MHB Help Proving Isomorphism of a group

[Bruce Wayne1]

Hi!

I'm trying to prove a cyclic group is isomorphic to ring under addition. What the strategy I would take? How would I get it started?

Here's what I know so far:

I need to meet 3 conditions-- 1 to 1, onto, and the operation is preserved. I also know that isomorphic means that the group is homomorphic with onto and 1 to 1.

 

[Evgeny.Makarov]

I'm trying to prove a cyclic group is isomorphic to ring under addition. What the strategy I would take? How would I get it started?

You need to come up with an explicit definition of an alleged isomorphism.

 

[Poirot1]

are you tring to prove a cyclic group G of order n is isomorphic to Z_n ={0,...,n-1}?Define $f(g)=g^n$ (mod n)

 

[Evgeny.Makarov]

are you tring to prove a cyclic group G of order n is isomorphic to Z_n ={0,...,n-1}?Define $f(g)=g^n$ (mod n)

I may be missing something. Is $f:G\to\mathbb{Z}_n$? Then what is $g^n\pmod{n}$?

 

[Bruce Wayne1]

are you tring to prove a cyclic group G of order n is isomorphic to Z_n ={0,...,n-1}?Define $f(g)=g^n$ (mod n)

Yes, exactly. I want to show that a cyclic group G of order n is isomorphic to Z_n.

I understand the concepts, and I know how to prove it for a relatively small n, but I haven't been able to find a completed proof online. I'd like to read it, and ask a few questions on the theory of why it works (should any arise).

 

[Evgeny.Makarov]

Since $G$ is cyclic of order $n$, there exists a $g\in G$ such that every $x\in G$ can be represented as $g^k$ for some $0\le k<n$. Therefore, we can define an $f:G\to\mathbb{Z}_n$ as follows: $f(g^k)=k$. It rests to show that $f$ is one-to-one, onto and a homomorphism.

 

[Poirot1]

I may be missing something. Is $f:G\to\mathbb{Z}_n$? Then what is $g^n\pmod{n}$?

yes sorry I mean f(g^k)=k (mod n), where g is the generator of G