A question of L. A. Shemetkov
Following P. Hall, we say that a finite group \(G\) satisfies the condition \(E_ \pi\) (respectively, \(E^ n_ \pi\), \(E^ z_ \pi\)) if it possesses a Hall \(\pi\)-subgroup (respectively, a nilpotent Hall \(\pi\)- subgroup, a Hall \(\pi\)-subgroup with cyclic Sylow subgroups). A group \(G\) satisfies the condition \(D_ \pi\) if all of its maximal \(\pi\)-subgroups are conjugate. If a normal subgroup \(K\) of a finite group \(G\) satisfies \(E^ n_ \pi\) and \(G/K\) satisfies \(D_ \pi\) and contains a soluble Hall \(\pi\)-subgroup then by P. Hall's theorem \(G\) satisfies \(D_ \pi\); moreover, by Shemetkov's theorem \(N_ K(M)\) satisfies \(E_ \pi\) for each soluble \(\pi\)-subgroup \(M\) of \(G\). In 1972 \textit{L. A. Shemetkov} proved that if a normal subgroup \(K\) of a finite group \(G\) satisfies \(E^ z_ \pi\) and \(G/K\) satisfies \(D_ \pi\) then \(G\) satisfies \(D_ \pi\); moreover, \(N_ K(M)\) satisfies \(E_ \pi\) for any \(\pi\)-subgroup \(M\) of \(K\) [see Mat. Sb., Nov. Ser. 76(118), No. 2, 271-287 (1968; Zbl 0212.049); and Dokl. Akad. Nauk BSSR 16, No. 10, 881-883 (1972; Zbl 0251.20018)]. Shemetkov's question: Prove that in the last theorem \(N_ K(M)\) satisfies \(E_ \pi\) for each soluble \(\pi\)- subgroup \(M\) of \(G\). In the present paper the author shows that the question of Shemetkov has a positive answer modulo the classification of finite simple groups.
scientific article; zbMATH DE number 536681
scientific article