On finite groups with \(p\)-decomposable cofactors of subgroups. (Q1929777)

The cofactor of a subgroup \(H\) of a finite group \(G\) is the quotient \(H/H_G\) where \(H_G=\bigcap_{x\in G}x^{-1}Hx\) is the core of the subgroup \(H\) in \(G\). Several authors study the influence of the structure of some cofactors of \(G\) on the structure of the group. \textit{Ya. G. Berkovich} [in Sib. Mat. Zh. 9, 243-248 (1968; Zbl 0197.30101)] obtained the solvability of every group all of whose maximal subgroups have nilpotent cofactors. \textit{S. M. Ethukova} and \textit{V. S. Monakhov} [``On finite groups with supersolvable subgroup cofactors'', Vetsi. Nats. Akad. Navuk Belarusi, Ser Fiz.-Mat. Nats. Navuk 4, 53-57 (2008)] studied groups for which the cofactors of the maximal subgroups are supersolvable. \textit{J. D. Dixon, J. Poland} and \textit{A. H. Rhemtulla} [in Math. Z. 112, 335-339 (1969; Zbl 0186.04004)] obtained that if \(p\) is a prime, \(p>2\), and the cofactors of every maximal subgroup of a group \(G\) are \(p\)-nilpotent, then \(G\) is \(p\)-solvable. If \(p\) is a prime, a group \(G\) is said to be \(p\)-decomposable if \(G=PO_{p'}(G)\) where \(P\) is a normal \(p\)-Sylow subgroup of \(G\). In this paper it is proved that if the cofactor of every maximal subgroup of a group \(G\) is \(p\)-decomposable, then \(G/F(G)\) is \(p\)-decomposable. This implies that every group all of whose maximal subgroups have nilpotent cofactors is metanilpotent.