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• W4385341330 abstract "We prove that if $L=mbox{}^2F_4(2^{2n+1})'$ and $x$ is a nonidentity automorphism of $L$ then $G=langle L,xrangle$ 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 $rgeqslant 5$. 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 ${}^3D_4(q)$." @default.
• W4385341330 created "2023-07-29" @default.
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• W4385341330 date "2023-07-29" @default.
• W4385341330 modified "2023-09-24" @default.
• W4385341330 title "On generations by conjugate elements in almost simple groups with socle ²𝐹₄(𝑞²)′" @default.
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• W4385341330 doi "https://doi.org/10.1515/jgth-2022-0216" @default.
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