In mathematics, specifically in group theory, an elementary abelian 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 n for n a non-negative integer. Here, Z/pZ denotes the cyclic group of order p, and the superscript notation means the n-fold direct product of groups. In general, a elementary abelian p-group is a direct sum of cyclic groups of order p. Presently, in the rest of this article, these groups are assumed finite.
Examples and properties
The elementary abelian group 2 has four elements:. Addition is performed componentwise, taking the result modulo 2. For instance,. This is in fact the Klein four-group.
In the group generated by the symmetric difference on a set, every element has order 2. Any such group is necessarily abelian because, since every element is its own inverse, xy = −1 = y−1x−1 = yx. Such a group, generalizes the Klein four-group example to an arbitrary number of components.
n is generated by n elements, and n is the least possible number of generators. In particular, the set, where ei has a 1 in the ith component and 0 elsewhere, is a minimal generating set.
Suppose Vn is an elementary abelian group. Since Z/pZ Fp, the finite field of p elements, we have V = nFpn, hence V can be considered as an n-dimensional vector space over the fieldFp. Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism Vn corresponds to a choice of basis. To the observant reader, it may appear that Fpn has more structure than the group V, in particular that it has scalar multiplication in addition to addition. However, V as an abelian group has a unique Z-module structure where the action of Z corresponds to repeated addition, and this Z-module structure is consistent with the Fp scalar multiplication. That is, c·g = g + g + ... + g where c in Fp gives V a natural Fp-module structure.
Automorphism group
As a vector space V has a basis as described in the examples, if we take to be any n elements of V, then by linear algebra we have that the mapping T = vi extends uniquely to a linear transformation of V. Each such T can be considered as a group homomorphism from V to V and likewise any endomorphism of V can be considered as a linear transformation of V as a vector space. If we restrict our attention to automorphisms of V we have Aut = = GLn, the general linear group of n × ninvertible matrices on Fp. The automorphism group GL = GLnacts transitively on V \ . This in fact characterizes elementary abelian groups among all finite groups: if G is a finite group with identity e such that Aut acts transitively on G \ , then G is elementary abelian. acts transitively on G \ , then all nonidentity elements of G have the same
A generalisation to higher orders
It can also be of interest to go beyond prime order components to prime-power order. Consider an elementary abelian group G to be of type for some prime p. A homocyclic group is an abelian group of type i.e. the direct product of n isomorphic cyclic groups of order m, of which groups of type are a special case.
Related groups
The extra special groups are extensions of elementary abelian groups by a cyclic group of order p, and are analogous to the Heisenberg group.
