Module Identifier MX30410 Module Title GROUPS Academic Year 2001/2002 Co-ordinator Dr T McDonough Semester Semester 1 Pre-Requisite MA11010 Mutually Exclusive MA20410 Course delivery Lecture 19 x 1 hour lectures Seminars / Tutorials 3 x 1 hour 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.

#### Aims

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:
• determine whether given algebraic systems are groups;
• determine whether given subsets of groups are subgroups;
• determine conjugates and centralisers of elements and subsets of groups;
• determine permutation representations of some elementary groups;
• calculate efficiently within groups of permutations, especially symmetric groups;
• prove propositions in group theory;
• describe the concepts of normal subgroup, homomorphism and quotient group;
• state and apply the fundamental homomorphism theorems.

#### Syllabus

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.

Books
** Recommended Text
R J B T Allenby. (1991) Rings, Fields and Groups. 2nd. Edward Arnold 0340544406
** Supplementary Text
J B Fraleigh. (1999) A First Course in Abstract Algebra. 5th. Addison-Wesley 0201474360
J A Green. (1995) Sets and Groups 1. 2nd. Chapman & Hall 0412708205
J F Humphreys. (1996) A Course in Group Theory. Oxford University Press 0198534531
C A Jordan and D A Jordan. (1994) Groups. Edward Arnold 034061045X
T A Whitelaw. (1983) An Introduction to Abstract Algebra. Blackie 021691437X