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

Fausto De Mari

Published 2016-12-05Version 1

It has been proved in [Arch. Math. (Basel) 86 (2006), p. 310-316] that any locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. This result is generalized here, in fact we prove that in a locally graded group $G$ the subgroup $\gamma_{k}(G)$ is finite when the set $\{\gamma_{k}(H)~|~H \ntriangleleft G \}$ is finite. Moreover, locally graded groups $G$ for which the set $\{\gamma_{k}(H)~|~H\ntriangleleft G,~H~$infinite$\}$ is finite are also completely described.