groups - Finite Groups and their Representations (...

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Unformatted text preview: Finite Groups and their Representations ( Mathematics 4H 19989 ) 17/7/2001 Dr A. J. Baker Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. E-mail address : a.baker@maths.gla.ac.uk URL : http://www.maths.gla.ac.uk/ ajb Contents Chapter 1. Linear and multilinear algebra 1 1. Basic linear algebra 1 2. Class functions and the CayleyHamilton Theorem 5 3. Separability 8 4. Basic notions of multilinear algebra 9 Chapter 2. Recollections and reformulations on basic group theory 13 1. The Isomorphism and Correspondence Theorems 13 2. Some definitions and notation 14 3. Group actions 15 4. The Sylow theorems 17 5. Solvable groups 17 6. Product and semi-direct product groups 18 7. Some useful groups 18 8. Some useful Number Theory 19 Chapter 3. Representations of finite groups 21 1. Linear representations 21 2. G-homomorphisms and irreducible representations 23 3. New representations from old 27 4. Permutation representations 29 5. Properties of permutation representations 31 6. Calculating in permutation representations 32 7. Generalized permutation representations 33 Chapter 4. Character theory 37 1. Characters and class functions on a finite group 37 2. Properties of characters 39 3. Inner products of characters 40 4. Character tables 43 5. Examples of character tables 46 6. Reciprocity formulae 51 7. Representations of semi-direct products 53 Chapter 5. Some applications to group theory 55 1. Characters and the structure of groups 55 2. A result on representations of simple groups 57 3. A Theorem of Frobenius 58 Chapter 6. Automorphisms and extensions 61 1. Automorphisms 61 2. Extensions 64 3. Classifying extensions [ optional extra material ] 67 3 4 CONTENTS Chapter 7. Some further applications 71 1. Fourier series and the circle group 71 CHAPTER 1 Linear and multilinear algebra In this chapter we will study the linear algebra required in representation theory. Some of this will be familiar but there will also be new material, especially that on multilinear algebra. 1. Basic linear algebra Throughout the remainder of these notes k will denote a field, i.e., a commutative ring with unity 1 in which every non-zero element has an inverse. Most of the time in representation theory we will work the field of complex numbers C and occasionally the field of real numbers R . However, a lot of what we discuss will work over more general fields, including those of finite characteristic such as Z /p for a prime p . Here, the characteristic of the field k is defined to be the smallest natural number p N such that p 1 = 1+ +1 = 0 if such a number exists then k is said to have finite characteristic ), otherwise it has characteristic 0. In the finite characteristic case, the characteristic is always a prime....
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groups - Finite Groups and their Representations (...

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