Let H={(n,n)|n is 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
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!
That isn't what I was saying. I'm saying that define n,n)\mapsto (n,0)" alt="\theta:H\to\mathbb{Z}\times\{0\}n,n)\mapsto (n,0)" />
Clearly this is injective since and surjective since if then . To see it's a homomorphism we note that .
So, . So now define z,0)\mapsto z" alt="\phi:\mathbb{Z}\times\{0\}\to\mathbb{Z}z,0)\mapsto z" />. This is clearly injective and surjective and . Thus, and so the conclusion follows by the transitivity of "is isomorphic to"
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!!!
This is not what you are required to show. In order to prove that f is operation conserving, prove that
This is not clear. First you should say that any element can be written as for some , say,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.
This is good.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.
You're sort of assuming what you are asked to prove here. You should let , write and reach thatShow 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!!!