Given a group G and a subgroup H, consider the normalizer N(H)

(a)Since the normalizer is the stabilizer of an element under a group action, we know that it is a subgroup. Give a direct proof of this fact, from the definition of the normalizer.(b)Show that N(H) is the largest subgroup of G which contains H, in which H is a normal subgroup. What is N(H) if H is a normal subgroup of G?

(c)Recall the subgroup C(H), the centralizer of H, Why is C(H) a normal subgroup of N(H)?

(d)Prove that N(H)/C(H) is isomorphic to a subgroup of Aut(H), the group of automorphisms of H