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Group Homomorphisms and order

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Let ψ: G→H be a homomorphism and let g ε G have finite order.
    a) Show that the order of ψ(g) divides the order of g


    3. The attempt at a solution
    I'm really lost here, but I'm guessing we can use the fact |ψ(g)| = {e,g...,g|g|-1}
    and ψ(g|g|-1) = ψ(g)ψ(g)ψ(g)ψ(g)ψ(g).... (|g|-1 times)
    I still have no idea where to start.
     
  2. jcsd
  3. Oct 11, 2011 #2

    micromass

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    First show that

    [tex]\psi(g)^{|g|}=e[/tex]

    Then use the general fact that if [itex]a^n=e[/itex], then |a| divides n.
     
  4. Oct 11, 2011 #3
    Ah ok but I would use eH?
    Also, if it is an isomorphism, could I show |ψ(g)| = |g|?
     
  5. Oct 11, 2011 #4

    micromass

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    Yes to both.
     
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