Modern pure mathematics extensively uses the classification of simple Lie algebras by roots systems and Dynkin diagrams. A gentler introduction to these diagrams comes from the classification of finite reflection groups, that is, finite subgroups of the group of orthogonal transformations of a Euclidean vector space generated by a set of reflections. For a 3MB project, the aim would be to go through the classification of these groups. For a 4MM project there are various ways it could be augmented, for example with Gabriel's theorem on representations of directed graphs.
Prerequisites:
A thorough understanding of level 1 group theory and level 2
linear algebra. The level 3 module on groups and symmetry,
perhaps taken concurrently, would also be an advantage.