# elementary abelian group

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 (http://planetmath.org/PGroup4) 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 $\kappa$.

Title elementary abelian group ElementaryAbelianGroup 2013-03-22 14:53:11 2013-03-22 14:53:11 yark (2760) yark (2760) 12 yark (2760) Definition msc 20F50 msc 20K10 elementary abelian Boolean group