Representations and Characters of Groups

by Gordon James , Martin Liebeck

• ISBN13: 9780521003926
• ISBN10: 052100392X
• Edition: 2nd
• Format: Paperback
• Copyright: 2001-11-05
• Publisher: Cambridge University Press

Summary

This 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.

Table of Contents

Preface	vii
Groups and homomorphisms	1	(13)
Vector spaces and linear transformations	14	(16)
Group representations	30	(8)
FG-modules	38	(11)
FG-submodules and reducibility	49	(4)
Group algebras	53	(8)
FG-homomorphisms	61	(9)
Maschke's Theorem	70	(8)
Schur's Lemma	78	(11)
Irreducible modules and the group algebra	89	(6)
More on the group algebra	95	(9)
Conjugacy classes	104	(13)
Characters	117	(16)
Inner products of characters	133	(19)
The number of irreducible characters	152	(7)
Character tables and orthogonality relations	159	(9)
Normal subgroups and lifted characters	168	(11)
Some elementary character tables	179	(9)
Tensor products	188	(22)
Restriction to a subgroup	210	(14)
Induced modules and characters	224	(20)
Algebraic integers	244	(19)
Real representations	263	(20)
Summary of properties of character tables	283	(5)
Characters of groups of order pq	288	(10)
Characters of some p-groups	298	(13)
Character table of the simple group of order 168	311	(11)
Character table of GL(2, q)	322	(15)
Permutations and characters	337	(11)
Applications to group theory	348	(13)
Burnside's Theorem	361	(6)
An application of representation theory to molecular vibration	367	(30)
Solutions to exercises	397	(57)
Bibliography	454	(1)
Index	455