Conditions for matchability in groups and vector spaces II

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.Comment: Comments are welcom

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

A matching from a finite subset $A$ of an abelian group to another subset $B$ is a bijection $f:A\rightarrow B$ with the property that $a+f(a)$ never lies in $A$. A matching is called acyclic if it is uniquely determined by its multiplicity function. Motivated by a question of E. K. Wakeford on canonical forms for symmetric tensors, the study of matchings and acyclic matchings in abelian groups was initiated by C. K. Fan and J. Losonczy in [16, 26], and was later generalized to the context of vector subspaces in a field extension [13, 1]. We discuss the acyclic matching and weak acyclic matching properties and we provide results on the existence of acyclic matchings in finite cyclic groups. As for field extensions, we completely classify field extensions with the linear acyclic matching property. The analogy between matchings in abelian groups and in field extensions is highlighted throughout the paper and numerous open questions are presented for further inquiry.Comment: 17 pages, minor corrections, subsection 2.2 is shortened. To appear in Journal of Algebr