A note on groups whose non-normal subgroups are either abelian or minimal non-abelian

A group G is called parahamiltonian if each non-normal subgroup of G is either abelian or minimal non-abelian. Thus all biminimal non-abelian groups are parahamiltonian, and the class of parahamiltonian groups contains the important class of metahamiltonain groups, introduced by Romalis and Sesekin about 50 years ago. The aim of this paper is to describe the structure of locally graded parahamiltonian groups.