1. Introduction: Together with Algebra II this unit is one of the main second year units in Pure Mathematics. It has the 1st year unit Linear Algebra II as prerequisite. Its aim is to introduce students to algebraic structures and to axiomatic methods in pure mathematics. Its main topic is Group Theory. It will introduce the principal notions of groups, subgroups, homomorphisms and factor groups. While it can be taken on its own, this unit in conjunction with Algebra II is the prerequisite for several third year Pure Mathematics courses.

2. Hours, Credit, Assessment: The course is a 10 UCU unit of 20 lectures with three support classes. Assessment is by examination (80%) and coursework (20%) via assessed homework.

3. Overview: This course introduces students to groups, which are among the most important algebraic structures and which appear in most branches of mathematics. Modern algebra bases itself on axiomatic systems where theories are developed from axioms. This is true particularly for Group Theory. One of its roots and influential in its development has been the study of geometrical transformations and symmetry. Group Theory is therefore very close to these fundamental mathematical notions. It is also a subject where theory and axiomatics interacts with intuition and insight in fascinating ways. Axiomatic methods are not always easy to get used to. Group Theory is a field where one can see their full power and at the same time be close to concrete applications.

The course starts with the definitions of groups, subgroups and cosets. Lagrange's Theorem, saying that the order of a subgroup of a finite group divides the order of the group, is proved by showing that cosets partition the group. In order to compare groups one needs the notion of homorphism and isomorphism. This leads to normal subgroups in a group and to quotient groups. Groups occur as symmetries of other mathematical objects and so there is a section introducing the notion of group action and its relationship to homomorphisms. Each of these concepts is explored in a variety of examples. One of the most useful theorems on group actions is the Orbit Stabilizer Theorem, which will be used to count orbits and patterns.

With the theory developed so far it is possible to classify all commutative groups generated by a finite number of elements and this result is one of the principal results in the course.

All of the following should be in the Library and there are many other books in the library which cover the material of the course.

I. N. Herstein "Topics in Algebra"

J.B. Fraleigh "A first course in abstract algebra"

R.A. Dean "Elements of abstract algebra"

Binary operations, groups. Elementary properties. Examples: (a) the symmetric group, cycle notation and working with permutations; (b) symmetries of some familiar objects in R3 ; (c) the rotational symmetries of the regular n-gon and the definition of the dihedral groups. (4 lectures)

Subgroups, the powers of an element, classification of cyclic groups. Cosets and the theorem of Lagrange. Homomorphism, isomorphisms, and normal subgroups. Factor groups and the first isomorphism theorem. Examples. (6 lectures)

Group actions. The connection between a group acting on a set X and the homomorphisms from the group into the symmetric group on X. Examples: actions on cosets, various actions of the dihedral groups. Orbits and the Orbit Stabilizer Theorem. Applications. ( 7 lectures)

The Smith canonical form and applications to finitely generated abelian groups. (3 lectures)