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 $p$ group for some prime $p$

Elementary abelian $2$ groups are sometimes called Boolean groups. A group in which every non-trivial element has order $2$ is necessarily Boolean, because abelianness is automatic: $xy=(xy)^{-1}=y^{-1}x^{-1}=yx$ There is no analogous result for odd primes, because for every odd prime $p$ there is a non-abelian group of order $p^3$ and exponent $p$

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