Elementary abelian $p$-group

Question

How can we show that if $N$ is abelian and $C_G(N)=N$, then $N$ is an elementary abelian $p$-group for some prime $p$?

Answer 1

Not true, consider an odd prime $p$, the nonabelian group of order $p^{3}$ and exponent $p^{2}$: $$ G = \langle a, b : a^{p^{2}} = 1, b^{p} = 1, b^{-1} a b = a^{1+p} \rangle $$ and let $N = \langle a \rangle$. Then $C_{G}(N) = N$, but $N$ is cyclic of order $p^{2}$.