Conditions for matchability in groups and field extensions II

TL;DR

This paper establishes new sufficient conditions for matchings in abelian groups and their linear analogues, classifies non-matchable subsets, and introduces concepts like Chowla subspaces, advancing the theoretical understanding of matchings in algebraic structures.

Contribution

It extends existing matching theory results by providing new conditions, classifying non-matchable subsets, and proposing conjectures related to Chowla subspaces and linear analogues.

Findings

Derived new sufficient conditions for matchings in abelian groups.

Classified subsets that cannot be matched within abelian groups.

Proposed conjectures on Chowla subspaces and their linear counterparts.

Abstract

We present sufficient conditions for the existence of matchings in abelian groups and their linear counterparts. These conditions lead to extensions of existing results in matching theory. Additionally, we classify subsets within abelian groups that cannot be matched. We introduce the concept of Chowla subspaces and formulate and conjecture a linear analogue of a result originally attributed to Y. O. Hamidoune [20] concerning Chowla sets. If proven true, this result would extend matchings in primitive subspaces. Throughout the paper, we emphasize the analogy between matchings in abelian groups and field extensions. We also pose numerous open questions for future research. Our approach relies on classical theorems in group theory, additive number theory and linear algebra. As the title of the paper suggests, this work is the second sequel to a previous paper [5] with a similar theme. This paper is self-contained and can be read independently.