I solved the 1st half of this question and I am stuck on the second part

Let $\displaystyle G$ be a group and let $\displaystyle Z_{1}$ be the center of $\displaystyle G$. For every $\displaystyle n\geq 1$, we'll define $\displaystyle Z_{n+1}=\{ x\in G|\forall y\in G, [x,y]\in Z_{n}\}$.

1. Show that for every $\displaystyle n\geq 1$, $\displaystyle Z_{n}\unlhd G$.

2. If $\displaystyle [G]=p^{n}$ ($\displaystyle p$ a prime number), show that $\displaystyle Z_{n}=G$

I showed part 1 by induction on $\displaystyle n$ and by showing that the center of $\displaystyle G/Z_{n}$ is isomorphic to $\displaystyle Z_{n+1}$ therfore it is normal in $\displaystyle G$.

For part 2, I'm thinking that since $\displaystyle Z_{n}\subseteq G$ then all that is left to show is that $\displaystyle [Z_{n}]=p^{n}$, but I can't figure out a way to do it.

Appreciate any ideas

SK