conjugacy class

**dori1123** · November 1st 2008, 09:35 PM

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 $g \in G$ .

**Opalg** · November 2nd 2008, 01:13 AM

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)...

**dori1123** · November 2nd 2008, 09:38 AM

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?