Representations and Characters of GroupsThis book provides a modern introduction to the representation theory of finite groups. Now in its second edition, the authors have revised the text and added much new material. The theory is developed in terms of modules, since this is appropriate for more advanced work, but considerable emphasis is placed upon constructing characters. Included here are the character tables of all groups of order less than 32, and all simple groups of order less than 1000. Applications covered include Burnside's paqb theorem, the use of character theory in studying subgroup structure and permutation groups, and how to use representation theory to investigate molecular vibration. Each chapter features a variety of exercises, with full solutions provided at the end of the book. This will be ideal as a course text in representation theory, and in view of the applications, will be of interest to chemists and physicists as well as mathematicians. |
Contents
Groups and homomorphisms | 1 |
Vector spaces and linear transformations | 14 |
Group representations | 30 |
FGmodules | 38 |
FGsubmodules and reducibility | 49 |
Group algebras | 53 |
FGhomomorphisms | 61 |
Maschkes Theorem | 70 |
Tensor products | 188 |
Restriction to a subgroup | 210 |
Induced modules and characters | 224 |
Algebraic integers | 244 |
Real representations | 263 |
Summary of properties of character tables | 283 |
Characters of groups of order pq | 288 |
Characters of some pgroups | 298 |
Schurs Lemma | 78 |
Irreducible modules and the group algebra | 89 |
More on the group algebra | 95 |
Conjugacy classes | 104 |
Characters | 117 |
Inner products of characters | 133 |
The number of irreducible characters | 152 |
Character tables and orthogonality relations | 159 |
Normal subgroups and lifted characters | 168 |
Some elementary character tables | 179 |
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Common terms and phrases
abelian group algebraic integer b¯¹ab b¹ab basis v1 calculate CG-homomorphism CG(gi CG(x character of G character table characters X1 classes of G column orthogonality relations complex numbers conjugacy classes conjugate Corollary deduce define direct sum eigenvalue eigenvectors element g elements of G endomorphism equivalent Example Let Exercises for Chapter finite group follows g₁ given group algebra group G group of order Hence homomorphism invertible involution irreducible CG-module irreducible characters irreducible representations isomorphic Ker 9 Let G linear characters linear transformation Maschke's Theorem matrix non-abelian non-zero normal subgroup prime number Proof Let Proposition Let Prove representation of G root of unity Schur's Lemma Show simple group subgroup of G subspace sum of irreducible symmetric symmetry group table of G U₁ v₁ vector space W₁ W₂ Z(CG Χι