Abstract:
Given a finite group G and a prime p, when can we find two Sylow p-subgroups that intersect trivially? If G is simple, then a theorem of Mazurov and Zenkov from 1996 shows that this question has a positive answer for every prime p, and their proof uses earlier work on defect groups of p-blocks for simple groups of Lie type.
In this talk, we will apply a probabilistic approach to study the intersections of randomly chosen Sylow subgroups. For non-alternating simple groups, we will use this tool to verify a very recent conjecture of Lisi and Sabatini on "simultaneous intersections" of Sylow subgroups. In addition, our approach allows us to give an independent proof of Mazurov-Zenkov for these groups. Moreover, by combining our results with earlier work of Kurmazov from 2013, we complete the proof of a strong form of a conjecture of Vdovin from 2002 on intersections of nilpotent subgroups of simple groups, which is stated as Problem 15.40 in the Kourovka Notebook.Joint work with Tim Burness (University of Bristol, UK).
