Every group of order $9$ is abelian: cyclic or elementary abelian.Suggest a slogan
Up to isomorphism, the groups of order \(9\) are \(\ZZ/9\ZZ\) and \((\ZZ/3\ZZ)^2\). Both are abelian.
A group \(G\) of order \(p^2\), \(p\) prime, has nontrivial center \(Z(G)\) by the class equation. If \(Z(G)\ne G\), then \(G/Z(G)\) has order \(p\) and is cyclic, which forces \(G\) to be abelian, a contradiction. So \(G\) is abelian, and the classification of finite abelian groups gives \(\ZZ/p^2\ZZ\) or \((\ZZ/p\ZZ)^2\).
What are the groups of order 9?