On maximal subgroups of minimax groups
It is known, that if \(G\) is a finitely generated soluble group such that \(G/\Phi(G)\) is nilpotent then \(G\) is nilpotent too, where \(\Phi(G)\) is the Frattini subgroup of the group \(G\). J. C. Lennox proved that if \(G\) is a finitely generated soluble group such that \(G/\Phi(G)\) is finite-by- nilpotent then \(G\) is finite-by-nilpotent. The main result of this paper is Theorem. Let \(G\) be a soluble residually finite minimax group. Then \(G\) is finite-by-nilpotent if and only if it has finitely many maximal subgroups which are not normal. Corollary. Let \(G\) be a soluble residually finite minimax group such that the factor- group \(G/\Phi(G)\) is finite-by-nilpotent. Then \(G\) is finite-by- nilpotent.
soluble residually finite minimax groups
scientific article; zbMATH DE number 817259