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Math Help - conjugacy class

  1. #1
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    conjugacy class

    If G is a group of odd order, show for any nonidentity element x \in G that x and x^{-1} are not conjugate in G.

    Definition: x and x^{-1} are conjugates if gxg^{-1}=x^{-1} for some g \in G.
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    If G is a group of odd order, show for any nonidentity element x \in G that x and x^{-1} are not conjugate in G.

    Definition: x and x^{-1} are conjugates if gxg^{-1}=x^{-1} for some g \in G.
    If gxg^{-1}=x^{-1} then (taking the inverse of both sides) gx^{-1}g^{-1}=x. Therefore g^2xg^{-2}= g(gxg^{-1})g^{-1} = x, g^3xg^{-3}=x^{-1}, and in fact g^nxg^{-n}=x^{-1} whenever n is odd (including when n is equal to the order of G)...
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    Quote Originally Posted by Opalg View Post
    If gxg^{-1}=x^{-1} then (taking the inverse of both sides) gx^{-1}g^{-1}=x. Therefore g^2xg^{-2}= g(gxg^{-1})g^{-1} = x, g^3xg^{-3}=x^{-1}, and in fact g^nxg^{-n}=x^{-1} whenever n is odd (including when n is equal to the order of G)...
    I don't understand how does this show that x and x^{-1} are not conjugates when |G|=n is odd?
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  4. #4
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    If |G|=n then g^n=e (the identity), so x=x^{-1}, or x^2=e. That in turn means that e=x^n=x.
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