Homework Help: Show the map is isomorphic

1. Nov 5, 2012

Zondrina

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.

2. Nov 5, 2012

Dick

That is pretty good. The DEFINITION of <a> is Za. So if r is in <a> then r=ka for some k in Z. So f(k)=r. It's kind of obviously surjective.

3. Nov 5, 2012

Zondrina

Yeah I didnt know whether I had to actually justify it or to say it was trivial. I guess I got my answer then.

Thanks

4. Nov 5, 2012

Dick

You ALWAYS have to justify it. Quoting a definition and showing why it works is a fine way to do that. Just saying 'it's trivial' is criminally lame. Don't do that! You didn't think it's trivial, why should someone else? Please forget my 'kind of obvious' comment.

Last edited: Nov 5, 2012