## MATH1022 Introductory Group Theory

### 10 creditsClass Size: 250

Module manager: Dr J Britnell
Email: j.r.britnell@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

### Pre-requisite qualifications

(MATH1015 & first semester of MATH1035) or (MATH1050 & MATH1060). Please note that MATH1060 may be taken at the same time as MATH1022.

This module is approved as an Elective

### Module summary

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.

### Objectives

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

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. Cyclic groups. Direct product of two groups. Product of two cyclic groups is cyclic if and only if their orders are coprime. Isomorphisms and homomorphisms. 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Tutorial 11 1.00 11.00 Private study hours 67.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated