Groups and Characters

218 Pages·1997·8.76 MB·English

by  Larry C. Grove

GROUPS AND CHARACTERS
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editor Emeritus: PETER HILTON
Editors: MYRON B. ALLEN III, DAVID A. COX,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
GROUPS AND CHARACTERS
LARRY C. GROVE
Department of Mathematics
University of Arizona
Tucson, Arizona
A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
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This text is printed on acid-free paper.
Copyright © 1997 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work beyond
that permitted by Section 107 or 108 of the 1976 United
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information should be addressed to the Permissions Department,
John Wiley & Sons, Inc., 605 Third Avenue, New York, NY
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Library of Congress Cataloging in Publication Data:
Grove, Larry C.
Groups and characters / Larry C. Grove.
p. cm. — (Pure and applied mathematics)
"A Wiley-Interscience publication."
Includes bibliographical references and index.
ISBN 0-471-16340-6 (cloth : alk. paper)
1. Group theory. I. Title. II. Series: Pure and applied
mathematics (John Wiley & Sons : Unnumbered)
QA174.2.G77 1997
512'.2—dc21 96-29776
CIP
10 9 8 7 6 5 4 3 21
Contents
Preface vii
1 Preliminaries
1.1 Some Notation and Generalities 1
1.2 Permutation Actions 3
1.3 Coset Enumeration 9
1.4 Semidirect Products 13
1.5 Wreath Products 14
1.6 Transitivity and Primitivity 16
1.7 Some Linear Algebra 20
2 Some Groups
2.1 Aut(Z ) 27
n
2.2 Metacyclic Groups 29
2.3 Sylow Subgroups of Symmetric Groups 30
2.4 Affine Groups of Fields 32
2.5 Finite Groups in 2 and 3 Dimensions 33
2.6 Some Linear Groups 39
2.7 Mathieu Groups 46
2.8 Symplectic Groups 52
3 Counting with Groups
3.1 Fixed Points and Orbits 61
3.2 The Cycle Index 64
3.3 Enumeration 68
3.4 Generating Functions 73
3.5 The Petersen Graph 75
4 Transfer and Splitting
4.1 Transfer and Normal Complements 77
4.2 Hall Subgroups 84
4.3 Mostly p-groups 89
v
vi CONTENTS
5 Representations and Characters
5.1 Representations 97
5.2 Characters 105
5.3 Contragredients and Products 121
6 Induction and Restriction
6.1 Modules 127
6.2 Induction 128
6.3 Normal Subgroups and Clifford Theory 136
6.4 Mackey Theorems 142
6.5 Brauer Theorems 147
7 Computing Character Tables
7.1 Burnside 151
7.2 Dixon 154
7.3 Schneider 158
8 Characters of S and A
n n
8.1 Symmetric Groups 161
8.2 Alternating Groups 168
9 Frobenius Groups
9.1 Frobenius Groups and Their Characters 171
9.2 Structure of Frobenius Groups 176
10 Splitting Fields
10.1 Splitting 183
10.2 The Schur Index 184
10.3 R versus C 190
Bibliography 203
Index 209
Preface
The present volume is intended to be a graduate-level text. It covers some
aspects of group theory, concentrating mainly, but not exclusively, on finite
groups.
The presentation has been strongly, and positively, influenced by a number
of earlier texts and monographs. Particular mention should be made of books
by Curtis and Reiner [18], Dornhoff [22], Feit [23], Gorenstein [29], Isaacs [39],
and Passman [50].
Chapters 1, 2, and 4 could serve as the text for a basic one-semester
course on group theory. Chapter 2 consists entirely of examples, so it could
in principal be omitted, but not without radically altering the flavor of the
undertaking. Chapter 3 is more easily omitted; on the other hand Chapter 3
is easy and fun, and it provides important applications to combinatorics.
Chapters 5 and 6 contain a basic introduction to ordinary character theory
— they do not depend heavily on the preceding four chapters. Chapters 7
through 10 can be read independently of each other in any order following
Chapter 6. In fact, Chapter 7 only requires Chapter 5. It should be noted,
though, that Chapter 9, on Frobenius groups, makes fairly heavy use of some
of the group theory from Chapter 4.
Some attention has been paid to computational aspects of the subject.
For example, the Schreier-Sims algorithm, Todd-Coxeter coset enumeration,
and various algorithms for calculating character tables are discussed, typically
in the context of the very powerful (and free!) computational group theory
package GAP.
It is assumed throughout that the reader has assimilated most of the
material from a standard first-year graduate abstract algebra course in a
U.S. university, such as in [31]. This includes elementary group theory, such
as Sylow theorems (although a proof is included in Chapter 1), presentations,
solvability and nilpotence, etc.; as well as basic facts about rings, modules,
and field extensions. It is important for the reader to have a reasonable
facility with linear algebra. Nevertheless, some basic linear algebra is included
in the text, on the grounds that it may not always be covered in standard
undergraduate courses.
vn
vin PREFACE
I wish to thank the faculty, students, and staff of Lehrstuhl D für Mathe-
matik at the RWTH in Aachen for support and stimulation during two sab-
batical leaves. Particular acknowledgment is in order for Professor Joachim
Neubiiser, the founding father of GAP; it was my privilege to attend many
of his wonderfully clear lectures on groups and representations.
My special thanks to Robert Beals and Olga Yiparaki for careful readings
of parts of the manuscript.
Larry C. Grove
The University of Arizona
August 1996
GROUPS AND CHARACTERS
Groups and Characters
by Larry C. Grove
Copyright © 1997 John Wiley & Sons, Inc.
Chapter 1
Preliminaries
In this chapter we present a variety of concepts and ideas, in part to establish
terminology, usage, and notation. Everything presented will appear in later
chapters.
It is assumed that the reader is familiar with the elementary group-
theoretical material normally covered in a standard first graduate-level alge-
bra course (in the United States), and also the material from a junior/senior-
level linear algebra course. Any gaps can be filled by browsing in one or more
of the many texts available for such courses.
1.1 Some Notation and Generalities
If G is a group and x, y € G we shall write xv to denote y~1xy and vx to
denote yxy'1. Note that xvz = (xv)z and yzx = y(zx) for all x,y ,z € G.
If G is a group and H Ç G is a subgroup we write H < G, or G £ H,
as usual. If H < G then a set T of (right) coset representatives for H in G
will be called a (right) transversal; we will always assume that T C\ H = 1,
i.e. that the representative of H itself is the identity element. Note that
|T| = [G:H], the index of H in G.
If T is a transversal for H in G we define a "transversal function" from G
to T, denoted x ►-► 3?, via x = t, where t G T and Hx = Ht, or equivalently
Hx C\T = {x}. Note that the notation depends completely on the choice of
a fixed transversal T; if S is any transversal then {s: s 6 S} = T.
The first proposition lists three trivial but important facts about the
transversal function.
Proposition 1.1.1 Suppose that T is a transversal for H <G. Then
1. x = x,
2. xx"1 6 H, and
1