MA9N8 Topics in Group Theory

Introductory description

Group theory is a fundamental area of pure mathematics that is motivated by and interrelated with many fields, from both within and outside mathematics. These include combinatorics, number theory, computer science, probability theory, geometry, as well as some branches of logic, algebra, and analysis. Understanding modern techniques in group theory will be useful and interesting for students from many of the backgrounds listed above.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The module will cover a range of advanced topics in group theory which may include

• Subgroup structure and extensions in finite groups;
• Finite simple groups;
• The theory of algebraic groups;
• Advanced representation theory;
• Permutation group theory;
• Growth and expansion in groups.

Learning outcomes

By the end of the module, students should be able to:

• be familiar with the classical structure theory in finite groups, including properties of maximal subgroups, the generalised Fitting subgroup, and the theory of group extensions.
• construct the finite simple groups of Lie type.
• be familiar with topics in algebraic groups such as connectedness, reductive groups, and Borel subgroups.
• know the classification of semisimple algebraic groups.
• be familiar with advanced topics in representation theory, such as the Artin-Wedderburn theorem, Schur indices, Mackey’s theorem and defect groups.
• know about primitive permutation groups and the O’Nan-Scott theorem.
• be familiar with properties of graphs associated to finite and/or algebraic groups, such as Cayley graphs and Schreier graphs
• be familiar with subgroup growth and word growth in groups.

Subject specific skills

Develop a deep understanding and applicability of the following topics:

• Subgroup structure and extensions of finite groups;
• Algebraic groups and finite groups of Lie type;
• Representation theory of finite groups;
• Structure of permutation groups;
• How both finite and infinite groups grow.

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)