Set-direct factorizations of groups (Q1799029)

In the main result of this paper (Theorem 1.2), the authors prove that for a general group $G$ and two normal subsets $X,Y\subseteq G$ every element $g\in G$ has a unique factorization $g=xy$ with $x\in X$ and $y\in Y$ (this means that $G$ is a set-direct product of $X$ and $Y$) if and only if $G$ is a central product of $\langle X\rangle$ and $\langle Y\rangle$ and for every $m\in\langle X\rangle$ and every $n\in\langle Y\rangle$ the subgroup $Z=\langle X\rangle\cap \langle Y\rangle$ is the set-direct product of $(m^{-1}X)\cap Z$ and $(n^{-1}Y)\cap Z$. Some particular cases, e.g. finite quasi-simple groups, are also studied. For instance, it is proved that simple groups have no non-trivial set-direct factorization.