Sylow p-subgroup is normal if and only if it is a characteristic subgroup Take finite group $G$ with Sylow p-subgroup $P$. Then, prove that
$P \unlhd G \iff P$ is a characteristic subgroup of $G$. 
I know how to show the $\impliedby$ direction (if characteristic, then normal) but can't think of how to prove $\implies$ (if normal, then characteristic). 

If a Sylow subgroup is normal, then it is the unique subgroup of that order, i.e. if $P\triangleleft G$ then there is only one subgroup of order $|P|$ which implies $P$ is characteristic since automorphisms preserve orders of subgroups.
