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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 Prerequisites
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 selfcontained 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 A_{5}; A_{5} is simple.
Reading list
Recommended:
C. R. Jordon and D. A. Jordon,
Groups,
Edward Arnold.
J. A. Green,
Sets and Groups: a First Course in Algebra,
RKP.
Walter Ledermann and Alan J. Weir,
Introduction to Group Theory, 2nd edition,
Addison Wesley Longman.
Background:
J. R. Durbin,
Modern Algebra: an Introduction, 3rd edition,
Wiley.
J. B. Fraleigh,
A First Course in Abstract Algebra, 5th edition,
AddisonWesley.
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