Possible Duplicate:
Prove that the center of a group is a normal subgroup
Suppose that $H$ is a normal subgroup of $G$. Prove that $C_{G}(H)$ is a normal subgroup of $G$, where $C_{G}(H)$ is the centralizer of $H$ in $G$.
I have proved that $C_{G}(H)$ is a subgroup but how do I prove that it is normal - is this not obvious by the definition of a centralizer?