Can someone help me with the definitions of centralizer and normalizer? My class doesn't use a book and I don't understand the definitions we were given.

Normalizer N(a)={x in G | xa=ax} if (G, *) is a group and a is in G.

Center of G=Z(G)={x in G | ax=xa , for all a in G}.

I need to prove that a is in Z(G) iff N(a)=G.