Algebraic structure → Group theory Group theory 

Infinite dimensional Lie group

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In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian pgroup) 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 pgroup.^{ [1] }^{ [2] } The case where p = 2, i.e., an elementary abelian 2group, is sometimes called a Boolean group.^{ [3] }
Every elementary abelian pgroup 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 nonnegative 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 nfold direct product of groups.^{ [2] }
In general, a (possibly infinite) elementary abelian pgroup 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.)
Presently, in the rest of this article, these groups are assumed finite.
Suppose V (Z/pZ)^{n} is an elementary abelian group. Since Z/pZF_{p}, the finite field of p elements, we have V = (Z/pZ)^{n}F_{p}^{n}, hence V can be considered as an ndimensional 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 (Z/pZ)^{n} corresponds to a choice of basis.
To the observant reader, it may appear that F_{p}^{n} 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 Zmodule structure where the action of Z corresponds to repeated addition, and this Zmodule structure is consistent with the F_{p} scalar multiplication. That is, c·g = g + g + ... + g (c times) where c in F_{p} (considered as an integer with 0 ≤ c < p) gives V a natural F_{p}module structure.
As a vector space V has a basis {e_{1}, ..., e_{n}} as described in the examples, if we take {v_{1}, ..., v_{n}} to be any n elements of V, then by linear algebra we have that the mapping T(e_{i}) = v_{i} 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) = { T : V→V  ker T = 0 } = GL_{n}(F_{p}), the general linear group of n × n invertible matrices on F_{p}.
The automorphism group GL(V) = GL_{n}(F_{p}) acts transitively on V \ {0} (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 \ {e}, then G is elementary abelian. (Proof: if Aut(G) acts transitively on G \ {e}, then all nonidentity elements of G have the same (necessarily prime) order. Then G is a pgroup. 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 primepower 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 (p^{k},p^{k},...,p^{k}) 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.
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