Set-direct factorizations of groups

We consider factorizations G=XY where G is a general group, X and Y are normal subsets of G and any g∈G has a unique representation g=xy with x∈X and y∈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 〈X〉 and 〈Y〉 and the central subgroup 〈X〉∩〈Y〉 satisfies certain abelian factorization conditions. We analyze some special cases and give examples. In particular, simple groups have no non-trivial set-direct factorization.