Two groups are said to be isomorphic if there exists a map f: G_{1}->G_{2} such that f is a bijection. Let G_{1} be (G,+) and let G_{2} be (H,*), where G and H are arbitrary non-empty sets and + and * are arbitrary operations on those sets. For g_{1}, g_{2} in G, f(g_{1} + g_{2})

= f(g_{1}) * f(g_{2}) => G_{1} and G_{2} are isomorphic.