On the sharp Baer--Suzuki theorem for $\pi$-radicals: sporadic 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 $r⩾5$. 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.

in Russian

References in corpus (1)

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

Cited by in corpus (1)

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