Groups with finiteness conditions on the lower central series of non-normal subgroups

Abstract

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group $G$ the subgroup ${\gamma }_k(G)$ is finite if the set $\{{\gamma }_k(H)|H⋪G\}$ is finite. Moreover, locally graded groups with finitely many $k$th terms of lower central series of infinite non-normal subgroups are also completely described.