Group Theory

Recommended Prerequisites

Knowledge of group theory obtained in a first undergraduate course of Algebra.

Teaching Methods

 The teaching/learning process will follow an interactive approach in which students will immediately apply the theory taught in solving exercises of various levels. The results will be discussed immediately.

Learning Outcomes

The aim of the unit is to equip students with the knowledge of finite group theory that is necessary to conduct research in computational algebra. In particular the course will cover the areas of finite solvable groups, nilpotent groups, the basic properties of finite p-groups, and will also introduce some classes of finite simple groups. In addition we will treat the fundamental concepts in permutation groups.
After completing this unit, the students should
● Appreciate the importance of finite group theory in abstract algebra;
● Be familiar with the most important classes of finite groups and have an understanding of the structure of important examples, such as alternating groups, symmetric groups, classical groups, etc;
be able to apply fundamental techniques of group theory to solve problems in other parts of algebra, such as ring theory, semigroup theory, loop theory, Galois theory, etc.

Work Placement(s)

No

Syllabus

1. Subgroup chains of finite groups: derived series, lower central series, upper central series.
2. Solvable and nilpotent groups; equivalent definitions.
3. Properties of nilpotent groups, finite p-groups.
4. Some classes of simple groups: alternating groups and projective classical groups. We will prove that these groups are simple.
5. Permutation groups. Transitive groups, primitive groups. Examples of primitive groups, affine groups, holomorph groups, wreath products. In this part we cover many classes of primitive groups that appear in the O’Nan-Scott Theorem. 

Assessment Methods

Assessment
Exam: 30.0%
Research work: 35.0%
Resolution Problems: 35.0%

Bibliography

Joseph J. Rotman, An Introduction to the Theory of Groups, Springer 1995.

Peter Cameron, Permutation Groups, CUP,  1999.

John Dixon and Brian Mortimer, Permutation Groups, Springer 1996.

Donald E. Taylor, The Geometry of the Classical Groups, Heldermann Verlag 1992.