Elementary abelian

In group theory, an elementary abelian group (or elementary abelian p-group) is an abelian group in which every nontrivial element has order p. The number p must be prime, and the elementary abelian groups are a particular kind of p-group. The case where p = 2, i.e., an elementary abelian 2-group, is sometimes called a Boolean group.

Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ)ⁿ for n a non-negative integer (sometimes called the group's rank). Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups.

In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p. (Note that in the finite case the direct product and direct sum coincide, but this is not so in the infinite case.)

Presently, in the rest of this article, these groups are assumed finite.

Suppose V ${\displaystyle \cong }$ (Z/pZ)ⁿ is an elementary abelian group. Since Z/pZ ${\displaystyle \cong }$ F_p, the finite field of p elements, we have V = (Z/pZ)ⁿ${\displaystyle \cong }$ F_pⁿ, hence V can be considered as an n-dimensional vector space over the field F_p. Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism V ${\displaystyle {\overset {\sim }{\to }}}$ (Z/pZ)ⁿ corresponds to a choice of basis.

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