On the sharp Baer--Suzuki theorem for the $\pi$-radical

TL;DR

This paper investigates a conjecture relating conjugacy classes and the $ ext{pi}$-radical in finite groups, confirming it for groups with specific simple nonabelian composition factors.

Contribution

It proves the conjecture for finite groups whose nonabelian composition factors are isomorphic to alternating, linear, and unitary simple groups.

Findings

Confirmed the conjecture for groups with alternating simple factors.

Confirmed the conjecture for groups with linear simple factors.

Confirmed the conjecture for groups with unitary simple factors.

Abstract

Let $\pi$ be a set of primes such that $|\pi|\geqslant 2$ and $\pi$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $O\pi(G)$ if and only if every $m$ elements of $D$ generate a $\pi$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.