<<

DE GRUYTER Roland Schmidt EXPOSITIONS IN 14 lattices of Groups de Gruyter Expositions in Mathematics 14

Editors

0. H. Kegel, Albert-Ludwigs-Universitat, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, The University of Melbourne, Parkville R. 0. Wells, Jr., Rice University, Houston de Gruyter Expositions in Mathematics

I The Analytical and Topological Theory of , K. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.) 2Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues

3The Stefan Problem, A. M. Meirmanov 4Finite Soluble Groups, K. Doerk, T. O. Hawkes

5The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin

6Contact and Linear Differential , V. R. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin

7Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, M. V. Zaicev

8Nilpotent Groups and their Automorphisms, E. I. Khukhro 9Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug 10The Link Invariants of the Chern-Simons Theory, E. Guadagnini

1 I Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao

12Moduli Spaces of Abelian Surfaces: Compactification, Degeneration, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub

13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Na- zarov, B. A. Plamenevsky Subgroup Lattices of Groups

by

Roland Schmidt

W DE G Walter de GruyterBerlin New York 1994 Author Roland Schmidt Mathematisches Seminar Universitat Kiel Ludewig-Meyn-StraBe 4 D-24098 Kiel Germany

1991 Mathematics Subject Classification: 20-01, 20-02 Keywords: , , subgroup lattice, projectivity,

® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data

Schmidt, Roland, 1944- Subgroup lattices of groups / by Roland Schmidt. p. cm. - (De Gruyter expositions in mathematics ; 14) Includes bibliographical references and index. ISBN 3-11-011213-2 1. .2. Lattice theory.I. Title.II. Series. QA174.2.S36 1994 512'.2 - dc20 94-20795 CIP

Die Deutsche Bibliothek - Cataloging-in-Publication Data

Schmidt, Roland: Subgroup lattices of groups / by Roland Schmidt. - Berlin ; New York : de Gruyter, 1994 (De Gruyter expositions in mathematics ; 14) ISBN 3-11-011213-2 NE: GT

© Copyright 1994 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting: ASCO Trade Ltd. Hong Kong. Printing: Gerike GmbH, Berlin. Binding: Liideritz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg. To Traute and AlexanderKatharinaViktoriaJohannesMaria

Preface

The central theme of this book is the between the structure of a group and the structure of its lattice of . The origin of the subject may be traced back to Dedekind, who studied the lattice of ideals in a of algebraic ; he discovered and used the modular identity, which is also called the Dedekind law, in his calculation of ideals. But the real history of the theory of subgroup lattices began in 1928 with a paper of Ada Rottlander which was motivated by the Galois cor- respondence between a field extension and its Galois group. Subsequently many group theorists have worked in this field, most notably Baer, Sadovskii, Suzuki, and Zacher. Since the only book on subgroup lattices is still Suzuki's Ergebnisbericht of 1956, there is a clear need for a book on this subject to describe more recent developments. We denote the subgroup lattice of a group G by L(G). If G and Gare groups, an from L(G) onto L(G) is called a projectivity from G to G. Since isomor- phic groups have isomorphic subgroup lattices, there is a correspondence between the of all groups and a certain class of (algebraic) lattices in which a group is sent to its subgroup lattice. This raises the following questions: (A) Given a class X of groups, what can we say about the subgroup lattices of groups in X? (B) Given a class IV of lattices, what can we say about the G such that L(G) a J? Since there are nonisomorphic groups having isomorphic subgroup lattices, we may further ask: (C) If X is a class of groups, what is the lattice-theoretic closure 1 of X? Here X is the class of all groups G for which there exists G e X such that L(G) is isomorphic to L(G). (D) Which classes of groups X satisfy _ X, that is, are invariant under pro- jectivities? If X consists of just one group, Problem (D) becomes the following: (E) Which groups G are determined by their subgroup lattice, that is are isomor- phic to every group G with L(G) isomorphic to L(G)? Every isomorphism between two groups induces a projectivity in a natural way; our final question is: (F) Which groups are not only determined by their subgroup lattice but even have the property that all their projectivities are induced by group ? These are the main problems in the field and they are studied in the first seven chapters of the book. In Chapter 8 we consider dualities between subgroup lattices and in Chapter 9 we briefly handle other lattices associated with a group, namely thelatticesof normal subgroups, composition subgroups, centralizers, and . viii Preface

Although the theory of subgroup lattices is just a small part of group theory, for reasons of space I have been unable to include all the contributions to this theory. I have tried to compensate for omissions by at least including statements and refer- ences wherever possible; also many interesting results are cited in the exercises at the end of each section. However, certain parts of the theory have not been covered at all. In writing what is after all a book on group theory I decided to omit those topics which would have led too far away from this subject. Therefore I do not cover embeddings of lattices in subgroup lattices, lattices of subsemigroups, homotopy properties of subgroup lattices, inductive groupoids, or anything about topological groups, although there are also interesting results in these fields. Furthermore I do not study between subgroup lattices. The reason is that when I started to write this book there had been no activity in this area for more than 30 years, so that Suzuki's book remained an adequate reference. This situation changed in 1985 and especially since 1990, but by then there was no room in the book for a chapter on lattice homomorphisms. Nevertheless I hope that this book will serve as a basic reference in the subject area, as a text for postgraduate studies, and also as a source of research ideas. It is intended for students who have some basic knowledge in group theory and wish to delve a little deeper into one of its many special areas. It is assumed that the reader is familiar with the basic concepts of group theory, but references are given for many well-known group theoretical facts. For this purpose I mainly use the books by Robinson [1982] and Huppert [1967]. I do not assume that the reader is familiar with lattice theory, but the basic facts are developed in § 1.1 and at the beginning of some other sections. For a thorough introduction into lattice theory the reader is referred to the books of Birkhoff, Crawley and Dilworth, or Gratzer. Since the book is intended as a textbook for students, I have included a large number of exercises of varying difficulty. While these exercises are not used in the main text they often give alternate proofs or contain supplementary results for which there was no room in the main text. It is a pleasure to thank the many students and colleagues who have read parts of the manuscript and made helpful suggestions. In particular, I mention O.H. Kegel and G. Zacher who have followed the whole project with much interest. Thanks are also due to Ms. J. Robinson for checking the English text and to Frau G. Christiansen for typing parts of the manuscript. Finally, I thank Dr. M. Karbe and the publisher for their courtesy and cooperation. Contents

Preface...... vii

Notation...... xiii

Chapter 1 Fundamental concepts

1.1 Basic concepts of lattice theory...... 1 1.2 Distributive lattices and cyclic groups...... 11 1.3 Projectivities...... 16 1.4 The group of autoprojectivities ...... 23 1.5 Power automorphisms ...... 29 1.6 Direct products...... 34

Chapter 2 Modular lattices and abelian groups...... 41 2.1 Modular lattices...... 41 2.2 P-groups...... 49 2.3 Finite p-groups with modular subgroup lattices...... 54 2.4 Groups with modular subgroup lattices...... 69 2.5 Projectivities of M-groups...... 88 2.6 Projectivities between abelian groups...... 101

Chapter 3 Complements and special elements in the subgroup lattice of a group...... 114 3.1 Groups with complemented subgroup lattices (K-groups)...... 115 3.2 Special complements...... 121 3.3 Relative complements...... 126 3.4 Neutral elements and related concepts...... 135 3.5 Finite groups with a partition ...... 145

Chapter 4 Projectivities and arithmetic structure of finite groups...... 157 4.1 Normal Hall subgroups ...... 158 4.2 Singular projectivities...... 166 4.3 OO(G), O°(G), and hypercentre...... 178 4.4 Abelian p-subgroups and projectivities...... 185 x Contents

Chapter 5 Projectivities and normal structure of finite groups...... 199 5.1 Modular subgroups of finite groups...... 200 5.2 Permutable subgroups of finite groups...... 216 5.3 Lattice-theoretic characterizations of classes of finite groups ...... 228 5.4 Projective images of normal subgroups of finite groups...... 235 5.5 Normal subgroups of p-groups with cyclic factor group ...... 243 5.6 Normalizers, centralizers and conjugacy classes...... 272

Chapter 6 Projectivities and normal structure of infinite groups...... 281 6.1 Subgroups of finite index...... 282 6.2 Permodular subgroups ...... 288 6.3 Permutable subgroups of infinite groups ...... 303 6.4 Lattice-theoretic characterizations of classes of infinite groups ...... 311 6.5 Projective images of normal subgroups of infinite groups and index preserving projectivities...... 321 6.6 The structure of N°/NU and N°/Na and projective images of soluble groups...... 330

Chapter 7 Classes of groups and their projectivities ...... 343

7.1 Free groups ...... 345 7.2 Torsion-free nilpotent groups...... 360 7.3 Mixed nilpotent groups ...... 374 7.4 Periodic nilpotent groups ...... 385 7.5 Soluble groups...... 395 7.6 Direct products of groups...... 406 7.7 Groups generated by involutions...... 421 7.8 Finite simple and lattice-simple groups...... 439

Chapter 8 Dualities of subgroup lattices...... 452 8.1 Abelian groups with duals...... 452 8.2 The main theorem...... 458 8.3 Soluble groups with duals...... 462 8.4 Finite groups with duals...... 468 8.5 Locally finite groups with duals...... 476

Chapter 9 Further lattices...... 485 9.1 Lattices of normal subgroups ...... 486 9.2 Lattices of subnormal subgroups...... 498 Contents xi

9.3 Centralizer lattices ...... 511 9.4 lattices...... 527

Bibliography...... 542 Index of Names ...... 561 Index of Subjects ...... 564

Notation

Throughout this list, G denotes a group, I a class of groups, L a lattice, p a prime, it a set of primes, n a natural number, x and y elements of G, u and v elements of L, and cp a projectivity fromG to a group G.

IV, fro, 7L set of natural numbers, nonnegative integers, integers 0, IB, C fieldof rational numbers, real numbers, complex numbers P set of all primes GF(p") field with p" elements Up group of p-adic units XS-= Y,X Y X is a , a proper subset of the set Y X\Y set of all elements in X which are not contained in Y IXI of the set X a divides b, a does not divide b, (a, b e 7L) (a, b) greatest common of a and b, (a, b e 7L) P nC P\(p), P\n H=K H is isomorphic with K DrX, direct product AeA

Lattice theory notation

S,n,U partial , intersection, in L ns,us greatest lower bound, least upper bound of the subset S , least, greatest element of L [u/v] interval of all w e L such that v< w 5 u u mod L u is a modular element of L C(L) set of all cyclic elements of L M" lattice of length 2 with n

Group theory notation

Y-'xY x-'x' where s is an element or an automorphism of G cyclic subgroup generated by x order of x (y°)-'(x°)-'(xy)°, amorphy of the map a xiv Notation

HS G, H < G H is a subgroup, a proper subgroup of G HgG H is a of G HgaG H is a of G H mod G H is a modular subgroup of G H pmo G H is a permodular subgroup of G H per G H is a permutable subgroup of G IG:HI index of the subgroup H in G group presented by generators X and relations R H wr K (standard restricted) of H with K Cr Gx cartesian product of groups Gx .SEA Aut G, Inn G automorphism group, group of G Pot G group of power automorphisms of G End G set of endomorphisms of G Hom (A, B) set of homomorphisms from A to B P(G) group of autoprojectivities of G PA(G), (G) group of all autoprojectivities of G induced by auto- morphisms, by inner automorphisms G" (x"lx a G> ((G), ("(G) (xeGlxP= 1>,(xeGlx°"= 1) if G is a p-group U"(G) GP" if G is a p-group Z(G) centre of G N(G) norm of G [X, S] ([x, s] I x e X, s E S> for subsets X of G and S of G or Aut G G'=[G,G] commutator subgroup of G Com G set of commutators of elements of G Z"(G), K"(G), Gc"' terms of the upper , the lower central series, the derived series of G Z.(G) hypercentre of G F(G) Fitting subgroup of G cD(G) of G Notation xv

F"(G), 1V"(G) terms of the upper Fitting series, the lower Fitting series of G R(G) Hirsch-Plotkin radical of G S(G) locally soluble radical of G G.' I-radical of G G, I-residual of G Op(G), O"(G) maximal normal p-subgroup, 7r-subgroup of G Op(G), O"(G) minimal normal subgroup of G with factor group a p- group, a n-group Gp p-component of the nilpotent torsion group G T(G) torsion subgroup of G Sylp(G) set of Sylow p-subgroups of G Exp G exponent of G 7r(G) set of all primes dividing the order of an element of G c(G) nilpotency class of G d(G) derived length of G 1p(G) p-length of G ro(G) torsion-free rank of the G LX class of all locally 1-groups W, 91, a classes of abelian, nilpotent, soluble groups P(n, p) P-groups Sym X, S. on the set X, on (1,...,n} Alt X, An on the set X, on (1, ... , n} C", CW of order n, of infinite order Cp= Prefer group, quasicyclic group D. of order n Q2 generalized group of order 2" GL(V), GL(n, F), GL(n, q) SL(V), SL (n, F), SL (n, q) PGL(n, q), PSL(n, q) projective general linear and projective special linear groups PSU(V), PQ(V), PSU(V) projective symplectic, orthogonal, unitary groups Sz(q) Suzuki groups L(G) subgroup lattice of G L"(G) set of all subgroups of G generated by n elements 92(G) lattice of normal subgroups of G R(G) lattice of composition subgroups of G (!(G) centralizer lattice of G 9R(G) coset lattice of G [X/Y]N, [X/Y]K, [X/Y]cinterval in 91(G), R(G), (1(G)

Chapter 1

Fundamental concepts

In this first chapter we briefly introduce the basic concepts of lattice theory and develop the elementary properties of subgroup lattices and projectivities of groups that will be needed later. In § 1.1, in addition to the purely lattice-theoretic concepts, we define the sub- group lattice L(G) of a group G, projectivities between groups, and introduce certain sublattices and meet-sublattices of L(G), namely the lattices of normal subgroups, composition subgroups, centralizers, and cosets of G. The main result of § 1.2 is Ore's theorem of 1938 which characterizes the groups with distributive subgroup lattices as the locally cyclic groups. It yields the basic fact that every projectivity maps cyclic subgroups to cyclic subgroups. So it seems rea- sonable to try to construct projectivities between two groups G and G using maps between the sets of cyclic subgroups or even element maps between G and G. This we study in § 1.3, and then prove Sadovskii's approximation theorems which enable us to extend isomorphisms inducing a given projectivity cp on certain sets of sub- groups or factor groups of G to an isomorphism inducing cp on the whole group. In § 1.4 we study the group P(G) of all autoprojectivities of G and introduce the subgroups PA(G) and PI(G) consisting of all autoprojectivities induced by automor- phisms and inner automorphisms, respectively. The of the natural map from Aut G to PA(G) is the group Pot G of all power automorphisms of G. In § 1.5 we prove Cooper's theorem that every power automorphism is central and determine the power automorphisms of abelian groups. Finally, in § 1.6 we investigate direct products of groups and prove Suzuki's theo- rem that the subgroup lattice of a group G is a direct product if and only if G is a direct product of coprime torsion groups. As a nice application we obtain that the number of groups whose subgroup lattice is isomorphic to a given finite lattice L is finite if and only if L has no chain as a direct factor.

1.1 Basic concepts of lattice theory

A or poset is a set P together with a < such that the following conditions are satisfied for all x, y, z e P: (1) x < X. (Reflexivity) (2) If x < y and y < x, then x = y. (Anti symmetry) (3) If x < y and y < z, then x < z. (Transitivity) 2 Fundamental concepts

An element x of a poset P is said to be a lower bound for the subset S of P if x < s for every s e S. The element x is a greatest lower bound of S if x is a lower bound of S and y < x for any lower bound y of S. By (2), such a greatest lower bound of S is unique if it exists; we denote it by inf S or n S. Similar definitions and remarks apply to upper bounds and the least upper bound; the latter is denoted by sup S or U S.

Lattices

A lattice is a partially ordered set in which every pair of elements has a least upper bound and a greatest lower bound. A partially ordered set in which every subset has a least upper bound and a greatest lower bound is called a complete lattice. We can also view lattices as with two binary operations. 1.1.1 Theorem. (a) Let (L, <) be a lattice and define the operations n and u on L (called intersection and union, respectively) by (4) x n y = inf {x, y} and x u y = sup {x, y}. Then the following properties hold for all x, y, z e L: (5) xny = y n x and x u y= y u x. (Commutativity) (6) (x n y) n z= x n (y n z) and (x u y) u z= x u (y u z). (Associativity) (7) x n (x u y) = x and x u (xny) = x. (Absorption identities) Furthermore, we have x < y if and only if x = x n y (or y = y u x). (b) Conversely, let L be a set with two binary operations n and u satisfying (5)-(7) and define the relation S on L by (8) x 5yif and only if x =xny. Then (L, <) is a lattice with x n y = inf{x, y} and x u y = sup{x, y} for all x, y eL. Proof. (a) Clearly, the operations defined in (4) are commutative. Also it is easy to see that (x n y) n z and x n (y n z) both are the greatest lower bound of {x, y, z}. Hence (x n y) n z = x n (y n z) and similarly (x u y) u z = x u (y u z).Furthermore x < y if and only if x = inf{x, y} = xny (or y = sup{y, x} = y u x). Since x

Since (xny)nx=xn(xny)=(xnx)ny=xny,xny

Subgroup lattice

If G is any group, the set L(G) of all subgroups of G is partially ordered with respect to set inclusion. Moreover any subset of L(G) has a greatest lower bound in L(G), the intersection of all its elements, and a least upper bound in L(G), the join of all its elements. Thus L(G) is a complete lattice, the subgroup lattice of G. We denote the operations of this lattice by n and u. So we shall write X n Y for the intersection, and X u Y but sometimes also for the join of the subgroups X and Y of G. Also, for a set .9' of subgroups of G, U .9' and

Minimal and maximal elements, chains and antichains

For elements x, y of a poset P we write x < y if x 5 y and x 0 y. If x < yand there is no element z e P such that x < z < y, then we say that x is covered by y or that y covers x. Let S be a subset of P and x e S. Then x is a minimal element of S if there is no s e S with s < x, and x is a least element of S if x 5 s for every s eS. Certainly the least element of S is a minimal element of S and is unique if it exists; on the other hand, it may happen that S contains many minimal elements but no least element. Similar definitions and remarks apply to maximal elements and the greatest element of S. We write 0 and I for the least and greatest elements of P, respectively (if they exist). An element of P that covers 0 is called an , an element that is covered by I is an antiatom of P. If L is a complete lattice, then inf L is the least and sup L the greatest element of L. In the subgroup lattice of the group G the trivial subgroup 1 is the least element, G is the greatest element, the minimal subgroups are the atoms and the maximal subgroups are the antiatoms of L(G). Two elements x and y in a poset P are called comparable if x < y or y 5 x. A subset S of P is a chain if any two elements in S are comparable, S is an antichain if no two different elements of S are comparable. The length of a finite chain S is ISI - 1. The poset P is said to be of length n, where n is a natural number, if there is a chain in P of length n and all chains in P are of length at most n. A poset P is of finite length if it is of length n for some n e N. Similarly P is said to be of width n if there is an antichain with n elements in P and all antichains in P have at most n elements. 4 Fundamental concepts

A poset P with the property that each of its nonempty subsets contains a maximal element is said to satisfy the maximal condition or the ascending chain condition. The second name comes from the following alternative formulation: P satisfies the maxi- mal condition if and only if it contains no infinite sequence of elements x1, x2, ... such that x1 < x2 < x3 < - The minimal condition or descending chain condition is defined correspondingly.

