Elementary abelian group

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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.)

Examples and properties

• 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, ..., e_n} , where e_i has a 1 in the ith component and 0 elsewhere, is a minimal generating set.
• Every finite elementary abelian group has a fairly simple finite presentation.

${\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{n}\cong \langle e_{1},\ldots ,e_{n}\mid e_{i}^{p}=1,\ e_{i}e_{j}=e_{j}e_{i}\rangle }$

Vector space structure

Suppose V ${\displaystyle \cong }$ (Z/pZ)ⁿ is a finite elementary abelian group. Since Z/pZ ${\displaystyle \cong }$ F_p, the finite field of p elements, we have V = (Z/pZ)ⁿ ${\displaystyle \cong }$ F_pⁿ, hence V can be considered as an n-dimensional 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 ${\displaystyle {\overset {\cong }{\to }}}$ (Z/pZ)ⁿ corresponds to a choice of basis.

To the observant reader, it may appear that F_pⁿ 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 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.

Automorphism group

As a finite-dimensional vector space V has a basis {e₁, ..., e_n} as described in the examples, if we take {v₁, ..., 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 p-group. It follows that G has a nontrivial center, which is necessarily invariant under all automorphisms, and thus equals all of G.)

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 (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.

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.

See also

• Elementary group
• Hamming space

References