On generations by conjugate elements in almost simple groups with socle $\mbox{}^2F_4(q^2)'$

TL;DR

This paper proves a property of automorphisms in the almost simple group ${}^2F_4(q^2)'$, and uses it to support a conjecture about the $ ext{pi}$-radical in finite groups, confirming it for many classes of simple groups.

Contribution

It establishes a key conjugacy property in ${}^2F_4(q^2)'$ groups and applies this to verify a conjecture on the $ ext{pi}$-radical for a broad class of finite groups.

Findings

Proved that certain automorphisms in ${}^2F_4(q^2)'$ have four conjugates generating the group.

Confirmed the $ ext{pi}$-radical conjecture for groups with specific simple composition factors.

Extended the conjecture's validity to many classes of finite groups.

Abstract

We prove that if $L=\mbox{}^2F_4(2^{2n+1})'$ and $x$ is a nonidentity automorphism of $L$ then $G=\langle L,x\rangle$ has four elements conjugate to $x$ that generate $G$. This result is used to study the following conjecture about the $\pi$-radical of a finite group: Let $\pi$ be a proper subset of the set of all primes and let $r$ be the least prime not belonging to $\pi$. Set $m=r$ if $r=2$ or $3$ and set $m=r-1$ if $r\geqslant 5$. Supposedly, an element $x$ of a finite group $G$ is contained in the $\pi$-radical $\operatorname{O}_\pi(G)$ if and only if every $m$ conjugates of $x$ generate a $\pi$-subgroup. Based on the results of this paper and a few previous ones, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups of type ${}^2B_2(2^{2n+1})$, ${}^2G_2(3^{2n+1})$, ${}^2F_4(2^{2n+1})'$, $G_2(q)$, or ${}^3D_4(q)$.