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**Haven** I've been racking my brain over this problem:

Let $\displaystyle G$ be a group of order $\displaystyle p^n$

Prove the Center of $\displaystyle G$ cannot have order $\displaystyle p^{n-1}$

Naturally, I assume for a contradiction that $\displaystyle |Z(G)| = p^{n-1}$

By Lagrange's Theorem, there are $\displaystyle \frac{|G|}{|Z(G)|} = \frac{p^n}{p^{n-1}} = p$ distinct left cosets of $\displaystyle Z(G)$

Let $\displaystyle a \in G$.

If $\displaystyle a \in Z(G)$ then $\displaystyle aZ(G) = Z(G)$

This is as far as I get and I get the contraditiction. Any help would be greatly appreciated