  The notion of acyclic matching property was provided by Losonczy and it was
proved that torsion-free groups admit this property. In this paper, we
introduce a duality of acyclic matching as a tool for classification of some
Abelian groups, moreover, we study matchings for vector spaces and give a
connection between matchings in groups and vector spaces. Our tools mix
additive number theory, combinatorics and algebra.
