Groups in Which Every Subgroup Is 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 BQF-group if there is a positive integer m such that every subgroup H of G contains a permutable subgroup K of G with |H : K| ≤ m. In this paper it is proved that any periodic locally graded BQF-group contains a quasihamiltonian subgroup of finite index. This result should be seen in relation with a theorem by Buckley, Lennox, Neumann, Smith and Wiegold concerning the corresponding problem when permutable subgroups are replaced by normal subgroups.