Zeros of irreducible characters in factorised groups

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 \(G\).