\(p\)-nilpotence of finite groups and minimal subgroups.
Let \(G\) be a finite group. A group \(G\) is called \(p\)-nilpotent if \(G_{p'}\) is normal in \(G\). It is proved that a finite group \(G\) is \(p\)-nilpotent if every minimal subgroup of \(P\cap O^p(G)\) is quasinormal in \(P\) and \(N_G(P)\) is \(p\)-nilpotent, and when \(p= 2\) either \([\Omega_2(P\cap O^p(G)),P]\leq\Omega_1(P\cap O^p(G))\) or \(P\) is quaternion free, where \(p\) is a prime dividing the order of \(G\) and \(P\) is a Sylow \(p\)-subgroup of \(G\). By using this result, the authors also obtain a series of corollaries for \(p\)-nilpotence and some results about solvability and supersolvability of finite groups.
scientific article; zbMATH DE number 2037074
scientific article