Groups with minimax commutator subgroup

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