Are groups of order $p^2$ abelian, $p$ prime?

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Okay so say you have a group of order p ( p a prime ) I know that this means that this group is isomorphic to $C_p$ ( the cyclic group of order p) and hence abelian. But what about groups of order $p^2$ are they isomorphic to $C_p\times C_p$ and also abelian because of this ?