . . . . . . . . . . . . . . . . . . 492A.18 Replacement theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 501A.19 New proof of Wards theorem on quaternion-free 2-groups . . . . . . . . 506A.20 Some remarks on automorphisms . . . . . . . . . . . . . . . . . . . . . 509A.21 Isaacs examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512A.22 Minimal nonnilpotent groups . . . . . . . . . . . . . . . . . . . . . . . 516A.23 Groups all of whose noncentral conjugacy classes have the same size . . 519A.24 On modular 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 522A.25 Schreiers inequality for ]-groups . . . . . . . . . . . . . . . . . . . . . 526A.26 ]-groups all of whose nonabelian maximal subgroups are either absolutelyregular or of maximal class . . . . . . . . . . . . . . . . . . . . . . . . 529Research problems and themes II . . . . . . . . . . . . . . . . . . . . . . . . 531Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594List of denitions and notationsSet theory[M[ is the cardinality of a set M (if G is a nite group, then [G[ is called its order).. M (. , M) means that . is (is not) an element of a set M. N _ M (N ,_ M)means that N is (is not) a subset of the set M; moreover, if M = N _ M we writeN c M. is the empty set.N is called a nontrivial subset of M, if N = and N c M. If N c M we say thatN is a proper subset of M.M N is the intersection and M LN is the union of sets M and N. If M. N are sets,then N M = {. N [ . , M] is the difference of N and M.Z is the set (ring) of integers: Z = {0. 1. 2. . . . ].N is the set of all natural numbers.Q is the set (eld) of all rational numbers.R is the set (eld) of all real numbers.C is the set (eld) of all complex numbers.Number theory and general algebra] is always a prime number. is a set of primes; tis the set of all primes not contained in .m, n, k, r, s are, as a rule, natural numbers.(m) is the set of prime divisors of m; then m is a -number.n; is the ]-part of n, nt is the -part of n.(m. n) is the greatest common divisor of m and n.m [ n should be read as: m divides n.m n should be read as: m does not divide n.List of denitions and notations ixGF(]n) is the nite eld containing ]nelements.F+is the multiplicative group of a eld F.L(G) is the lattice of all subgroups of a group G.If n = ]11 . . . ]kk is the standard prime decomposition of n, then z(n) =Pki=1i.GroupsWe consider only nite groups which are denoted, with a pair exceptions, by uppercase Latin letters.If G is a group, then (G) = ([G[).G is a ]-group if [G[ is a power of ]; G is a -group if (G) _ .G is, as a rule, a nite ]-group.H _ G means that H is a subgroup of G.H < G means that H _ G and H = G (in that case H is called a proper subgroupof G). {1] denotes the group containing only one element.H is a nontrivial subgroup of G if {1] < H < G.H is a maximal subgroup of G if H < G and it follows from H _ M < G thatH = M.H E G means that H is a normal subgroup of G; moreover, if, in addition, H = Gwe write H {1].H is a minimal normal subgroup of G if (a) H E G; (b) H > {1]; (c) N < G andN < H implies N = {1]. Thus, the group {1] has no minimal normal subgroup.G is simple if it is a minimal normal subgroup of G (so [G[ > 1).H is a maximal normal subgroup of G if H < G and GH is simple.The subgroup generated by all minimal normal subgroups of G is called the socle ofG and denoted by Sc(G). We put, by denition, Sc({1]) = {1].NG(M) = {. G [ .-1M. = M] is the normalizer of a subset M in G.CG(.) is the centralizer of an element . in G : CG(.) = {: G [ :. = .:].CG(M) =Tx CG(.) is the centralizer of a subset M in G.If _ T and . T E G, then CG(T) = H, where H = CG(T).x Groups of prime power orderwr T is the wreath product of the passive group and the transitive permutationgroup T (in what follows we assume that T is regular); T is called the active factor ofthe wreath product). Then the order of that group is [[|B|[T[.Aut(G) is the group of automorphisms of G (the automorphism group of G).Inn(G) is the group of all inner automorphisms of G.Out(G) = Aut(G)Inn(G), the outer automorphism group of G.If a. b G, then ab= b-1ab.An element . G inverts a subgroup H _ G if hx= h-1for all h H.If M _ G, then 'M) = '. [ . M) is the subgroup of G generated by M.Mx= .