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| {{Group theory sidebar |Basics}}
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| In [[mathematics]], the '''wreath product''' of [[group theory]] is a specialized product of two groups, based on a [[semidirect product]]. Wreath products are an important tool in the classification of [[permutation group]]s and also provide a way of constructing interesting examples of groups.
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| Given two groups ''A'' and ''H'' there exist two variations of the wreath product: the '''unrestricted wreath product''' ''A'' Wr ''H'' (also written ''A''≀''H'') and the '''restricted wreath product''' ''A'' wr ''H''. Given a set Ω with an [[group action|''H''-action]] there exists a generalisation of the wreath product which is denoted by ''A'' Wr<sub>Ω</sub> ''H'' or ''A'' wr<sub>Ω</sub> ''H'' respectively.
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| == Definition ==
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| Let ''A'' and ''H'' be groups and Ω a set with ''H'' [[group action|acting]] on it. Let ''K'' be the [[Direct product of groups|direct product]]
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| : <math>K \equiv \prod_{\omega \,\in\, \Omega} A_\omega</math>
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| of copies of ''A''<sub>ω</sub> := ''A'' indexed by the set Ω. The elements of ''K'' can be seen as arbitrary sequences (''a''<sub>ω</sub>) of elements of ''A'' indexed by Ω with component wise multiplication. Then the action of ''H'' on Ω extends in a natural way to an action of ''H'' on the group ''K'' by
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| : <math> h (a_\omega) \equiv (a_{h^{-1}\omega})</math>.
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| Then the '''unrestricted wreath product''' ''A'' Wr<sub>Ω</sub> ''H'' of ''A'' by ''H'' is the [[semidirect product]] ''K'' ⋊ ''H''. The subgroup ''K'' of ''A'' Wr<sub>Ω</sub> ''H'' is called the '''base''' of the wreath product.
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| The '''restricted wreath product''' ''A'' wr<sub>Ω</sub> ''H'' is constructed in the same way as the unrestricted wreath product except that one uses the [[Direct sum of groups|direct sum]]
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| : <math>K \equiv \bigoplus_{\omega \,\in\, \Omega} A_\omega</math>
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| as the base of the wreath product. In this case the elements of ''K'' are sequences (''a''<sub>ω</sub>) of elements in ''A'' indexed by Ω of which all but finitely many ''a''<sub>ω</sub> are the [[identity element]] of ''A''.
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| The group ''H'' [[Group action|acts]] in a natural way on itself by left multiplication. Thus we can choose Ω := ''H''. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by ''A'' Wr ''H'' and ''A'' wr ''H'' respectively. We say in this case that the wreath product is '''regular'''.
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| == Notation and Conventions ==
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| The structure of the wreath product of ''A'' by ''H'' depends on the ''H''-set Ω and in case Ω is infinite it also depends on whether one uses the restricted or unrestricted wreath product. However, in literature the notation used may be deficient and one needs to pay attention on the circumstances. | |
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| * In literature ''A''≀<sub>Ω</sub>''H'' may stand for the unrestricted wreath product ''A'' Wr<sub>Ω</sub> ''H'' or the restricted wreath product ''A'' wr<sub>Ω</sub> ''H''.
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| * Similarly, ''A''≀''H'' may stand for the unrestricted regular wreath product ''A'' Wr ''H'' or the restricted regular wreath product ''A'' wr ''H''.
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| * In literature the ''H''-set Ω may be omitted from the notation even if Ω≠H.
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| * In the special case that ''H'' = ''S''<sub>''n''</sub> is the [[symmetric group]] of degree ''n'' it is common in the literature to assume that Ω={1,...,''n''} (with the natural action of ''S''<sub>''n''</sub>) and then omit Ω from the notation. That is, ''A''≀''S''<sub>n</sub> commonly denotes ''A''≀<sub>{1,...,''n''}</sub>''S''<sub>''n''</sub> instead of the regular wreath product ''A''≀<sub>''S''<sub>''n''</sub></sub>''S''<sub>''n''</sub>. In the first case the base group is the product of ''n'' copies of ''A'', in the latter it is the product of [[Factorial|''n''!]] copies of ''A''.
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| == Properties ==
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| * Since the finite direct product is the same as the finite direct sum of groups it follows that the unrestricted ''A'' Wr<sub>Ω</sub> ''H'' and the restricted wreath product ''A'' wr<sub>Ω</sub> ''H'' agree if the ''H''-set Ω is finite. In particular this is true when Ω = ''H'' is finite.
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| * ''A'' wr<sub>Ω</sub> ''H'' is always a [[subgroup]] of ''A'' Wr<sub>Ω</sub> ''H''.
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| * Universal Embedding Theorem: If ''G'' is an [[Group extension|extension]] of ''A'' by ''H'', then there exists a subgroup of the unrestricted wreath product ''A''≀''H'' which is isomorphic to ''G''.<ref>M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. Szeged 14, pp. 69-82 (1951)</ref>
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| * If ''A'', ''H'' and Ω are finite, then
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| :: |''A''≀<sub>Ω</sub>''H''| = |''A''|<sup>|Ω|</sup>|''H''|.<ref>Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)</ref>
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| == Canonical Actions of Wreath Products ==
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| If the group ''A'' acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which ''A'' Wr<sub>Ω</sub> ''H'' (and therefore also ''A'' wr<sub>Ω</sub> ''H'') can act.
