Elementary abelian group

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In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group.^[1][2] A group for which p = 2 (that is, an elementary abelian 2-group) is sometimes called a Boolean group.^[3]

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.^[2]

In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p.^[4] (Note that in the finite case the direct product and direct sum coincide, but this is not so in the infinite case.)