On the sharp Baer--Suzuki theorem for $\pi$-radicals: sporadic groups

TL;DR

This paper investigates a conjecture relating conjugacy classes and $ ext{pi}$-radicals in finite groups, confirming it for groups with nonabelian composition factors as sporadic or alternating groups, thus extending the sharp Baer--Suzuki theorem.

Contribution

It proves the conjecture for finite groups with nonabelian composition factors being sporadic or alternating groups, broadening the theorem's applicability.

Findings

Confirmed the conjecture for groups with sporadic composition factors.

Extended the sharp Baer--Suzuki theorem to new classes of finite groups.

Provided a characterization of $ ext{pi}$-radicals in these groups.

Abstract

Let $\pi$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m = r$ if $r = 2$ or $3$ 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 the $\pi$-radical $\mathrm{O}_\pi(G)$ of $G$ 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 sporadic or alternating groups.