SEMINAR: Groups and Combinatorics Seminar, Generation of finite groups

Time and place: 3pm Friday 23 January in Blakers LT.

Speaker: Colva Roney-Dougal (University of St Andrews)

Title: Generation of finite groups.

Abstract: There is a beautiful result, due to Liebeck, Shalev, and others, that the probability that two random elements of a finite simple group G generate the whole of G tends to 1 as the order of G tends to infinity. Many variations on this result are known -- for example, we could insist that one of the two elements lay in a specified conjugacy class, or had a specified order -- and I shall discuss a few of them, together with some open problems. Far less is known about random generation of arbitrary non-finite groups, but I'll briefly discuss this too. If time permits, I'll finish with a discussion of the Product Replacement Algorithm, which is used to make random elements of a group, and some open problems relating to this.