Introduction to Group Theory

The present book is the outcome of a one-semester lecture course which the author has given frequently during the last three decades. The course has been gradually modified over the years in accordance with changing outlook and with the steadily increasing sophistication of the audience, third- and fourth-year honours classes in several universities in Australia and in Canada. Out of the conviction that no branch of Mathematics can be mastered by memorizing facts and methods I have tried from the beginning to make the subject interesting for the reader. Clearly one cannot hope to please everybody in this respect. I have sought, how ever, to attract the reader's interest by including a number of dis cussions and examples which either have interested me on occasion or resulted from questions of students; some sections and examples have been taken from work done by now eminent mathematicians in an early period of their career, assuming that such selections will appeal to younger readers. So J hope that this book will be found to be a reasonably modern, although not conventional, text proposing an amount of material most of which can be dealt with in a half-year's lecture course. After studying the book the reader should be able to tackle those problems in group theory which are scattered in the problem sections of the American Mathematical Monthly and other similar periodicals.

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I Definition of a Group and Examples.1 The abstract group and the notion of group isomorphism.2 Groups of mappings. Permutations. Cayley's theorem.3 Arithmetical groups.4 Geometrical Groups.- II Subsets, Subgroups, Homomorphisms.1 The algebra of subsets in a group.2 A subgroup and its cosets. Lagrange's theorem.3 Homomorphisms, normal subgroups and factor groups.4 Transformation. Conjugate elements. Invariant subsets.5 Correspondence theorems. Direct products.6 Double cosets and double transversals.- III Automorphisms and Endomorphisms.1 Groups of automorphisms. Characteristic subgroups.2 The holomorph of a finite group. Complete groups.3 Group extensions.4 A problem of Burnside: Groups with outer automorphisms leaving the classes invariant.5 Endomorphisms and operators.- IV Finite Series of Subgroups.1 The fundamental concepts of lattice theory.2 Lattices of subgroups.3 The theory of O. Schreier.4 Central chains and series.- V Finite Groups and Prime Numbers.1 Permutation groups.2 Sylow's theorems.3 Finite nilpotent groups.4 The structure of finite abelian groups.- Appendix Hints or Solutions to Some of the Exercise Problems.- Author index.