# Course Identification

Representations and Characters of Finite groups
20234072

## Lecturers and Teaching Assistants

Dr. Josephine Shamash
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## Course Schedule and Location

2023
Second Semester
Tuesday, 09:15 - 11:00, Ziskind, Rm 155
18/04/2023
21/07/2023

## Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 2.00 points

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## Prerequisites

2 years of undergraduate course in Algebra, including Group Theory or Linear Algebra and Algebra Through Examples.

20

English

## Attendance and participation

Expected and Recommended

Numerical (out of 100)

15%
85%

Take-home exam

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-
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## Syllabus

The theory of the representations and characters of finite groups is central both to algebra and group theory. The tools that the theory develops enable us to reach a deeper understanding of the structure of a finite group. We shall study classical ordinary, or complex, representations and characters, and then develop briefly the main theorems and ideas of modular representations and characters. Modular representation theory deals with representations of finite groups over a field of prime characteristic, and their connections and relations with the ordinary representations. The resulting beautiful theory which was developed by Brauer, gives additional insights into the structure of the group.

Group algebras, Maschke's theorem, structure of finite-dimensional semi-simple algebras over a field, centre of the group algebra. Ordinary (complex) irreducible representations and characters of finite groups, character tables, orthogonality relations. Central characters and algebraic integers. Burnside's theorem. Induced characters, Frobenius reciprocity. Modular representations of finite groups.Brauer characters, blocks and decomposition numbers. Defect groups, the Brauer correspondence, Brauer's first main theorem. Blocks of cyclic defect and Brauer trees, Dade's theorem.

## Learning Outcomes

Upon successful completion of the course students will be able to:

1. Demonstrate proficiency in calculating the ordinary and modular characters of finite groups.
2. Comprehend the connections and relations between the ordinary and modular character theory of a finite group.
3. Apply the theorems and methods of character theory to obtain insights, understanding and deep theorems in the structure of finite groups

1. G. James, M. Liebeck: Representations and Characters of Groups, Cambridge University Press 1993, 2001
2. I.M. Isaacs: Characters of Finite Groups
3. B. Huppert: Characters of Finite Groups
4. D.M. Goldschmidt: Lectures on character theory, Publish or Perish Inc.

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