Group theory is one of the great simplifying and unifying ideas in modern mathematics. It was introduced in order to understand the solutions to polynomial equations, but only in the last one
hundred years has its full significance, as a mathematical formulation of symmetry, been understood. It plays a role in our understanding of fundamental particles, the structure of crystal lattices and the geometry of molecules.
In this module we build on material from Linear Algebra II and will begin by revising the simple axioms satisfied by groups and begin to develop basic group theory by reference to some elementary
examples. We will analyse the structure of 'small' finite groups, and examine examples arising as groups of permutations of a set, symmetries of regular polygons and regular solids, and groups of
matrices. We will develop the notions of homomorphism, normal subgroups and quotient groups and study the First Isomorphism Theorem and its application.
We will also examine how the notion of a permutation group can be generalized to that of a group action on a set, and will show how to use this in certain counting problems arising in combinatorics. We will also see how to use group actions to prove strong results about the structure of finite groups. We shall study Sylow’s Theorems and some of their applications.