SEMINAR: Groups and Combinatorics Seminar: Idempotent generation and road closures

Time and place: 16:00 Friday 23 September in Blakers LT

Speaker: Peter Cameron (St. Andrews)

Title: Idempotent generation and road closures

Abstract: This is part of a big project linking the properties of a transformation semigroup to those of the permutation group forming its group of units. The particular question I am discussing is this: For which permutation groups $G$ is it the case that, for any transformation $a$ of rank 2, the semigroup $ angle G,a angle etminus G$ is generated by its idempotents? Such a group must be primitive, but not all primitive groups have this property. It turns out to be equivalent to the property that, for any orbital graph for $G$ (regarded as a road network), if we close the edges in a block of imprimitivity for the action of $G$, the network remains connected. We have a conjecture about the primitive groups which satisfy this condition. This is joint work with João Arajúo (Lisbon).