  A matching in a group G is a bijection f from a subset A to a subset B in G
such that af(a) does not belong to A for all a in A. The group G is said to
have the matching property if, for any finite subsets A,B in G of same
cardinality with B avoiding 1, there is a matching from A to B.
  Using tools from additive number theory, Losonczy proved a few years ago that
the only abelian groups satisfying the matching property are the torsion-free
ones and those of prime order. He also proved that, in an abelian group, any
finite subset A avoiding 1 admits a matching from A to A.
  In this paper, we show that both Losonczy's results hold verbatim for
arbitrary groups, not only abelian ones. Our main tools are classical theorems
of Kemperman and Olson, also pertaining to additive number theory, but
specifically developped for possibly nonabelian groups.
