Module overview
This module develops the theory of groups beyond the finite setting studied in MATH2003. While many familiar results in algebra concern finite structures, a large part of modern mathematics studies infinite groups, whose behaviour is richer and often governed by geometry rather than counting.
We begin with structural examples, including finitely generated abelian groups and their classification, before introducing free groups and the combinatorial viewpoint of group theory via words and relations. This leads to geometric and graphical methods, including Cayley graphs and Stallings graphs, which allow algebraic problems to be studied using topology and geometry.
The module then explores how groups arise as symmetry groups of spaces. In particular, we study isometry groups of Euclidean, spherical and hyperbolic geometries, illustrating how algebra encodes geometric structure and how geometry constrains algebra.
Throughout the course, the emphasis is on developing intuition for infinite groups and on techniques that appear across modern areas of mathematics such as geometric group theory, topology and algebra.