DOCSLIB.ORG

Subgroup Lattices of Groups De Gruyter Expositions in Mathematics 14

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Subgroup Lattices of Groups De Gruyter Expositions in Mathematics 14 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
Load more

Recommended publications
  • ORTHOGONAL GROUP of CERTAIN INDEFINITE LATTICE Chang Heon Kim* 1. Introduction Given an Even Lattice M in a Real Quadratic Space
    JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 20, No. 1, March 2007 ORTHOGONAL GROUP OF CERTAIN INDEFINITE LATTICE Chang Heon Kim* Abstract. We compute the special orthogonal group of certain lattice of signature (2; 1). 1. Introduction Given an even lattice M in a real quadratic space of signature (2; n), Borcherds lifting [1] gives a multiplicative correspondence between vec- tor valued modular forms F of weight 1¡n=2 with values in C[M 0=M] (= the group ring of M 0=M) and meromorphic modular forms on complex 0 varieties (O(2) £ O(n))nO(2; n)=Aut(M; F ). Here NM denotes the dual lattice of M, O(2; n) is the orthogonal group of M R and Aut(M; F ) is the subgroup of Aut(M) leaving the form F stable under the natural action of Aut(M) on M 0=M. In particular, if the signature of M is (2; 1), then O(2; 1) ¼ H: O(2) £ O(1) and Borcherds' theory gives a lifting of vector valued modular form of weight 1=2 to usual one variable modular form on Aut(M; F ). In this sense in order to work out Borcherds lifting it is important to ¯nd appropriate lattice on which our wanted modular group acts. In this article we will show: Theorem 1.1. Let M be a 3-dimensional even lattice of all 2 £ 2 integral symmetric matrices, that is, ½µ ¶ ¾ AB M = j A; B; C 2 Z BC Received December 30, 2006. 2000 Mathematics Subject Classi¯cation: Primary 11F03, 11H56.
    [Show full text]
  • Frattini Subgroups and -Central Groups
    Pacific Journal of Mathematics FRATTINI SUBGROUPS AND 8-CENTRAL GROUPS HOMER FRANKLIN BECHTELL,JR. Vol. 18, No. 1 March 1966 PACIFIC JOURNAL OF MATHEMATICS Vol. 18, No. 1, 1966 FRATTINI SUBGROUPS AND φ-CENTRAL GROUPS HOMER BECHTELL 0-central groups are introduced as a step In the direction of determining sufficiency conditions for a group to be the Frattini subgroup of some unite p-gronp and the related exten- sion problem. The notion of Φ-centrality arises by uniting the concept of an E-group with the generalized central series of Kaloujnine. An E-group is defined as a finite group G such that Φ(N) ^ Φ(G) for each subgroup N ^ G. If Sίf is a group of automorphisms of a group N, N has an i^-central series ι a N = No > Nt > > Nr = 1 if x~x e N3- for all x e Nj-lf all a a 6 £%f, x the image of x under the automorphism a e 3ίf y i = 0,l, •••, r-1. Denote the automorphism group induced OR Φ(G) by trans- formation of elements of an £rgroup G by 3ίf. Then Φ{£ίf) ~ JP'iΦiG)), J^iβiG)) the inner automorphism group of Φ(G). Furthermore if G is nilpotent9 then each subgroup N ^ Φ(G), N invariant under 3ίf \ possess an J^-central series. A class of niipotent groups N is defined as ^-central provided that N possesses at least one niipotent group of automorphisms ££'' Φ 1 such that Φ{βίf} — ,J^(N) and N possesses an J^-central series. Several theorems develop results about (^-central groups and the associated ^^-central series analogous to those between niipotent groups and their associated central series.
    [Show full text]
  • ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆.
    [Show full text]
  • ON the INTERSECTION NUMBER of FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler
    University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-14-2019 ON THE INTERSECTION NUMBER OF FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Algebra Commons, and the Discrete Mathematics and Combinatorics Commons Recommended Citation Bautista Serrano, Humberto, "ON THE INTERSECTION NUMBER OF FINITE GROUPS" (2019). Math Theses. Paper 9. http://hdl.handle.net/10950/1332 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. ON THE INTERSECTION NUMBER OF FINITE GROUPS by HUMBERTO BAUTISTA SERRANO A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Kassie Archer, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler April 2019 c Copyright by Humberto Bautista Serrano 2019 All rights reserved Acknowledgments Foremost I would like to express my gratitude to my two excellent advisors, Dr. Kassie Archer at UT Tyler and Dr. Lindsey-Kay Lauderdale at Towson University. This thesis would never have been possible without their support, encouragement, and patience. I will always be thankful to them for introducing me to research in mathematics. I would also like to thank the reviewers, Dr. Scott LaLonde and Dr. David Milan for pointing to several mistakes and omissions and enormously improving the final version of this thesis.
    [Show full text]
  • 7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
    7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes.
