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Inmathematics, specifically ingroup theory, anelementary abelian group is anabelian group in which all elements other than the identity have the sameorder. This common order must be aprime number, and the elementary abelian groups in which the common order isp are a particular kind ofp-group.^[1][2] A group for whichp = 2 (that is, an elementary abelian 2-group) is sometimes called aBoolean group.^[3]

Every elementary abelianp-group is avector space over theprime field withp elements, and conversely every such vector space is an elementary abelian group.By theclassification of finitely generated abelian groups, or by the fact that every vector space has abasis, every finite elementary abelian group must be of the form (Z/pZ)ⁿ forn a non-negative integer (sometimes called the group'srank). Here,Z/pZ denotes thecyclic group of orderp (or equivalently the integersmodp), and the superscript notation means then-folddirect product of groups.^[2]

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

• 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 theKlein four-group.
• In the group generated by thesymmetric 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 byn elements, andn is the least possible number of generators. In particular, the set{e₁, ...,e_n}, wheree_i has a 1 in theith component and 0 elsewhere, is a minimal generating set.
• Every finite elementary abelian group has a fairly simplefinite presentation:

  ${\displaystyle (\mathbb{Z}/p\mathbb{Z}{)}^n\cong ⟨e_1,\dots ,e_n\mid e_i^p=1,e_i{e}_j=e_j{e}_i⟩}$

SupposeV${\displaystyle \cong }$ (Z/pZ)ⁿ is a finite elementary abelian group. SinceZ/pZ${\displaystyle \cong }$F_p, thefinite field ofp elements, we haveV = (Z/pZ)ⁿ${\displaystyle \cong }$F_pⁿ, henceV can be considered as ann-dimensionalvector space over the fieldF_p. Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphismV${\displaystyle \stackrel{\cong }{\to }}$ (Z/pZ)ⁿ corresponds to a choice of basis.

To the observant reader, it may appear thatF_pⁿ has more structure than the groupV, in particular that it has scalar multiplication in addition to (vector/group) addition. However,V as an abelian group has a uniqueZ-module structure where the action ofZ corresponds to repeated addition, and thisZ-module structure is consistent with theF_p scalar multiplication. That is,c⋅g =g + g + ... + g (c times) wherec inF_p (considered as an integer with 0 ≤ c < p) givesV a naturalF_p-module structure.

As a finite-dimensional vector spaceV has a basis {e₁, ...,e_n} as described in the examples, if we take {v₁, ...,v_n} to be anyn elements ofV, then bylinear algebra we have that the mappingT(e_i) =v_i extends uniquely to a linear transformation ofV. Each suchT can be considered as a group homomorphism fromV toV (anendomorphism) and likewise any endomorphism ofV can be considered as a linear transformation ofV as a vector space.

If we restrict our attention toautomorphisms ofV we have Aut(V) = {T :V →V | kerT = 0 } = GL_n(F_p), thegeneral linear group ofn × n invertible matrices onF_p.

The automorphism group GL(V) = GL_n(F_p) actstransitively onV \ {0} (as is true for any vector space). This in fact characterizes elementary abelian groups among all finite groups: ifG is a finite group with identitye such that Aut(G) acts transitively onG \ {e}, thenG is elementary abelian. (Proof: if Aut(G) acts transitively onG \ {e}, then all nonidentity elements ofG have the same (necessarily prime) order. ThenG is ap-group. It follows thatG has a nontrivialcenter, which is necessarily invariant under all automorphisms, and thus equals all ofG.)

It can also be of interest to go beyond prime order components to prime-power order. Consider an elementary abelian groupG to be oftype (p,p,...,p) for some primep. Ahomocyclic group^[5] (of rankn) is an abelian group of type (m,m,...,m) i.e. the direct product ofn isomorphic cyclic groups of orderm, of which groups of type (p^k,p^k,...,p^k) are a special case.

Theextra special groups are extensions of elementary abelian groups by a cyclic group of orderp, and are analogous to theHeisenberg group.

• Elementary group
• Hamming space

• Abelian group theory
• Finite groups
• P-groups

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