Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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
up vote 1 down vote accepted

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.

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.