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 Gi\Zi ⊆ Mi ⊆ Gi, 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).
ASJC Scopus subject areas
- Algebra and Number Theory