An Exposition on Artin–Brauer Induction Theorems: The Characterization of Linear Representations of Finite Groups Open Access

Abstract

This thesis presents an exposition of the Artin–Brauer Induction Theorems in the representation theory of finite groups. Both theorems address the fundamental problem of describing characters of a finite group in terms of characters induced from its subgroups. Artin’s theorem expresses characters as rational linear combinations of induced characters from cyclic subgroups, while Brauer’s theorem strengthens this result by providing an expression with integer coefficients using elementary subgroups. This work develops necessary preliminaries on representation theory and expands on intermediate arguments to provide a more explicit and accessible treatment of the proofs. An application of Brauer's theorem on symmetric group $S_4$ is presented at the end. 

Table of Contents

1 Introduction ...................................................................... 1

 1.1 Historical Context and Motivation ................................... 1

 1.2 Overview ....................................................................... 2

2 Preliminaries ..................................................................... 3

 2.1 Representations and Characters ...................................... 3

  2.1.1 Subrepresentations and Irreducible Representations ..... 5

  2.1.2 Characters .................................................................. 7

  2.1.3 Decomposition of Representations ............................... 14

 2.2 Induced Representations ................................................. 17

 2.3 Ring of Virtual Characters ............................................... 21

 2.4 Elementary Groups ......................................................... 24

3 Main Theorems .................................................................. 28

 3.1 Proof of Artin’s Theorem ................................................. 29

  3.1.1 Proof of Theorem 3.1.1 ................................................ 31

 3.2 Proof of Brauer’s Theorem ............................................... 33

  3.2.1 Proof of Theorem 3.2.1 ................................................ 35

  3.2.2 Proof of Theorem 3.2.2 ................................................ 36

4 Application of Brauer's Theorem ......................................... 42

 4.1 Characters of Cyclic Groups ............................................. 43

 4.2 Symmetric Group $S_4$ as an Example ............................. 44

Bibliography ......................................................................... 48

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Representation Theory Artin–Brauer Induction Theorems Finite Groups
Committee Chair / Thesis Advisor	Venapally, Suresh, Emory University
Committee Members	Pohoata, Cosmin, Emory University Borthwick, David, Emory University Silversmith, Rob, Emory University Raman, Parimala, Emory University

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