  We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$
are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$
with $x\in X$ and $y\in Y$. This definition coincides with the customary and
extensively studied definition of a direct product decomposition by subsets of
a finite abelian group. Our main result states that a group $G$ has such a
factorization if and only if $G$ is a central product of $\left\langle
X\right\rangle $ and $\left\langle Y\right\rangle $ and the central subgroup
$\left\langle X\right\rangle \cap \left\langle Y\right\rangle $ satisfies
certain abelian factorization conditions. We analyze some special cases and
give examples. In particular, simple groups have no non-trivial set-direct
factorization.

Group Theory