The question of finding sets of monomials which are removable from a generic homogeneous polynomial through a linear change in its variables was raised by E. K. Wakeford in 1916. This linear algebra question motivated J. Losonczy to define the concept of acyclic matchings in Z n, and later in abelian groups. In this paper, we give a constructive approach to study the acyclic matchings in cyclic groups. We also introduce the notion of weakly matched subsets and investigate its relation with matchings in abelian groups.