Elementary abelian group
4843202
225189275
2008-07-12T10:43:31Z
Richard Pinch
1902818
unreferenced
In [[group theory]] an '''elementary abelian group''' is a finite [[abelian group]], where every nontrivial element has order ''p'' where ''p'' is a prime.
By the [[classification of finitely generated abelian groups]], every elementary abelian group must be of the form
:(''Z''/''pZ'')<sup>''n''</sup>
for ''n'' a non-negative integer. Here ''Z/pZ'' denotes the [[cyclic group]] of order ''p'' (or equivalently the integers [[Modular arithmetic|mod]] ''p''), and the notation means the ''n''-fold Cartesian product.
== Examples and properties ==
* The elementary abelian group (''Z''/2''Z'')<sup>2</sup> 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'')<sup>''n''</sup> is generated by ''n'' elements, and ''n'' is the least possible number of generators. In particular the set {''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>} where ''e''<sub>''i''</sub> has a 1 in the ''i''th component and 0 elsewhere is a minimal generating set.
* Every elementary abelian group has a fairly simple [[Presentation of a group|finite presentation]].
:: (''Z''/''pZ'')<sup>''n''</sup> <math>\cong</math> < ''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub> | ''e''<sub>''i''</sub><sup>''p''</sup> = 1, ''e''<sub>''i''</sub>''e''<sub>''j''</sub> = ''e''<sub>''j''</sub>''e''<sub>''i''</sub> >
== Vector space structure ==
Suppose ''V'' = (''Z''/''pZ'')<sup>''n''</sup> is an elementary abelian group. Since ''Z''/''pZ'' <math>\cong</math> ''F''<sub>''p''</sub>, the [[finite field]] of ''p'' elements, we have ''V'' = (''Z''/''pZ'')<sup>''n''</sup> <math>\cong</math> ''F''<sub>''p''</sub><sup>''n''</sup>, hence ''V'' can be considered as an ''n''-dimensional [[vector space]] over the field ''F''<sub>''p''</sub>.
To the observant reader it may appear that F<sub>''p''</sub><sup>''n''</sup> 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''<sub>''p''</sub> scalar multiplication. That is, ''c''·''g'' = ''g'' + ''g'' + ... + ''g'' (''c'' times) where ''c'' in ''F''<sub>''p''</sub> (considered as an integer with 0 ≤ ''c'' < ''p'') gives ''V'' a natural ''F''<sub>''p''</sub>-module structure.
== Automorphism group ==
As a vector space ''V'' has a basis {''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>} as described in the examples. If we take {''v''<sub>1</sub>, ..., ''v''<sub>''n''</sub>} to be any ''n'' elements of ''V'', then by [[linear algebra]] we have that the mapping ''T''(''e''<sub>''i''</sub>) = ''v''<sub>''i''</sub> 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 [[automorphism|automorphisms]] of ''V'' we have Aut(''V'') = { ''T'' : ''V'' -> ''V'' | ker ''T'' = 0 } = GL<sub>''n''</sub>(''F''<sub>''p''</sub>), the [[general linear group]] of ''n'' × ''n'' invertible matrices on F<sub>''p''</sub>.
==References==
{{unreferenced|date=July 2008}}
[[Category:Abelian group theory]]
[[Category:Finite groups]]
{{algebra-stub}}