Conditions for matchability in groups and field extensions

The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [26] which was tackled in 1996 [13]. In this paper, we prove a sufficient condition for the existence of matchings in abelian groups which leads to some generalizations of the existing results in the theory of matchings in the group setting. We also formulate and prove linear analogues of results concerning matchings, along with a conjecture that, if true, would extend the primitive subspace theorem. We discuss the dimension m-intersection property for vector spaces and its connection to matching subspaces in a field extension, and we prove the linear version of an intersection property result of certain subsets of a given set.