MC242 Introduction to Groups

Next: MC243 Aspects of Linear Up: Year 2 Previous: MC241 Linear Algebra

MC242 Introduction to Groups

Credits: 10

Convenor: Dr. J. C. Ault

Semester: 2

Prerequisites:	essential: MC144, MC145, MC147	desirable: MC241
Assessment:	Computer project and course work: 20%	One and a half hour exam in May/June: 80%

Lectures:	18	Classes:	5
Tutorials:	none	Private Study:	47
Labs:	5	Seminars:	none
Project:	none	Other:	none
Total:	75

Explanation of Pre-requisites

Ideas from MC144, MC145, MC147 and MC241 are used in the construction of examples. In particular, the concepts of sets, mappings and Cartesian products from MC144 and of modular arithmetic and algebraic structures from MC145 are used extensively. Basic concepts from matrix algebra are needed from MC147 and (to a lesser extent) MC241.

Course Description

This module provides an introduction to group theory through a combination of abstract theory and specific examples of groups. Group theory can be thought of as the measurement of symmetry, both of geometrical objects and of other structures and this will be a significant theme of the lectures.

Aims

The main aim is to lay down a firm foundation in the basic concepts of the theory ready for development in later (third and fourth year) modules, but it is also intended that this course should be self-contained and lead to its own interesting conclusions.

Objectives

It is intended that the workshops will encourage experimentation with the examples of small groups provided and that the projects will give an opportunity for practice with the presentation of abstract mathematical material. You will be expected to become familiar with various particular examples of groups and to know how to prove some of the theorems. You should also be able to apply your newly learned knowledge in unfamiliar but similar situations.

Syllabus

The course begins with the general concept of a binary operation and the various properties that such an operation may enjoy leading to the definition of a group. These ideas are investigated in greater detail through use of the computer package ``Exploring Small Groups''. Apart from those provided by the package, there is a standard list of examples derived from symmetries of geometrical shapes, matrices and modular arithmetic. These are used to illustrate the results and ideas introduced in later chapters.

Topics covered include: subgroups and generators; the lattice of subgroups; cyclic (sub)groups; the order of an element; cosets and Lagrange's Theorem; the index equation; conjugation; centre; class equation; normal subgroups and factor groups; the symmetric and alternating groups; cycle notation for permutations; conjugate permutations; the class equation of A₅; A₅ is simple.

Reading list

Background:

J. R. Durbin, Modern Algebra: an Introduction, 3rd edition, Wiley.

J. B. Fraleigh, A First Course in Abstract Algebra, 5th edition, Addison-Wesley.

I. N. Herstein, Topics in Algebra, 2nd edition, Wiley.

Details of Assessment

Assessment is based on three components - a one and a half hour examination in May/June (80%), 4 problem sheets set in alternate weeks (10%) and two computer assignments (1 short, 1 longer) (10%).

The examination paper will have four questions all carrying the same weight and full marks will be obtained for correct answers to three. Questions on the paper will include proofs of elementary results. Familiarity with the examples of groups met in the course will be expected and you should be able to apply the methods learnt.

Next: MC243 Aspects of Linear Up: Year 2 Previous: MC241 Linear Algebra

S. J. Ambler
11/20/1999