CONTENTS Introduction xi CHAPTER I. Groups and Counting Principles 1 1 Groups 1 2 G-spaces 2 3 Direct and semidirect products 5 4 Finite groups of rotations 11 5 The Platonic groups 13 6 The Sylow theorems 16 7 Counting and group structure 18 CHAPTER II. Fundamentals of Group Representations 21 1 Definition and unitarity 21 2 Irreducibility and complete reduction 23 3 The group algebra and the regular representations 25 4 Schur's lemma 27 5 Tensor products 29 6 Complex conjugate representations Quaternionic representations 30 7 One-dimensional representations 34 CHAPTER III. Abstract Theory of Representations of Finite Groups 35 1 Orthogonality relations and the first fundamental relation 36 2 Characters, class functions, and conjugacy classes 39 3 One-dimensional representations 42 4 The dimension theorem 43 5 The theorem of Frobenius and Schur 47 Appendix to III.5—Representations on real and quaternionic vector spaces 50 6 Representations and group structure 55 7 Projections in the group algebra 56 8 Fourier analysis 57 9 Direct products 59 10 Restrictions 59 11 Subgroups of index 2 60 12 Examples 62
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