Isomorphisms and projectivities

Let L and L be lattices. A map a: L -. L is called a if (10) (xny)° = x°ny° and (xuy)° = x°uy° for all x, y c L. A homomorphism is an isomorphism if it is bijective, and an isomor- phism of a lattice with itself is called an automorphism. We write L - L if L and L are isomorphic. Finally, we introduce a shorter name for isomorphisms between sub- group lattices. If G and G are groups, an isomorphism from L(G) to L(G) is called a projectivity from G to G. We also say that G and G are lattice-isomorphic if there exists a projectivity from G to G. In order to show that a bijective map between two lattices is an isomorphism it suffices to prove that it has one of the two properties in (10) or that it preserves the order relations of the lattices. In addition, every isomorphism preserves union and intersection of arbitrary subsets.

1.1.2 Theorem. Let a be a bijective map of a lattice L to a lattice L. Then the follow- ing properties are equivalent. (a) For all x,yEL,x

Proof. Let x, y c- L. By 1.1.1, x < y if and only if x n y = x (or x u y =y). It follows that (b) (and also (c)) implies (a). Conversely, if a satisfies (a) and if S is a subset of L such that z = n S exists, then z° is a lower bound of S° and every lower bound of S° is of the form w° where w is a lower bound of S. Since z is the greatest lower bound of S, w < z and hence w° < z°. This shows that z° is a greatest lower bound of S° and that (n S)° = n S°. In particular (b) holds. The corresponding result for the least upper bound follows similarly.

An easy, but useful, consequence of this result is that a projectivity of a group is determined by its action on the cyclic subgroups.

1.13 Corollary. If q and 0 are projectivities of a group G such that X1 = X'' for every cyclic subgroup X of G, then cp = 0. 1.1 Basic concepts of lattice theory 5

Proof. Let H < G. Then by 1.1.2, HW _ (xU(x>)4= zU(x)- = xU = H.

Hasse diagram

Every finite poset P, in particular every finite lattice, may be represented by a dia- gram. Represent each element of P by a point in the plain in such a way that the point py associated with an element y lies above the point associated with x when- ever x < y. Then, whenever y covers x, connect the points px and p, by a line segment. We call the resulting figure a Hasse diagram of P. It is easy to see that it is possible to construct such a diagram. For the finite poset P has a maximal element z, P\ {z} is a smaller poset, and therefore, by induction, has a diagram; now choose a point p= above the points of this diagram and connect it with all the points px associated to elements x covered by z. Figure 1 shows the Hasse diagrams of all lattices with at most five elements.

A, A, A3 A, B, A5 B5 C5 D5 E5

Figure 1

Sublattices

A subset of a lattice is called a sublattice if it is closed with respect to the operations n and u defined in I.I.I. It is evident that a sublattice is a lattice relative to the induced operations. Examples of sublattices of a lattice L are, for x, y E L, the sets S = {x, y, x n y, x u y} or, if x < y, the intervals

[y/x] = {z E Llx < z < y}.

We mention two sublattices of the subgroup lattice of a group G.

Lattice of normal subgroups

SJR(G) = {NON L:3 G}. 6 Fundamental concepts

Lattice of composition subgroups

We say that the subgroup H of G is subnormal in G, and we write H g a G, if there are a nonnegative n and a series (11) H=H, =G of subgroups of G. We say that H is a composition subgroup of G, if there exists a series (11) for H in which all the factor groups H./H;+1 are simple; we then call n the type of H and (11) a composition series from G to H of length n. Clearly a composi- tion subgroup is always subnormal, but the converse is false.

1.1.4 Lemma. If H is a composition subgroup of type n in G, H < K < G and K :9:9 G, then there exist composition series of lengths less than n from G to K and from K to H.

Proof. Let K = K. a Krn_1 9 g KO = G and let (11) be a composition series from G to H. Then (12) =G is a normal series from G to H. By the Schreier refinement theorem, the normal series (11) and (12) have isomorphic refinements. There is no proper refinement of the composition series (11). Hence the isomorphic refinement of (12) is a composition series from G to H of length n. Since H < K < G, the two parts from G to K and from K to H of this series are composition series of lengths less than n. El

1.15 Theorem (Wielandt [1939]). Let G be a group. The set $1(G) of all composition subgroups of G is a sublattice of the subgroup lattice L(G).

Proof. Let A, B e R(G). Then there are composition series

and =G from G to A and B, respectively. We use induction on min(m, n) to prove that A n B e R(G), and we may assume that m 5 n. If m = 0, then A n B = B e R(G);so let m >_ 1. Since Al SA1uB=A1uB,9A1uBn_1 =G and G/A1 is simple, we have either Al u B = Al or Al u B = G. In the first case, Al n B = B; in the second, Al n B a B and B/A1 n B G/A1 is simple. In both cases, Al n B e R(G) and 1.1.4 yields that Al n B e R(A1). By induction A n B = A n (A1 n B) e R(A1), and so A n B e R(G) since G/A1 is simple. We use induction on the ordered pair (max(m, n), min(m, n)) to prove that A U B e R(G). Since this is clear if m = 0 or n = 0, we may assume that m, n >_ 1 and that the join of two composition subgroups of types r and s in a group is again a composition subgroup whenever r 5 s < max(m, n) or s = max(m, n) and r < min(m, n). Suppose 1.1 Basic concepts of lattice theory 7 that A is not normal in A v B. Then there exists b e B such that A6 # A. Since A1 :Q G, Ab < A, and there are composition series of lengths m - 1 < max(m, n) from A, to A and to Ab. By induction A u Ab e $1(A,) and hence A u Ab 9 9 G. By 1.1.4 there is a composition series of length less than m from G to A u A6. By induction A u B = (A u Ab) u B E R(G), as desired. So we may assume that A 9 A u B and, similarly, B 9 A u B. Again by induction A._, v B E R(G). If Am_, u B # G, then by 1.1.4 there exist composition series of lengths less than m and n from Am_, u B to A and B, respectively; by induction A u B E R(Am_, U B) and therefore A u B E R(G). So we may finally assume that Am_, u B = G and, similarly, A u G. Then A 9 G since A is normalized by Am_, and B. Similarly B:9 G and so AuB9G.By 1. 1.4, A u B e R(G).

Lemma 1.1.4 shows that in a group G with a composition series (from G to 1) every subnormal subgroup is a composition subgroup. Hence in these groups, in particular in finite groups, subnormal and composition subgroups coincide. There- fore, in this case, R(G) is also called the lattice of subnormal subgroups of G. For subnormal subgroups A and B in an arbitrary group G, it is clear that A n B 9 9 G, but A u B in general is not subnormal in G. The reader will find examples and a thorough investigation into the join problem in the book by Lennox and Stonehewer [1987].

Meet-sublattices

Let L be a lattice and M a subset of L which is a lattice relative to the induced partial order. Clearly, M need not be a sublattice of L; the subgroup lattice of a group G as a subset of the lattice of all subsets of G is an example. If n and n denote the intersection in L and M, respectively, then for x, y e M, x A y is a lower bound of {x, y} in L and so x A y S x n y. We call M a meet-sublattice of L if x A y= xn y for all x, y E M, and M is a complete meet-sublattice of L if M and L are complete lattices and for every subset S of M the greatest lower bounds of S in M and L coincide. Actually, one need not require that M is a lattice here.

1.1.6 Lemma. If M is a subset of a complete lattice L such that for every subset S of M the greatest lower bound n S of sin L is contained in M, then M is a complete meet-sublattice of L.

Proof. For a subset S of M let S* be the set of upper bounds of S in M. By hypothe- sis, inf S* E M. Every s E S is a lower bound of S*, hence s S inf S*. Thus inf S* is an upper bound of S and then, certainly, the least upper bound of S in M. Clearly, inf S is the greatest lower bound of S in M. Hence M is a complete lattice and the greatest lower bounds of S in M and L coincide.

If G is a group, L(G) is an example of a complete meet-sublattice of the lattice of all subsets of G. We introduce two further lattices of this kind. 8 Fundamental concepts

Centralizer lattice

Let (1(G) be the set of all centralizers of subgroups of G. Since for Hi 5 G,

in C6(Hi) =Ca Hi I e I(G),

(4;(G) is a complete meet-sublattice of L(G); in general it is not a sublattice (see Exercise 7). We call E(G) the centralizer lattice of G. The least element of (f(G) is the centre Z(G), the greatest element is G.

Coset lattice

Let 91(G) be the set of all right cosets of subgroups of G together with the 0. If 9 is a nonempty subset of 91(G), then either n .9' = 0 e 91(G) or there exists an element x en .. In the latter case, 9 = {HixIi e I} for certain subgroups Hi of G, and hence n.5" = (n Hi) x e 91(G). If .5" is the empty subset of 91(G), then n .5" = G E 91(G). By 1.1.6, 91(G) is a complete meet-sublattice of the lattice of all subsets of G. Since Hx = x(x-'Hx) for all H 5 G and x e G, every right coset is a left coset, and conversely. So 91(G) is the set of all (right or left) cosets in G and we call 91(G) the coset lattice of G. The subgroups of G are precisely the cosets containing the coset 1. Hence L(G) is the interval [G/1] in 91(G).

Direct products

Let (LA)AE Abe a family of lattices. The direct product, L = Dr Lx, is the lattice whose AEA underlying set is the cartesian product of the sets Lx, that is the set of all functions f defined on A such that f (A) e Lx for all ). a A, and whose partial order is defined by the rule that f 5 g if and only if f(A) < g(A) for all A E A. Clearly, f n g and f ug are the functions mapping each A to f(2) n g(A) and f(l) u g(.1), respectively, so that L is a lattice in which intersection and union are performed componentwise. It follows that L satisfies a given lattice identity if and only if each Lx satisfies this identity. If A is finite, say A = { I_-, n} for some n e ICI, we also use the simpler n-tupel notation for the elements of L. Thus, in this case,

L= L,x xL. = and (x...... x.) 5 if and only if xi < yi for all i = 1, ..., n. We shall need the following simple result which clarifies the structure of a direct product of lattices with greatest and least elements.

1.1.7 Lemma. Let (LA)AEA be a family of lattices, and put L = Dr L.J. Suppose that L leA contains a least element 0 and a greatest element I. If A a A, then 0(2) is the least 1.1 Basic concepts of lattice theory 9 and I(A) the greatest element of LA. Define fA e L by ft(µ) = 0(µ) for A # µ E A and f(A) = 1(A1). Then (a) U f,,=land Uf, nf,=0, veA vEA\{.Z} (b) [ ft/0] z LA, and (c) [x u y/O] [x/O] x [y/0] if x e [fx/O], y e [ f,/O] and A µ E A.

Proof. For z e LA, the axiom of choice implies that there is a function f E L with f(A) = z. From 0S f 5 I it follows that 0(A) S z S 1(.1). This shows that 0(.1) is the least and 1(2) the greatest element of LA. Since union and intersection in L are performed componentwise, (a) is clear. The map a: Lx - [ fZ/0] defined by z°(v) = 0(v) for A # v e A and z°(.1) = z clearly is an isomorphism; thus (b) holds. Finally, the map

T: [x/0] x [y/O] - [x v y/0]; (u, v) i-- w where w(1) = u(.1), w(µ) = v(µ) and w(v) = 0(v) for v e A with A # v # z also is an isomorphism, and this implies (c).

Embedding of lattices in subgroup lattices

We mention without proof three results on subgroup lattices that have little connec- tion with the topics treated in this book. They show how complex subgroup lattices can be.

1.1.8 Theorem (Whitman [1946]). Every lattice is isomorphic to a sublattice of the subgroup lattice of some group.

The reader can find a proof of this result in Whitman's paper or in the book by Gratzer [1978]. Whitman first shows that every lattice L is isomorphic to a sub- lattice of a partition lattice of some set X; this lattice then is isomorphic to a sub- lattice of the subgroup lattice of the symmetric group Sym X on X (see Exercise 9). The set X constructed by Whitman in general is infinite even when L is finite. However, with a different construction, it is possible to keep the set X finite in this case.

1.1.9 Theorem (Pudlak and Tuma [1980]). Every finite lattice is isomorphic to a sublattice of the subgroup lattice of some .

The situation is more complex if one tries to replace "sublattice" by "interval" in these statements. An element c in a complete lattice L is called compact if whenever S c L and c <_ U S there exists a finite subset T S S with c :!g U T; and a lattice is called algebraic if it is complete and each of its elements is a join of compact 10 Fundamental concepts elements. Now if H is a subgroup of a group G and g e G, then

1.1.10 Theorem (Tuma [1989]). Every algebraic lattice is isomorphic to an interval in the subgroup lattice of some group.

The finite analogue, however, is an open problem even for the simplest lattices, for example, for the lattice M of length 2 with n atoms. It is easy to construct finite (soluble) groups with such an interval if n - 1 is a prime power (see Exercise 11) and the lattices M7 and M11 occur in the alternating group A31 (see Feit [1983], Palfy [1988]). However, if n > 13 and n - 1 is not a prime power, it is not known whether M,. is an interval in the subgroup lattice of some finite group.

Exercises

1. LetLbealatticeandletx,y,zeL. (a) If y with o(a) = 8, o(b) = 2 and the group

1.2 Distributive lattices and cyclic groups

In this section we are going to study subgroup lattices of cyclic groups. Since these turn out to be distributive, we shall first determine all groups with distributive subgroup lattices.

Distributive lattices

A lattice L is called distributive if for all x, y, z e L the distributive laws hold: (1) xu(ynz)=(xuy)n(xuz), (2) xn(yuz)=(xny)u(xnz).

1.2.1 Remark. Since distributivity is a property defined by identities, sublattices and direct products of distributive lattices are distributive. Furthermore, a lattice L is already distributive if it satisfies one of the two distributive laws. For example, if (1) holds in L, then for x, y, z e L we have

(xny)u(xnz) _ ((xny)ux)n((xny)uz) = xn(zu(xny))

= xn((zvx)n(zuy)) = xn(yuz); thus (2) holds. The other implication follows similarly.

It is easy to see that every chain is a . We shall also need the following distributive lattices. For a, b c No, the set of nonnegative integers, write a 5'b if b divides a. Clearly, < is a partial order on No for which sup{a,b} is the and inf{a, b} is the of a and b. Thus (NO, S') is a lattice with greatest element 1 and least element 0. We denote this lattice by T., and for n e N, we write T, for the interval [1/n] in T.. Then T is the lattice of all of n. The reader may check that T. is distributive; this will, however, also follow from 1.2.2 and 1.2.3. As sublattices of T., all the T,. are distributive as well.

72

Figure 2: T72 12 Fundamental concepts

Cyclic groups

Let n be a natural number or the symbol oo. We write n e N u {oo} to express this and we denote the cyclic group of order n by C. Let C = . When n = oo, it is well-known that for every r e N there is a subgroup of index r in , namely , again . Hence, if we define v: T,. - L( for all r e T,,, then v is bijective. Furthermore, for r, s e T., we have r <'s if and only if s divides r, and this holds if and only if <

1.2.2 Lemma. If n e N u {oo}, then L(C,,) ^- T,,.

Groups with distributive subgroup lattices

A group G is called locally cyclic if every finite subset of G generates a cyclic sub- group. Equivalently we can say that

1.2.3 Theorem (Ore [1938]). The subgroup lattice of a group G is distributive if and only if G is locally cyclic.

Proof. Suppose first that L(G) is distributive and let a, b e G. We have to show that n is centralized by a and b, n S Z( = and so by (1),

n ) = () n () =

Thus n . Then ab = a'c implies that ). If one of the three subgroups A n , B n , C n , B n

)(Bn C n

Again it follows that x e A(B n Q. Thus AB n AC :!5; A(B n C), as required.

1.2.4 Corollary. The subgroup lattice of a finite group G is distributive if and only if G is cyclic.

Ore's theorem, and perhaps even more its corollary, is one of the most beautiful results on two of the basic problems mentioned in the preface: Which class of groups is the class of all groups with a given lattice property and, conversely, which lattice property characterizes a given class of groups. Here an interesting class of lattices, the finite distributive lattices, belongs to a simple class of groups, the finite cyclic groups. This example is rather atypical. We shall see that nice classes of lattices often lead to fairly complicated classes of groups and, similarly, it is usually very difficult or even impossible to characterize a given interesting class of groups.

Lattice-theoretic characterizations of cyclic groups

Using Ore's theorem, however, it is not difficult to characterize the class of cyclic groups.

1.2.5 Theorem. The group G is cyclic if and only if its subgroup lattice L(G) is distrib- utive and satisfies the maximal condition.

Proof. It is well-known and easy to prove that a group whose subgroup lattice satisfies the maximal condition is finitely generated. Thus if L(G) is distributive and satisfies the maximal condition, then G is locally cyclic, by Ore's theorem, and finitely generated; hence G is cyclic. Conversely, if G is cyclic, then L(G) is distribu- tive, by Ore's theorem. Furthermore, if 1 < H < G, then [G/H] is finite. So, clearly, L(G) satisfies the maximal condition.

1.2.6 Corollary (Baer [1939a]). Let G be a group. Then GC. if and only if L(G) T..

Proof. If G C., then L(G) Tom, by Lemma 1.2.2. Conversely, let L(G) T.. By 1.2.2, L(G) L(C.). By 1.2.5, L(G) is distributive and satisfies the maximal condi- tion and therefore G is cyclic. Clearly, I GI is infinite.

