In 1955, Marshall Hall Jr and Lowell Paige conjectured that the Cayley table of a finite group G has an orthogonal mate if and only if the Sylow 2-subgroups of G are trivial or non-cyclic. This was proved in 2009 by the combined efforts of Stuart Wilcox, Anthony Evans and John Bray, but the final step (involving the sporadic group J4) was never published.
Last year, an application emerged, a proof that primitive permutation groups of simple diagonal type with more than two factors in the socle are non-synchronizing. A paper has appeared on the arXiv including both a description of Bray’s argument for J4 and a proof of the application (and a little bit more).
Starting from a minimal amount of background, Peter Cameron (St Andrews) will discuss these matters in this seminar which will be held in Lecture Theatre D.