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.)
(Z/pZ)n\cong\langlee1,\ldots,en\mid
| p | |
| e | |
| i |
=1, eiej=ejei\rangle
Suppose V
\cong
\cong
\cong
\overset{\cong}{\to}
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 (vector/group) 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 (c times) where c in Fp (considered as an integer with 0 ≤ c < p) gives V a natural Fp-module structure.
As a finite-dimensional 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(ei) = vi extends uniquely to a linear transformation of V. Each such T can be considered as a group homomorphism from V to V (an endomorphism) 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(V) = = GLn(Fp), the general linear group of n × n invertible matrices on Fp.
The automorphism group GL(V) = GLn(Fp) acts transitively on V \ (as is true for any vector space). This in fact characterizes elementary abelian groups among all finite groups: if G is a finite group with identity e such that Aut(G) acts transitively on G \ , then G is elementary abelian. (Proof: if Aut(G) acts transitively on G \ , then all nonidentity elements of G have the same (necessarily prime) order. Then G is a p-group. It follows that G has a nontrivial center, which is necessarily invariant under all automorphisms, and thus equals all of G.)
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 (p,p,...,p) for some prime p. A homocyclic group[5] (of rank n) is an abelian group of type (m,m,...,m) i.e. the direct product of n isomorphic cyclic groups of order m, of which groups of type (pk,pk,...,pk) are a special case.
The extra special groups are extensions of elementary abelian groups by a cyclic group of order p, and are analogous to the Heisenberg group.