On generations by conjugate elements in almost simple groups with socle ²𝐹₄(𝑞²)′

Abstract

We prove that if L = F 4 2 ⁢ ( 2 2 ⁢ n + 1 ) ′ L={}^{2}F_{4}(2^{2n+1})^{\prime} and 𝑥 is a nonidentity automorphism of 𝐿, then G = ⟨ L , x ⟩ G=\langle L,x\rangle has four elements conjugate to 𝑥 that generate 𝐺. This result is used to study the following conjecture about the 𝜋-radical of a finite group. Let 𝜋 be a proper subset of the set of all primes and let 𝑟 be the least prime not belonging to 𝜋. 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 𝑥 of a finite group 𝐺 is contained in the 𝜋-radical O π ⁡ ( G ) \operatorname{O}_{\pi}(G) if and only if every 𝑚 conjugates of 𝑥 generate a 𝜋-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 ⁢ ( 2 2 ⁢ n + 1 ) {}^{2}B_{2}(2^{2n+1}) , G 2 2 ⁢ ( 3 2 ⁢ n + 1 ) {}^{2}G_{2}(3^{2n+1}) , F 4 2 ⁢ ( 2 2 ⁢ n + 1 ) ′ {}^{2}F_{4}(2^{2n+1})^{\prime} , G 2 ⁢ ( q ) G_{2}(q) , or D 4 3 ⁢ ( q ) {}^{3}D_{4}(q) .