Heilbronn Algebra Seminar - Tim Burness
| Dates: | 28 October 2025 |
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| Times: | 14:00 - 15:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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| Speaker: | Tim Burness |
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Title: Simple groups, nilpotent subgroups and their intersections
Abstract:
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 \cap H^x = O_p(G) for some element x in G, where O_p(G) is the intersection of all the Sylow p-subgroups of G. And for an arbitrary finite group G, Zenkov (1996) uses CFSG to show that H \cap H^x \cap H^y = O_p(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 \cap H^x = 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 an interesting conjecture of Lisi and Sabatini (2025) on “synchronised intersections" of Sylow subgroups. Our method yields a new proof of the Mazurov-Zenkov theorem for these groups, and we are also 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 \cap K^x = 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. (2025) and Eberhard (2025) on symmetric and alternating groups.
This is joint work with Hongyi Huang (SUSTech, China).
Speaker
Tim Burness
Organisation: University of Bristol
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Frank Adams 1
Alan Turing Building
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