Table Of Content

Example o f Groiro Examples of Groups This book is volume 1 of the series Examples of Mathematical Structures Examples of Groups Michael Weinstein Polygonal Publishing House 80 Passaic Avenue Passaic NJ, 07055 Copyright © 1977 by Polygonal Publishing House All rights of publication reserved Paste-up and mechanicals by Jose Garcia Artwork by Dorothy Brosterman Cover design by Janet Wallach Library of Congress Cataloging in Publication Data Weinstein, Michael 1945“ Examples of groups. Bibliography : p. Includes indexes. 1. Groups, Theory of. I. Title. QA171.W39 512*.22 76-51379 Printed in the United States of America Dedicated to Belle, Abe, and Jerry who helped a lot Contents Logical Dependence of Chapters........................................................................viii Preface ............................................................................................................. ix Part One Materials and Constructions Chapter 1 Constructions............................................................................... . 1 Chapter 2 Free Groups and Presentations.......................................................53 Chapter 3 Matrices..........................................................................................64 Part Two Examples Introduction to Part Two..................................................................................98 Chapter 4 Finite Groups.................................................................................99 Chapter 5 Infinite Abelian Groups...............................................................159 Chapter 6 Infinite Nonabelian Groups..........................................................206 Appendixes A.l Referenced Theorems...........................................................................259 A.2 Direct Factors and the ® Notation......................................................260 A.3 The Axiom of Choice and Zorn’s Lemma............................................262 A.4 Abelian Groups....................................................................................263 A.5 The Ascending Central Series and Nilpotent Groups...........................265 A.6 The Derived Series and Solvable Groups...............................................270 A.l Hypercyclic Groups.............................................................................272 A.8 Semidivisible, Divisible, and p-divisible Groups.....................................274 A.9 Lattices and Subgroup Lattices.............................................................276 A. 10 Miscellaneous.......................................................................................281 References.......................................................................................................286 Hints to Some of the Exercises.........................................................................287 Glossary and Index of Definitions....................................................................293 Notation Index................................................................................................297 Index of Examples Arranged by Group Properties..........................................301 Index of Some Interesting Examples...............................................................303 Index of Counterexamples.............................................................................305 Logical Dependence of Chapters and Where to Find Things It is recommended that this book be read “grasshopper” style, rather than from cover to cover. The reader, depending on his tastes, can begin reading in Chapters 4, 5, or 6 and turn to the earlier chapters and to the appendixes when he needs to. As can be seen from the chart, the appendixes provide background material necessary for the other chapters. It is suggested, therefore, that the reader scan the appendixes to see what topics are covered there before he begins reading elsewhere. Many terms are used in the book that are not assumed to be known to the reader. Their definitions can be found in the glossary. The notation index per forms a similar function for notation. Preface “Theories change but the groups remain.” Marshall Hall Jr. and James K. Senior [15] Why study examples? Several reasons come to mind. For the mathematician doing research, exam ples are all but indispensable to his work. To begin with, the direction of his re search is guided by a thorough examination of all the pertinent examples he can get his hands on. Only after these examples are analyzed does he attempt to formulate their common properties into some sort of a theorem, and then attempt to prove the theorem. For the student of group theory, regardless of his level, examples help to clar ify and justify the definitions and theorems of the subject he is studying. Take, for instance, the definition of solvable group. As soon as the student sees this he will want to know the answers to at least two questions (a) Are there such things? (b) Are there groups that are nonsolvable? These questions are more than just reasonable, they seem necessary to an under standing of the concept. Both questions must be answered by examples. A third use of examples, related but not identical to the previous two, is the idea of a counterexample. Counterexamples are given, and rightfully so, a much more honored position in mathematices than the two kinds of examples prev iously discussed. Where the examples a research mathematician uses to formulate a theorem he later proves take a back seat to the theorem itself, and where the examples a student (or his instructor) proffers to answer questions such as (a) and (b) are of a rather personal and short-lived nature, a counterexample is as much a result in the theory of groups as a theorem is, and occupies a similarly high position. Theorems and counterexamples are simply results in opposite di rections. Theorems assert that certain propositions about groups are true while counterexamples assert that other propositions are false. This book was written with the dual purposes in mind of providing examples that serve to illustrate various group-theoretical concepts, and providing counter examples. One further topic requires some explanation. If certain methods and materials can be used to supply an example with interesting properties, it is not unreason able to hope that a subtle change in these methods or materials will result in a group with equally interesting but somewhat different properties. It is not un reasonable to hope that the methods can be abstracted into a theory which yields not one but a number of different examples. When such a procedure is effected, it is called a construction. Constructions will be described abstractly, that is without regard to what material groups go into the construction, in Chap ter 1, then particularized when they are needed in the later chapters.