Thus the infinite cyclic group C. is determined by its subgroup lattice: it is the only group G such that L(G) L(CD). For finite cyclic groups the situation is differ- ent since for m, m' a N, the lattices T. and T", may be isomorphic although m 96 m'. 14 Fundamental concepts

If m = p;'... p;ris the prime power decomposition of m, then every divisor of m has the form pi'...p;" with 0 < ki < n; for i = 1,..., r.The map o defined by

k, k,. a k. k,. 'P1 Pr) =(P1,.,Pr ) is an isomorphism from Tm to the direct product of the Tp.,,, and these TP., are chains of lengths ni. Furthermore, Ti,, contains exactly r antiatoms, namely p,, ..., Pr, and to each pi exactly one chain of maximal length through 1 and p,, namely El/p"'] of length ni. So if m' e R such that Tm. T. and r, say, is an isomorphism from T. to T,,.,, then the p; are distinct primes and m' = (pi)', ... (p; )"r.

1.2.7 Theorem (Baer [1939a]). Let n1,..., nr E N.The group G is cyclic of order pi' ... p;r with distinct primes pi if and only if L(G) is a direct product of chains of lengths n l, ..., nr.

Proof. If G is cyclic of order m = p;'... p,r,then L(G) T. is a direct product of chains of lengths nl, ..., nr. Conversely, suppose that L(G) is such a direct product, let q1,..., qr bedistinct primes and put m' = qi'...q;r. Then L(G) Tm. is dis- tributive and finite. By 1.2.5, G is cyclic. Thus Tm. L(G) = Tic,, and hence IGI = pi' ... p" with distinct primes A.

1.2.8 Corollary. Let n1, ... , nr E Fand let pl,..., prbe distinct primes. If G is a cyclic group of order p;' ... p;, and G is any group, then L(G)L(G) if and only if G is cyclic of order qi'... q;rwith distinct primes q1,..., qr.

Thus the finite cyclic groups are far away from being determined by their sub- group lattices. Indeed, to every finite cyclic group C,, there exist infinitely many nonisomorphic groups C with L(C) However, all these groups are cyclic. And that is the important fact. If H is a subgroup of a group G, L(H) is the interval [H/1] in L(G). Hence it is possible to decide in the lattice L(G) whether a given subgroup H of G is cyclic or not.

Lattice-theoretic characterizations and invariance under projectivities

Let us pause for a general remark on the study of classes of groups and their lattices. Ore's theorem, or rather Theorem 1.2.5, is our first example of a lattice-theoretic characterization of a class I of groups. By this we mean the specification of a lattice- theoretic property, that is a class 9) of lattices, such that for every group G, (3) G E L if and only if L(G) E 9). We call a class L of groups invariant under projectivities if for every projectivity between two groups G and G, (4) G E L implies 6 E X. 1.2 Distributive lattices and cyclic groups 15

Suppose that a class I of groups has a lattice-theoretic characterization J. Then if G E I and cp is a projectivity from G to G, L(G) L(G) c J and hence G E X, by (3). So we have the following.

1.2.9 Lemma. Let X be a class of groups. If X has a lattice-theoretic characterization, then I is invariant under projectivities.

The converse of this statement is also true. If X is invariant under projectivities, then the class IV of all lattices isomorphic to some L(G) with G E £ satisfies (3). However, we are looking for a lattice-theoretic description of this class J as, for example, in Theorem 1.2.5. This result together with 1.2.9 yields the following.

1.2.10 Theorem. Every projectivity maps cyclic groups onto cyclic groups.

If H is a subgroup of G and cp is a projectivity from G to G, then the restriction cPH of p to [H/1] = L(H) clearly is a projectivity from H to H`°; we say that cp induces the projectivity (PH in H. Theorem 1.2.10 shows that if H is cyclic, so is H`° since it is the of H under the projectivity (PH.

Finite and locally finite groups

We conclude this section with a first application of our results on subgroup lattices of cyclic groups. Let I be a class of groups. We say that G is locally an X-group, if every finitely generated subgroup of G lies in X. We write LX for the class of all locally X-groups.

1.2.11 Lemma. Let X be a class of groups. If X is invariant under projectivities, then so is LX.

Proof. Let q be a projectivity from G to G and suppose that G E LX. Let H =

H4'=(u...u )w=4U...u4= u ... u -

Thus H4' is finitely generated and hence H`0 E X, since G E LX. Now cp-' induces a projectivity from H4' to H, and since I is invariant under projectivities, H E 1. Thus G E LX.

1.2.12 Theorem. A group is finite if and only if its subgroup lattice is finite. Hence every projectivity maps finite groups onto finite groups and locally finite groups onto locally finite groups.

Proof. If JGI is finite, then G has only finitely many subgroups. Conversely, suppose that L(G) is finite. If C is a cyclic subgroup of G, then L(C) is an interval in L(G), 16 Fundamental concepts hence finite; by 1.2.2, ICI is finite. Since G possesses only finitely many such sub- groups and every element of G is contained in one of them, I G Iis finite. By 1.2.9, the class of all finite groups is invariant under projectivities. Then 1.2.11 implies that the class of locally finite groups has the same property.

Exercises

1. Let p be a prime and write C,,for the Priifer group of type p°°, that is CP. = A/7L where A is the additive group of rational numbers whose denominator is a power of p. Show that L(CP=) is a chain. 2. If L(G) is a chain, show that G- CP., where p e P and n e N u {oo}. 3. (Ore [1938]) The elements x, y of a lattice L are said to be a n-distributive ("meet- distributive") pair in L if for all z e L the distributive law zn(xuy)=(znx)u(zny)

holds. Show that the subgroups A, B form a n-distributive pair in L(G) if and only if for every c e A u B, in neither A nor B, there-exist coprime natural num- bers m and n such that cm e A and c" E B. 4. If A, B is a n-distributive pair in L(G) with A n B a A u B, show that A and B are normal subgroups of A u B. (Hint: For a e A, B = (B n A) u (B n B°) = B° since A, B and A, Ba are n-distributive pairs.) 5. Use Exercises 3 and 4 to give an alternate proof of Theorem 1.2.3. 6. A n-distributive pair x, y is said to be trivial if x < y or y < x. Show that L(G) has only trivial n-distributive pairs if and only if every element of G has prime power order. 7. (Bruno [1972]) If A, B is a n-distributive pair in L(G), show that A, B is a u-distributive pair in L(A u B), that is satisfies Cu(AnB)=(CuA)n(CuB)

for all C e L(A u B). Also show that the converse is false. 8. Let £ be a class of groups and suppose that'd is a class of lattices such that (3) holds. Find a lattice-theoretic characterization of the class Ll of all locally X- groups.

1.3 Projectivities

In this section we shall present some general methods to construct projectivities between groups and to find isomorphisms that induce a given projectivity. 1.3 Projectivities 17

Construction of projectivities

It is often possible to work with element maps. For such a map a: G - G and a subset X of G we define X° = {x°jx c X}. If a is bijective and satisfies (1) X < G if and only if X° < G for all subsets X of G, then by 1.1.2, the map

(2)i7: L(G) - L(G); H - H° is a projectivity from G to G. We call it the projectivity induced by Q, and, more generally, we say that a given projectivity cp is induced by an element map a between the two groups if H10 = H° for all H < G. A useful concept in the study of element maps between groups is the amorphy of a map. It measures the deviation from homomorphy. To be precise, if a is an element map from G to G, then the map

(3) a: G xG _9 G; (x,Y)i--,(Y°)-1(x°)-'(xy)° is called the amorphy of Q. Clearly, (xy)° = x°y°a(x, y) and so a is a homomorphism if and only if a(x, y) = 1for all x, y c G. Furthermore, we have the associativity identities

(4) a(x, y)='a(xy, z) = a(y, z)a(x, yz) for all x, y, z e G.

For, we can write

((xy)z)° = (xy)°z°a(xy, z)= x°y°a(x,Y)z°a(xY, z) and

(x(Yz))° = x°(Yz)°a(x,Yz) = x°y°z°a(Y,z)a(x,Yz), and the associative law in G implies (4). If a: G - G isbijective and induces a projectivity, then (xy)° E ° _ ° u ° = and hence also a(x,y) E . We show that, conversely, this property and the corresponding one for o--' suffice to guarantee that a induces a projectivity. This will be our most useful tool in constructing projectivities.

1.3.1 Theorem. Let a- be a bijective map from G to G such that (5) (xy)° c for all x, y c G, and (6) (uv)°-' E for all u, v c- G. Then the map Q: L(G) -- L(G) defined in (2) is a projectivity from G to G.

Proof. Let H < G and let x°, y° E H°. By (6), (x°y°)°-' e < H and hence x°y° E H. If we can show that also (x°)-' E H°, then H° is a subgroup of G. By 18 Fundamental concepts symmetry then also K°-' < G for every K S G. Thus a satisfies (1) and 6 is a projec- tivity. So it remains to show that (x°)-' a H. To do this we study an arbitrary pair of elements a, b e G with (a°)-' = b° = u, say. By an obvious induction we get from (5), (7) (a")° a for all n e NI, and from (6), (u")°-' e and (u-")°e , that is. (8) u" e ° and u-" e °. Again by (5), (ab)° is contained in = and hence in ° or in °, by (8). Thus ab lies in or in and so we get the following. (9) If a, b e G such that (a°)-' = b°, then a e or b e . We want to show that = in (9). For this purpose we suppose that 0 and, without-loss of generality, we may assume that < . Then (7) and (8) show that s and that a, b and u all have infinite order. Write (b-')° = v. Then (9), applied to u, v e G and v-', yields that u e or v e . Suppose first that v e and let v = u' with r e Z. Since o is bijective, Irl > 2. By (7), (b-")° a for all n e N and so c . Thus = . Hence for every i e RI there exist k, t e 7L such that u" = (bk)° and (b -')a = u', and we can choose i so that I k I > 2 and I t I > 2. Now (9), applied to u'", u' e G and a -', shows thatu'" a . In both cases, d = (r, t) # 1 and (bl'l)° a . Since also (b-")° e < for every n e NI, we get by (5)

u = b° = (bIkIb-(Ik1-1))°a is similar. Here < , let u = vS where Isi > 2. Again by (7), c S . Hence for i e N there exist k, t e 7L such that (bk)° = vs" and (b-k)° = v', and we can choose i so that Ikl > 2 and Iti >_ 2. Now replace u by v and r by s to get the same contradiction as before. Thus we have shown that = in (9) and hence that (a°)-' e '. Return- ing to the element x e H we see that (x°)-' e " s H°, as required.

For n e N let L1(G) be the set of all subgroups of the group G that can be gener- ated by n elements. Since any group is the join of its cyclic subgroups, every projec- tivity is determined by its action on the set L1(G). In general the most accessible subgroups of a group are the cyclic ones. Therefore it is important to know under which conditions one can extend a bijection between L1(G) and L1(G) to a projec- tivity from G to G.

1.3.2 Theorem. Let G and G be groups and suppose that r is a bijective map from the set L1(G) of all cyclic subgroups of G to L1(G) such that for all X, Y, Z e L1(G), (10) X 5 if and only if XT < . For H < G let HO be the set union of all the XT where X e L1(H). Then 9 is a projectivity from Gto G. 1.3 Projectivities 19

Proof. We first show that H9' is a subgroup of G. So let a, b e H9. Then there exist Y, Z e L 1(H) such that a e Y` and b e Z'. Since T is surjective, there is an X E L, (G) such that X` = ' c- L1(H'') and hence x e H`0*; thus H < H". Conversely, the definitions of f and p imply that to every y e H1*0 there exist a L1(H9) and Z E L1(H) such that y e '-' < Z, by (10). Thus y e H and so H'P" = H. Similarly, Kd'' = K for all K S G. It follows that q' and i are bijective maps and they certainly preserve inclusion. By 1.1.2, cp is a projectivity.

It does not suffice to assume in Theorem 1.3.2 that T is an isomorphism between the partially ordered sets (by inclusion) L1(G) and L 1(G). This can be seen by taking any two groups of the same order p" and exponent p, p a prime, that are not lattice isomorphic. One needs the subgroups generated by two elements in (10). But for n > 2, every isomorphism between L"(G) and L"(G) can be extended to a projec- tivity. This result, of course, is most interesting for n = 2; however, it seems possible that one knows L"(G) and L"(G) for some n > 2 without being able to decide which subgroups are cyclic or lie in L2(G).

1.3.3 Theorem (Poland [1985]). Let G and G be groups and let n e N with n > 2. If a is a bijective map from L"(G) to L"(G) such that for all X, Y e L"(G), (11) X < Y if and only if X ° 5 Y°, then there exists a unique extension p of a to a projectivity from G to

Proof. We first show that a maps cyclic subgroups of G to cyclic subgroups of G. So let H S G be cyclic. Then L"(H°)L"(H) = L(H) and therefore L"(H°) ^- T. where m = I H I e N u {cc}, by 1.2.2. In particular, L"(H°) satisfies the maximal condition and hence there exists a maximal cyclic subgroup of H°. Suppose that H° # , take y e H'\ and put Z = n # 1. Thus in any case, the interval [Z/ is the only complement to ; it follows that . Thus 9 Z and Z/ is a cyclic normal subgroup of K with cyclic factor group and hence K e L2(Z) c L"(Z). This shows that L(Z) = L"(Z) -- T and by 1.2.5, Z is cyclic, contrary to the maximality of . We have shown that H° is cyclic. Since a-' satisfies the assumptions of the theorem with G and G interchanged, a-' maps cyclic subgroups of G to cyclic subgroups of G. The restriction r of a to L, (G), therefore, is a bijective map from L1(G) to L1(G). For k < n and X1,..., Xk E L1(G), by (11) we have X, = X° S

Now for X, Y, Z E L,(G), (11) and (12) imply that X < if and only if X' = X° S . By 1.3.2 there is a projectivity cp from G to G whose restriction to L,(G) is T. By (12), cp is an extension of a and it is unique since every projectivity is determined by its action on L,(G).

We shall quite often use 1.3.1 or 1.3.2 to construct projectivities. Theorem 1.3.3, however, is more of theoretical interest. Nevertheless it seems worthwhile to investi- gate more closely these partially ordered sets L°(G), in particular, L2(G).

Induced projectivities

An isomorphism a from G to G satisfies (1) and therefore induces a projectivity. It is one of the main problems about subgroup lattices to decide under what conditions a given projectivity cp is induced by a group isomorphism a. Clearly, it suffices to know that cp and a operate in the same way on L, (G). We shall need the following more general result.

1.3.4 Lemma. If cp is a projectivity from G to G and a: G -+ G is a homomorphism such that X° = X" for every cyclic subgroup X of G, then a is an isomorphism inducing cp.

Proof. If X is a cyclic subgroup of the kernel of o, then X`0 = X° = 1 and so X = 1. Thus u is injective. Let H S G. Since a is a homomorphism, He < G. Hence by 1.1.2,

(P H`0 = U J=U = U ° = H°. \XE H XEH XEH In particular, G° = G. Thus a is surjective and induces cp.

The fact that a projectivity maps cyclic groups onto cyclic groups makes it possi- ble to construct element maps. A general result in this direction is the following.

1.3.5 Theorem._A projectivity cp from a group G to a group G is induced by a bijective map from G to G if and only if I '* I = I I for every g c- G.

Proof. The condition is clearly necessary. Conversely, let cp be a projectivity sat- isfyingI " I = I Ifor all g e G. For any cyclic group C let C* be the set of generators of C. If C < G is cyclic, then so is C", by 1.2.10, and since IC°I = ICI, we have I(CP)*I = I C*I; let ac be a bijection from C* to (CP)*. Forge G define g° = g where C = . Then, clearly, a is injective. If H < G and g e H, then < H10 and hence g° a H`0; conversely, for y e H0 the preimage 'P-' = D is cyclic and therefore y = x= x° for some x e D < H. Thus He = H`0. This shows that o is bijective and that cp is induced by o. 11 The disadvantage of this theorem is that the map a constructed in its proof does not have any nice properties. It is much more interesting to know, for example, that 1.3 Projectivities 21 a given projectivity is induced by an isomorphism. We present two general results due to Sadovskii that reduce this problem to the corresponding problem for finitely generated groups.

Sadovskii's approximation theorems

A family $ of subgroups of a group G is called a local system of subgroups of G if (13) for any X, Y E . there exists Z E S/ such that X u Y < Z, and (14) every element of G is contained in some X E Y.

1.3.6 Theorem (Sadovskii [1941]). Let cp be a projectivity from a group G to a group G and let. ° be a local system of subgroups of G. For every X E. let cpx be the projectivity induced by cp in X, and Ax be the set of all isomorphisms from X to X" that induce qx. If Ax is nonempty and finite for all X E., then cp is induced by an isomorphism from G to G.

Proof. Recall that a directed set is a partially ordered set in which every pair of elements has an upper bound. By (13), . is a directed set and (Ax)xE ,is a family of sets indexed by 9. For each pair (X, Y) of elements of .' such that X < Y, we define a mapping irxy: Al - Ax in the following way. Take a E Ay. Then o induces cpy, so the restriction 6jx of a to X induces cpx, and aE Ax; now we can define 7rxy(a) _ aI x. Clearly, these mappings satisfy the following conditions: (15) For every X E ., nxx is the identity mapping of Ax. (16) If X < Y < Z, then rtxz = trxytryz Since every Ax is nonempty and finite, the inverse limit A = lim Ax of the family (Ax)x,.v with respect to the mappings ttxy is not empty (see Bourbaki [1968], Chap- ter III, § 7.4). An element of A is an element of the cartesian product of the sets Ax, hence a function f defined on . such that f(X) E Ax for all X E 9, with the property that (17) f(X) = nxy(f(Y)) = f(Y)Ix for X < Y. We take such an f and define a mapping p: G G in the following way. For g e G there exists X E 9 such that g c- X, and we let gP = gf(x). If in addition g c- Y E 9, then there exists Z E. such that X u Y < Z, and hencegf(X) = gf(Z)_ gf(I), by (17). Thus p is well defined. For x, y e G there are X, Y, Z E .' such that x c X, y c Y and X u Y < Z. Since p1z = f(Z) is an isomorphism inducing the pro- jectivity cpz, (xy)P = xPyP and P = ". By 1.3.4, p is an isomorphism induc- ing cp. O

If H = ', i = 1, ... , n. Since ' has only a finite number of 22 Fundamental concepts generators and since a is determined by the images of the xi, there are only finitely many isomorphisms inducing 0. So we get the following corollary to Theorem 1.3.6 which usually suffices for the applications.

