# Gwybodaeth Modiwlau

Module Identifier
MA30110
Module Title
Group Theory
2021/2022
Co-ordinator
Semester
Semester 1
Pre-Requisite
Other Staff

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   (Written Examination)  100%
Supplementary Exam 2 Hours   (Written Examination)  100%

### Learning Outcomes

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.

### Brief description

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.

### Aims

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.

### Content

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.

### Notes

This module is at CQFW Level 6