On matching property for groups and field extensions

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 eld extensions satisfy this property.

On matchable subsets in abelian groups and their linear analogues

“…In this section, we introduce matching matrices in abelian groups and prove that the matching matrix of an acyclic matching is invertible. Then we introduce the weak acyclic matching property in abelian groups and provide an open problem similar to those in [1] regarding characterizing prime numbers in terms of acyclicity. Next, we define the linear analogue of matching matrices in field extensions and we conjecture the linear version of matching matrices in the field setting.…”

“…But characterizing the acyclic matching property in finite groups of prime order is still an unsolved problem. It is shown in [1] that there are infinitely many prime p for which Z/pZ does not satisfy the acyclic matching property. In Section 5, we will employ a MATLAB code to numerically check for which values of p, Z/pZ have the acyclic matching property.…”

“…Back to Losonczy's result ([17, Theorem 4.1]), every torsion-free abelian group has the acyclic matching property. Combining an observation from [1] which states that there are infinitely many prime p such that Z/pZ does not have the acyclic matching property with the fact that the weak acyclic matching property does not imply the acyclic matching property, one can probably find a prime p such that Z/pZ does not have the weak acyclic matching property. Conjecture 2.…”

“…Remark 1. Note that the techniques used in the proof of the existence of infinitely many prime p with Z/pZ not having the acyclic matching property mentioned in [1] might not be useful to apply for Conjecture 2 as the set…”

“…An acyclic matching from A to B is defined to be a strong matching ϕ : A → B such that for any strong matching ψ : A → B that is equivalent to ϕ, one has ϕ = cψ, for some constant c ∈ K. Finally, it is said that the extension K ⊂ L possesses the linear acyclic matching property if for every pair A and B of nonzero n-dimensional K-subspaces of L with AB ∩ A = {0}, there is at least one acyclic matching from A to B. The main result concerning the linear acyclic matching property is proven in [1,Theorem 4.5], where it is shown that any purely transcendental field extension has the linear acyclic matching property. The existence of acyclic matching is verified by the necessary condition if the following conjecture holds.…”