Thread: conjugacy class

1. conjugacy class

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

Definition: $\displaystyle x$ and $\displaystyle x^{-1}$ are conjugates if $\displaystyle gxg^{-1}=x^{-1}$ for some $\displaystyle g \in G$.

2. Originally Posted by dori1123
If $\displaystyle G$ is a group of odd order, show for any nonidentity element $\displaystyle x \in G$ that $\displaystyle x$ and $\displaystyle x^{-1}$ are not conjugate in $\displaystyle G$.

Definition: $\displaystyle x$ and $\displaystyle x^{-1}$ are conjugates if $\displaystyle gxg^{-1}=x^{-1}$ for some $\displaystyle g \in G$.
If $\displaystyle gxg^{-1}=x^{-1}$ then (taking the inverse of both sides) $\displaystyle gx^{-1}g^{-1}=x$. Therefore $\displaystyle g^2xg^{-2}= g(gxg^{-1})g^{-1} = x$, $\displaystyle g^3xg^{-3}=x^{-1}$, and in fact $\displaystyle g^nxg^{-n}=x^{-1}$ whenever n is odd (including when n is equal to the order of G)...

3. Originally Posted by Opalg
If $\displaystyle gxg^{-1}=x^{-1}$ then (taking the inverse of both sides) $\displaystyle gx^{-1}g^{-1}=x$. Therefore $\displaystyle g^2xg^{-2}= g(gxg^{-1})g^{-1} = x$, $\displaystyle g^3xg^{-3}=x^{-1}$, and in fact $\displaystyle g^nxg^{-n}=x^{-1}$ whenever n is odd (including when n is equal to the order of G)...
I don't understand how does this show that $\displaystyle x$ and $\displaystyle x^{-1}$ are not conjugates when $\displaystyle |G|=n$ is odd?

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