Group Theory

On successful completion of this module students should be able to:

1. Determine whether given algebraic systems are groups;
2. Determine whether elements and subsets of a group possess a variety of properties;
3. State and prove some fundamental theorems, selected from the isomorphism theorems, the orbit-stabilizer theorem and the theorems of Lagrange, Cayley and Sylow;
4. Compute with permutations in the context of the symmetric groups and of the alternating groups
5. Identify group actions and to make use of the orbit-stabilizer theorem, for example in applications to counting with the Burnside lemma
6. Solve problems in group theory by selecting and applying appropriate theorems and techniques from the general theory.

The concept of a group occurs naturally in situations involving symmetry or in which some quantity is being preserved; for example, various letters such as A, S and I possess different numbers of symmetries and rigid motions preserve distance. This module will introduce the notion of a group as an algebraic object defined by a simple set of axioms. Various techniques for describing groups will be studied. The principal structure theorems for finite groups will be described and applied in a variety of group theoretic contexts.

To provide a deeper understanding of the concepts and techniques of abstract algebra, introduced in module MA20310, by focusing on the group concept, starting with an axiomatic development of group theory, establishing a structure theory, mainly in the context of finite groups, and giving brief illustrations of a selection of applications of group theory.

1. Fundamentals: Definitions and examples. Presentations of groups. Elementary consequences of the definitions. Subgroups. cosets. Lagrange's theorem.
2. Basic structure theory: Normal subgroups and factor groups. Direct products. Homomorphisms. The isomorphism theorems. Automorphism groups.
3. Permutation groups: Symmetric groups. Cycle decomposition, Cayley's theorem. Orbits. Stabilizers. The orbit-stabilizer theorem, Burnside lemma and applications to counting.
4. Local structure theory: p-subgroups. The Sylow theorems. Classifying groups of small order.

This module is at CQFW Level 6