Representation Theory of Finite Groups and its Applications Open Access

Xu, Siwei (Spring 2022)

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In this paper, we give an exposition of the representation theory of finite groups: character theory, and Frobenius-Schur descent of complex representations to real ones. We also give the applications of representation theory in proofs to the following three theorems: Burnside theorem, on the degree of α+β, Eckmann’s proof on Hurwitz’s theorem.

Table of Contents

0. Introduction 2

1. Preliminaries 2

1.1. Sylow Theorems 2

1.2. p-groups 3

1.3. Nilpotent and Solvable Groups 3

1.4. Algebraic Integer 4

1.5. Field Extensions and Galois Theory 5

1.6. Linear Algebra/Spectral Theorem 6

2. Representation Theory of Finite Groups 6

2.1. Introduction 6

2.2. Character 8

2.3. Real Representation 12

3. Burnside’s Theorem 17

4. On the Degree of α + β 21

5. Eckmann’s proof on Hurwitz’s Theorem 23

References 28

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