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內容簡介

Representation theory is concerned with the ways of writing a groupas a group of matrices. Not only is the theory beautiful in its own right，but it also provides one of the keys to a proper understanding of finitegroups. For example， it is often vital to have a concrete description of aparticular group； this is achieved by finding a representation of thegroup as a group of matrices. Moreover， by studying the differentrepresentations of the group， it is possible to prove results which lieoutside the framework of representation theory. One simple example： allgroups of order p2 （where p is a prime number） are abelian； this can beshown quickly using only group theory， but it is also a consequence ofbasic results about representations.

內頁插圖

目錄

Preface
1 Groups and homomorphisms
2 Vector spaces and linear transformations
3 Group representations
4 FG-modules
5 FG-submodules and reducibility
6 Group algebras
7 FG-homomorphisms
8 Maschke's Theorem
9 Schur's Lemma
10 Irreducible modules and the group algebra
11 More on the group algebra
12 Conjugacy classes
13 Characters
14 Inner products of characters
15 The number of irreducible characters
16 Character tables and orthogonality relations
17 Normal subgroups and lifted characters
18 Some elementary character tables
19 Tensor products
20 Restriction to a subgroup
21 Induced modules and characters
22 Algebraic integers
23 Real representations
24 Summary of properties of character tables
25 Characters of groups of order pq
26 Characters of some p-groups
27 Character table of the simple group of order 168
28 Character table of GL（2， q）
29 Permutations and characters
30 Applications to group theory
31 Burnside's Theorem
32 An application of representation theory to molecular vibration
Solutions to exercises
Bibliography
Index

前言/序言

　　We have attempted in this book to provide a leisurely introduction tothe representation theory of groups. But why should this subjectinterest you？　
　　Representation theory is concerned with the ways of writing a groupas a group of matrices. Not only is the theory beautiful in its own right，but it also provides one of the keys to a proper understanding of finitegroups. For example， it is often vital to have a concrete description of aparticular group； this is achieved by finding a representation of thegroup as a group of matrices. Moreover， by studying the differentrepresentations of the group， it is possible to prove results which lieoutside the framework of representation theory. One simple example: allgroups of order p2 （where p is a prime number） are abelian； this can beshown quickly using only group theory， but it is also a consequence ofbasic results about representations. More generally， all groups of order （p and q primes） are soluble； this again is a statement purely aboutgroups， but the best proof， due to Burnside， is an outstanding exampleof the use of representation theory. In fact， the range of applications ofthe theory extends far beyond the boundaries of pure mathematics， andincludes theoretical physics and chemistry - we describe one suchapplication in the last chapter.　The book is suitable for students who have taken first undergraduatecourses involving group theory and linear algebra. We have included twopreliminary chapters which cover the necessary background material.The basic theory of representations is developed in Chapters 3-23， andour methods concentrate upon the use of modules； although this accordswith the more modem style of algebra， in several instances our proofsdiffer from those found in other textbooks. The main results are elegantand surprising.

群的錶示與群的特徵（第2版） [Representations and Characters of Groups] 下載 mobi epub pdf txt 電子書

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