Groups with many quasihamiltonian-by-finite subgroups

A group G is said quasihamiltonian if XY=YX for all subgroups X and Y of G. In this paper we study locally graded groups in which all proper subgroups are quasihamiltonian-by-finite. We prove that such groups are quasihamiltonian-by-finite or periodic. Moreover, we investigate the behaviour of groups of infinite rank in which all proper subgroups of infinite rank are quasihamiltonian-by-finite, within a suitable universe of generalised solubility condition. A result for uncountable groups of cardinality ℵ without simple homomorphic images of cardinality ℵ in which all proper large subgroups are quasihamiltonian-by-finite is also obtained.