On commutativity and finiteness in groups

The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f: H -> K which fixes the identity and such that h commutes with its image h (f) for all h a H. The present paper gives support to conjectures about the nilpotency of groups generated by two weakly commuting finite abelian groups H, K.