On matchings in groups (Q1271154)

The matching property for lattices is generalized for Abelian groups. Let \(A\) and \(B\) be nonempty finite subsets of an additive Abelian group \(G\) that satisfy \(|A|=|B|\) and \(0\not\in B\). A bijection \(\pi\colon A\to B\) is called a matching if \(a+\pi(a)\not\in A\) for all \(a\in A\). Given such a bijection \(\pi\), \(m_\pi\colon G\to\mathbb{Z}\) is defined by \(m_\pi(g)=\#\{a\in A\mid a+\pi(a)=g\}\), and \(\pi\) is called acyclic if for any matching \(\tau\colon A\to B\) with \(m_\tau=m_\pi\) one has \(\tau=\pi\). \(G\) possesses the (cyclic) matching property if for every pair \(A\), \(B\) of nonempty finite subsets satisfying \(|A|=|B|\) and \(0\not\in B\), there exists at least one (cyclic) matching from \(A\) to \(B\). It is proved that for any Abelian group \(G\) and any nonempty finite subset \(A\) of \(G\setminus\{0\}\), there exists at least one matching \(\pi\colon A\to A\); an Abelian group possesses the matching property if and only if it is torsion-free or cyclic of prime order; a torsion-free Abelian group possesses the acyclic matching property.