1.3.7 Corollary. Let cp be a projectivity from G to G and let . be a local system of finitely generated subgroups of G. If cp is induced by an isomorphism on every X E .9', then cp is induced by an isomorphism on G.

We now prove a result dual to Theorem 1.3.6. Clearly, if cp is a projectivity from G to G and N a G such that N" a G, then the map

(P": L(G/N) --+ L(G/N"); H/N i--. H10/N11 is a projectivity from GIN to G/N'; we call gyp" the projectivity induced by q in G/N.

1.3.8 Theorem (Sadovskii [1965a]). Let p be a projectivity from a group G to a group G and let 9 be a family of normal subgroups of G such that (18) for any X, Y e. there exists Z e So with Z < X n Y, and (19) for every g e G there exists X e .So such that n X = 1. For every X e., suppose that X`0 g G and let Ax be the set of all isomorphisms from G/X to G/X0 that induce the projectivity tpx. If A. is nonempty and finite for all X e .9', then cp is induced by an isomorphism from G to G.

Proof. Let X, Y e 5° and define X* = {g e GI n X = 1). Every a e Ax induces a map a* from X* to G. For, if g e X*, then (gX)° e X`°/X9', and since such that (gX)° = g'X°. Now we set g= g' and define Bx = {a*Ia e Ax}. If Y < X and /3 e A1, then the isomorphism /3: G/Y 6/Y`° induces in the natural way an isomorphism y : G/X --+ 6/X`°. Since /3 induces qp on G/Y, it follows that y induces q on G/X and hence y e Ax. Now Y < X implies X* < Y*, and for g e X*, gQ' is an element of satisfying (gX)7 = gP*X''. Thus gfl = gY' and hence y* = fl*Ix.. Let us write X <' Y if and only if Y < X. Then (Y, <') is a directed set, by (18), and we have just shown that for X <' Y there exists a map 7[,y: By -+ Bx defined by nxr(a) = aI x. for all o e By.. Clearly, these maps have the properties (15) and (16) with respect to and since every B. is finite and nonempty, the inverse limit B = lim B. with respect to the mappings nxy is not empty. An element f e B has the property that f(X) e Bx and (20) f(X) = nxr(f(Y)) = f(Y)lx. if X <' Y, i.e. Y<_ X. We take such an f and define p: G - G in the obvious way: if g e G, then by (19) there exists X e. such that g e X*, and we define g° = gf(X). If also g e Y*, then there exists Z e . such that Z < X n Y, and hence gf(x) = gf(Z)= gf(I), by (20). Thus p is well defined. Let x, y e G and consider the amorphy a(x,y) = (y°)-`(x°)-`(xy)° e G. Since cp is a projectivity, there exists g e G such that , and our assumptions on . yield the existence of a subgroup 1.4 The group of autoprojectivities 23

Z E Y such that x, y, xy, g are all contained in Z*. Then f(Z) = a* for some a e AZ and, using the definitions of p and a*, we get = " and

(xy)°Z`° = (xY)a*Z*= (xYZ)° = (xZ)°(yZ)° =x' yPZ'D.

Therefore a(x, y) c -Z". On the other hand a(x, y) E and

Also in this theorem the assumption that Ax is nonempty and finite may be replaced by the condition that G/X is finitely generated and cp is induced by some isomorphism on G/X.

Exercises

1. Let a: G - G be a map, a the amorphy of a and write A = and B = . Show that A a B and that the map a* defined by x= xA for all x a G is an epimorphism from G to B/A. 2. (Gaschutz and Schmidt [1981]) Let G be a finite group, a a permutation of G and a the amorphy of a. Show that the following are equivalent: (a) (Hx)° = H°x° and (xH)° = x°H° for all H < G and x e G, (b) a(x,y) e for all x, y e G. 3. Let a be a bijective map from G to G such that (a) (xy-' )° E for all x, y c- G, and (b) (uv-')°-' E

1.4 The group of autoprojectivities

Let G be a group. A projectivity from G to G is called an autoprojectivity of G; the group of all autoprojectivities of G, that is of all automorphisms of L(G), is denoted by P(G). Clearly, every automorphism of G induces an autoprojectivity, as does every element g a G via the inner automorphism induced by g. We collect these simple facts in the following theorem and introduce important subgroups of G, Aut G and P(G). 24 Fundamental concepts

1.4.1 Theorem. Let G be a group. (a) The map p: Aut G -+ P(G) defined by H" = Ha for H < G, a e Aut G is a homo- morphism. The kernel of p is the group

Pot G = {aeAutGJH2=H for all H

N(G) = n NG(H), H

0 P (G)

Aut G PA(G)

PI (G) n G t

1 Pot G N(G)

Z (G) -0 1

1 6 Figure 3

Proof. Let a, /3 E Aut G and H < G. Then t' is the projectivity induced by Of and H1afi'° = Hae = Ha°P°. Hence p is a homomorphism. For g c G we have H9'° = g -'Hg = H9", and therefore rl = 7rp is the product of two homomorphisms. Thus rl is also a homomorphism. The remaining assertions are trivial or follow from the isomor- phism theorem for groups. El

If two isomorphisms a and z induce the same projectivity between two groups G and G, then o r' is a power automorphism of G, and conversely. Therefore,

1.4.2 Corollary. If a projectivity of a group G is induced by an isomorphism, itis induced by exactly I Pot GI isomorphisms. 1.4 The group of autoprojectivities 25

There is a large class of groups in which the homomorphisms p and ry described in Theorem 1.4.1 are injective.

1.4.3 Theorem (Baer [1956], Suzuki [1951a]). If G is a group with Z(G) = 1, then also N(G) = I and Pot G = 1, hence PI(G) G and PA(G) Aut G.

Proof. This is an immediate consequence of a more general result on power auto- morphisms of groups, so we shall postpone the proof until 1.5.2 below. But there is a short proof for finite groups which we want to present here. So let G be finite and suppose first that N(G) : 1. Then there exists a nontrivial Sylow p-subgroup P of N(G). Since N(G) normalizes every subgroup, P 9 N(G) and hence P a G. So if Sc Sylo(G), then P a S and P n Z(S) 1; let 1 a E P n Z(S). Then S< CG(a) and for primes q # p, P normalizes every Sylow q-subgroup Q of G and hence QP = Q x P. Therefore also Q < CG(a) and CG(a) = G, contrary to Z(G) = 1. Thus N(G) = 1 and PI(G) G, by 1.4.1. The second assertion now follows from the fact that G(Inn G) = 1if G is a group with trivial centre. Indeed, Pot G n Inn G = N(G)" = 1, by 1.4.1, and since Pot G and Inn G are normal subgroups of Aut G, they centralize each other. There- fore, if aEPotGandx,geG,

g lxag = xaa' = X9,a = (g-lxg)a = (ga)-'xaga and so gag-l centralizes xa. Since x was arbitrary, gag-' E Z(G) = 1 and hence ga = g. Thus Pot G = 1 and PA(G) Aut G.

In the remainder of this section, we shall study L(G) and P(G) for some well- known groups G; in particular we want to determine the subgroups PA(G) and P1(G) of P(G). Since every automorphism of a cyclic group is a power automor- phism, 1 for all n c N u {cc}. The results of Section 1.2 yield the descrip- tion of given in Exercises 1 and 2. The next example will explain those parts of our terminology and notation that originated in projective geometry.

Elementary ahelian groups

Let p be a prime, n a natural number and let G be an of order p". Then it is well-known that we can regard G as a vector space of dimension n over GF(p), the field with p elements, and that L(G) is the projective geometry of this vector space. An automorphism of the group G is an invertible linear map of the vector space G onto G, and an autoprojectivity of G is simply a projectivity of the projective geometry L(G) onto itself in the sense of projective geometry. It follows that PA(G)PGL(n,p). If n = 2, then every nontrivial, proper subgroup of G has order p and hence L(G)\{G, 1 } is an antichain with p + I elements. Every permuta- tion of these elements yields an autoprojectivity of G, and therefore P(G) is isomor- phic to the symmetric group Sp+1 on p + 1 letters. If n >_ 3, the Fundamental Theo- rem of Projective Geometry (see, for example, Baer [1952], p. 44) tells us that every 26 Fundamental concepts autoprojectivity of G is induced by a semi-linear mapping of the vector space G. But since GF(p) has only the trivial automorphism, this mapping is linear, i.e. an auto- morphism. Thus we have the following result.

1.4.4 Theorem. Let G be an elementary abelian group of order p", p a prime, n e N. Then PA(G) PGL(n, p); and P(G) = PA(G) for n >- 3, but P(G) ^-- SP+i if n = 2.

In Section 2.6 we shall show, without using the Fundamental Theorem of Projec- tive Geometry, that for a much larger class of abelian groups all autoprojectivities are induced by automorphisms.

Groups of orderpq

Let p and q be primes such that q I p - 1 and let H be a nonabelian group of order pq. Then by Sylow's theorem there are p subgroups of order q and one subgroup of order p in H and so L(H) consists of H, 1 and an antichain with p + 1 elements. Therefore, again, P(H)Sp+1 whereas PA(H) Aut H and PI(H) H, by 1.4.3. Thus PA(H) and PI(H) are subgroups of orders p(p - 1) and pq, respectively, in Sp+1 This shows that in general PA(G) and PI (G) are not distinguished subgroups of P(G); in fact they are not even normal in P(G). However, this is not very surprising if, as is the case here, the subgroup lattice is structurally small, in particular if it does not determine the group. The situation is different in our next example.

The alternating group A4

Let G be the alternating group of degree 4. Then G has 3 subgroups of order 2, generating the Klein 4-group V, and 4 maximal subgroups of order 3. It is easy to see that G is determined by its subgroup lattice. Indeed let G be a group with isomorphic subgroup lattice; then G operates faithfully by conjugation on the set F of those atoms of this lattice which are also antiatoms. Hence G is a subgroup of S4 and it follows that GG. Let A be the set of those atoms of L(G) that are not antiatoms, and let S, T be the kernels of the actions of P(G) on T, A, respectively. zx Figure 4: L(A4) 1.4 The group of autoprojectivities 27

Clearly, P(G)=SxT=S3xS4.

By 1.4.3, PA(G) Aut G. An automorphism a of G fixing all H E I operates on each of these groups in the same way as on G/V. Hence a either centralizes or inverts all H E r; and since one can easily check that the latter is impossible, a = 1. Thus PA(G) n S = 1 and PA(G) is isomorphic to a subgroup of T ^- S4. On the other hand, S4 operates faithfully by conjugation on its normal subgroup A4 and induces an S3 on V. It follows that PA(G) S4 and PA(G)S = P(G) = PA(G)T. An element of S normalizing PA(G) would centralize PA(G) and therefore also PA(G)T = P(G); but Z(S) = 1. Thus N,,(G)(PA(G)) = PA(G) and PA(G) is one of 6 conjugate sub- groups of P(G).

Dihedral 2-groups

The results obtained so far show an obvious general phenomenon. The more struc- ture the subgroup lattice of a group has, the more likely it is that it determines group-theoretic invariants. Compare, for example, the situation for A4 and S4 (see Exercise 3) or the cases n = 2 and n > 3 in 1.4.4. The dihedral 2-groups are an exception to this rule. Let n>3 and let G = D2" _ be the dihedral group of order 2". Since (x'y)2 = x'x -' = 1 for all i c N, every element in G \ is an involution; in particular, is the only cyclic of G. It follows that is invariant under P(G) and that P(G) operates faithfully on the set A of subgroups of order 2 not contained in . Every power automorphism has to centralize all these involutions and therefore Pot G = 1 and PA(G) Aut G. If n = 3, it is easily seen that

P(D8) = PA(D8) Aut D8 nt D8.

Figure 5: L(D1e) 28 Fundamental concepts

For n > 4 there are exactly three maximal subgroups in G, namely and two subgroups, M1 and M2, say, isomorphic to D2" Since M1 n M2 = , P(G) contains a subgroup isomorphic to P(M1) X P(M2), and because there are even automorphisms of G interchanging M, and M2, P(DAI = 21P(D2"-,)12. By an obvi- ous induction we get I 22"-'-1 for all n > 4. Since this is the highest power of 2 dividing (2"-')! and IAA = 2"-', P(D2.) is isomorphic to a Sylow 2-subgroup of the symmetric group S2"-1. Every automorphism of G is determined by the images of y and xy, and for these there are 2"-' and 2n-2 involutions respectively available. Thus IAutGI = 22n-3 so that PA(D2") is properly contained in P(D2.,) for all n > 4.

Quaternion groups

Let n > 3 and let G = Q2" = be the gener- alized of order 2". Then Z(G) is the only minimal subgroup of G and G/Z(G) ^ - D2" Thus P(Q2") ^- P(D2"-,), and since every automorphism of G/Z(G) can be lifted to an automorphism of G.

PA(Q2") PA(D2"-,) ^, Aut D2.,-,.

Figure 6: L(Q16) In particular, P(Q,) = PA(Q,) ^- S31 P(Q16) = PA(Q16) De and PA(Q2") < P(Q2") for n > 5. For every homomorphism a: G -+ Z(G), the map a: G -i G defined by g° = gg° for g e G is a power automorphism of G. It follows that I Pot G 1 > 4, and since G is generated by two elements of order 4, 1 Pot G I = 4. Finally, we mention that because the quaternion groups are the only noncyclic finite groups with just one minimal subgroup, they are determined by their subgroup lattices. We leave it as an exercise to show that this also holds for the dihedral 2-groups.

Exercises 1. Show that P(C.) ^ Sym P. 2. Let m e N and for i e t\J define n; to be the number of primes p such that p' is the maximal power of p dividing m. Show that P(C,") Dr S. (where of course, So is the trivial group). ' E " 1.5 Power automorphisms 29

3. Draw a Hasse diagram of L(S4) and show that P(S4) = PA(SO) ^ S4. 4. Give details for those parts in the discussion of the dihedral and quaternion groups that have only been sketched in the text. 5. Show that D2., is determined by its subgroup lattice for n > 2. 6. Let p be an odd prime and let G be the nonabelian group of order p3 and exponent p. Determine P(G), PA(G) and Pot G.

1.5 Power automorphisms

In this section we shall prove Cooper's theorem that every power automorphism of a group is central. This will have a number of interesting consequences. First we give an elementary result.

1.5.1 Theorem. Let G be a group. Then Pot G is abelian.

Proof. For g c G let F(g) = {a e Pot GIga = g}. Then F(g) a Pot G and Pot G/F(g), being isomorphic to a subgroup of Aut(g)., is abelian. Since n F(g) = 1, Pot G is abelian. a E c

Commutators

For elements x, y e G and a E Aut G, [x,y] = x-'y-'xy = x-'xy is the commutator of x and y and, similarly, [x, a] = x-'x° is the commutator of x and a. Then if X s G and S is a subset of G or Aut G, [X, S] = ([x, s] Ix e X, s e S). Finally,for elements x e G and s...... s" of G or Aut G we define inductively

[x, s,,.-.,s"] = [[x,s"--.,s"-,],s"] for n > 2; for subsets X of G and S...... S. of G or Aut G we define [X, S,_., S"] similarly. For the convenience of the reader we list the basic properties of commutators which we shall require. So let x, y e G, s and t elements of G or Aut G, X, Y, Z subgroups of G, S a subset of G or Aut G and let n e 101, m e Z. Then (1) [x y, s] = [x, s]y,[ y, s] and [x, st] = [x, t] [x, s]', (2) [xm, s] = [x, s]m if [x, s] and x commute, (3) [x, sm] = [x, s]m if [x, s]S = [x, s], (4) (xy)" = x"y"[y, x]()) if [x, y] commutes with x and y, (5) [x, y]° = [x°, y°] for every homomorphism a of G, (6) [X, S]'r = [X, S] and [X, Y] a X u Y, (7) [X, Y, Z] = I = [Y, Z, X]implies[Z, X, Y] = 1. 30 Fundamental concepts

The proofs of these results for s, t e G and S 9 G can be found in Robinson [1982], Chapter 5; for s, t e Aut G and S c Aut G, they are similar.

Cooper's theorem

15.2 Theorem (Cooper [1968]). Let G be a group. Then [G, Pot G] 5 Z(G).

This result has a number of immediate consequences. First of all, we get the

Proof of Theorem 1.4.3. If G is a group with Z(G) = 1, then [G, Pot G] = 1 and thus Pot G = 1. Hence also N(G) = 1, by 1.4.1, (c). Then 1.4.1, (a) and (b) show that PA(G) Aut G and PI (G) G.

Furthermore, we can show that power automorphisms centralize not only G/Z(G) but also the commutator subgroup G' of G. In addition they commute with the inner automorphisms. Finally, we get a result of Schenkman's [1960] that the norm of a group G is contained in the second centre Z2(G).

15.3 Corollary. If G is a group, then (a) [G', Pot G] = 1, (b) [Inn G, Pot G] = 1, (c) N(G) < Z2(G).

Proof. Let g, x e G and a e Pot G. By 1.5.2 there exist elements z, w e Z(G) such that = gz and x2 = xw. Hence [g, x]° = [ga, xa] = [gz, xw] = [g, x] and this implies (a). As in 1.4.1 let 7C be the natural homomorphism from G to Aut G. Then

(g-1xg)Q (9a)-lxaga x° = = = z-1g-lxagz = xaa` and (b) follows. Finally, if g e N(G), then g" e Pot G and hence [x, g] = [x, g'] e Z(G), by 1.5.2. Thus g e ZZ(G) and (c) holds.

