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 ${\gamma_{k}(H)\;|\;H\ntriangleleft G}$ 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.