  We formulate and prove matroid analogues of results concerning matchings in
groups. A matching in an abelian group $(G,+)$ is a bijection $f:A\to B$
between two finite subsets $A,B$ of $G$ satisfying $a+f(a)\notin A$ for all
$a\in A$. A group $G$ has the matching property if for every two finite subsets
$A,B \subset G$ of the same size with $0 \notin B$, there exists a matching
from $A$ to $B$. In [19] it was proved that an abelian group has the matching
property if and only if it is torsion-free or cyclic of prime order. Here we
consider a similar question in a matroid setting. We introduce an analogous
notion of matching between matroids whose ground sets are subsets of an abelian
group $G$, and we obtain criteria for the existence of such matchings. Our
tools are classical theorems in matroid theory, group theory and additive
number theory.