Proof of Theorem 1.5.2. Let x, y e G and a e Pot G. We have to show that [x, a, y] = 1. Since xa e and hence (10) [X, a, Y] e

In particular, [x, a, y] commutes with [x, a] and y. Therefore the commutator formulas (8), (2) and (3) yield (11) [xm, a, y"] = [[x, a]m, y"] = [x, a, y]""" for all m, n e Z. This implies [y -1, a, x -1 ] = [y, a, x] and so we finally get (12) [x, a, y] [y, a, x] = 1. Letx°=x',ya=ysand(yx)a=(y")` where r, s,te7L.Then (y`)"=(yx)`=(yx) (ys)xr-'. ys2 (ya)"° = (ys)" and hence yS = (y`)x'-' and y' = It follows that =(,S)2 =

((Y`)x, ((Y`)""-.,.Y .)2 ((YS)',, ,)2and this yields1 = [x" = (Y`S)"" = = (YS2)x 2,YS2] = [[x,a]r-l,YS2]. Using (11), we get (13) [x, a, Y](r-1)S2 = 1. Now suppose that the theorem is false. Then there exist elements x, y e G and a e Pot G such that [x, a, y] # 1. If x or y had infinite order, then so would [x, a, y], by (10), and (13) would imply that r = 1. But then [x, a] = xr-1 = 1 and [x, a, y] = 1, a contradiction. So x and y have finite order and we can choose x, y e G with [x, a, y] # 1 such that o(x) + o(y) is minimal. By (12), also [y, a, x]1 and so without loss of generality we may assume that o([x, a]) z off y, a]). If p is a prime divisor of o(x) or o(y), then by our choice of x and y, [x°, a, y] = 1 or [x, a, y°] = 1; in both cases [x, a, y]" = 1, by (11). Hence o(x) and o(y) are powers of the same prime p and (14) [x, a, y]° = 1. Again let r, s e Z such that xa = x' and ya = ys. Since a is an automorphism, s * 0 (mod p). So from (13) we get [x, a, y]'-1 = 1 and then (12) and (11) yield

1 = [Y,a,x]'-1 = [Y,a,x'-1]= [[Y,a],[x,a]].

Thus A = < [x, a], [y, a] > is an abelian p-group and [x, a] is an element of maximal order in A. Such an element generates a direct factor (see, for example, the more general Lemma 2.3.11), that is we have A = ([x, a] > x B where B:!5; A. Now [y, a] e A and so there exist b e B and n e 7L such that [y, a] = [x, a] -"b. Thus (15) <[x, a]> n <[x, a]"[Y, a] > = 1. Since < [x, a, y] > is a minimal and < [x, a] > a nontrivial subgroup of the cyclic p- group , we have [x, a, y] e < [x, a] >. Hence there exists j e 7L such that [x, a, y] _ {x,a]J;letm=n(1 - j). Then by (1), -"jx"y, rx mY, a] = [x a] = [x -"', a]x"y [x"Y, a]

=([x,a]-"1)x"1[x",a]y[Y,a](by(8)and(1))

_ [x, a, y] `[x", a] [x", a, y] [y, a](since [x, a, y] e Z())

_ [x, a]"[y, a](by (8) and (11)). 32 Fundamental concepts

Now if [xmy, a] # 1, then <[x, a] > and <[xmy, a]> are nontrivial p-subgroups of and , respectively, and they have trivial intersection by (15). Since is a cyclic p-group and therefore contains only one minimal subgroup, it follows that

Power automorphisms of abelian groups

Cooper's theorem does not say anything about abelian groups. However, it is not difficult to determine all power automorphisms of these groups. We call a power automorphism a of G universal if there exists an integer n such that x° = x" for all x e G. We first remark that every power automorphism of a finitely generated abelian group is universal. More generally we show the following.

15.4 Lemma. Let A be a finitely generated abelian group with A = x x . If an automorphism a of A fixes ,..., and , then there exists an integer n such that a° = a"for all a e A.

Proof. By hypothesis there exist integers n, n, such that a; = a;' and

-a -n- aln an an1' ...ranom - = (a1... ar= (a1 ... ar r It follows that a;=a;' =a'fori= 1,..., rand then a' = a" for allaeA.

An abelian torsion group A is the direct product of its primary components AP and therefore Pot A is the cartesian product of the groups Pot AP where p runs through P. So let A be an abelian p-group and let a e Pot A. Consider first the case Exp A = pk, and take an x e A with o(x) = pk. Then a induces an automorphism in and so there is an integer n such that 0 < n < pk, (n, p) = 1 and x° = x". For every a e A, a induces a power automorphism in and by 1.5.4 there exists an integer m such that a° = a' and xa = x'. It follows that m = n (mod pk) and a2 = a". Hence a is universal and Pot A ^_- Aut . Now suppose that Exp A is infinite and consider KIk(A) = {a e Al a" = 11 for k c N. Then at is universal on ((A), so there exist integers nk such that 0 < nk < pk and a° = a"k for all a E i2k(A). Since i2k(A) 5 ilk+1(A), we have a"k = a° = a""' and there- fore nk = nk+1 (mod pk). Hence there exist unique integers r, such that 0 < ro < p, k-1 0< r, < p for i E 101 and nk = rip ; thus a determines a p-adic r = rip'. i=o i=o Conversely, such a p-adic unit yields a power automorphism of A.

1.55 Lemma. Let A be an abelian p-group and let r be a p-adic unit, r = j rip', say, i=o where rie7L,0

k-1 define a° = an,, where nk = rip'. Then a is a power automorphism of A; we shall i=o write a' for a°.

Proof. First of all, a° is well-defined since the ri are uniquely determined by r. Furthermore n; = nk (mod pJ) for j S k and therefore a° = a"k for all a e S2k(A). Thus or induces a power automorphism on every S2k(A). It follows that a is a power automorphism of A.

Now if Exp A is infinite, the correspondence between power automorphisms and p-adic units described above clearly is bijective and multiplicative. Thus we have the following result.

15.6 Theorem (Robinson [1964]). Let A be an abelian torsion group and for p e P let AP be its p-component. (a) Pot A _- Cr Pot Ap. peP (b) If Exp AP is infinite, then Pot AP ^-- Up, the of p-adic units; in fact, the map sending r e Up to the power automorphism or defined in 1.5.5 is an isomorphism. (c) If Exp AP = pk is finite, then every power automorphism of AP is universal and Pot AP ^-- Aut Cpk, the multiplicative group of units of the ring 7L/pk7L (or of p-adic units modulo pk).

For nonperiodic abelian groups the situation is much simpler.

15.7 Theorem. If A is an abelian group with elements of infinite order, then IPotAI=2anda°=a-1for1 #aePotAandallaeA.

Proof. Suppose that a e Pot A and let x e A with o(x) = oc. Since a induces an automorphism in , we have x° = x" where n e { + 1, - 1}. If a e A, then a induces a power automorphism in which is universal, by 1.5.4. Thus a" = a" for all a e A. The result follows.

Power automorphisms of nonabelian groups

Here we only need the following result; it is an immediate consequence of Cooper's theorem.

15.8 Theorem. if G is a nonabelian group generated by elements of infinite order, then Pot G = 1.

Proof. Suppose that 1 # a e Pot G. Since G is generated by elements of infinite order, there exists x e G such that o(x) = oo and x" # x, and hence x" = x-1. By 1.5.2, x-2 = [x, a] e Z(G). Therefore, if g c G, is abelian and by 1.5.4, a induces a 34 Fundamental concepts universal power automorphism in this group. Thus g° = g-' for all g e G. But this implies that G is abelian, a contradiction. El

There are also nonabelian groups possessing nontrivial power automorphisms. Universal power automorphisms exist in metacyclic groups (see Exercise 1) and, more generally, in certain groups with modular subgroup lattices (see 2.3.10). Exam- ples of nonuniversal power automorphisms can be found in the quaternion groups and in the groups of Exercise 2.

Exercises

1. If G = 2 and n >_ 2, show that a: G -+ G defined by x2 = x'+P"-' for x e G is an automorphism of G. 2. Let G be a finite p-group and let1 < K < H < G such that K S for all xEG\H. (a) If a: G -+ K is a homomorphism with H < Ker a, show that a: G - G defined by x° = xx° for x e G is a power automorphism of G. (b) If all elements in G \ H have the same order and if I G : HI > p2, show that there exist power automorphisms of G that are not universal. (Groups with these properties can be found in Huppert [1967], p. 334.) 3. (Baer [1934]) If N(G) contains an element of infinite order, show that N(G) _ Z(G). 4. (Cooper [1968]) Let G be a nonabelian group with elements of infinite order and define W(G) = . Show that every power automorphism of G centralizes W(G) and G/W(G). 5. (Huppert [1961]) If G is a nonabelian finite p-group, show that Pot G is a p- group.

1.6 Direct products

In this section we shall study the subgroup lattice of a ; we also investigate the structure of a group whose subgroup lattice decomposes into a direct product.

The subgroup lattice of the direct product of two groups

If G = H x K, then in general L(G)L(H) x L(K); see G = CP x CP for p e P. However, it is possible to construct L(G) out of L(H), L(K) and all isomorphisms between sections of H and K. This was discovered by Goursat as early as 1890. 1.6 Direct products 35

1.6.1 Theorem. Let H, K < G and G = H x K. (a) For U < G and x c- UK n H define xa = {y c- KIxy c- U}. Then a is an epimor- phism from UK n H onto UH n K/U n K with kernel U n H. (b) Conversely, if U1 < H, W:9 UZ < K and a is an epimorphism from U, onto U,/W, then

U = D(U,,a) = {xyIx c- U,,y c- xa} is a subgroup of G with U, = UK n H, UZ = UH n K, W= U n K and Ker a = UnH. (c) For U,, V, < H and epimorphisms a, fi as in (b), we have D(U,,a) < D(V,,fJ) if and only if U, 5 V1 and xa9x0forall xeU,.

Proof. (a) Let U, = {x c- HIxy E U for some y c- K} be the set of H-components of elements of U. If x c- U, and y c- K such that xy c- U, then x = (xy)y-' E UK n H; conversely, if x c- UK n H, x = uk with u c- U and k c- K, say, then u = xk-' and hence x c- U,. Thus U, = UK n H and, similarly, UH n K = U2 is the set of K- components of elements of U. For x c- UK n H, therefore, xa is a nonempty subset of UH n K. Furthermore, if y, z c xa, then xy and xz are elements of U and so 1z Y-= (xy)-'xz c- U n K. Hence y and z lie in the same coset of U n K, and if w is another element of this coset y(U n K), then w = yu with u c- U n K implies xw = xyu c- U, that is w c- xa. Thus we have shown that x° E U2/U n K. For x,, x2 E U, and y; E x; we have x1xzyly2 = x1y1xzy2 E U and hence yty2 E (xlx2)Therefore (xt x2)a = y1 y2(U n K) = xi xz and a is an epimorphism. Finally, x c- Ker a if and only if xa = U n K, i.e. 1 E x. It follows that Ker a = U n H. (b) If x1, x2 E U1 and y; E x; ,then y, y2-' E xi(xz)-' _ (x,xZ'r and hence (xty1)(xzy2)-' = x,x2'y1y2' E U. Therefore U is a subgroup of G and U, is its set of H-components. Thus U, = UK n H and a is the epimorphism defined in (a) for the subgroup U of G. The other assertions in (b) now follow from (a). (c) If U=D(U,,a)

The isomorphism theorem yields U1/U n HU2/U n K in 1.6.1 (b). Therefore we are interested in direct products of isomorphic groups. These are characterized by the existence of diagonals. Here, a subgroup D of G = H x K such that DH = G = DK and D n H = 1 = D n K is called a diagonal in G (with respect to H and K).

1.6.2 Theorem. Let H, K < G and G = H x K. If 6: H - K is an isomorphism, then

D(S) = D(H, S) = {xxalx c- H} is a diagonal in G (with respect to H and K). Conversely, if D is a diagonal in G (with respect to H and K), then there exists a unique isomorphism S: H --- K such that D = D(S). Thus there is a bijection between diagonals (with respect to H and K) and isomorphisms of H to K, and also, if H K, between diagonals and automorphisms of H. 36 Fundamental concepts

Proof. If 6: H - K is an isomorphism and U = D(H, 6), then it follows from 1.6.1 (b) that UnH=1=UnK,H=U1

Ul X U2/Up = Ul U0/U0 X U2 U0/U0.

G

Figure 7 Furthermore, UUl >_ U2, UU2 >_ Ul and so UUl = Ul x U2 = UU2. Now Dedekind's law yields that Ul U0 n U = (U1 n U)UO = (U n H) U0 = U0 and, simi- larly, U2 U0 n U = U0. Hence U/Uo is a diagonal in Ul x U2/Uo with respect to Ul Uo/U0 and U2 Uo/U0. It is easy to see that the corresponding isomorphism b: Ul U0/U0 -+ U2 U0/U0 is the one induced by a, that is (xUo)" = x2U0 for x c U1.

Groups with decomposable subgroup lattices

Our description of the subgroup lattice of G = H x K shows that L(G) L(H) x L(K) if every epimorphism a in 1.6.1 (a) is trivial. Clearly, this holds if (and only if) the elements in H and K have finite, coprime orders. It is easy to see that this result is also true for more than two factors. Let us say that the groups Gx (A E A) are coprime if every G,A is a torsion group and (o(x), o(y)) = 1 for all x E G,l, y E Gµ with A :0 µ; for finite groups this is equivalent to (IGAl, 1 for a. µ. 1.6 Direct products 37

1.6.4 Lemma. If G = Dr Gx with coprime groups Gx (A e A), then L(G) = Dr L(GA); AEA AEA in fact, the map r: L(G) -+ Dr L(GA) defined by HT(A) = H n Gx for A e A and H < G AEA is an isomorphism.

Proof. Let L = Dr L(GA). Recall that L is the set of all functions f defined on A f(A)AEA such that e L(Gx) for all A e A; hence H' is a well-/defined element of L. Let p: L - L(G) be defined by f ° = Dr f(.1). Then f °T(A) = I Dr f(µ) I n Gx = for AEA \UEA / all A e A and hence pr is the identity on L. On the other hand, if H < G and x e H, then x is the product of certain xx e Gx and since the Gx are coprime, x = xxyx where xxyx = yxxx, o(xA) = n, o(yx) = m and (n,m) = 1. There exist integers s,t such that ns + mt=1; thus x""=xx`yx`=xx-'-=xx. Hence xxeH for all A and H= Dr (H n Gx)= H`°. Thus rp is the identity on L(G). It follows that T is bijective AEA and by 1.1.2, r is an isomorphism.

We come now to the main result of this section. We show that any group with decomposable subgroup lattice is a direct product of coprime groups.

1.65 Theorem (Suzuki [1951a]). Let G be a group such that L(G) Dr Lx where AEA

(Lx)xE Ais a family of lattices, I A I > 2 and I L.I > 2 for all A e A; write L = Dr L.and IEA suppose that or: L(G) --+ L is an isomorphism. For A e A let Ox be the least and Ix the greatest element of Lx, define ft e L by fx(µ) = Oµ for A # µ e A and fx(A) = Ix and, finally, let Gx be the subgroup of G with G7 = fx. Then G = Dr Gx, the groups Gx (A a A) are coprime and L(GA) ^- Lx for all A e A. AEA

Proof. Clearly, G° is the greatest and 1° the least element 0 of L, so that by 1.1.7, every Lx has a least and a greatest element and L(GA) [fx/0]L. For A # µ e A, 1 x e Gx and l y e Gµ, furthermore

L( u

) = [0 u 0/0] [a/0] x [0/0] L() x L().

By 1.2.3, L() and L() are distributive, as is their direct product. Therefore is locally cyclic, and hence cyclic. Since x and y generate this cyclic group and x # 1 y, and have finite, coprime indices in it, and as n = 1, o(x) and o(y) are finite and coprime. This shows that the groups Gx (A e A) are coprime and that they centralize each other. By 1.1.7, G = M a A> and A µ e A> n Gx = 1. It follows that Gx g G and G = Dr Gx. AEA

Suzuki's theorem and Lemma 1.6.4 have a number of important consequences. In conjunction with Lemma 1.2.9 they show that the class of nontrivial direct products of coprime groups is invariant under projectivities. Moreover, the direct factors are mapped onto direct factors. 38 Fundamental concepts

1.6.6 Theorem. Let G = Dr G, with coprime subgroups Gk (A e A) and let cp be a AEA projectivity from G to a group G. Then G = Dr Gx and the subgroups Gf (A a A) are coprime. A E A

Proof. Let T: L(G) - Dr L(GA) be the isomorphism defined in 1.6.4. Then a = cp-1T CA is an isomorphism from'I L(G) to Dr L(GA). In the notation of 1.6.5, fA(A) = GA _ AEA G1(A) and f1(µ) = 1 = for A µ e A. Hence fA = Gx and so (G9)' = ft. Now all the assertions follow from 1.6.5.

We mention the following special case of 1.6.6 since we shall often use the result in this form.

1.6.7 Corollary. Let H and K be coprime subgroups of a group G such that [H, K] = 1. If cp is a projectivity from G to some group d, then (HK)' = H' x K' and H`0, K" are coprime.

Proof. Apply 1.6.6 to the projectivity induced by cp in HK = H x K.

Note that neither of the properties [H, K] = 1 and H, K coprime in 1.6.7, is preserved under projectivities. This, for example, is shown by the projectivities be- tween the elementary abelian group of order 9 and the nonabelian group of order 6. As a consequence of 1.6.7 we get a more general result in this direction.

1.6.8 Corollary. Let it be a set of primes and suppose that H and K are subgroups of a group G such that H is generated by 7t-elements, K by n'-elements and [H, K] = 1. If 9 is a projectivity from G to some group G, then [H', K'] = 1.

Proof. For every 7t-subgroup X of H and every n'-subgroup Y of K, [X`0, Y"] = 1, by 1.6.7. Since H' is generated by these X', it follows that [H', Y'] = 1, and since K' is generated by these Y', [H', K'] = 1.

We have shown that L(G) is decomposable if and only if G is a'nontrivial di- rect product of coprime groups. For finite groups this property is inherited by the Frattini factor group. We remind the reader that the Frattini subgroup fi(G) of an arbitrary group G is defined to be the intersection of all the maximal subgroups of G, with the stipulation that it shall equal G if G should have no maximal subgroups. If G is finite, then D(G) is nilpotent and cD(G)H < G for every proper subgroup H of G.

