On the intersections of nilpotent subgroups in simple groups

TL;DR

This paper proves a conjecture about the existence of elements in simple groups that minimize intersections of Sylow subgroups, using probabilistic methods, and extends results to nilpotent subgroups in simple groups.

Contribution

It confirms the Lisi-Sabatini conjecture for non-alternating simple groups and completes the proof of Vdovin's conjecture, also establishing a stronger intersection property for nilpotent subgroups.

Findings

Proved Lisi-Sabatini conjecture for all non-alternating simple groups.

Completed proof of Vdovin's conjecture on nilpotent subgroup intersections.

Analyzed probabilities of trivial intersections of Sylow p-subgroups in simple groups.

Abstract

Let $G$ be a finite group and let $H_p$ be a Sylow $p$-subgroup of $G$. A recent conjecture of Lisi and Sabatini asserts the existence of an element $x \in G$ such that $H_p \cap H_p^x$ is inclusion-minimal in the set $\{H_p \cap H_p^g \,:\, g \in G\}$ for every prime $p$. For a simple group $G$, in view of a theorem of Mazurov and Zenkov from 1996, the conjecture implies the existence of an element $x \in G$ with $H_p \cap H_p^x = 1$ for all $p$. In turn, this statement implies a conjecture of Vdovin from 2002, which asserts that if $G$ is simple and $H$ is a nilpotent subgroup, then $H \cap H^x = 1$ for some $x \in G$. In this paper, we adopt a probabilistic approach to prove the Lisi-Sabatini conjecture for all non-alternating simple groups. By combining this with earlier work of Kurmazov on nilpotent subgroups of alternating groups, we complete the proof of Vdovin's conjecture. Moreover, by combining our proof with earlier work of Zenkov on alternating groups, we are able to establish a stronger form of Vdovin's conjecture: if $G$ is simple and $A,B$ are nilpotent subgroups, then $A \cap B^x = 1$ for some $x \in G$. To obtain these results, we study the probability that a random pair of Sylow $p$-subgroups in a simple group of Lie type intersect trivially, complementing recent work of Diaconis et al. and Eberhard on symmetric and alternating groups.