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