On mathching property for groups and vector spaces

TL;DR

This paper establishes a sufficient condition for the existence of matchings in groups and their linear analogues, generalizing previous results and exploring properties in algebraic structures like division rings and field extensions.

Contribution

It introduces a new criterion for matchings in groups and linear spaces, along with the concept of relative matchings and their behavior under homomorphisms, extending the theory.

Findings

Identifies prime numbers p where Z/pZ lacks the acyclic matching property.

Proves a sufficient condition for matchings in groups and linear spaces.

Shows that purely transcendental field extensions satisfy the linear acyclic matching property.

Abstract

In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms. We also present infinite families of prime numbers p such that Z/pZ does not have the acyclic matching property. Finally, we introduce the linear version of acyclic matching property and show that purely transcendental field extensions satisfy this property