Matchings in arbitrary groups

Shalom Eliahou, Cedric Lecouvey

Published 2006-11-20Version 1

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.