Counterexample for the normalizer being a normal subgroup Let $G$ be a group, and $H$ a subgroup.
$H$'s normalizer is defined: $N(H):=\{g\in G| gHg^{-1}=H \}$. 
Prove $N(H)$ is a normal subgroup of G, or give counterexample.
Intuitively it seems to me that this claim is wrong, however, I'm having trouble with finding a counterexample.
Thans in advance for any assistance! 

The normaliser of any proper nontrivial subgroup of a simple group would work too. Can you see why?
