Conditions for matchability in groups and field extensions

TL;DR

This paper explores conditions for matchability in abelian groups and field extensions, introducing linear analogues of known results, proposing conjectures, and analyzing intersection properties related to matchings in algebraic structures.

Contribution

It formulates and proves linear analogues of matching results, discusses the dimension intersection property, and proposes a conjecture extending the primitive subspace theorem.

Findings

Characterization of unmatchable subsets in abelian groups

Linear analogues of matching theorems proved

A conjecture extending the primitive subspace theorem

Abstract

The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [24] which was tackled in 1996 [10]. 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.