1. The problem statement, all variables and given/known data Suppose H is an infinite cyclic subgroup of Z. Show that H and Z are isomorphic. 2. Relevant equations We know that any infinite cyclic group H isomorphic to Z. H = <a> ≠ <0> |a| = ∞ 3. The attempt at a solution Define f : Z → H | f(k) = ak for all k in Z. We want to show that f is injective, surjective and operation preserving. ( Note, I think that map is correct. Since Z is an additive group in this case i think ak is the same as saying ka ) Case : Injectivity. Suppose that f(k) = f(q) for integers k and q. We want to show k = q. So : f(k) = f(q) ka = qa (k-q)a = 0 We know that |a| = ∞ and (k-q) is in Z. Hence our equation reduces to k-q = 0 and thus k=q. Therefore f is injective. Case : Surjectivity. Pick r in H. Then f(k) = ka = r for some integer k. ( Having a bit of trouble with this one ). So : Case : Operation preserving ( Homomorphic ). To show f is a homomorphism, we must show that f(k+q) = f(k) + f(q) for integers k and q. So : f(k+q) = ka + kq = f(k) +f(q) Hence f is a homomorphism. Now if I could get a bit of help showing that f is surjective, that will mean f is a bijective homomorphism aka an isomorphism and thus Z≈H and H≈Z.