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 G0 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.