Conditions for matchability in groups and field extensions

Abstract

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.

Keywords:

• Dimension m-intersection property
• field extension
• matchable subsets
• primitive subspace

2020 Mathematics Subject Classifications:

• Primary: 05D15
• Secondary: 11B75
• 20D60
• 12F10

Acknowledgements

We are deeply grateful to Shira Zerbib and Khashayar Filom for their constant encouragement, generosity, and for many insightful conversations. We would like to thank anonymous referees for their useful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Iowa State University Dean's High Impact Award for undergraduate summer research in mathematics.