Module description
In this course you will
- review the basic properties of groups;
- work with cyclic groups, permutation groups, dihedral groups, equivalence classes, cosets, Lagrange's theorem, and direct product groups;
- be introduced to quotient groups, construct the groups of low order, learn about the conjugation map, and construct conjugacy classes;
- meet the classical matrix groups, which are examples of continuous (or Lie) groups;
- work with group homomorphisms, isomorphisms, automorphisms, normal subgroups, kernels of homomorphisms, and prove and make extensive use of the group homomorphism theorem (also known as the first isomorphism theorem);
- learn about the semi-direct product and semi-direct product groups;
- construct and investigate the Euclidean group;
- investigate the geometric structure of some of the classical matrix groups, in particular SU(2)and SO(3);
- work with group actions on sets, stabilisers and orbits; and
- prove the Sylow theorems.
Syllabus:
Finite groups of low order and the classical matrix groups.
Assessment details
Written examination.
Semester 1 only students will be set an alternative assessment in lieu of in-person exams in January.
Full year students will complete the standard assessment.
Educational aims & objectives
To provide an advanced understanding of group theory covering groups of finite order and classical matrix groups.
Learning outcomes
To be able to construct and classify finite groups of low order; to be able to use the homomorphism theorem, the direct product and the semi-direct product to construct groups; to be able to work with the classical matrix groups; to understand the structure of SU(2) and SO(3); to study the isomorphisms of Euclidean spaces.
Teaching pattern
Three hours of lectures and one hour of tutorial per week throughout the term.
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists