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.

Keywords

• minimax group
• commutator subgroup
• torsion-free rank