Autumn semester

Unit(s) of Assessment	Weighting Towards Module Mark( %)
Coursework:   1 test                             1 take home assignment	12.5%     12.5%
Exam:  End of semester exam	75%
Qualifying Condition(s)

MS103 Linear Algebra

The objectives of the lecture course are threefold: to introduce the axiomatic approach to group theory, to show that group theory is the natural language of symmetry, and to cover a range of fundamental pure and applied topics in the theory of groups such as subgroups, morphisms, generators, quotient and product groups, cosets, conjugacy and actions.

At the end of the module the student should have a basic understanding of the abstract concept of a group and its role in analysing symmetry; a familiarity with the basic groups such as dihedral, cyclic, and orthogonal groups, and an understanding of the analysis and structure of groups.

The contents of the lectures will include:

Introduction: 

symmetry plays a central role in mathematics as well as in nature, art and science. Examples of "symmetric objects" are polygons, spheres, platonic solids, lattices and planar tilings. The language of symmetry is group theory. The module will develop and study the axiomatic theory of groups; including such issues as group actions, subgroups, generators, cosets, homomorphisms, conjugacy, commutivity, and quotient or product constructions. Examples of groups to be considered are cyclic groups, dihedral groups, and orthogonal groups.

Subgroups: 

the concept of a proper subgroup will be introduced and special subgroups such as normal subgroups, the centre of a group, isotropy subgroups, as well as the concept of a generator and the role of Lagrange's theorem, will be presented.

Quotient and product constructions: 

direct products, semi-direct products and quotient groups will be introduced, as well as the role of cosets.

Morphisms: 

the concepts of isomorphism, homomorphism and automorphism will be introduced and applied to the study of groups.

Groups: 

examples of groups to be studied are the dihedral groups, reflection groups, cyclic groups, the orthogonal groups, as well as an introduction to continuous transformation groups.

Teaching is by lectures and example classes. Learning takes place through lectures, exercises, preparation for tests and background reading. The book by Armstrong (1988) is followed quite closely in the lectures, and it is recommended (but not mandatory) that the students purchase it.

3 contact hours per week for 10 weeks

M.A. Armstrong, Groups and Symmetry, Springer-Verlag, (1988).

R.P. Burn, Groups: A Path to Geometry, Cambridge University Press, (1985).

J.A. Green, Sets and Groups, Routledge & Kegan Paul, (1965).

31 July 2007