1. ## 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 $g \in G$.

2. Originally Posted by dori1123
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)...

3. Originally Posted by Opalg
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?

4. If $|G|=n$ then $g^n=e$ (the identity), so $x=x^{-1}$, or $x^2=e$. That in turn means that $e=x^n=x$.