Let G be a finite group and let x be in G.
Prove that if $\displaystyle g\in N_G(<x>)$, then $\displaystyle gxg^{-1}=x^a$ for some $\displaystyle a\in\mathbb{Z}$.
$\displaystyle N_G(<x>)=\{g\in G:g<x>g^{-1}=x\}$
Let $\displaystyle |x|=n$
$\displaystyle <x>=\{e,x,x^2,\cdots x^{n-1}\}$
Now what?