1.6.9 Theorem. Let G be a finite group. Then the following properties are equivalent. (a) L(G) is directly decomposable. (b) There exist nontrivial subgroups H and K of G such that G = H x K and (IHI, IKI) = 1. (c) L(G/b(G)) is directly decomposable. 1.6 Direct products 39

Proof. That (a) implies (b) follows from 1.6.5. If (b) holds, then G/I(G) is the direct product of HD(G)/1(G) and K b(G)/(D(G) and these groups are coprime. Since every prime divisor of IGI also divides IG/O(G)I (see Robinson [1982], p. 263), both factors are nontrivial. Now (c) follows from 1.6.4. Finally, let L(G/b(G)) be decomposable, G/I(G) = H,/CD(G) x H2/I(G) where lb(G) < Hi (i = 1, 2) and (IH,/cD(G)I, H2/D(G)I) = 1. If Di is the product of those Sylow p-subgroups of (D (G) for which p divides IHi/D(G)I, then D; g G (i = 1, 2), as characteristic subgroups of b(G). Furthermore, D(G) = D, x D2 since every prime divisor of ID(G)I also divides IG/D(G)I. Thus D, is a normal of H2. By the Schur-Zassenhaus theorem there exists a complement G2 to D, in H2 and all these complements are conjugate in H2. Since D, and H2 are normal subgroups, G operates by conjugation on the set S2 of these complements and H2 is transitive on Q. By the Frattini argument, G = NG(G2)H2. As H2 = G2D G = NG(G2)D, < NG(G2)CD(G) and hence G = NG(G2) since 'D(G) is the set of nongenerators of G. Thus G2 a G with IG21 = IH2: D11 = IH2/c(G)IID21 and similarly, we get a normal sub- group G, of G such that IG11 = IHi/c(G)IID,1. Then (IG,1,1G21) = 1, 1G111G21 = IGI and so G = G, x G2. Now (a) follows from 1.6.4.

The number of groups with a given subgroup lattice

We finish this chapter with an application of our results on direct products. In 1.2.8 we saw that for every finite cyclic group G there exist infinitely many nonisomorphic groups that are lattice isomorphic to G. By 1.6.4 and 1.6.5, this still holds for finite groups G whose subgroup lattice possesses a chain as direct factor; for then G = Cp x K where p E P, n e R, (p, IKI) = 1, and every group G= Cq., x K with q e P and (q, I K 1) = 1 is lattice isomorphic to G. However, we can show that these are the only finite groups with this property.

1.6.10 Theorem (Suzuki [1951a]). Let L be a finite lattice that has no chain as a direct factor. Then there are only finitely many nonisomorphic groups whose subgroup lattices are isomorphic to L. More precisely: If 1 is the length and w the width of L (see Section 1.1), then IGI < w'" for every group G with L(G) ^- L.

Proof. It is clear that the first assertion follows from the second since there are only finitely many groups of order at most w''". To prove the second assertion, we shall show that p :!g w for every prime divisor p of IGI. Since a maximal subgroup of a p-group has index p, the subgroup lattice of a group of order p" contains a chain of length n. Therefore it will follow that I P 1 < p' < w' for every Sylow p-subgroup P of G, and since there are at most w primes p < w, we get IGI < (w')w = w''". So let p be a prime divisor of I G I and let P be a Sylow p-subgroup of G. If P is not cyclic, then P contains at least p + 1 maximal subgroups and these form an an- tichain in L(G). If P is not normal in G, then, by Sylow's theorem, the conjugates of P form an antichain containing at least p + 1 elements. In both cases p < w. So we may suppose that P is a cyclic normal subgroup of G. By the Schur-Zassenhaus theorem, P has a complement K in G. If K were normal in G, then 1.6.4 would imply 40 Fundamental concepts that L(G) -- L(P) x L(K), which is impossible since L(P) is a chain. Thus K is not normal in G and the conjugates of K form an antichain with p" (n c N) elements. But then p < w, as asserted.

Clearly, our bound on IGI is not best possible. It seems worthwhile to look for better bounds and also to ask how many groups G satisfy L(G) - L for a given lattice L. Some remarks on the first problem and a characterization of infinite groups with subgroup lattice of finite width can be found in Brand] [1988].

Exercises

1. Let G = H x K and let U be a subgroup of G. Show that the following properties are equivalent (the U, are defined as in 1.6.3). (a) U a G. (b) [U H] < U n H and [U2, K] < U n K. (c) Uo a G and U/Uo 5 Z(G/U0). 2. (Rose [1965]) Let G = H x K, H ne K and let D be a diagonal in G with respect to H and K. Show that [G/D] is isomorphic to the lattice of normal subgroups of H. 3. Show that for A, µ E A with A µ, the groups G,, and G defined in 1.6.5 form a n-distributive pair in L(G) and use Exercises 1.2.3 and 1.2.4 to give an alternate proof of Theorem 1.6.5. 4. Let G, be a group having only finitely many nonisomorphic projective images- call them G1, ..., G,-and suppose that cp is a projectivity from G = G, x x G, to some group G. If Gr 9 G for all i = 1, ..., r, show that G f-- G. 5. (Brandl [1988]) Let p be a prime, n c N. Show that there are only finitely many noncyclic finite p-groups whose subgroup lattices have width n. Chapter 2

Modular lattices and abelian groups

In this chapter we are going to study subgroup lattices of abelian groups. Since these are modular, we shall more generally investigate groups with modular subgroup lattices, called M-groups for short. Unlike the groups with distributive subgroup lattices, these do not form a nice class of groups. However, there are close connec- tions between M-groups and abelian groups. In § 2.1 we give the basic properties of modular lattices and introduce one of the most important concepts of this book, the modular elements of a lattice. These will be used later to study projective images of normal subgroups of groups. In 1939 Baer determined the projective images of an elementary abelian p-group. These were called P-groups by Suzuki [1951a], who proved that every projective image of an arbitrary noncyclic finite p-group either is a p-group or a P-group. These results are fundamental for the study of projectivities between finite groups; they are proved in § 2.2. In 1941 and 1943 Iwasawa investigated the locally finite M-groups and those with elements of infinite order; he showed that they are metabelian and determined their exact structure. His results are presented in §§ 2.3 and 2.4, together with the theorem of Schmidt [1986] which characterizes the periodic M-groups and thus completes the characterization of groups with modular subgroup lattices. The main results of § 2.5 are two theorems of Baer [1944] and Sato [1951]. These assert, respectively, that every nonhamiltonian locally finite p-group with modular sub- group lattice, and every M-group with elements of infinite order admit a projectivity onto a suitable abelian group. In the p-group case, there exists a crossed isomorphism which even induces an isomorphism between the coset lattices of the two groups. In §2.6 we prove two important theorems of Baer [1939a] on projectivities be- tween abelian groups. He showed that every projectivity between two abelian p- groups is induced by an isomorphism if these groups contain with every element of order p" at least three independent elements of this order. Also, if G is an abelian group with two independent elements of infinite order, then every projectivity of G to an arbitrary group is induced by an isomorphism. Finally, we study projectivities between abelian groups of torsion-free rank 1.

2.1 Modular lattices

A lattice L is called modular if for all x, y, z e L the modular law holds: (1) Ifx

2.1.1 Remarks. Let L be a lattice and let x, y, z c L. (a) If x

It is easy to see that the only nonmodular lattice with 5 or less elements is the one called E5 in Section 1.1. This lattice can be used to characterize modularity.

2.1.2 Theorem. The lattice L is modular if and only if it does not contain a sublattice isomorphic to E5.

Proof. If L is modular, then every sublattice of L is modular and therefore cannot be isomorphic to E5. Conversely, we have to show that every nonmodular lattice L contains a sublattice isomorphic to E5. Since L is not modular, there exist x, y, z c L with x

a

Figure 8: E5 it follows that b n d = a and c u d = e. Thus S is a sublattice of L and as b

We shall also consider the modular law for single elements of a lattice. Thus we come to one of the fundamental concepts of this book. 2.1 Modular lattices 43

Modular and permutable subgroups

We say that the element m of the lattice L is modular in L, and write m mod L, if (2) xu(mnz)=(xum)r zforallx,zELwithx z, and (3) mu(ynz)=(muy)nzfor ally,zeLwithm

2.1.3 Theorem (Ore [1937]). Let G be a group. (a) If NaG,then NH=HN for all H

Proof. If N g G, then Nx = xN for all x E G and (a) holds. To prove (b), first let X< Z< G.Then,clearly, X u (M n Z) < (X u M) n Z.Furthermore,if g c- (X u M) n Z, then, since X u M= XM, there exist x e X and m c- M such that g = xm. Since X < Z, we have m = x-' g E Z and hence g = xm c X u (M nZ). Thus X u (M n Z) _ (X u M) n Z and (2) holds. Now let Y, Z < G with M < Z. Then Mu(YnZ)<(MAY)nZandfrom g=mye(MuY)nZand mEM

Following Stonehewer [1972] we call a subgroup M of G permutable in G, and write M per G, if MH = HM for all subgroups H of G. These permutable subgroups had been introduced by Ore [1937] who had called them "quasinormal". Theorem 2.1.3 shows that a normal subgroup is permutable and a permutable subgroup is modular in G. Thus a normal subgroup is modular in G and the modular laws (2) and (3) are the main properties of a normal subgroup that are visible in the subgroup lattice. We shall use this fact later in our study of images of normal subgroups under projectivities. In this chapter we investigate the connections between commutativity of the group and modularity of its subgroup lattice. Here Theorem 2.1.3 yields the following.

2.1.4 Theorem. The lattice of normal subgroups of an arbitrary group and the sub- group lattice of an abelian group are modular.

Proof. By 2.1.3, every normal subgroup of a group G is modular in L(G) and hence also in 91(G) since this is a sublattice of L(G). Thus 91(G) is modular. If G is abelian, then L(G) = 91(G). 44 Modular lattices and abelian groups

Properties of modular elements

We give a useful characterization of modular elements in arbitrary lattices, which leads to an important property of elements in modular lattices.

2.1.5 Theorem. The following properties of the element m of a lattice L are equivalent. (a) m is modular in L. (b) For every a c- L, the map pa,m: [a/a n m] --+ [a u m/m]; x " x u m is an isomorphism. (c) For every ac-L, the map Y'a.m: [a u m/m] -+ [a/a n m];z H z n aisan isomorphism. (d) For every a E L, coa.mll/am =idiaianmj and Y'a.mcoa.m =

Proof. (a)(d). Let a E L. If x c- [a/a n m], then by (2),

xv°m'k°^" = (x u m) n a= x u (m n a) = x and if z E [a u m/m], then by (3),

z0-9- _ (z n a) u m = (m u a) n z = z.

Thus cpa.mqia.m = idlaiatm) and Y'a.m(pa.m = idiajmim]. (d) = (a). Let x, z c- L such that x < z. Then x u (m n z) E [z/z n m] and hence

xu(mnz) = (m n z)) u n z = (xum)nz, that is, (2) holds. If y, z c- L such that m z, then (m u y) n z E [y u m/m] and hence (muy)nz = ((muy)nz)'Oy.-wy.- _ (((muy)nz)ny)um = (ynz)um.

Hence (3) also holds and m is modular in L. To prove the equivalence of (b), (c) and (d), let a e L and write (p, = (P and 4/a.m= 0. (b). (d). Let x c- [a/a n m]. Then x < (x u m) n a and hence x u m < ((x u m) n a) u m. The other inclusion is trivial since m and (x u m) n a are contained in x u m. So we get x°=xuIn =((xum)na)um=((xum)na)".

Since cp is injective, it follows that x = (x u m) n a = x'". Now let z e [a u m/m]. Then (z n a) u m < z and since cp is surjective, there exists b e [a/a n m] such that z=b"=bum. Thenb

(c) (d). For x E [a/a n m], again, x < (x u m) n a. Since qi is surjective, there exists c E [a u m/m] such that x= c o= c n a. Then x u m< c and (x u m) n a< c n a = x. Thus x = (x um)na = xl*O. Now let zn [a um/m]. Then (za) urn

Theorem 2.1.5 shows that if m is a modular element of a lattice L, then (4) [a u m/m] [a/a n m] for every a E L. But this condition alone does not imply that m is modular in L. This is shown by the following example. Take a circle K in the euclidean , choose two points I and 0 on K dividing the circle into two arcs and for x, y E K, define x < y if and only if x and y are on the same arc and x is nearer to 0 than y or x = y. Let a, m E K; then both [a u m/m] and [a/a n m] consist of a single element if a < m, and are

I=aum

0=anm

Figure 9 isomorphic to the real if am. So m satisfies (4), but, clearly, m is not modular in K if 0 V: m v: 1. If L is a finite lattice, it is easy to see that m is modular in L if it satisfies (4) (see Exercise 2). However, we shall not need this fact since we shall always use the mappings cpa., and t/ia,, to prove (4). We collect the main inheritance properties of modular elements.

2.1.6 Theorem. Let L be a lattice, m, n, a c L and suppose that m is modular in L. (a) m n a is modular in [a/0] = {x E Lix < a}. (b) m u a is modular in [I/a] = {y E LIa < y}. (c) If m < n and n is modular in [I/m], then n is modular in L. (d) If n and m are modular in L, then so is m vn. _ (e) If u is an isomorphism from L to a lattice L, then m° is modular in L.

Proof. (a) Let b e [a/0]. If X E [b/b n (m n a)] = [b/b n m], then by (2),

x`°b.- = x v (m n a) = (x u m) n a = xa1b.-O..M. 46 Modular lattices and abelian groups

Hence cpb,m,.,° is the product of (pb,m and the restriction of 'n,m to [b u m/m]. Since these two maps are isomorphisms, (p6,mnn is also an isomorphism and by 2.1.5, m n a is modular in [a/0]. (b) Letc e [I/a].Ifz c- [c u (m u a)/m u a] = [c u m/m u a],thenz z n c = Thus oc,m° is the restriction of 'tlmto [cum/m u a] and hence an isomorphism. By 2.1.5, m u a is modular in [I/a]. (c) Let d e L. Then for z e [d u n/n], we have z k&n= z n d = z n (d u m)n d z note that the right-hand side is well-defined because m S z n (d u m) < d u m. Since Iidum,n and the restriction of Od,m to [d u m/(d u m) n n] are isomor- phisms, Iid,n is also an isomorphism. By 2.1.5, n is modular in L. (d) By (b), m u n is modular in [I/n] and therefore by (c), m u n is modular in L. (e)is trivial. Note that m n n in general is not modular in L if m and n are modular elements in L. This, for example, is shown by the lattice represented by the Hasse diagram of Figure 10. But there are also simple examples in subgroup lattices (see Exercise 3).

Figure 10

It is well-known (see Robinson [1982], p. 66) that the class of groups satisfying the maximal condition (or other chain conditions) is closed with respect to forming extensions. This is a more general property of intervals in lattices. 2.1.7 Lemma. Let a, b, m be elements of a lattice L such that b < m S a and suppose that m is modular in L. If [a/m] and [m/b] satisfy the maximal condition, then so does [alb].

Proof. Let (c;)j, N be an ascending chain in [a/b]. Then (c; u m);,-:Nand (c; n m),EN are ascending chains in [a/m] and [m/b], respectively. Since these two intervals satisfy the maximal condition, there exists a natural number r such that c, u m = csumand c,nm=cSnmfor all s _! r. Then c, :!9 c, and by (2), c,=c,u(mnc,)=c,u(mnc5)=(c,um)nc5=(c,um)nc,=c5 for all s > r. Thus the chain (c1)1N is also finite.

Semimodular lattices

A lattice L is called upper semimodular if for all x, y e L, (5) x u y covers x whenever y covers x n y; 2.1 Modular lattices 47 and L is called lower semimodular if for all x, y a L, (6) y covers x n y whenever x u y covers x. By 2.1.5, a satisfies (5) and (6) and is therefore upper and lower semimodular. The following theorem shows that for subgroup lattices, property (6) is more important.

2.1.8 Theorem. The lattice of composition subgroups of an arbitrary group and the subgroup lattice of a finite are lower semimodular.

Proof. Let G be a group and let X, Y be composition subgroups of G such that X u Y covers X in the lattice R(G) of composition subgroups of G. By 1.1.4, there exists a composition series from X u Y to X. Clearly, all members of this series belong to R(G) and since X u Y covers X in R(G), the series must have length 1. So X a X u Y and X u Y/X is simple. Hence also X n Y g Y and Y/X n Y is simple. Again by 1.1.4, if H e R(G) such that X n Y< H < Y, then there exists a composi- tion series from Y to H and it follows that H = X n Y. So Y covers X n Y in R(G) and R(G) is lower semimodular. In a finite nilpotent group G, every subgroup is subnormal and hence L(G) = R(G).

The Jordan-Dedekind chain condition

We say that a lattice L satisfies the Jordan-Dedekind chain condition if for every x, y e L with x < y there exists a finite maximal chain x = xo < < xr = y from x to y and all maximal chains from x to y have the same length l(x, y).

2.1.9 Theorem. Every upper or lower semimodular lattice L of finite length satisfies the Jordan-Dedekind chain condition.

Proof. We give a proof in the case that L is lower semimodular; the proof for upper semimodular lattices is similar. If x, y e L with x < y, then there exists a finite maximal chain from x to y since L has finite length. We prove by induction on n the following assertion which then, clearly, will imply the Jordan-Dedekind chain condi- tion for L. (*) If x, y a L with x < y and if there exists a maximal chain from x to y of length n, then every maximal chain from x to y has length n. This is clear if n = 1, that is if y covers x. So assume that (*) is true for n - 1, let x = xo < < x = y be a maximal chain of length n and suppose that x = yo < .. < y, = y is another maximal chain. If yrn_1, then by induction, n - 1 = m - 1. So assume that yrn_1 and let z = n yrn_1. Then y = u and z is covered by x,,_1 and yrn_1 since L is lower semimodular. There- chain, then x =xo < < xi_1 are maximal chains from x to xn_1. By the induction assump- tion, n - 1 = k + 1. Thus also x = zo < < zk < y._, is a maximal chain of length 48 Modular lattices and abelian groups n - 1 and the induction assumption applied to this chain and to x = yo < < yin_1 yields n - 1 = m - 1. Thus n = m and (*) holds.

2.1.10 Theorem. The following properties of a lattice L of finite length are equiva- lent. (a) L is modular. (b) L is upper and lower semimodular. (c) L satisfies the Jordan-Dedekind chain condition and 1(x, x u y) = !(x n y, y) for all x, y e L.

Proof. That (a) implies (b) is clear. If (b) holds, then L satisfies the Jordan-Dedekind chain condition as we have just shown. Let x = xo < < x, = x u y be a maximal chain. Then

(7) For everyi e {0, ... , n - 11,since xi+1covers xi,eitherxi n y = xi+1 n y or xi u (xi+1 n y) = xi+1 and then xi n y = xi n (xi+1 n y) is covered by xi+1 n y sinceL is lower semimodular. Therefore (7) yields a maximal chain from x n y to y of length at most n; that is, we have 1(x, x u y) > 1(x n y, y). The other inequality fol- lows similarly from the upper semimodularity of L. Thus (c) holds. Finally, suppose that L satisfies (c) but is not modular. Then by 2.1.2, there exists a sublattice {a, b, c, d, e} of L isomorphic to Es. If a < b < c < e and a < d < e, then

l(a, b) = l(d n b, b) = l(d, d u b) = l(d, e); similarly, l(a, c) = l(d, e). Thus l(a, b) = l(a, c), but this is impossible since b < c. This contradiction shows that (c) implies (a).

