Solubilizers in profinite groups

TL;DR

This paper investigates the measure-theoretic properties of solubilizers in profinite groups, linking their size to the structure of chief factors, and proves a related conjecture for alternating groups.

Contribution

It proves a conjecture about the measure of solubilizers in profinite groups specifically for alternating groups, reducing the problem to finite simple groups.

Findings

Proved the conjecture for alternating groups.

Reduced the main conjecture to a finite simple group problem.

Work in progress for groups of Lie type.

Abstract

The solubilizer of an element $x$ of a profinite group $G$ is the set of the elements $y$ of $G$ such that the subgroup of $G$ generated by $x$ and $y$ is prosoluble. We propose the following conjecture: the solubilizer of $x$ in $G$ has positive Haar measure if and only if $g$ centralizes "almost all" the non-abelian chief factors of $G$. We reduce the proof of this conjecture to another conjecture concerning finite almost simple groups: there exists a positive $c$ such that, for every finite simple group $S$ and every $(a,b)\in (Aut(S)\setminus \{1\}) \times Aut(S)$, the number of $s$ is $S$ such that $\langle a, bs\rangle $ is insoluble is at least $c|S|$. Work in progress by Fulman, Garzoni and Guralnick is leading to prove the conjecture when $S$ is a simple group of Lie type. In this paper we prove the conjecture for alternating groups.