Understanding the Building Blocks of Symmetry

Mathematicians use the concept of a group to describe symmetry in an abstract way. Just as every whole number can be broken down into prime numbers, every finite group can be built from basic components called finite simple groups. The Classification Theorem of Finite Simple Groups, a major achievement in modern mathematics, identifies all such basic components. However, to apply this knowledge to concrete scientific problems, we often need to understand more about how these components behave, and that’s the focus of this project.

The team will investigate how these simple groups interact with the smaller structures inside them, by studying associated constructions such as double cosets and suborbits. Although this area of Pure Mathematics research is a classical topic in group theory, many cases remain unsolved, especially for the exceptional groups of Lie type. These groups and their `twisted' versions form eight of the 18 infinite families of finite simple groups. They stand out as an `exception’ with distinctive properties, which makes them a particularly interesting object of study.

The team plans to tackle these open questions using an innovative approach that combines mathematical theory and advanced computational tools, areas in which the Monash Warwick Alliance team is especially experienced. Joining them is former Monash Warwick Alliance PhD student Eileen Pan, whose thesis explored one such infinite family of simple groups, adding valuable insight to the project.

This collaboration forms a strong partnership for advancing knowledge and developing new ideas. By combining expertise from Monash University and the University of Warwick, the project will foster research at the frontiers of pure mathematics and lay the groundwork for continued collaborations.