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

TL;DR

This paper generalizes known results about locally graded groups with finitely many derived subgroups, showing conditions under which certain lower central series subgroups are finite and providing a complete description of related group structures.

Contribution

It extends previous theorems by characterizing when the $k$th lower central series subgroup is finite based on the finiteness of certain non-normal subgroup series.

Findings

$\gamma_{k}(G)$ is finite if finitely many $\gamma_{k}(H)$ for non-normal $H$

Locally graded groups with finitely many $k$th terms of infinite non-normal subgroups are fully described

Generalizes results about finite-by-abelian groups with specific subgroup conditions

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.