Exercises

1. Show that a modular subgroup of a group G is in general not permutable, and a permutable subgroup is in general not normal in G. 2. (Schenke [ 1986]) Let L be a lattice and let m e L such that [a u m/m] [a/a n m] for every a e L. If [I/m] is finite, show that m is modular in L. 3. Let p be an odd prime, H = the nonabelian group of order p3 and exponent p2, K = cyclic of order p2 and let G = H x K. Find two modular subgroups of G whose intersection is not modular in G. 4. Show that a lattice all of whose sublattices are lower semimodular is modular. Deduce that sublattices of lower semimodular lattices are in general not lower semimodular. 5. Let L be the direct product of lattices Lx. Show that L is lower semimodular if and only if every Lx is lower semimodular. 6. Show that for an arbitrary lattice L, no two of the conditions (a)-(c) in Theorem 2.1.10 are equivalent. 2.2 P-groups 49

7. The element m of the lattice L is called semimodular in L if it satisfies (2). Show that (a) m is semimodular in L if and only if (p",ml//a,m = id["i"nm, for all a e L; (b) if m is semimodular in L, then 0m,"(pm," = id[.,,.,.] for all a e L; (c)if m and n are semimodular in L, then [m u n/n] [n/m n n]; (d) an antiatom of L is semimodular in L if and only if it is modular in L. 8. Let L be a lattice, m, n, a e L, and suppose that m is semimodular in L. Show that (a) m n a is semimodular in [a/0]; (b) if m is modular in L, m < n and n is semimodular in [I/m], then n is semi- modular in L; (c)if m 5 n and n is modular in [I/m], then n in general is not semimodular in L; (d) if [I/m] is a chain and m < n, then n is semimodular in L.

2.2 P-groups

By 1.6.6, every projective image of a periodic abelian group G is the direct product of the images of the primary components of G. However, in general these images are neither abelian nor primary: the nonabelian group of order pq, p and q primes with q dividing p - 1, and the elementary abelian group of order p2 have isomorphic subgroup lattices. This is the first of a series of examples that we shall now investi- gate more closely.

The classes P(n,p)

Let p be a prime and n >- 2 be a cardinal number. We say that a group G belongs to the class P(n, p) if G is either elementary abelian of order p", or a of an elementary abelian normal subgroup A of order p"-1 by a group of prime order q # p which induces a nontrivial power automorphism on A. Here, if n is infinite, by an elementary abelian group of order p" or p"-1we shall mean a direct product of n cyclic groups of order p. We call G a P-group if G e P(n, p) for some prime p and some cardinal number n >- 2.

2.2.1 Remarks. Let G be a nonabelian P-group, so G = A with an elementary abelian p-group A and an element t of order q that induces a nontrivial power automorphism on A. By 1.5.6, t is universal on A, that is, there exists an integer r such that (1) t-tat=a'for all aeA. Since t 0 CG(A) and t9 = 1, (2) r # 1 (mod p) and r9 = 1 (mod p). Hence q divides p - 1. In particular, the classes P(n, 2) only contain the elementary abelian groups of order 2". For p > 2, however, to every prime divisor q of p - 1, 50 Modular lattices and abelian groups there exists an integer r satisfying (2); then the semidirect product G = A of order q where t-1 at = a' for all a e A is a nonabelian P-group. Any two such groups with the same A and q (for different r) are isomorphic, as the reader can easily verify. Thus for p > 2, the classes P(n, p) contain the elementary abelian group of order p", and, for every prime divisor q of p - 1, exactly one nonabelian P-group with elements of order q; if n is finite, the order of this group is p"-' q.

We want to show that these P(n, p) are complete classes of lattice-isomorphic groups. For this purpose and for later use, we collect some simple properties of P-groups.

2.2.2 Lemma. Let G be a nonabelian P-group and suppose that p, q and A are as in 2.2.1. (a) G' = A = CG(a) for all 1 0 a e A. (b) Z(G) = 1. (c) The normal subgroups of G are G and the subgroups of A. (d) Every element x e G\A has order q. Hence the subgroups of order q generate G.

Proof. (a) Since G/A is cyclic, G' < A. If 1 a e A, in the notation of 2.2.1, then [a, t] = a'-' and since r # 1 (mod p). Hence < G' and a 0 Z(G). Thus G' = A and C6(a) = A since A is an abelian maximal subgroup of G. (b) By (a), Z(G) < CG(A) = A and a 0 Z(G) for all 1 rA a E A. Thus Z(G) = 1. (c) Since A is abelian and t induces a power automorphism on A, every subgroup of A is normal in G. If N:9 G and NA, then AN = G and GINA/A n N is abelian. It follows that A = G' < N and hence N = G. (d) If x e G\A, then G = A). By (a), A n = 1. Hence

2.2.3 Theorem (Baer [1939a]). For every prime p and every cardinal number n > 2, all groups in P(n, p) are lattice-isomorphic.

Proof. We show that every nonabelian group G E P(n, p) is lattice-isomorphic to the elementary abelian group in P(n,p). So let q, r and G = A has order q and therefore contains exactly one element out of each coset of A, in particular out of At. Hence there exists exactly one element a e A such that = . We define the map r: L1(G) - L1(G) by ' = if x e A, and = if = A. Since every cyclic subgroup of the elementary abelian group G = A x <1) which is not contained in A also contains exactly one element of the form at with a E A, we see that r is bijective. By 1.3.2, we have to show that for all X, Y, Z E L1(G), (3) X <

This is clear if Y and Z are contained in A since T is the identity on L,(A). If Y < A and Z $ A, that is, Y = and Z = with b, c e A, then n A = Ysince every subgroup of A is normal in G. Similarly, n A = Y`. Thus, if X < A, then X < if and only if X < Y, and this is the case if and only if X'< Y`, Z`>. For X $ A, that is, X = with a e A, we have X < if and only if at e n At = ( n A)ct = ct, i.e. a e c; moreover, this is the case if and only if U` = < x = < YT, Z`>. Finally, if neither Y = nor Z = is contained in A, then with W = < A we have = (W, Z> and = = . Thus (3) also holds in this case as we have just shown.

We now want to show that the classes P(n, p) are closed under projectivities. For the case n = 2 of this assertion we prove a more general result that will also be used in the next section.

2.2.4 Lemma. Let G be a finite group and suppose that M and N are two different minimal subgroups of G which are modular in G. Then I M u NI = pq where p and q are primes.

Proof. We may assume that G = M u N. If H is a maximal subgroup of G, then H cannot contain both M and N. So if M H, say, then H n M = 1, H u M = G and, by 2.1.5,

[H/1] = [H/H n M][H u M/M] = [N u M/M][N/N n M] = [N/1].

Hence H is a minimal subgroup of G. If H a G, then I HI = p and I G/HI = q are primes and I GI = pq. So assume that H is not normal in G. Then H = NG(H) and the conjugates of H in G together contain I G : HI (I HI- 1) = IGI - I G :HI nontrivial elements, that is, at least i IGI but less than IGI - 1. Since H is not normal, G is not cyclic and therefore every element of G is contained in a maximal subgroup of G. Hence there exists a maximal subgroup K of G that is not conjugate to H. But then K a G and IGI = pq since otherwise the conjugates of K would contain at least i I GI further nontrivial elements of G.

2.2.5 Theorem (Baer [1939a]). For every prime p and every cardinal number n-2, the class P(n, p) is invariant under projectivities.

Proof. Let cp be a projectivity from G to G and suppose that G E P(n, p). If n = 2, then the nontrivial proper subgroups of G form an antichain with p + 1 elements. By 2.2.4, 1 GI = qr where q and r are primes with q > r, say. Since the antichain contains more than 2 elements, G is not cyclic and hence elementary abelian of order q2 (if q = r) or nonabelian of order qr (if q r). In both cases, G has exactly q + 1 minimal subgroups. It follows that q = p and G E P(2, p). Now let n > 3. By 2.2.3, we may assume that G is elementary abelian. So if x and y are nontrivial elements of G such that , then ' and ' are cyclic of 52 Modular lattices and abelian groups order p, and is elementary abelian of order p2. By the case n = 2, already settled,

(4) E P(2, p). In particular, if P is the set of p-elements in G, then P 1 and we show that P is an elementary abelian normal subgroup of G. For 1 x c- P, is cyclic of order p and hence also o(x) = p. If x, y E P with o(x) = p = o(y) and : , then by (4), E P(2, p). And since a nonabelian group in P(2, p) has only one subgroup of order p, I = p2. Thus xy-' E P and xy = yx. This also holds if = and hence P is an abelian subgroup of G. Since it is the set of all elements of order p or 1 in G, it is clearly normal and elementary abelian. Now P and P° are lattice-isomorphic elementary abelian p-groups. If P = Dr Px .ZEA where I P I = p for all A, then P' = U Px and Px n U P", =(Pxn U Pµ ) = l for A E A all A. Hence P`° = Dr Px and PO P. Therefore if P = G, we are finished. So assume A E A that P G. Since G/P does not contain elements of order p, 16/P01 = p by the case n = 2, which is already settled. Thus P`0 and P are elementary abelian of order p"-' andI G : PI = q for some prime q. Let t E G with o(t) = q. Then G = P. For 1 x E P, E P(2, p) by (4) and therefore g . Thus t induces a non- trivial power automorphism on P and G E P(n, p).

Projective images of locally finite p-groups

By 2.2.3 and 2.2.5, the class P(n, p) is precisely the class of projective images of the elementary abelian group of order p". So for p > 2, this group has projective images that are not even primary groups. We show that the elementary abelian groups are the only finite p-groups with this property. 2.2.6 Theorem (Suzuki [1951a]). Let p be a prime, n a natural number, G a group of order p", and suppose that cp is a projectivity from G to some group G. If 161 IGI, then either (a) G is cyclic and G is cyclic of order q" where q is a prime different from p, or (b) G is elementary abelian, n >_ 2 and G is a nonabelian P-group of order p"-'q where q is a prime dividing p - 1.

Proof. We use induction on n. Since IGI IGI, G is not a p-group. If G is cyclic, then by 1.2.8, (a) holds. In particular our assertion is correct for n = 1 and we may assume that n > 2 and that G is not cyclic. Then by the Burnside Theorem, G/cD(G) is elementary abelian of order p' where m >_ 2. Since cb(G)`' = cb(G), cp induces a projectivity from G/cD(G) to 61(l)(6) and by 2.2.5, G/cb(G) E P(m, p). Hence there exists a subgroup P of G such that cD(G) < P and I P'/D(G)J = p'-' By the induction assumption, P'0 is a p-group or lies in P(k, p) for some k; in the latter case by 2.2.2, D(G) is a p-group as a normal subgroup of P°. Hence in both cases P" is a p-group. Since G is not a p-group, OD(G) is nonabelian of order p"'-' q where q is a prime dividing p - 1 and P'0 is the Sylow p-subgroup of G. 2.2 P-groups 53

1

Figure 11

If (D(G) = 1, then (b) holds and we are finished. So suppose for a contradiction that D6(G) # 1. Let Q < G such that IQ"I = q, take a maximal subgroup R of G containing Q and put D = P n R. Since P`°Q° = G, R 0 P and so G/D is elementary abelian of order p2. Hence there exists a maximal subgroup_S of G such that D < S and R S : P. Since P4' is the only Sylow p-subgroup of G, R0 and SP are not p-groups; on the other hand, p divides the order of cD(G) < R'O n S". By induction, R and S are elementary abelian. This implies that D 5 Z(G) and that P is abelian as a cyclic extension of the central subgroup D. So for x, y E G, we have [y, x] e G' S D 5 Z(G) and by (4) of 1.5,

(xy)" = x"y"[y,x](z) = xryr since D is elementary abelian and p > 2. Therefore the set of elements of order dividing p is a subgroup of G containing R and S, hence is G. It follows that P is elementary abelian. Since P' is a p-group, it is also elementary abelian, and, by Maschke's Theorem, P" is completely reducible under the action of Q'P. Hence there exists a Q'-invariant subgroup KP of G such that P' = K0 x cD(G). Then K0Q' is a proper subgroup of G, but every maximal subgroup of G containing K°Q0 also contains KocF(G)Qq' = P'Q' = G. This contradiction shows that D(G) = 1 and (b) holds.

The corresponding result for locally finite groups is an easy consequence.

2.2.7 Corollary. Let p be a prime and let G be a locally finite p-group. If there exists a projective image of G that is not a p-group, then G is locally cyclic (that is G- Cp for some n e f\l u { oo }) or elementary abelian. Proof. Let cp be a projectivity from G to G where G is not a p-group and suppose that G is not elementary abelian. Then there exist elements x, y, z c- G such that w is not a p-group and is not elementary abelian. Now if g1,..., g E G, then H = is a finite p-group, and since x, y, z e H, we see that I H' : IHI and H is not elementary abelian. By 2.2.6, H is cyclic and thus G is locally cyclic. 54 Modular lattices and abelian groups

As another consequence of Suzuki's theorem it is possible to describe the projec- tive closure of the class of finite nilpotent groups, that is, the smallest class of groups which is invariant under projectivities and contains all finite nilpotent groups (see Exercise 7). Nilpotent torsion groups will be handled in Section 7.4.

Exercises

1. Show that there is at most one P-group of a given finite order (see Remark 2.2.1). 2. Give an alternate proof of Theorem 2.2.3. With the notation used there show the following. (a) Every subgroup H of G that is not contained in A has the form H = A, where A, = Hr'A < A2 if and only if A, < A2 and a, a2 E A2' (c) The map cp: L(G) - L(G) defined by H0 = H for H < A and (A, )"_ A, for A, < A and a E A is a projectivity from G to G. 3. Let p be a prime and 2 < n c N. If H and K are nonabelian groups in P(n, p) and P is a p-group, show that P x H and P x K are lattice-isomorphic. 4. Let G be a finite group such that any two different maximal subgroups of G intersect trivially. Show that G is cyclic of prime power order or a semidirect product of an elementary abelian normal subgroup A by a group of prime order operating irreducibly on A. 5. Give an alternate proof of Theorem 2.2.6 (avoiding Maschke's Theorem). Show by induction that IG : D(G)l = p2 and JD(G)J = p if D(G) # 1. Then study the conjugacy classes of subgroups of order p of P`0. 6. Illustrate Theorem 2.2.6 by the following examples. That is, without using 2.2.6, show that for primes p and q with qlp - 1, the groups G and G are not lattice- isomorphic if _ (a) G = Cpl x CP and G is the semidirect product of C,,2 by Cq to an automor- phism of order q of CPz, _ (b) G is nonabelian of order p3 and exponent p, and G = CP x H where H is nonabelian of order pq, and (c) G is as in (b), G =

2.3 Finite p-groups with modular subgroup lattices

In this section and the one following we wish to determine all groups with modular subgroup lattices, called M-groups for short. We start with the most difficult part of this program and describe first the finite p-groups with this property. So, in this 2.3 Finite p-groups with modular subgroup lattices 55 section, p is a prime and every p-group considered is finite. Our aim is to prove the following result.

2.3.1 Theorem (Iwasawa [1941]). A finite p-group G has modular subgroup lattice if and only if (a) G is a direct product of a quaternion group Q8 of order 8 with an elementary abelian 2-group, or (b) G contains an abelian normal subgroup A with cyclic factor group G/A; further there exists an element b c G with G = A and a positive integer s such that b-'ab=a1+F` for all aEA,with s>_2incase p=2.

The proof of Iwasawa's theorem will occupy nearly the whole section. In 2.3.3 we shall present a useful criterion for a p-group to have modular subgroup lattice. From this it will easily follow that the groups in 2.3.1 are M-groups. The converse, how- ever, is much deeper. We first study p-groups with modular subgroup lattices in which the quaternion group Q8 is involved, and show in 2.3.8 that they are the groups in (a) of Theorem 2.3.1. Recall that a group is called hamiltonian if itis nonabelian and all of its subgroups are normal. By 2.1.4, these groups have modular subgroup lattices, so the hamiltonian p-groups have to show up in Iwasawa's theo- rem. We shall see in 2.3.12 that they are also exactly the groups in (a) of Theorem 2.3.1. Therefore it remains to consider p-groups with modular subgroup lattices in which Q8 is not involved, or which are not hamiltonian. We call these M*-groups and establish a number of important properties that will also be used in later sections, the most noteworthy being 2.3.10, 2.3.11, 2.3.14 and 2.3.16. These proper- ties, of course, can also be proved as soon as one knows that the groups satisfy (b) of Theorem 2.3.1; however, they are not obvious consequences of Iwasawa's theorem and we need them in the proof of it. This proof will finally distinguish the two cases that there exists a cyclic normal subgroup X in the M*-group G such that X J = Exp G, in which case we get the desired structure for G quite easily (2.3.15), and that there is no such X, where the computations become a little more involved (2.3.18). In both cases additional information will be obtained on how to choose the sub- group A, the element b and the integer s in Iwasawa's theorem. That these are not uniquely determined by G is shown by the abelian groups, for example. There any integer s > 2 with ps > Exp G and every subgroup A and element b with G = A will satisfy (b) of 2.3.1. We start with an elementary remark. If H and K are subgroups of a p-group G such that [H u K/K] [K/H n K], then I H u K: K I= p" = I K: H n K I, where n is the length of any of the two isomorphic intervals, and hence HK = H U K = KH. So 2.1.5 and 2.1.3 yield the following result.

2.3.2 Lemma. A finite p-group has modular subgroup lattice if and only if any two of its subgroups permute.

The groups of order p3 are well-known. The dihedral group and the nonabelian group of exponent p for p > 2 are generated by two elements of order p and there- fore, by 2.2.4, cannot be M-groups. In the other groups of order p3 any two elements 56 Modular lattices and abelian groups of order p commute and every subgroup of order p2 is normal. By 2.3.2, these groups are M-groups. For an arbitrary p-group, the sections of order p' decide whether it has modular subgroup lattice or not.

2.3.3 Lemma. Let G be a finite p-group. Then G has modular subgroup lattice if and only if each of its sections of order p' does. Therefore if G is not an M-group, then there exist subgroups H, K of G with K g H such that H/K is dihedral of order 8 or nonabelian of order p' and exponent p for p > 2.

