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Concept
Elementary abelian group
Summary
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. A group for which p = 2 (that is, 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.)
In the rest of this article, all groups are assumed finite.
The elementary abelian group (Z/2Z)2 has four elements: {(0,0), (0,1), (1,0), (1,1)} . Addition is performed componentwise, taking the result modulo 2. For instance, (1,0) + (1,1) = (0,1). This is in fact the Klein four-group.
In the group generated by the symmetric difference on a (not necessarily finite) set, every element has order 2. Any such group is necessarily abelian because, since every element is its own inverse, xy = (xy)−1 = y−1x−1 = yx. Such a group (also called a Boolean group), generalizes the Klein four-group example to an arbitrary number of components.
(Z/pZ)n is generated by n elements, and n is the least possible number of generators. In particular, the set {e1, ..., en} , where ei has a 1 in the ith component and 0 elsewhere, is a minimal generating set.
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