|Co-ordinator||Dr T P McDonough|
|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%|
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 of mathematics, however there are also applications outside of mathmatics, 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.
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.
** Recommended Text
Allenby, R B J T. (1991) Rings, Fields and Groups. 2nd edition. Edward Arnold
** Supplementary Text
Durbin, J R. (2000) Modern Algebra. 4th edition. John Wiley and Sons, Inc.
Fraleigh, J B. A First Course in Abstract Algebra. 5th edition. Addison-Wesley
Jordan, C R & Jordan, D A. Groups. Edward Arnold
Green, J A. Sets and Groups I. Routledge
Whitelaw, T A. An Introduction to Abstract Algebra. Blackie