On matchable subsets in abelian groups and their linear analogues

Abstract

In this paper, we introduce the notions of matching matrices in groups and vector spaces, which lead to some necessary conditions for existence of acyclic matching in abelian groups and its linear analogue. We also study the linear local matching property in field extensions to find a dimension criterion for linear locally matchable bases. Moreover, we define the weakly locally matchable subspaces and we investigate their relations with matchable subspaces. We provide an upper bound for the dimension of primitive subspaces in a separable field extension. We employ MATLAB coding to investigate the existence of acyclic matchings in finite cyclic groups. Finally, a possible research problem on matchings in n-groups is presented. Our tools in this paper mix combinatorics and linear algebra.

Results and questions on matchings in groups and vector subspaces of fields

“…Paper [1] shows that for primes p with p ≡ −1 (mod 8) the group Z/pZ does not have the acyclic matching property via exhibiting an explicit subset of Z/pZ that does not admit any acyclic matching onto itself. Based on experimental evidence, it is conjectured in [3] that Z/pZ does not admit the acyclic matching property for any prime p > 5. We shall prove the following theorems on the existence of matchings between certain subsets of a cyclic group of prime order.…”

“…In view of the discussion above, an abelian group G is said to admit the weak acyclic matching property if there exists an acyclic matching between any two subsets A and B of G that have the same cardinality and satisfy A ∩ (A + B) = ∅. Any cyclic group Z/nZ of order smaller than 23 satisfies the weak acyclic matching property, but the existence of infinitely many cyclic groups Z/pZ of prime order with this property is an open question [3].…”

Discussion on the Structure of the Reduced Residue Class Additive Group in Cyclic Group

“…The additive group [1][2][3] of the residue class ring [4][5] mod m is, up to isomorphism, the cyclic group of order m generate, say, by the residue class 1 mod m. For finite group, the direct sum decomposition [6] with respect to the prime power appearing in m is simply a special case of the basis theorem for finite abelian groups, according to the sum of cyclic subgroups of prime power order. Here, however, the direct sum of each cyclic subgroup belongs to the same prime number, rather than the direct sums of the cyclic subgroups themselves, which are uniquely determined.…”

On linear version of an elementary group theory result

“…Motivated by the aforementioned theorem on the number of generating elements of Z/p r Z, one naturally inquires about the size of a primitive subspace A. In [2] it is proven that dim…”

“…In this paper, we will generalize the primitive subspace theorem. We will also give a positive answer to the question inquired as Remark 5 in [2] whether (2) holds in case the base field is finite, for some special cases.…”