Course Syllabus for

# Group and Ring Theory Grupp- och ringteori

## MATP33F, 7.5 credits

Valid from: Spring 2021
Decided by: Professor Thomas Johansson
Date of establishment: 2020-09-24

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: MATP33
Teaching language: English

## Aim

The course aims to provide, in comparison with the course Algebraic structures, a deeper understanding of group theory and ring theory as a basis for further studies in algebraic subject areas, and to provide general mathematical knowledge.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to, in detail, explain the concepts, theorems and methods included in the course,
• be able to identify the most important theorems in the course and present their proofs.

Competences and Skills

For a passing grade the doctoral student must

• in connection with problem solving be able to demonstrate the ability to integrate knowledge from the different parts of the course,
• be able to independently identify problems that can be solved by methods that are part of the course and use appropriate solution methods,
• be able to explain the solution to a mathematical problem within the course framework, in speech and in writing, logically coherently and with adequate terminology.

Judgement and Approach

For a passing grade the doctoral student must be able to argue for the importance of group theory and ring theory as tools in other areas such as algebraic geometry and algebraic number theory, and discuss their limitations.

## Course Contents

• Groups: Permutation groups. Burnside's lemma with application to Pólya arithmetic. Sylow's theorems. Symmetric and alternating groups. The structure of finitely generated Abelian groups. • Rings: Noetherian and Artinian rings and modules. Artin-Wedderburn's theorem. Finitely generated modules over a principal ideal domain with application to Jordan's normal form of matrices. • Linear algebra: Multilinear mappings. Tensor products.

## Course Literature

Bhattacharya, P. B., Jain, S. K. & Nagpaul, S. R.: Basic Abstract Algebra. Cambridge University Press, 1994. ISBN 9780521466295.

## Instruction Details

Types of instruction: Lectures, seminars

## Examination Details

Examination formats: Written exam, oral exam