On the intersections of Sylow subgroups in almost simple groups

TL;DR

This paper completes the classification of almost simple groups where Sylow p-subgroups have non-trivial intersections under conjugation, using probabilistic methods and fixed point ratio estimates.

Contribution

It finalizes Zenkov's classification by employing probabilistic techniques to analyze Sylow subgroup intersections in almost simple groups.

Findings

Identifies conditions under which Sylow p-subgroups intersect non-trivially.

Completes the classification for all cases, including those over finite fields with special properties.

Uses fixed point ratio estimates to analyze subgroup intersections.

Abstract

Let $G$ be a finite almost simple group and let $H$ be a Sylow $p$-subgroup of $G$. As a special case of a theorem of Zenkov, there exist $x,y \in G$ such that $H \cap H^x \cap H^y = 1$. In fact, if $G$ is simple, then a theorem of Mazurov and Zenkov reveals that $H \cap H^x = 1$ for some $x \in G$. However, it is known that the latter property does not extend to all almost simple groups. For example, if $G = S_8$ and $p=2$, then $H \cap H^x \ne 1$ for all $x \in G$. Further work of Zenkov in the 1990s shows that such examples are rare (for instance, there are no such examples if $p \geqslant 5$) and he reduced the classification of all such pairs to the situation where $p=2$ and $G$ is an almost simple group of Lie type defined over a finite field $\mathbb{F}_q$ and either $q=9$ or $q$ is a Mersenne or Fermat prime. In this paper, by adopting a probabilistic approach based on fixed point ratio estimates, we complete Zenkov's classification.