# Thread: Proving isomorphism to Zn

1. ## Proving isomorphism to Zn

I have proven cyclic group to be a group, but now must prove cyclic group order n to be isomorphic to Zn. I realize I must show homomorphism, injection and surjection. But not struggling with initial equivalences. Please help.

2. ## Re: Proving isomorphism to Zn

Let $\displaystyle a$ be a generator of your cyclic group $\displaystyle G$, can you find a function such that $\displaystyle f:G \to \mathbb{Z}_n$ and $\displaystyle f(a)=1$ such that $\displaystyle f$ is an homomorphism?

3. ## Re: Proving isomorphism to Zn

are you suggesting a concrete f like f(x)=x+5 or generalized like f(x)=ix+j.

4. ## Re: Proving isomorphism to Zn

Something concrete, in terms of things we know must exist (like generator "a").
Try a^n

5. ## Re: Proving isomorphism to Zn

Originally Posted by MAdone
are you suggesting a concrete f like f(x)=x+5 or generalized like f(x)=ix+j.
????

Suppose the operation of your group is given by $\displaystyle +$, if an $\displaystyle f$ as I described existed, what would be the image of $\displaystyle ka=a+a+a\hdots a$ ($\displaystyle k$ summands) under $\displaystyle f$, and $\displaystyle a$ is a generator so...

6. ## Re: Proving isomorphism to Zn

all elements of the cyclic group

7. ## Re: Proving isomorphism to Zn

Originally Posted by Jose27
Let $\displaystyle a$ be a generator of your cyclic group $\displaystyle G$, can you find a function such that $\displaystyle f:G \to \mathbb{Z}_n$ and $\displaystyle f(a)=1$ such that $\displaystyle f$ is an homomorphism?
MAdone, this is really the whole of the idea.

if $\displaystyle f:G \to \mathbb{Z}_n$ is to be a homomorphism, we must have f(a*a) = f(a) + f(a) = 1+1. so there is really only "one" essential way to define f:

$\displaystyle f(a^k) = k$. that f is a homomorphism follows immediately from the laws of exponents.

since G is finite it suffices to show that f is surjective (clearly |G| = $\displaystyle |\mathbb{Z}_n|$ = n), and thus f is an isomorphism.