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> **Possible Duplicate:**  
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> [Normal and central subgroups of finite $p$-groups](https://math.stackexchange.com/questions/83841/normal-and-central-subgroups-of-finite-p-groups) 
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I want to show that if $O(G)=p^{n}$ then $Z(G)\neq \{e\}$, where $p$ is a prime number and $Z(G)=\{a\in G | ax=xa, \forall x\in G\}$, which is also known as a center of the group $G$.
I think I have to use the Lagrange theorem which state that in a finite group $G$, if $H$ is a subgroup of $G$ then $O(H)|O(G)$. But i don't get any right approach to use this to prove the given result.