An elementary abelian group is an abelian group in which every non-trivial element has the same finite order. It is easy to see that the non-trivial elements must in fact be of prime order, so every elementary abelian group is a -group for some prime .

Elementary abelian -groups are sometimes called Boolean groups. A group in which every non-trivial element has order is necessarily Boolean, because abelianness is automatic: . There is no analogous result for odd primes, because for every odd prime there is a non-abelian group of order and exponent .

Let be a prime number. Any elementary abelian -group can be considered as a vector space over the field of order , and is therefore isomorphic to the direct sum of copies of the cyclic group of order , for some cardinal number . Conversely, any such direct sum is obviously an elementary abelian -group. So, in particular, for every infinite cardinal there is, up to isomorphism, exactly one elementary abelian -group of order .