On product-equality-preserving mappings in groups

Let G, H be groups, M ⊆ G. A mapping f: M → H is called a Product-equality-preserving (PEP) mapping iff it satisfies ∀a,b,c,d ∈ M [ab = cd ⇒ f(a)f(b) = f(c)f(d)]. (*)_M THEOREM. Let G₁,G₂ be nonabelian groups with centers Z₁,Z₂ and let M₁,M₂ satisfy G_i\Z_i ⊆ M_i ⊆ G_i, i = 1,2. Let f: M₁ → M₂ be a PEP mapping that maps M₁ onto M₂. Then there is an epimorphism φ: G₁ → G₂ and υ ∈ Z₂ such that for every x ∈ M₁, f(x) = υφ(x).