13:00 Tim Burness
Simple groups, nilpotent subgroups and their intersections
Let G be a finite group, let p be a prime and let H be a Sylow p-subgroup. Problems concerning the intersections of Sylow subgroups have been studied for many decades. For example, a theorem of Ito from 1958 shows that if G has odd order, then H ∩ Hx = Op(G) for some element x in G, where Op(G) is the intersection of all the Sylow p-subgroups. And for an arbitrary finite group G, Zenkov (1996) uses CFSG to show that H ∩ Hx ∩ Hy = Op(G) for some x, y in G. In the special case where G is a (non-abelian) simple group, the main result is due to Mazurov and Zenkov (1996), who showed that H ∩ Hx = 1 for some x in G. Their proof of the latter result uses earlier work from the 1980s on defect groups of p-blocks for simple groups of Lie type.
In this talk, I will present a probabilistic approach to study the intersections of randomly chosen Sylow p-subgroups. For non-alternating simple groups, we will use this method to verify a remarkable conjecture of Lisi and Sabatini (2025) on “synchronised intersections" of Sylow subgroups. Our method also yields a new proof of the Mazurov-Zenkov theorem for these groups, and we are able to complete the proof of a strong form of a conjecture of Vdovin from 2002 on intersections of nilpotent subgroups of simple groups: if G is simple and H,K are nilpotent subgroups, then H ∩ Kx = 1 for some element x in G. Along the way, we establish new asymptotic results on the probability that two random Sylow p-subgroups in a simple group of Lie type have trivial intersection, complementing recent work of Diaconis et al. and Eberhard on symmetric and alternating groups.
This is joint work with Hongyi Huang (SUSTech, China).
14:00 Scott Harper
Probabilistic generation of finite groups
Liebeck and Shalev (1995) proved Dixon's conjecture that two randomly chosen elements of a simple group almost surely generate it. Which other families of groups have this property? In this talk, I will discuss recent work that extends Liebeck and Shalev's result to all groups with a unique chief series. I'll discuss the structure of these groups and show that there is a rich array of examples of them. I'll also discuss an application to profinite groups.
This is joint work with Martyn Quick (St Andrews).
15:00 Refeshments (preceded by a photo)
15:45 Colva Roney-Dougal
Minimal generating sets of permutation groups
The product replacement algorithm is a way to make random elements of a finite group. Introduced by Leedham-Green and collaborators in the 1990s, it has been observed to work far better then theoretical estimates would suggest, and there are a range of conjectures concerning this difference. One useful parameter in the analysis is m(G): the largest size of a generating set for a group G such that no proper subset generates. In 2000, Whiston showed that if G is the symmetric group on n points then m(G) = n-1, and that every proper subgroup H of G satisfies m(H) < n-1. I’ll present very recent work giving tight bounds on m(H) for all subgroups H of Sn, together with some consequences for product replacement.
This is joint work with Peter Brooksbank, Maria Elisa Fernandez, Scott Harper and Dimitri Leemans.
16:45 Reception
18:30 Dinner
All talks will be held in Lecture Theatre WG05 in the Aston Webb Building.
Directions The Aston Webb Building is the enormous semicircular building dominating campus. Enter through the main entrance in the middle of the semicircle, opposite the clock tower and below the carved figures of Plato, Newton, etc (see R38 on the map). Turn right and go straight down the corridor until the very end. The lecture theatre is on the right (see R39 on the map).