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2007/08 Undergraduate Module Catalogue

MATH1022 Introductory Group Theory

10 creditsClass Size: 250

Module manager: Dr R. Marsh
Email: marsh@maths.leeds.ac.uk

Taught: Semester 2 View Timetable

Year running 2007/08

Pre-requisite qualifications

The pre-requisites for this module are either (MATH1015 & MATH1035) or (MATH1050 & MATH1060).

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

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

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

Delivery typeNumberLength hoursStudent hours
Lecture221.0022.00
Tutorial111.0011.00
Private study hours67.00
Total Contact hours33.00
Total hours (100hr per 10 credits)100.00

Methods of assessment


Coursework
Assessment typeNotes% 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 typeExam duration% of formal assessment
Standard exam (closed essays, MCQs etc)2 hr 00 mins85.00
Total percentage (Assessment Exams)85.00

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

Reading list

The reading list is available from the Library website

Last updated: 18/03/2008

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