Suppose H is an infinite cyclic subgroup of Z. Show that H and Z are isomorphic.
We know that any infinite cyclic group H isomorphic to Z.
H = <a> ≠ <0>
|a| = ∞
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.