Peter Cameron

The geometry of diagonal groups

This research was begun by Cheryl Praeger and Csaba Schneider as an attempt to find a geometric structure on which the simple diagonal groups in the O'Nan--Scott Theorem act. Subsequently, when Rosemary Bailey and I were involved, it broadened to give an axiomatic definition of a class of "diagonal semilattices": in two dimensions these are just Latin squares, but in higher dimensions a group naturally emerges. Our approach includes all groups, not necessarily finite or simple. There is an application to the synchronization problem for finite primitive groups.