-1M. = {.x[ . M] for . G and M _ G... .| = .-1.-1.. = .-1.+is the commutator of elements .. . of G. If M. N _ Gthen M. N| = '.. .| [ . M. . N) is a subgroup of G.o(.) is the order of an element . of G.An element . G is a -element if (o(.)) _ .G is a -group, if (G) _ . Obviously, G is a -group if and only if all of itselements are -elements.Gtis the subgroup generated by all commutators .. .|, .. . G (i.e., Gt= G. G|),G(2)= Gt. Gt| = Gtt= (Gt)t, G(3)= Gtt. Gtt| = (Gtt)tand so on. Gtis calledthe commutator (or derived) subgroup of G.Z(G) =TxG CG(.) is the center of G.Zi(G) is the i -th member of the upper central series of G; in particular, Z0(G) = {1],Z1(G) = Z(G).Ki(G) is the i -th member of the lower central series of G; in particular, K2(G) = Gt.We have Ki(G) = G. . . . . G| (i _ 1 times). We set K1(G) = G.If G is nonabelian, then n(G)K3(G) = Z(GK3(G)).M(G) = '. G [ CG(.) = CG(.;) is the Mann subgroup of a ]-group G.Syl;(G) is the set of ]-Sylow subgroups of an arbitrary nite group G.Sn is the symmetric group of degree n.An is the alternating group of degree n;n is a Sylow ]-subgroup of S;n.GL(n. J) is the set of all nonsingular n n matrices with entries in a eld J, then-dimensional general linear group over J, SL(n. J) = { GL(n. J) [ det() =1 J], the n-dimensional special linear group over J.List of denitions and notations xiIf H _ G, then HG =TxG .-1H. is the core of the subgroup H in G andHG=T1_1EG N is the normal closure or normal hull of H in G. Obviously,HG E G.If G is a ]-group, then ]b(x)= [G : CG(.)[; b(.) is said to be the breadth of . G,where G is a ]-group; b(G) = max {b(.) [ . G] is the breadth of G.(G) is the Frattini subgroup of G (= the intersection of all maximal subgroups of G),({1]) = {1], ]d(G)= [G : (G)[.Ii = {H < G [ (G) _ H. [G : H[ = ]i], i = 1. . . . . d(G), where G > {1].If H < G, then I1(H) is the set of all maximal subgroups of H.exp(G) is the exponent of G (the least common multiple of the orders of elements ofG). If G is a ]-group, then exp(G) = max {o(.) [ . G].k(G) is the number of conjugacy classes of G (= G-classes), the class number of G.1x is the G-class containing an element . (sometimes we also write cclG(.)).Cn is the cyclic group of order m.Gnis the direct product of m copies of a group G. T is the direct product of groups and T. + T is a central product of groups and T, i.e., + T = T with . T| = {1].E;m = Cn; is the elementary abelian group of order ]n. G is an elementary abelian]-group if and only if it is a ]-group > {1] and G coincides with its socle. Next, {1]is elementary abelian for each prime ].A group G is said to be homocyclic if it is a direct product of isomorphic cyclic sub-groups (obviously, elementary abelian ]-groups are homocyclic).ES(m. ]) is an extraspecial group of order ]12n(a ]-group G is said to be extraspe-cial if Gt= (G) = Z(G) is of order ]). Note that for each m N, there are exactlytwo nonisomorphic extraspecial groups of order ]2n1.S(]3) is a nonabelian group of order ]3and exponent ] > 2.A special ]-group is a nonabelian ]-group G such that Gt= (G) = Z(G) iselementary abelian. Direct products of extraspecial ]-groups are special.D2n is the dihedral group of order 2m. m > 2. Some authors consider E22 as thedihedral group D4.Q2m is the generalized quaternion group of order 2n_ 23.SD2m is the semidihedral group of order 2n_ 24.M;m is a nonabelian ]-group containing exactly ] cyclic subgroups of index ].xii Groups of prime power ordercl(G) is the nilpotence class of a ]-group G.dl(G) is the derived length of a ]-group G.CL(G) is the set of all G-classes.A]-group of maximal class is a nonabelian group G of order ]nwith cl(G) =...</p>