Prove that is a normal subgroup of and that is the largest subgroup of containing as a normal subgroup.
I want to add more to what NonCommAlg said. You need to understand the meaning of "largest". This problem is saying that if is a subgroup of so that is a normal subgroup of then i.e. " is the largest". Show that any element in must be automatically be an element of to complete the proof. That is all you need to show, now follow the definition of normalizer.