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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
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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).

Conclude that there exist elementary abelian groups which are cyclic groups of a prime order; but not every elementary abelian group is a cyclic group of a prime order.

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$\mathbb Z_3\times\mathbb Z_3$ is elementary abelian but not cyclic.

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