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Doubt about abelian conjugacy class

  1. Mar 24, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm kinda lost in the concept of conjugate elements in group theory. It says that a element "h conjugate by g" is:

    g-1×h×g = hg

    Then it says that if the group is abelian h = hg

    2. Relevant equations
    Abelian group: a*b = b*a

    3. The attempt at a solution
    I don't get why the fact that the group is abelian (a*b = b*a) leads to the conclusion that h = hg

    g-1×h×g = g×h×g-1 why does it lead to h. I know it implies g×g-1 = e. But why wouldn't g-1×g = e aswell then leading to every non-abelian group also having the property h = hg?

    PS: Sorry my english.
     
  2. jcsd
  3. Mar 24, 2015 #2
    if the group is abelian then it's possible to swap the ##g^{-1}## & ##h## or the ##h## & ##g## to get either ##g^{-1}g## or ##gg^{-1}## leaving h by itself.
     
  4. Mar 24, 2015 #3
    got it! thanks fourier
     
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