Groups whose finite homomorphic images are metahamiltonian

A group G is metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a finitely generated soluble group is metahamiltonian if and only if all its finite homomorphic images are metahamiltonian; the behaviour of soluble minimax groups with metahamiltonian finite homomorphic images is also investigated. Moreover, groups satisfying the minimal condition on non-metahamiltonian subgroups are described