Frequency (statistics)

In group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; it is a particular kind of p-group.

By the classification of finitely generated abelian groups, every 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 notation means the n-fold Cartesian product.

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 mod 2. For instance, (1,0) + (1,1) = (0,1).

• (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 elementary abelian group has a fairly simple finite presentation.

(Z/pZ)ⁿ $\cong$ < e₁, ..., e_n | e_i^p = 1, e_ie_j = e_je_i >

Vector space structure

Suppose V $\cong$ (Z/pZ)ⁿ is an elementary abelian group. Since Z/pZ $\cong$ F_p, the finite field of p elements, we have V = (Z/pZ)ⁿ $\cong$ 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 $\stackrel{\sim }{\to }$ (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 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.

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 (of rank n) is an abelian group of type (p^e,p^e,...,p^e) i.e. the direct product of n isomorphic groups of order p^e.

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.