Is Phi an Isomorphism in an Abelian Group?

AI Thread Summary
In an abelian group G, the map phi defined by x -> x^n is established as a homomorphism. The kernel of phi is identified as the set of elements g in G such that the order of g divides n. Additionally, phi is shown to be an isomorphism if n is relatively prime to the order of G. The discussion highlights the importance of definitions in approaching the problem. Overall, the thread emphasizes the relationship between the properties of the group and the map phi.
b0mb0nika
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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...:-(
 
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If you're using |g| to mean the order of g in G, then you've made a typo in part (a).
 
yes i did..
|g| divides n
sorry about that :)
 
Well, when you have no clue where to begin, the definitions are often a very good place to start.
 
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.
 

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