# Elementary abelian vs. cyclic groups

Sorry if this question sounds naive to someone. What is (are) the difference(s) between an elementary abelian group and a cyclic group of a prime order? Thanks

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Definition: An elementary abelian group is a finite group where every non-trivial element has order $p$ for some prime $p$. Taking that on to Mariano Suárez-Alvarez's answer gives you what you need. –  user1729 Aug 18 '13 at 16:46

By the classification of finitely generated abelian groups, every elementary abelian group must be of the form $$(Z/pZ)^n$$ for n a non-negative integer.
Here, $Z/pZ$ denotes the cyclic group of order $p$ (or equivalently the integers$\mod p$), and the notation means the $n$-fold Cartesian product (ref. wiki). On the other hand every group of prime order is cyclic (ref. for instance here or here).
$\mathbb Z_3\times\mathbb Z_3$ is elementary abelian but not cyclic.