    [Show full text]
  • Topological Duality and Lattice Expansions Part I: a Topological Construction of Canonical Extensions
    TOPOLOGICAL DUALITY AND LATTICE EXPANSIONS PART I: A TOPOLOGICAL CONSTRUCTION OF CANONICAL EXTENSIONS M. ANDREW MOSHIER AND PETER JIPSEN 1. INTRODUCTION The two main objectives of this paper are (a) to prove topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. The paper is first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper. Regarding objective (a), consider the following simple question: Is there a subcategory of Top that is dually equivalent to Lat? Here, Top is the category of topological spaces and continuous maps and Lat is the category of bounded lattices and lattice homomorphisms. To date, the question has been answered positively either by specializing Lat or by generalizing Top. The earliest examples are of the former sort. Tarski [Tar29] (treated in English, e.g., in [BD74]) showed that every complete atomic Boolean lattice is represented by a powerset. Taking some historical license, we can say this result shows that the category of complete atomic Boolean lattices with complete lat- tice homomorphisms is dually equivalent to the category of discrete topological spaces. Birkhoff [Bir37] showed that every finite distributive lattice is represented by the lower sets of a finite partial order. Again, we can now say that this shows that the category of finite distributive lattices is dually equivalent to the category of finite T0 spaces and con- tinuous maps.
    [Show full text]
  • Finite Groups Whose «-Maximal Subgroups Are Subnormal
    FINITE GROUPS WHOSE «-MAXIMAL SUBGROUPS ARE SUBNORMAL BY AVINO'AM MANN Introduction. Dedekind has determined all groups whose subgroups are all normal (see, e.g., [5, Theorem 12.5.4]). Partially generalizing this, Wielandt showed that a finite group is nilpotent, if and only if all its subgroups are subnormal, and also if and only if all maximal subgroups are normal [5, Corollary 10.3.1, 10.3.4]. Huppert [7, Sätze 23, 24] has shown that if all 2nd-maximal subgroups of a finite group are normal, the group is supersolvable, while if all 3rd-maximal sub- groups are normal, the group is solvable of rank at most 2 (see below for the defi- nitions of the terms employed). Continuing this, Janko [8] has determined all nonsolvable finite groups, all of whose 4-maximal subgroups are normal, and also proved some results on the structure of solvable groups with the same property. Later, he has also determined all finite simple groups, all of whose 5-maximal subgroups are normal [9]. The present paper deals with similar problems. Our main concern is with the class of groups defined in the title. In §2, we derive some simple properties of these groups. For «^5, we extend the aforementioned results of Huppert and Janko to cover also our case (i.e., replacing normality by subnormality), and prove some results under slightly more general assumptions. In §3 we study solvable groups with subnormal «-maximal subgroups. If the order of the group in question has enough distinct prime divisors, the structure of the group is very limited.
    [Show full text]
  • (Hereditarily) Just Infinite Property in Profinite Groups
    Inverse system characterizations of the (hereditarily) just infinite property in profinite groups Colin D. Reid October 6, 2018 Abstract We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are ‘universal’ in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. This is a corrected and revised version of [8]. 1 Introduction Notation. In this paper, all groups will be profinite groups, all homomorphisms are required to be continuous, and all subgroups are required to be closed; in particular, all references to commutator subgroups are understood to mean the closures of the corresponding abstractly defined subgroups. For an inverse system Λ= {(Gn)n>0, ρn : Gn+1 ։ Gn} of finite groups, we require all the homomorphisms ρn to be surjective. A subscript o will be used to indicate open inclusion, for instance A ≤o B means that A is an open subgroup of B. We use ‘pronilpotent’ and ‘prosoluble’ to mean a group that is the inverse limit of finite nilpotent groups or finite soluble groups respectively, and ‘G-invariant subgroup of H’ to mean a subgroup of H normalized by G. A profinite group G is just infinite if it is infinite, and every nontrivial normal subgroup of G is of finite index; it is hereditarily just infinite if in addition every arXiv:1708.08301v1 [math.GR] 28 Aug 2017 open subgroup of G is just infinite. At first sight the just infinite property is a qualitative one, like that of simplicity: either a group has nontrivial normal subgroups of infinite index, or it does not.
    [Show full text]
  • Finite Groups with Pro-Normal Subgroups
    FINITE GROUPS WITH PRO-NORMAL SUBGROUPS T. A. PENG The purpose of this paper is to study a class of finite groups whose subgroups of prime power order are all pro-normal. Following P. Hall, we say that a subgroup H of a group G is pro-normal in G if and only if, for all x in G, H and 771 are conjugate in (77, 77x), the subgroup generated by 77 and 77*. We shall determine the structure of all finite groups whose subgroups of prime power order are pro-normal. It turns out that these groups are in fact soluble ¿-groups. A ¿-group is a group G whose subnormal subgroups are all normal in G. The struc- ture of finite soluble ¿-groups has been determined by Gaschiitz [l]. Are soluble ¿-groups precisely those groups whose subgroups of prime power order are all pro-normal? The answer is yes. Thus our study furnishes another characterization of soluble /-groups. Except for the definitions given above, our terminology and notation are standard (see, for example, M. Hall [2]). We first prove two general facts about pro-normal subgroups. Lemma 1. Let H be a pro-normal subgroup of a group G and let K be a subgroup of G containing 77. Then H is subnormal in K if and only if H is normal in K. Proof. Let H be subnormal in K. We may assume that there is a subgroup Loi G such that 77 is normal in L and L is normal in K. Let x be any element of K.
