## MATH1022 Introductory Group Theory

### 10 creditsClass Size: 250

Taught: Semester 2 View Timetable

Year running 2006/07

### Pre-requisites

MATH1200 and MATH1011, or equivalent.

This module is approved as an Elective

### Module summary

Prerequisite: MATH1015 and Semester 1 of MATH1035 or equivalent.Informal description: Group theory may be regarded as an abstract study of symmetry. Thus for a typical geometrical figure, its degree of symmetry may be captured by the corresponding group, certainly how many symmetries there are, but also, precisely how they interact (the "structure" of the group). Groups play a central role in mathematics and its applications. This course treats the basic theory as far as Lagrange's theorem (the order of a subgroup divides the order of the group) and quotient groups. See the schools website or contact: a.slomson@leeds.ac.uk for more information.

### Objectives

To give a coherent treatment of the fundamental ideas of group theory. To illustrate how the groups corresponding to various geometrical shapes and patterns correspond to their degree and type of symmetry. On completion of this module, students should be able to: a) determine whether or not a given structure is a group; b) describe groups of rotations and isometries, and to identify their subgroups; c) perform computations in finite cyclic groups, and relate this to calculations involving congruences; d) calculate using permutations, and determine the order of a permutation; e) list the families of cosets of various groups of small order.

### Syllabus

Group theory may be regarded as an abstract study of symmetry. Thus for a typical geometrical figure, its degree of symmetry may be captured by the corresponding group, certainly how many symmetries there are, but also, precisely how they interact (the `structure' of the group). Groups play a central role in mathematics and its applications. This course treats the basic theory as far as Lagrange's theorem (the order of a subgroup divides the order of the group) and quotient groups. The presentation mixes formal development with many examples. The topics covered are: Definitions and examples of groups. Basic terminology. Symmetries of geometrical figures, and isometries. Multiplicative group of units of Z / n Z. Additive groups. Subgroups. Subgroup criterion. Examples. Order of an element. Powers of elements of finite order. Order of xy where x and y commute and have coprime orders. Cyclic groups. Direct product of two groups. Product of two cyclic groups is cyclic ? their orders are coprime. Isomorphisms and homomorphisms. Revision of equivalence relations (for those who didn't take MATH1200). Cosets and Lagrange's theorem. Order of an element of a finite group divides the order of the group. Groups of order p, 4. Intersections of subgroups. Subgroups of the dihedral groups of order 6 and 8. Fermat's little theorem. Permutations. Cycle notation. Order of a permutation. Even and odd permutations. Sn- and An. Conjugacy, centre, normal subgroups. Centralizer of an element. Number of conjugates = index of centralizer. Conjugacy classes in Sn. Quotient groups and the first isomorphism theorem.

### Teaching methods

Lectures (22 hours) and tutorials (11 hours).

### Methods of assessment

2 hour written examination at end of semester (85%), coursework (15%).