elementary abelian group

An elementary abelian group is an abelian groupMathworldPlanetmath 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-1x-1=yx. There is no analogous result for odd primes, because for every odd prime p there is a non-abelian groupMathworldPlanetmath of order p3 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 isomorphicPlanetmathPlanetmathPlanetmath to the direct sumMathworldPlanetmath of κ copies of the cyclic groupMathworldPlanetmath of order p, for some cardinal numberMathworldPlanetmath κ. Conversely, any such direct sum is obviously an elementary abelian p-group. So, in particular, for every infiniteMathworldPlanetmath cardinal κ there is, up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, exactly one elementary abelian p-group of order κ.

Title elementary abelian group
Canonical name ElementaryAbelianGroup
Date of creation 2013-03-22 14:53:11
Last modified on 2013-03-22 14:53:11
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Definition
Classification msc 20F50
Classification msc 20K10
Defines elementary abelian
Defines Boolean group