Groups And Symmetries

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.

 

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

Module description disclaimer

King’s College London reviews the modules offered on a regular basis to provide up-to-date, innovative and relevant programmes of study. Therefore, modules offered may change. We suggest you keep an eye on the course finder on our website for updates.

Please note that modules with a practical component will be capped due to educational requirements, which may mean that we cannot guarantee a place to all students who elect to study this module.

Please note that the module descriptions above are related to the current academic year and are subject to change.