About Sylow p-subgroup Let $G$ be a group and $P$ a Sylow $p$-subgroup of $G$. I know that $P$ consists of all $p$-elements in $N_{G}(P)$. My question is the following:
If $P$ is any $p$-subgroup of $G$, and if there is no $p$-element of $G-P$ normalizes $P$ in $G$, then is it true that $P$ is a Sylow $p$-subgroup of $G$? If it is not, maybe someone knows a counter example. Thanks!

If $P$ is a $p$-subgroup, but not Sylow, then it is contained in a Sylow one $P_1$. Then there is a subgroup $P_2<P_1$ such that $[P_2:P]=p$ and $P$ is normal in $P_2$.
