Representation Theory of Finite Groups and its Applications Open Access

Abstract

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

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Representation Theory
Committee Chair / Thesis Advisor	Borthwick, David, Emory University Parimala, Raman, Emory University Zureick-Brown, David, Emory University

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