  Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a
strict $n$-split decomposition if it can be partitioned as the disjoint union
of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$,
such that $|B_i| > 1$ for each $i$ and within each set $B_i$, no two distinct
elements commute. We show that every finite nonabelian group has a strict
$n$-split decomposition for some $n$. We classify all finite groups $G$, up to
isomorphism, which have a strict $n$-split decomposition for $n = 1, 2, 3$.
Finally, we show that for a nonabelian group $G$ having a strict $n$-split
decomposition, the index $|G:A|$ is bounded by some function of $n$.
