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单词 ElementaryAbelianGroup
释义

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.

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更新时间:2025/5/4 8:12:09