On product-equality-preserving mappings in groups (Q1922485)

Let \(G_1\) and \(G_2\) be non-abelian groups with centres \(Z_1\) and \(Z_2\), respectively, and let \(M_1\) and \(M_2\) be subsets of \(G_1\) and \(G_2\), respectively, such that \(G_i\setminus Z_i\subseteq M_i\) (\(i=1,2\)). A map \(f:M_1\to M_2\) is said to be a PEP map of \(M_1\) in \(M_2\) (product equality preserving map) if from \(ab=cd\) it follows that \(f(a)f(b)=f(c)f(d)\) for all \(a,b,c,d\) in \(M_1\). The main result of this paper states that, if \(f\) is a PEP map of \(M_1\) onto \(M_2\), there exist an epimorphism \(\varphi:G_1\to G_2\) and an element \(v\) in \(Z_2\) such that \(f(x)=v\varphi(x)\) for every \(x\) in \(M_1\).