Summer Semester 2020


  • May 7th, 15:00 (CEST time): Cheryl Praeger (The University of Western Australia)

Title: Diagonal Structures and Permutation Groups

Permutation groups are often best understood by the structures they preserve on the underlying point set. For example, several families of maximal subgroups of finite symmetric groups arise as stabilisers of subsets, or partitions, or Cartesian decompositions of the point set. Among the finite primitive groups, one family (apart from the almost simple groups) resisted such a description until now, namely the maximal simple diagonal groups. We introduce the concept of a diagonal structure associated with any group T, finite or infinite, and any integer m at least 2. We prove that the stabilizer of the diagonal structure is precisely the corresponding diagonal group, and that this group is primitive essentially when T is characteristically simple.

This work began as a collaboration with Csaba Schneider, and over the past year the collaboration includes also R. A. Bailey and Peter Cameron. We have introduced also a diagonal graph, built from a collection of Hamming graphs: its automorphism group is the diagonal group. And there are close connections with Latin squares, and with certain kinds of higher dimensional Latin hypercubes. We continue to learn more about the combinatorial links.