Groups with minimax commutator subgroup

Abstract

A result of Dixon‎, ‎Evans and Smith shows that if $G$ is a locally (soluble-by-finite) group whose proper subgroups are (finite rank)-by-abelian‎, ‎then $G$ itself has this property‎, ‎i.e‎. ‎the commutator subgroup of $G$ has finite rank‎. ‎It is proved here that if $G$ is a locally (soluble-by-finite) group whose proper subgroups have minimax commutator subgroup‎, ‎then also the commutator subgroup $G'$ of $G$ is minimax‎. ‎A corresponding result is proved for groups in which the commutator subgroup of every proper subgroup has finite torsion-free rank.‎