Sorry, I meant for all g in G. That is, for any n that is not in any N(g). I think thats true, since if n is in N(g), ng = gn for some g in G, so the result is trivial.
I think you're confused (or perhaps I am: go figure!) : you define the centralizer of the element (try no to use the letter N for this since it is used for another thing, the normalizer). But an element belongs to the centralizer of each and every element of G! So it can't be that an element in the center is NOT in some element's centralizer
Your answer is obvious. It's certainly true, but I hardly see why they'd want do highlight this particular point. But it is the most logical choice.