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| 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]].
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| By the [[classification of finitely generated abelian groups]], every elementary abelian group must be of the form
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| :(''Z''/''pZ'')<sup>''n''</sup>
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| 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.
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| == Examples and properties ==
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| * 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).
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| * (''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.
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| * Every elementary abelian group has a fairly simple [[Presentation of a group|finite presentation]].
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| :: (''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> >
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| == Vector space structure ==
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| 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.
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| 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. | |
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| == Automorphism group ==
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| 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.
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| 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>.
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| == A generalisation to higher orders ==
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| 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>''.
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| == Related groups ==
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| 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]].
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| ==References==
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| <references/>
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| [[Category:Abelian group theory]]
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| [[Category:Finite groups]]
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| [[Category:P-groups]]
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Greetings! I am Marvella and I really feel comfortable when people use the full title. My working day job is a meter reader. Years in the past we moved to Puerto Rico and my family members enjoys it. To collect badges is what her family and her enjoy.
My homepage :: http://Srgame.co.kr/qna/21862