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 [On canonical forms. Proc London Math Soc (2). 1920;18:403–410] which was tackled in Fan and Losonczy [Matchings and canonical forms for symmetric tensors. Adv Math. 1996;117(2):228–238]. In this paper, we first discuss unmatchable subsets in abelian groups. Then we 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. [ABSTRACT FROM AUTHOR]