DE GRUYTER Roland Schmidt EXPOSITIONS IN MATHEMATICS 14 Subgroup 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 Semigroups, 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 Geometry and Linear Differential Equations, 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 Field 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: Group, lattice, subgroup lattice, projectivity, duality ® 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. Group theory.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 translation 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 relation between the structure of a group and the structure of its lattice of subgroups. The origin of the subject may be traced back to Dedekind, who studied the lattice of ideals in a ring of algebraic integers; 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 isomorphism 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 class 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 class of groups 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 isomorphisms? 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 cosets. 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 homomorphisms 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