Module Identifier MX30410  
Module Title GROUPS  
Academic Year 2000/2001  
Co-ordinator Dr T P McDonough  
Semester Semester 1  
Pre-Requisite MA11010  
Mutually Exclusive MA20410  
Course delivery Lecture   19 x 1hour lectures  
  Seminars / Tutorials   3 x 1hour example classes  
Assessment Exam   2 Hours (written examination)   100%  
  Resit assessment   2 Hours (written examination)   100%  

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.

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:

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.

Reading Lists
** Recommended Text
R J B T Allenby. Rings, Fields and Groups. Edward Arnold
** Supplementary Text
J B Fraleigh. A First Course in Abstract Algebra. 5th. Addison-Wesley
J A Green. Sets and Groups 1. Routledge
J F Humphreys. A Course in Group Theory. Oxford University Press
C A Jordan and D A Jordan. Group Theory. Edward Arnold
T A Whitelaw. An Introduction to Abstract Algebra. Blackie