Abstract: We prove that if L = F 4 2 a (2 2 a n + 1) ′ L={}^{2}F_{4}(2^{2n+1})^{\prime} and is a nonidentity automorphism of L, then G = ⟨ L, x ⟩ G=\langle L,x\rangle has four elements conjugate to that generate a. This result is used to study the following conjecture about the-radical of a finite group. Let be a proper subset of the set of all primes and let r be the least prime not belonging to. Set m = r m=r if r = 2 r=2 or 3 and m = r - 1 m=r-1 if r ≥ 5 r\geqslant 5. Supposedly, an element of a finite group a is contained in the-radical O π (G) \operatorname{O}_{\pi}(G) if and only if every m conjugates of generate a-subgroup. Based on the results of this and previous papers, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, unitary simple group, or to one of the groups of type B 2 2 a (2 2 a n + 1) {}^{2}B_{2}(2^{2n+1}), G 2 2 a (3 2 a n + 1) {}^{2}G_{2}(3^{2n+1}), F 4 2 a (2 2 a n + 1) ′ {}^{2}F_{4}(2^{2n+1})^{\prime}, G 2 a (q) G_{2}(q), or D 4 3 a (q) {}^{3}D_{4}(q).

Cite: Revin D.O. , Zavarnitsine A.V.
On generations by conjugate elements in almost simple groups with socle ${}^2F_4( ^2)′$
Journal of Group Theory. 2024. V.27. N1. P.119-140. DOI: 10.1515/jgth-2022-0216 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Dec 28, 2022
Published online:	Jul 29, 2023
Published print:	Jan 8, 2024

Identifiers:

Web of science:	WOS:001039289000001
Scopus:	2-s2.0-85167437716
Elibrary:	62331174
OpenAlex:	W4385341330

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Scopus	2
Web of science	2
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