04548 - Groups and Symmetry [Armstrong].pdf

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M.A. Armstrong
Groups and Symmetry
With 54 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo Hong Kong
M.A. Armstrong
Department of Mathematical Sciences
University of Durham
Durham DHI 3LE
England
Editorial Board
J.H. Ewing
Department of
Mathematics
Indiana University
Bloomington, IN 47405
U.S.A.
F.W. Gehring
Department of
Mathematics
University of Michigan
Ann Arbor, MI 48019
U.S.A.
P.R. Halmos
Department of
Mathematics
Santa Clara University
Santa Clara, CA 95053
U.S.A.
Mathematics Subject Classifications (1980): 20-0 I, 20F32
Library of Congress Cataloging-in-Publication Data
Armstrong, M.A. (Mark Anthony)
Groups and symmetry! M.A. Armstrong.
(Undergraduate texts in mathematics)
p.
cm.
Bibliography: p.
Includes index.
I. Groups, Theory of. 2. Symmetry groups.
I.
TItle
QAI7l.A76 1988
512'.2-dcI9
87-37677
Cover art taken from
Ornamental Design
by Claude Humbert
Switzerland.
©
Office du Livre, Freibourg,
©
1988 by Springer-Verlag New York Inc.
All rights reserved. This work may not
be
translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010,
U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec-
tion with any form of information storage and retrieval, electronic adaptation, computer soft-
ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to
be
taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
Printed and bound by R.R. Donnelley
&
Sons, Harrisonburg, Virginia.
Printed in the United States of America.
98765432
ISBN 0-387-96675-7 Springer-Verlag New York Berlin Heidelberg
ISBN 3-540-96675-7 Springer-Verlag Berlin Heidelberg New York
For Jerome and Emily
The beauty of a snow crystal depends on its mathematical regularity and
symmetry; but somehow the association of many variants of a single
type, all related but no two the same, vastly increases our pleasure and
admiration.
0'
ARCY
THOMPSON
(On Growth and Form,
Cambridge, 1917.)
En generalje crois que les seules structures mathematiques interessantes,
dotees d'une certaine legitimite, sont celles ayant une realisation na-
turelle dans Ie continu .... Ou reste, cela se voit tres bien dans des
theories purement algebriques comme la theorie des groupes abstraits
ou on a des groupes plus ou moins etranges apparaissant comme des
groupes d'automorphismes de figures continues.
RENE THOM
(Paraboles et Catastrophes,
Flammarion, 1983.)
Preface
Numbers measure size,
groups measure symmetry.
The first statement comes
as no surprise; after all, that is what numbers "are for". The second will be
exploited here in an attempt to introduce the vocabulary and some of the
highlights of elementary group theory.
A word about content and style seems appropriate.
In
this volume, the
emphasis is on
examples
throughout, with a weighting towards the symmetry
groups of solids and patterns. Almost all the topics have been chosen so as to
show groups in their most natural role, acting on (or permuting) the members
of a set, whether it be the diagonals of a cube, the edges of a tree, or even some
collection of subgroups of the given group. The material is divided into
twenty-eight short chapters, each of which introduces a new result or idea.
A glance at the Contents will show that most of the mainstays of a "first
course" are here. The theorems of Lagrange, Cauchy, and Sylow all have a
chapter to themselves, as do the classification of finitely generated abelian
groups, the enumeration of the finite rotation groups and the plane crystallo-
graphic groups, and the Nielsen-Schreier theorem.
I have tried to be informal wherever possible, listing only significant results
as theorems and avoiding endless lists of definitions. My aim has been to write
a book which can be read with or without the support of a course of lectures.
It is not designed for use as a dictionary or handbook, though new concepts
are shown in bold type and are easily found in the index. Every chapter ends
with a collection of exercises designed to consolidate, and in some cases fill
out, the main text.
It
is essential to work through as many of these as possible
before moving from one chapter to the next. Mathematics is not for spectators;
to gain in understanding, confidence, and enthusiasm one has to participate.
As prerequisites I assume a first course in linear algebra (including matrix
multiplication and the representation of linear maps between Euclidean

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