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Homework Help: Abelian group isomorphism help

  1. Dec 5, 2004 #1
    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...:-(
     
    Last edited: Dec 6, 2004
  2. jcsd
  3. Dec 5, 2004 #2

    Hurkyl

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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).
     
  4. Dec 5, 2004 #3
    yes i did..
    |g| divides n
    sorry about that :)
     
  5. Dec 5, 2004 #4

    Hurkyl

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    Well, when you have no clue where to begin, the definitions are often a very good place to start.
     
  6. Dec 6, 2004 #5
    A. Let [itex]x\in G[/itex]. Then [itex] x^n=e \Leftrightarrow O(x)|n \Rightarrow Ker(\phi)=\{g\in G| O(g)|n\} [/itex]

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