Problem with an exercise about maximal subgroups in nilpotent group

Question

I'm trying to resolve the following exercise.

Let be $G$ a nilpotent group, A a maximal abelian and normal subgroup in G. Then $C_G(A)=A$.

Now my question is: if I choose $G=\mathbb{Z}$ and $A=p\mathbb{Z}$, they satisfy the hypothesis but I have that $C_G(A)=G$.

Where is my mistake? Is there a mistake in the text exercise, since this seems to be an easy counterexample?