Two groups are said to be isomorphic if there exists a map f: G1->G2 such that f is a bijection. Let G1 be (G,+) and let G2 be (H,*), where G and H are arbitrary non-empty sets and + and * are arbitrary operations on those sets. For g1, g2 in G, f(g1 + g2)
= f(g1) * f(g2) => G1 and G2 are isomorphic.