Let $G$ be a finite group and $x\in G$ be an element of order $p$ ($p$ prime). Suppose that $x\in P$, where $P$ is some $p$-Sylow subgroup of $G$.
I could not prove the following:
$x$ is not conjugate in $G$ to any other element of $P$ if and only if $x$ commutes with none of its conjugates in $G$ other than itself.
Is it easy? Does it depend on some group theory result?