Is Phi an Isomorphism in an Abelian Group?

[b0mb0nika]

let G be an abelian group, and n positive integer
phi is a map frm G to G sending x->x^n
phi is a homomorphism

show that
a.)ker phi={g from G, |g| divides n}
b.) phi is an isomorphism if n is relatively primes to |G|

i have no clue how to even start the prob...:-(

 

[Hurkyl]

If you're using |g| to mean the order of g in G, then you've made a typo in part (a).

 

[b0mb0nika]

yes i did..
|g| divides n
sorry about that :)

 

[Hurkyl]

Well, when you have no clue where to begin, the definitions are often a very good place to start.

 

[Palindrom]

A. Let $x\in G$. Then $x^n=e \Leftrightarrow O(x)|n \Rightarrow Ker(\phi)=\{g\in G| O(g)|n\}$

B. I have to go, I might come back later if it isn't solved by then.