SEMINAR: Groups and Combinatorics Seminar: Almost synchronising groups

Time and place: 15:00 Friday 02 October in Blakers LT.

Speaker: Gordon Royle (University of Western Australia)

Title: Almost synchronising groups.

Abstract: A permutation group G on a set X is said to be synchronising if for every transformation f of X that is not a permutation, the semigroup generated by G and f contains a constant transformation. A synchronising group is necessarily primitive, but there are examples of non-synchronising primitive groups. For every non-synchronising primitive group, there is a uniform transformation witnessing the fact that the group is non-synchronising, and non-synchronising primitive groups for which every witness is uniform are called almost synchronising. Araujo, Bentz and Cameron conjectured that all primitive groups are either synchronising or almost synchronising, but then shortly afterwards found counterexamples to their own conjecture. In this talk, I will discuss subsequent work and remaining open questions on the topic of almost synchronising groups.