EPSRC award – The Binary Actions of Finite Groups

Dr Nick Gill and collaborators have been awarded a £420,312 EPSRC Standard Research grant for the project "The Binary Actions of Finite Groups." 

The project summary reads as follows.

This research project is in the field of group theory which is the mathematical study of symmetry. A faithful group action is binary if and only if it is isomorphic to the action on vertices of the automorphism group of a homogeneous edge-coloured graph.

The notion of a binary action was introduced by the model theorist, Gregory Cherlin, who described an "organising principle" for the universe of finite permutation groups. From this point of view, binary actions form the most basic examples of such actions. 

Still, basic or not, 25 years later we still do not know the binary actions of the finite groups. A major breakthrough in this regard was achieved when 3 members of the proposed research team were able to complete a classification of the finite primitive binary permutation groups. Primitive groups are a special class of permutation groups and Cherlin himself gave a conjecture as to what the binary ones should look like. It took 20 years, but we now know that Cherlin's conjecture was correct. 

The challenge now is to extend this classification to all finite permutation groups. If we are able to do this, then the full force of the model theory developed by Cherlin to understand finite permutation groups will be able to come into play. The proposed research proposes to approach this challenge by utilising the Classification of Finite Simple Groups (CFSG).