  Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if
there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$
of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper
investigates problems relating to group with abelian partitions. Among other
results, we show that every finite group is isomorphic to a subgroup of a group
with an abelian partition and also isomorphic to a subgroup of a group with no
abelian partition. We also find bounds for the minimum number of partitions for
several families of groups which admit abelian partitions -- with exact
calculations in some cases. Finally, we examine how the size of partitions with
the minimum number of parts behaves with respect to the direct product.
