On the intersections of Sylow subgroups in almost simple groups
Timothy C. Burness, Hong Yi Huang
TL;DR
The paper completes the classification of almost simple groups $G$ with a Sylow $p$-subgroup $H$ for which $H\cap H^x\neq 1$ for all $x\in G$ by proving $b(G,H)=2$ in the remaining open cases identified by Zenkov, specifically for $p=2$ and Lie-type socles over fields with $q=9$ or $q$ a Mersenne or Fermat prime. It advances the probabilistic fixed-point ratio method (Lovász–Liebeck–Shalev framework) to bound the base size via $Q(G,H)\le\widehat{Q}(G,H)$, leveraging detailed involution-size bounds and strategic subgroup containment to bound fixed-point ratios. The work combines rigorous analytic bounds with targeted Magma computations for small groups to close all remaining cases, yielding a complete classification and a corollary description for triples of primary subgroups with nontrivial intersection under conjugacy. The results have implications for base sizes and subgroup depth in almost simple groups, and they provide a robust methodological template for similar base-size classifications in other families of 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 $p \geqslant 5$) 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.