  In this paper we prove a sufficient condition for the existence of matchings
in arbitrary groups and its linear analogue, which lead to some generalizations
of the existing results in the theory of matchings in groups and central
extensions of division rings. We introduce the notion of relative matchings
between arrays of elements in groups and use this notion to study the behavior
of matchable sets under group homomorphisms. We also present infinite families
of prime numbers p such that Z/pZ does not have the acyclic matching property.
Finally, we introduce the linear version of acyclic matching property and show
that purely transcendental field extensions satisfy this property
