The structure of groups whose subgroups are permutable-by-finite

A subgroup H of a group G is said to be permutable if HX=XH for each subgroup X of G, and the group G is called quasihamiltonian if all its subgroups are permutable. We shall say that G is a QF-group if every subgroup H of G contains a subgroup K of finite index which is permutable in G. It is proved that every locally finite QF-group contains a quasihamiltonian subgroup of finite index. In the proof of this result a theorem by Buckley, Lennox, Neumann, Smith and Wiegold, concerning the corresponding problem when permutable subgroups are replaced by normal subgroups, is used: if G is a locally finite group such that every subgroup contains a G-invariant subgroup of finite index, then G contains an abelian subgroup of finite index.