On the intersections of Sylow subgroups in almost simple groups

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 $p⩾5$) 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.

26 pages; to appear in the Journal of Algebra