Suppose that $\displaystyle G$ is a group with $\displaystyle |G|=p^3$ where $\displaystyle p$ is prime. Prove that $\displaystyle |Z(G)|\ne p^2$.
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If so, then |G/Z(G)|=p, and so G/Z(G) is cyclic. But G/Z(G) cyclic implies G is abelian, a contradiction.
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