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| * The '''imprimitive''' wreath product action on Λ×Ω.
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| : If ((''a''<sub>ω</sub>),''h'')∈''A'' Wr<sub>Ω</sub> ''H'' and (λ,ω')∈Λ×Ω, then
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| :: <math>((a_{\omega}), h) \cdot (\lambda,\omega') := (a_{h(\omega')}\lambda, h\omega')</math>.
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| * The '''primitive''' wreath product action on Λ<sup>Ω</sup>.
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| : An element in Λ<sup>Ω</sup> is a sequence (λ<sub>ω</sub>) indexed by the ''H''-set Ω. Given an element ((''a''<sub>ω</sub>), ''h'') ∈ ''A'' Wr<sub>Ω</sub> ''H'' its operation on (λ<sub>ω</sub>)∈Λ<sup>Ω</sup> is given by
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| :: <math>((a_\omega), h) \cdot (\lambda_\omega) := (a_{h^{-1}\omega}\lambda_{h^{-1}\omega})</math>.
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| == Examples ==
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| * The [[Lamplighter group]] is the restricted wreath product ℤ<sub>2</sub>≀ℤ.
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| * ℤ<sub>m</sub>≀''S''<sub>''n''</sub> ([[Generalized symmetric group]]).
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| : The base of this wreath product is the ''n''-fold direct product
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| :: ℤ<sub>''m''</sub><sup>''n''</sup> = ℤ<sub>''m''</sub> × ... × ℤ<sub>''m''</sub>
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| : of copies of ℤ<sub>''m''</sub> where the action φ : ''S''<sub>''n''</sub> → Aut(ℤ<sub>''m''</sub><sup>''n''</sup>) of the [[symmetric group]] ''S''<sub>''n''</sub> of degree ''n'' is given by
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| :: φ(σ)(α<sub>1</sub>,..., α<sub>''n''</sub>) := (α<sub>σ(1)</sub>,..., α<sub>σ(''n'')</sub>).<ref>J. W. Davies and A. O. Morris, "The Schur Multiplier of the Generalized Symmetric Group", J. London Math. Soc (2), 8, (1974), pp. 615-620</ref>
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| * ''S''<sub>2</sub>≀''S''<sub>''n''</sub> ([[Hyperoctahedral group]]).
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| : The action of ''S''<sub>''n''</sub> on {1,...,''n''} is as above. Since the symmetric group ''S''<sub>2</sub> of degree 2 is [[Group isomorphism|isomorphic]] to ℤ<sub>2</sub> the hyperoctahedral group is a special case of a generalized symmetric group.<ref>P. Graczyk, G. Letac and H. Massam, "The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution", J. Theoret. Probab. 18 (2005), no. 1, 1-42.</ref>
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| * Let ''p'' be a [[Prime number|prime]] and let ''n''≥1. Let ''P'' be a [[Sylow theorems|Sylow ''p''-subgroup]] of the symmetric group ''S''<sub>''p''<sup>''n''</sup></sub> of degree ''p''<sup>''n''</sup>. Then ''P'' is [[Group isomorphism|isomorphic]] to the iterated regular wreath product ''W''<sub>''n''</sub> = ℤ<sub>''p''</sub> ≀ ℤ<sub>''p''</sub>≀...≀ℤ<sub>''p''</sub> of ''n'' copies of ℤ<sub>''p''</sub>. Here ''W''<sub>1</sub> := ℤ<sub>''p''</sub> and ''W''<sub>''k''</sub> := ''W''<sub>''k''-1</sub>≀ℤ<sub>''p''</sub> for all ''k''≥2.<ref>Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)</ref><ref>L. Kaloujnine, "La structure des p-groupes de Sylow des groupes symétriques finis", Annales Scientifiques de l'École Normale Supérieure. Troisième Série 65, pp. 239–276 (1948)</ref>
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| * The [[Rubik's Cube group]] is a subgroup of small index in the product of wreath products, (ℤ<sub>3</sub>≀''S''<sub>8</sub>) × (ℤ<sub>2</sub>≀''S''<sub>12</sub>), the factors corresponding to the symmetries of the 8 corners and 12 edges.
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| == References ==
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| {{Reflist}}
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| == External links ==
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| * [http://planetmath.org/encyclopedia/WreathProduct.html PlanetMath page]
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| * [http://www.encyclopediaofmath.org/index.php/Wreath_product Springer Online Reference Works]
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| * [http://www.abstractmath.org/Papers/SAWPCWC.pdf Some Applications of the Wreath Product Construction]
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| [[Category:Group theory]]
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| [[Category:Permutation groups]]
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| [[Category:Binary operations]]
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The writer is known by the title of Numbers Lint. My working day job is a meter reader. The favorite pastime for my kids and me is to play baseball and I'm attempting to make it a occupation. South Dakota is exactly where I've always been residing.
Review my site over the counter std test (click through the next website page)