Pure Maths Colloquium: Rosemary Bailey and Peter Cameron

Diagonal groups form an important class of permutation groups, arising as one of four basic classes in the description of primitive permutation groups given by the O’Nan–Scott theorem. In joint work with Cheryl Praeger and Csaba Schneider, we have produced a combinatorial description of objects (which we call “diagonal semilattices”) whose automorphism groups (in dimension at least 3) are diagonal groups. The groups emerge naturally from combinatorial assumptions. The proof is by induction on the dimension; starting the induction in 3 dimensions is the hardest part of the argument, and involves characterising a certain class of Latin cubes as sets of partitions.