Proof. The second assertion follows from the first as our remarks on groups of order p3 show. And since sublattices of modular lattices are modular, one part of the first assertion is trivial. So we have to show that G has modular subgroup lattice if every section of order p3 does. We choose a minimal counterexample G to this statement. Then every proper section of G is an M-group. Since G is not an M-group, there exist subgroups H and K of G with HK KH; choose H and K so that IHI IKI is minimal. If H, and K1 are maximal subgroups of H and K, respectively, then H1K1 = K1H1, H1K = KH1 and HK1 = K1H. For X = HK1 and Y = H1K, it follows that HK = HH1 K 1 K = HK1 H1 K = X Y and KH = KH1 K 1 H = YX. Thus X Y 0 YX, and furthermore

IX:X r Yl <_ IHK,:H1K11 = IH:HnH1K1I

Similarly I Y: X n YI S p and hence X n Y:9 X v Y. Since all proper sections of G are M-groups, X u Y = G and X n Y = 1. In particular, IXI = I YI = p. If N is a minimal normal subgroup of G, then, since INI = p and XY YX, we see that XN/N and YN/N are different subgroups of order p generating the M-group GIN. By 2.2.4, IG/NI = p2 and hence IGI = p'. But then by assumption, G is an M-group, a contradiction.

As an easy application of 2.3.3 we get the following criterion.

2.3.4 Lemma. Let G be a finite p-group and let A be an abelian subgroup of G such that every subgroup of A is normal in G and G/A is cyclic. If p = 2 and Exp A > 4, assume further that there exist subgroups A2 < A, of A with A, /A2 ^- C4 and [A,,G] < A2. Then G is an M-group.

Proof. We use induction on IGI. Since G/A is cyclic, there exists b E G with G = A (b). By 1.5.4, b induces a universal power automorphism on A; that is, there exists an integer k such that b-1 ab = a' for all a e A. If p = 2 and Exp A > 4, our assump- tion implies that k =_ 1 (mod 4) and so every element of A with order at most 4 lies in the centre of G. Thus if H is a proper subgroup of G, every subgroup of A n H is normal in H, and H/A n H AH/A is cyclic; also, if p = 2 and Exp A n H > 4, there exist 1 = A2 < Al < A n H such that A 1 C4 and [A1, H] 5 [A G] = 1, By in- duction, L(H) is modular. So if G were not an M-group, then by 2.3.3 there would exist a normal subgroup N of G with GIN D8 or GIN nonabelian of order p' and exponent p for p > 2. Since G/AN is cyclic, we have I G : AN I < p in both cases. But 2.3 Finite p-groups with modular subgroup lattices 57 every subgroup of AN/N is normal in GIN, and this implies that p = 2 and AN/N is the cyclic subgroup of order 4 in the dihedral group GIN. Hence A/A n N C4 and by assumption, [A, G] < A n N. It follows that [AN, G] 5 N, a contradiction. Thus G is an M-group.

It will follow immediately from Lemma 2.3.4 that the groups in Theorem 2.3.1 are M-groups. We turn to the proof of the more difficult part of this theorem. Recall the definition, for an arbitrary p-group G and i e N, of the characteristic subgroups

O,(G)=andJ(G)=; we put f2(G) = Q1(G). By 1.2.7, Q,(G)0 = fl,(G) and Ui(G)'P = Ji(G) for every projectivity cp from G to a p-group G.

23.5 Lemma. If G is a finite p-group with modular subgroup lattice, then (1(G) is elementary abelian and Q,(G) = {x E GIxP' = 1} for all i E N.

Proof. Let P = {x E G i xP = 11. If x and y are nontrivial elements of P with , then by 2.3.2, is elementary abelian of order p2 and hence xy_'E P and xy = yx. This clearly also holds if = . Thus P is an abelian subgroup of G, and therefore ((G) = P is elementary abelian. This is the first assertion of the lemma and also the case i = 1 of the second. So assume that this second statement is true for i - 1. Then for x, y e G with xP' = 1 = yP', we have x"'-', y"'-' E ((G), and by induction applied to G/11(G), we get (xy)P'-' E ((G). It follows that (xy)P' = 1 and that fQi(G) only contains elements of order at most p'.

We first determine the 2-groups with modular subgroup lattices in which the quaternion group Q8 is involved. We start with a property of modular 2-groups that will be needed later.

23.6 Lemma. If G is a finite 2-group with modular subgroup lattice, then (2(G) normalizes every subgroup of G, that is f22(G) < N(G), the norm of G.

Proof. We use induction on IGI. Let H be a nontrivial subgroup of G and let Z = be a normal subgroup of order 2 of H. By 2.3.2, if x E f22(G), then I : I S 2 and hence ' = . If z did not centralize x, then xZ = x-' and would be a dihedral group of order 8; but this is not an M-group. Therefore is abelian and it follows that Z a Hc22(G). The group H02(G)/Z has smaller order than G, hence by induction, H/Z is normalized by f22(Hf22(G)/Z) and this group contains (2(G)Z/Z. Thus (12(G) normalizes H.

23.7 Lemma. If G is a finite M-group of exponent 4, then G is abelian or G = Q x A where Q- Q8 and A is elementary abelian.

Proof. Suppose that G is not abelian, let x, y e G with xy yx and put Q = . By 2.3.6, every subgroup of G is normal in G. Hence 1 # [x, y] E n and so 58 Modular lattices and abelian groups

I I < 8. Since the dihedral group D. is not an M-group, Q- Q8. As a subgroup of Pot Q, G/CG(Q) has order at most 4. On the other hand QCG(Q)/CG(Q) = Q/Z(Q) is of order 4. Thus G = QCG(Q). Force CG(Q), we have (xc) = x-'c : xc and since o(xc) < 4, it follows that x-'c = (xcv = (xc)-' = x-'c-'. Thus c2 = 1 and CG(Q) is elementary abelian. Then G = Q x A where A is a complement to Z(Q) in CG(Q)

2.3.8 Theorem. Let G be a finite 2-group with modular subgroup lattice. If the quaternion group Q8 is involved in G, that is,if there exist K a H < G with H/K Q8, then G = Q x A where Q ^-' Q8 and A is elementary abelian.

Proof. We use induction on I GI to show that Exp G = 4. Then the theorem will follow from 2.3.7. So suppose that Exp G > 4 and take u e G with o(u) = 8. Assume first that there exists a proper subgroup of G in which Q8 is involved. Then by induction, Q8 is also a subgroup of G and hence of i22(G). By 2.3.7, S22(G) = Q x A where Q Q8 and A is elementary abelian. Since u2 E S22(G), there exist x e Q and a c A with o(x) = 4 and u2 = xa. Let Q = . Then u4 = x2 = y2, hence a and, by 2.3.6 applied to /, it follows that u = . Since o(y) = 4, y"2 = y; on the other hand, y" = y"° = y-', a contradiction. We are left with the case that Q8 is not involved in any proper subgroup of G. Since Q8 is involved in G, there exists N a G with GIN Q8. As Exp G > 4, N :A 1 and we take a minimal normal subgroup Z of G contained in N. By induction, Q8 is a subgroup of G/Z and, since it cannot be a proper subgroup, G/Z ^- Q8. Thus Z = N and IGI = 16. Since L(D8) is not modular and Q8 is not a subgroup of G, every subgroup of order 8 of G is abelian. Since GIN Q8, we see that N < and is the intersection of abelian maximal subgroups of G; thus :!5; Z(G). On the other hand there exists v e G with GIN = and u° = u-'z for some z e N, and it follows that (u2)" = u-2. This contradiction shows that Exp G = 4.

M*-groups

We say that a finite p-group G is an M*-group if L(G) is modular and the quaternion group Q8 is not involved in G. Thus for p > 2, every p-group with modular subgroup lattice is an M*-group. Furthermore, every subgroup and every factor group of an M*-group is an M*-group. We want to show that every M*-group satisfies (b) of Theorem 2.3.1; in view of 2.3.8 this will finish the proof of that theorem. We start with a useful remark on cyclic permutable subgroups of maximal order in arbitrary p-groups.

2.3.9 Lemma. If X is a cyclic permutable subgroup of the p-group G such that I X I= Exp G, then f2(X) < Z(G).

Proof. Let y c- G and Y = . If y E X, then y clearly centralizes f2(X). So assume that y X and let N be a maximal subgroup of X Y containing X. Then N a X Y, 2.3 Finite p-groups with modular subgroup lattices 59 hence Xy < N and therefore JX:XnX =IXX':Xyl

Since I XI = ExpG, we have I Y: X n YI < I X I and it follows that X n Xy 1. But then the minimal subgroup of X n X' is S2(X) and also Q(Xy), and we get that 0(X) = Q(Xy) = S2(X)y. Since Q(X) is of order p,itis centralized by y. Thus S2(X) < Z(G).

2.3.10 Lemma. Let G be an M*-group of exponent p". Then the map u defined by x° = x°"-' for x e G is a homomorphism from G to i2(Z(G)).

Proof. That x° e S2(Z(G)) for all x e G follows from 2.3.9. So we have to show that (xy)""-' = x°"-'y°"-' for all x, y e G, and we do this by induction on n. The case n = 1 being trivial, assume first that n = 2. Then if p = 2, G is abelian by 2.3.7. For p > 2, we let H = and show that H' < Q(H) n Z(H). This is clear if IHI < p3. And if JHI > p3, then o(x) = o(y) = p2 and thus S2()Q() < Z(H) by 2.3.9. Furthermore H/S2()i2() is generated by two elements of order p and hence is abelian. Therefore once again H' < f2(H) n Z(H) and by (4) of 1.5,

(xy)" = x"y"[y,x](2)= x"y".

Finally, let n >_ 3 and assume that the formula is correct for n - 1. By 2.3.5, G/f2(G) is an M*-group of exponent p"-' and it follows that (xy)p°-2 = xp°-2ypi-'z where z e Q(G). Since x""-', yp"-' and z are contained in the M*-group Q2(G) of exponent p2, the first part of our proof yields that

(xy)p" l (x"" 2y". xp^-'yp-'Zp = xp- 'yp-'. = 2Z)" =

The last two results suggest studying elements of maximal order in M*-groups. In abelian p-groups, a fundamental property of such elements is that they generate direct factors. A similar result holds in M*-groups.

2.3.11 Lemma. Let X be a cyclic subgroup of the M*-group G such thatI X j = Exp G. If H < G with H n X = 1, then there exists a complement C of X in G containing H, that is, a subgroup C of G such that G = XC, X n C = 1 and H < C.

Proof. We use induction on I G1. If H contains a nontrivial normal subgroup N of G, then X N/N is cyclic of maximal order in GIN and X N n H = N(X n H) = N. By induction there exists a subgroup C of G containing N and H such that G = XNC and XNnC= N.HenceG = XCand XnC

Hamiltonian p-groups

2.3.12 Theorem. All the subgroups of a finite p-group G are normal if and only if G is abelian or G = Q x A where Q ^- Q8 and A is an elementary abelian 2-group.

Proof. If G is such a direct product, then the elements of order 2 in G are contained in Z(G) and subgroups of exponent 4 contain Z(Q) = G'. Hence every subgroup of G is normal. Conversely, suppose that G is a hamiltonian p-group. We show by induction that G has the desired structure. By 2.1.4, G is an M-group. If G were an M*-group, then for a cyclic subgroup X of maximal order in G, by 2.3.11 there would exist a subgroup C of G such that G = X x C. Since G is not abelian, C would be hamiltonian and, by induction, Q8 would be involved in C, a contradiction. Thus G is not an M*-group and by 2.3.8, G has the desired structure.

The structure of arbitrary hamiltonian groups is well-known (see Robinson [1982], p. 139), and is described in Exercise 1. Theorems 2.3.8 and 2.3.12 show that a p-group is an M*-group if and only if it is a nonhamiltonian M-group. So these two notions are the same and we introduce a third, related one.

Iwasawa triples

The triple (A, b, s) is called an Iwasawa triple for the p-group G if A is an abelian normal subgroup of G, b e G and s is a positive integer which is at least 2 in case p = 2 such that G = A and b-lab = a1+p° for all a c A. To finish the proof of Iwasawa's theorem, we have to show that every M*-group possesses an Iwasawa triple. We start with a sufficient condition for the existence of such a triple.

2.3.13 Lemma. Let A be a subgroup of the M*-group G such that every subgroup of A is normal in G. (a) Then A is abelian and G/CG(A) is cyclic. (b) If Exp A # 4 and B is a cyclic subgroup of G, then there exist a generator b of B and an integer s such that (A, b, s) is an Iwasawa triple for AB. 2.3 Finite p-groups with modular subgroup lattices 61

Proof. By 2.3.12, A is abelian since QS is not involved in G. Furthermore, the elements of G induce power automorphisms on A which are universal by 1.5.4. So if x e A with o(x) = Exp A = p", then CG(A) = CG() and hence G/CG(A) is isomor- phic to a subgroup of Aut(x). If p > 2, Aut (x) is cyclic of order pn-1(p - 1) and for s = 1,..., n,the sub- group of order p"-' of Aut(x) is generated by the automorphism a with x° = xl+°' (see Huppert [1967], p. 84). In particular, (a) holds in this case. Furthermore, IB/CB((x))l = p"-' for some s e {1,...,n}, and hence there exists b e B such that B = and xb = xl+P. Since b operates as a universal power automorphism on A and o(x) = Exp A, it follows that ab = a1+p' for all a e A. Thus (A, b, s) is an Iwasawa triple for AB. Now assume that p = 2 and n >_ 3. Then Aut n (g) S (x4). By 2.3.6 applied to G/ 2. Finally, if p = 2 and n 5 2, then lAutl < 2. In particular, G/CG(A) is cyclic. If Exp A # 4, it follows that A < Z(G) and then for every generator b of B, (A, b, 2) is an Iwasawa triple for AB.

If Exp A = 4 in 2.3.13, A need not be a member of an Iwasawa triple for AB (see Exercise 4). It is not difficult to determine the structure of AB in this case (see Exercise 5). However, we shall not need this in our proof of Iwasawa's theorem since we can use 2.3.6 if Exp A = 4.

2.3.14 Lemma. Let G be an M*-group, X a cyclic subgroup of G with I X I = Exp G and suppose that H < G such that H n X = 1. (a) If Hx = H, then every subgroup of H is normal in G. (b) H/HG is cyclic.

Proof. (a) Let X = and p" = o(x) = Exp G. By 2.3.11 there exists a complement C to X in G with H S C. Every g e G therefore has the form g = x'y where i e N, y e C, and it follows that Ha = H'' S C. Thus the normal closure HG of H is con- tained in C. Therefore, if K 5 H and g e G with n C = 1, then

K = K((g) n HG) = K_ XC = G. (b) Since our assumptions are inherited by G/HG, we may assume that HG = 1. Then by (a), CH(X) = 1 and hence NH(X) is cyclic by 2.3.13, (a). On the other hand, 62 Modular lattices and abelian groups every minimal subgroup K of H normalizes X since I X K: X I= I K I= p. Thus H has only one minimal subgroup and therefore is cyclic since QB is not involved in G..

2.3.15 Theorem. Let G be an M*-group and suppose that X is a cyclic normal sub- group of G withI X I = Exp G. Then every subgroup of C6(X) is normal in G and G/CG(X) is cyclic. If G = CG(X)B where B is cyclic, then there exists a generator b of B and an integer s such that (CG(X), b, s) is an Iwasawa triple for G.

Proof. By 2.3.13, G/CG(X) is cyclic. If C is a complement to X in G, then CG(X) = XC n CG(X) = X x Co where Co = C n CG(X). Let g e G and X = . Since X 9 G, there exists an integer r such that xa = x'. For c e Co take y E X with o(y) = o(c). Then = x is abelian and < yc> n X = 1 = n X. By 2.3.14, and are normal subgroups of G. Since y c- X a G, also 9 G and by 1.5.4, g induces a universal power automorphism in . Hence ca = c' and it follows that as = a' for all a e CG(X) = X x Co. Thus every subgroup of CG(X) is normal in G. By 2.3.13, G has the desired structure except, possibly, when Exp CG(X) = 4. But then Exp G = o(x) = 4, G is abelian by 2.3.7 and the assertion holds trivially.

Of course, CG(X) is abelian in 2.3.15. In particular, as a useful consequence, we note the following.

2.3.16 Corollary. If G is an M*-group containing an element z e Z(G) with o(z) _ Exp G, then G is abelian.

Theorem 2.3.15 shows how to find an Iwasawa triple in the M*-group G if there exists a cyclic subgroup X of maximal order that is normal in G: one can take CG(X) as the abelian normal subgroup A. If there is no such X, we cannot describe A so easily. Furthermore, we have to do some unpleasant calculations.

2.3.17 Lemma. Let L = be an M*-group, X = , o(a) < o(x) = Exp L and suppose that ax = a'+psxo wherex0 e S2(X) and s is an integer which is at least 2 in case p = 2. (a) If X is not normal in L, then R+1(X) 5 Z(L) and there exists x1 E Sts+1(X) such that d" = d'+e` and L = where d = ax1. (b) Suppose that there exists u e L such that L = X , X n = 1and 9 L. Then Sts+,(X) 5 Z(L), u e Q5+1(X) and ux = u'+e.

Proof. (a) First note that by 2.3.9, xo E Z(L) and therefore ax' = for all i e N. Indeed, this is clear by assumption if i = 1; and if it is true for some i e N, then

axi+i = (ax)cl+papx0 = all+p"1,+ix0i+1

Let o(x) = p". Since X is not normal in L, ae' o X and hence ps < o(a) < p", that '= is, s < n - 1. It follows that (1 + p'')e"-'-' ° 1 (modp"-') and therefore ax°" ' ,-lxo" a('+e`)° '' = a. Thus Sts+1(X) = 5 Z(L). Finally, x0 = x`e"-' for 2.3 Finite p-groups with modular subgroup lattices 63 some i e NJ and we put x, = x'P"-'-'. Then for d = ax,, we clearly have L = (x, a> _ (x, d > and

x"'"-.-' al+P.xfP. (axl)l+t dl+p d' = (ax'D^ ')x = = = (b) By (a), I,+1(X) < Z(L) if X is not normal in L; and if X a L, then L = X x is even abelian. Since L/(u) X is cyclic, [a,x] = aP'xo e and hence aP'+` _ (aP'xo )P a n . Therefore I : I= I : (a> n I<< ps+l and so < C,+, (X). Let u, a (u> and x, a 52,+, (X) such that a = ul x- j'. Then L = X