Frequency (statistics): Difference between revisions
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The | 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|''p''-group]]. | ||
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 (sometimes called the group's ''rank''). 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'' <math>\cong</math> (''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>. Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism ''V'' <math>\overset{\sim}{\to}</math> (''Z''/''pZ'')<sup>''n''</sup> corresponds to a choice of basis. | |||
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 (mathematics)|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]]s 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>. | |||
== 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'''<ref>{{cite book|last=Gorenstein|first=Daniel|title=Finite Groups|publisher=Harper & Row|location=New York|year=1968|chapter=1.2|pages=8|isbn=0-8218-4342-7}}</ref> (of rank ''n'') is an abelian group of type (''p<sup>e</sup>'',''p<sup>e</sup>'',...,''p<sup>e</sup>'') i.e. the direct product of ''n'' isomorphic groups of order ''p<sup>e</sup>''. | |||
== Related groups == | |||
The [[extra special group]]s are extensions of elementary abelian groups by a cyclic group of order ''p,'' and are analogous to the [[Heisenberg group]]. | |||
==References== | |||
<references/> | |||
[[Category:Abelian group theory]] | |||
[[Category:Finite groups]] | |||
[[Category:P-groups]] | |||
Revision as of 14:18, 15 November 2013
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, ..., en}, where ei 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.
Vector space structure
Suppose V (Z/pZ)n is an elementary abelian group. Since Z/pZ Fp, the finite field of p elements, we have V = (Z/pZ)n Fpn, hence V can be considered as an n-dimensional vector space over the field Fp. 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 Fpn 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 Fp scalar multiplication. That is, c·g = g + g + ... + g (c times) where c in Fp (considered as an integer with 0 ≤ c < p) gives V a natural Fp-module structure.
Automorphism group
As a vector space V has a basis {e1, ..., en} as described in the examples. If we take {v1, ..., vn} to be any n elements of V, then by linear algebra we have that the mapping T(ei) = vi 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 } = GLn(Fp), the general linear group of n × n invertible matrices on Fp.
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[1] (of rank n) is an abelian group of type (pe,pe,...,pe) i.e. the direct product of n isomorphic groups of order pe.
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.
References
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534