General description
Symmetry is one of the central themes of modern mathematics. This module deals with symmetry in an algebraic context, namely groups. Groups have applications in almost all branches mathematics, however there are also applications outside of mathematics, for example in particle physics and crystallography. The module will start with concrete examples of groups. Then more general and powerful abstract notions are introduced which will show how many problems involving these special cases may be dealt with in a unified way.

Aims
To provide an introduction to abstract algebra by developing elementary aspects of theory of groups. To show how a variety of examples, from disparate areas, may be dealt with in a unified way by the development of an abstract theory which embraces them.

Learning outcomes
On completion of this module, a student should be able to:

• determine whether given algebraic systems are groups;
• determine whether given subsets of groups are subgroups;
• determine conjugates and centralisers of elements and subsets of groups;
• determine permutation representations of some elementary groups;
• calculate efficiently within groups of permutations, especially symmetric groups;
• prove propositions in group theory;
• describe the concepts of normal subgroup, homomorphism and quotient group;
• state and apply the fundamental homomorphism theorems;
• state the Sylow theorems and apply them to determining the structure of groups of a given finite order.

Syllabus
1. EXAMPLES OF GROUPS: Matrix groups, symmetry groups, groups given by generators and relations, cyclic groups.
2. SUBGROUPS: Criteria for subgroups. Special subgroups. Lagrange' theorem - cosets, index of subgroups, order of elements.
3. PERMUTATION GROUPS: Cycle notation for permutations. Regular representation. Cayley's theorem. Orbits. Stabilisers. Symmetric groups.
4. HOMOMORPHISMS: Normal subgroups, quotient groups, homomorphisms and isomorphisms. Isomorphism theorems. Automorphism groups.
5. SYLOW SUBGROUPS: Existence, number and conjugacy properties.