Geodesic
On the sharp Baer-Suzuki theorem for the $\pi$-radical of a~finite group
Sbornik. Mathematics, Tome 214 (2023) no. 1, pp. 108-147

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Let $\pi$ be a proper subset of the set of prime numbers. Denote by $r$ the least prime not contained in $\pi$ and set $m=r$ for $r=2$ and $3$ and $m=r-1$ for $r\geqslant5$. The conjecture under consideration claims that a conjugacy class $D$ of a finite group $G$ generates a $\pi$-subgroup of $G$ (equivalently, is contained in the $\pi$-radical) if and only if any $m$ elements of $D$ generate a $\pi$-group. It is shown that this conjecture holds if every non-Abelian composition factor of $G$ is isomorphic to a sporadic, an alternating, a linear, or a unitary simple group. Bibliography: 49 titles.
Keywords: simple linear groups, simple unitary groups, $\pi$-radical of a group, Baer-Suzuki $\pi$-theorem. 
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     title = {On the sharp {Baer-Suzuki} theorem for the $\pi$-radical of a~finite group},
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N. Yang; Zh. Wu; D. O. Revin; E. P. Vdovin. On the sharp Baer-Suzuki theorem for the $\pi$-radical of a~finite group. Sbornik. Mathematics, Tome 214 (2023) no. 1, pp. 108-147. http://geodesic.mathdoc.fr/item/SM_2023_214_1_a4/