    [Show full text]
  • Groups with Identical Subgroup Lattices in All Powers
    GROUPS WITH IDENTICAL SUBGROUP LATTICES IN ALL POWERS KEITH A. KEARNES AND AGNES´ SZENDREI Abstract. Suppose that G and H are groups with cyclic Sylow subgroups. We show that if there is an isomorphism λ2 : Sub (G × G) ! Sub (H × H), then there k k are isomorphisms λk : Sub (G ) ! Sub (H ) for all k. But this is not enough to force G to be isomorphic to H, for we also show that for any positive integer N there are pairwise nonisomorphic groups G1; : : : ; GN defined on the same finite set, k k all with cyclic Sylow subgroups, such that Sub (Gi ) = Sub (Gj ) for all i; j; k. 1. Introduction To what extent is a finite group determined by the subgroup lattices of its finite direct powers? Reinhold Baer proved results in 1939 implying that an abelian group G is determined up to isomorphism by Sub (G3) (cf. [1]). Michio Suzuki proved in 1951 that a finite simple group G is determined up to isomorphism by Sub (G2) (cf. [10]). Roland Schmidt proved in 1981 that if G is a finite, perfect, centerless group, then it is determined up to isomorphism by Sub (G2) (cf. [6]). Later, Schmidt proved in [7] that if G has an elementary abelian Hall normal subgroup that equals its own centralizer, then G is determined up to isomorphism by Sub (G3). It has long been open whether every finite group G is determined up to isomorphism by Sub (G3). (For more information on this problem, see the books [8, 11].) One may ask more generally to what extent a finite algebraic structure (or algebra) is determined by the subalgebra lattices of its finite direct powers.
    [Show full text]
  • The M\" Obius Number of the Socle of Any Group
    Contents 1 Definitions and Statement of Main Result 2 2 Proof of Main Result 3 2.1 Complements ........................... 3 2.2 DirectProducts.......................... 5 2.3 The Homomorphic Images of a Product of Simple Groups . 6 2.4 The M¨obius Number of a Direct Power of an Abelian Simple Group ............................... 7 2.5 The M¨obius Number of a Direct Power of a Nonabelian Simple Group ............................... 8 2.6 ProofofTheorem1........................ 9 arXiv:1002.3503v1 [math.GR] 18 Feb 2010 1 The M¨obius Number of the Socle of any Group Kenneth M Monks Colorado State University February 18, 2010 1 Definitions and Statement of Main Result The incidence algebra of a poset P, written I (P ) , is the set of all real- valued functions on P × P that vanish for ordered pairs (x, y) with x 6≤ y. If P is finite, by appropriately labeling the rows and columns of a matrix with the elements of P , we can see the elements of I (P ) as upper-triangular matrices with zeroes in certain locations. One can prove I (P ) is a subalgebra of the matrix algebra (see for example [6]). Notice a function f ∈ I(P ) is invertible if and only if f (x, x) is nonzero for all x ∈ P , since then we have a corresponding matrix of full rank. A natural function to consider that satisfies this property is the incidence function ζP , the characteristic function of the relation ≤P . Clearly ζP is invertible by the above criterion, since x ≤ x for all x ∈ P . We define the M¨obius function µP to be the multiplicative inverse of ζP in I (P ) .
    [Show full text]
  • Nilpotent Groups
    Chapter 7 Nilpotent Groups Recall the commutator is given by [x, y]=x−1y−1xy. Definition 7.1 Let A and B be subgroups of a group G.Definethecom- mutator subgroup [A, B]by [A, B]=! [a, b] | a ∈ A, b ∈ B #, the subgroup generated by all commutators [a, b]witha ∈ A and b ∈ B. In this notation, the derived series is given recursively by G(i+1) = [G(i),G(i)]foralli. Definition 7.2 The lower central series (γi(G)) (for i ! 1) is the chain of subgroups of the group G defined by γ1(G)=G and γi+1(G)=[γi(G),G]fori ! 1. Definition 7.3 AgroupG is nilpotent if γc+1(G)=1 for some c.Theleast such c is the nilpotency class of G. (i) It is easy to see that G " γi+1(G)foralli (by induction on i). Thus " if G is nilpotent, then G is soluble. Note also that γ2(G)=G . Lemma 7.4 (i) If H is a subgroup of G,thenγi(H) " γi(G) for all i. (ii) If φ: G → K is a surjective homomorphism, then γi(G)φ = γi(K) for all i. 83 (iii) γi(G) is a characteristic subgroup of G for all i. (iv) The lower central series of G is a chain of subgroups G = γ1(G) ! γ2(G) ! γ3(G) ! ··· . Proof: (i) Induct on i.Notethatγ1(H)=H " G = γ1(G). If we assume that γi(H) " γi(G), then this together with H " G gives [γi(H),H] " [γi(G),G] so γi+1(H) " γi+1(G).
    [Show full text]