Abstract
Abstract We prove that if L = F 4 2 âą ( 2 2 âą n + 1 ) âČ L={}^{2}F_{4}(2^{2n+1})^{\prime} and đ„ is a nonidentity automorphism of đż, then G = âš L , x â© G=\langle L,x\rangle has four elements conjugate to đ„ that generate đș. This result is used to study the following conjecture about the đ-radical of a finite group. Let đ be a proper subset of the set of all primes and let đ be the least prime not belonging to đ. Set m = r m=r if r = 2 r=2 or 3 and m = r â 1 m=r-1 if r â©Ÿ 5 r\geqslant 5 . Supposedly, an element đ„ of a finite group đș is contained in the đ-radical O Ï âĄ ( G ) \operatorname{O}_{\pi}(G) if and only if every đ conjugates of đ„ generate a đ-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 âą ( 2 2 âą n + 1 ) {}^{2}B_{2}(2^{2n+1}) , G 2 2 âą ( 3 2 âą n + 1 ) {}^{2}G_{2}(3^{2n+1}) , F 4 2 âą ( 2 2 âą n + 1 ) âČ {}^{2}F_{4}(2^{2n+1})^{\prime} , G 2 âą ( q ) G_{2}(q) , or D 4 3 âą ( q ) {}^{3}D_{4}(q) .
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