  An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there
exists an irreducible character $\chi$ of $G$ such that $\chi(g)=0$; in this
case, $g$ is also called a zero of $G$. The aim of this paper is to obtain
structural properties of a factorised group $G=AB$ when we impose some
conditions on prime power order elements $g\in A\cup B$ which are
(non-)vanishing in $G$.
