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 -group (http://planetmath.org/PGroup4) 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 .