Let H={(n,n)|nis an element of Z (integers)}.Prove (Z+Z)/H is isomorphic to Z.

The hint I was given was that my mapping should involve subtraction. I am totally clueless on this one and any help would be of great benefit to me.

Thanks

Printable View

- Apr 28th 2010, 06:57 PMwutangisomorphism question
Let H={(n,n)|n

**i**s an element of Z (integers)}**.**Prove (Z+Z)/H is isomorphic to Z.

The hint I was given was that my mapping should involve subtraction. I am totally clueless on this one and any help would be of great benefit to me.

Thanks - Apr 28th 2010, 07:07 PMtonio
- Apr 28th 2010, 07:09 PMDrexel28
- Apr 29th 2010, 12:37 AMSwlabr
This does note work. You are (I believe) claiming, essentially, that when . It is (again, I believe) true when everything is finite and abelian. However, take and .

and so these groups are clearly not isomorphic.

I know that what you claim is not true as I once claimed it was true too! - Apr 29th 2010, 08:56 AMDrexel28
That isn't what I was saying. I'm saying that define

Clearly this is injective since and surjective since if then . To see it's a homomorphism we note that .

So, . So now define . This is clearly injective and surjective and . Thus, and so the conclusion follows by the transitivity of "is isomorphic to" - Apr 29th 2010, 12:26 PMtonio
- Apr 29th 2010, 04:41 PMDrexel28
- Apr 29th 2010, 05:06 PMwutang
Here is my proof.

Let f: Z+Z to Z where f(a,b)=a-b. NTS that F is an epimorphism.

f is obviously a well defined function.

Operation preserving: let x,y be elemnts of Z+Z. The f(x,y)=x-y = f(x)-f(y). Thus f is operation preserving.

Onto- Let z-y be an element of Z and let z,y be elements of Z+Z. Then

f(z,y)=z-y. Thus f is onto.

show that H is contained in Ker f.

Let x be an element of H. Then x=(n,n) for some n that is an element of Z. Then (n,n) is an element of kerf since f(n,n)=0. Thus x is an element of ker f and H is contained in Ker f.

Show that Ker f is contained in H.

Let x be an element of Ker f. Then x=(n,n) for some n that is an element of Z since f(n,n)=0. Then (n,n) is an element of H by the definition of H. Thus x is an element of H. So ker f is contained in H.

Therefore by th 1st isomorphism theroem, (Z+Z)/H is isomorphic to Z.

P.S. I will be getting my labtop back soon, so I wil be able to write in LaTex again!!! - Apr 29th 2010, 05:38 PMDefunkt
This is not what you are required to show. In order to prove that f is operation conserving, prove that

Quote:

Onto- Let z-y be an element of Z and let z,y be elements of Z+Z. Then

f(z,y)=z-y. Thus f is onto.

Quote:

show that H is contained in Ker f.

Let x be an element of H. Then x=(n,n) for some n that is an element of Z. Then (n,n) is an element of kerf since f(n,n)=0. Thus x is an element of ker f and H is contained in Ker f.

Quote:

Show that Ker f is contained in H.

Let x be an element of Ker f. Then x=(n,n) for some n that is an element of Z since f(n,n)=0. Then (n,n) is an element of H by the definition of H. Thus x is an element of H. So ker f is contained in H.

Quote:

Therefore by th 1st isomorphism theroem, (Z+Z)/H is isomorphic to Z.

P.S. I will be getting my labtop back soon, so I wil be able to write in LaTex again!!!

- Apr 30th 2010, 12:37 AMSwlabr
- Apr 30th 2010, 06:53 AMDrexel28