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{{Group theory sidebar |Basics}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


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''&nbsp;Wr&nbsp;''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''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' or ''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H'' respectively.
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== Definition ==
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


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]]
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


: <math>K \equiv \prod_{\omega \,\in\, \Omega} A_\omega</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


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
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


: <math> h (a_\omega) \equiv (a_{h^{-1}\omega})</math>.
==Demos==


Then the '''unrestricted wreath product''' ''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' of ''A'' by ''H'' is the [[semidirect product]] ''K''&nbsp;⋊&nbsp;''H''. The subgroup ''K'' of ''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' is called the '''base''' of the wreath product.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


The '''restricted wreath product''' ''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H'' is constructed in the same way as the unrestricted wreath product except that one uses the [[Direct sum of groups|direct sum]]


: <math>K \equiv \bigoplus_{\omega \,\in\, \Omega} A_\omega</math>
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


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''.
==Test pages ==


The group ''H'' [[Group action|acts]] in a natural way on itself by left multiplication. Thus we can choose Ω&nbsp;:=&nbsp;''H''. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by ''A''&nbsp;Wr&nbsp;''H'' and ''A''&nbsp;wr&nbsp;''H'' respectively. We say in this case that the wreath product is '''regular'''.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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*[[Help:Formula]]


== Notation and Conventions ==
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
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.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
* In literature ''A''≀<sub>Ω</sub>''H'' may stand for the unrestricted wreath product ''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' or the restricted wreath product ''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H''.
 
* Similarly, ''A''≀''H'' may stand for the unrestricted regular wreath product ''A''&nbsp;Wr&nbsp;''H'' or the restricted regular wreath product ''A''&nbsp;wr&nbsp;''H''.
 
* In literature the ''H''-set Ω may be omitted from the notation even if Ω≠H.
 
* In the special case that ''H''&nbsp;=&nbsp;''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''.
 
== Properties ==
 
* Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted ''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' and the restricted wreath product ''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H'' agree if the ''H''-set Ω is finite. In particular this is true when Ω = ''H'' is finite.
 
* ''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H'' is always a [[subgroup]] of ''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H''.
 
* 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>
 
* If ''A'', ''H'' and Ω are finite, then
 
:: |''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>
 
== Canonical Actions of Wreath Products ==
 
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''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' (and therefore also ''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H'') can act.
 
* The '''imprimitive''' wreath product action on Λ×Ω.
 
: If ((''a''<sub>ω</sub>),''h'')∈''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' and (λ,ω')∈Λ×Ω, then
 
:: <math>((a_{\omega}), h) \cdot (\lambda,\omega') := (a_{h(\omega')}\lambda, h\omega')</math>.
 
* The '''primitive''' wreath product action on Λ<sup>Ω</sup>.
 
: 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''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' its operation on (λ<sub>ω</sub>)∈Λ<sup>Ω</sup> is given by
 
:: <math>((a_\omega), h) \cdot (\lambda_\omega) :=  (a_{h^{-1}\omega}\lambda_{h^{-1}\omega})</math>.
 
== Examples ==
 
* The [[Lamplighter group]] is the restricted wreath product ℤ<sub>2</sub>≀ℤ.
 
* ℤ<sub>m</sub>≀''S''<sub>''n''</sub> ([[Generalized symmetric group]]).
 
: The base of this wreath product is the ''n''-fold direct product
 
:: ℤ<sub>''m''</sub><sup>''n''</sup> = ℤ<sub>''m''</sub> × ... × ℤ<sub>''m''</sub>
 
: of copies of ℤ<sub>''m''</sub> where the action φ&nbsp;:&nbsp;''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
 
:: 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>
 
* ''S''<sub>2</sub>≀''S''<sub>''n''</sub> ([[Hyperoctahedral group]]).
 
: 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>
 
* 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>
 
* 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>)&nbsp;× (ℤ<sub>2</sub>≀''S''<sub>12</sub>), the factors corresponding to the symmetries of the 8 corners and 12 edges.
 
== References ==
 
{{Reflist}}
 
== External links ==
* [http://planetmath.org/encyclopedia/WreathProduct.html PlanetMath page]
* [http://www.encyclopediaofmath.org/index.php/Wreath_product Springer Online Reference Works]
* [http://www.abstractmath.org/Papers/SAWPCWC.pdf Some Applications of the Wreath Product Construction]
 
 
[[Category:Group theory]]
[[Category:Permutation groups]]
[[Category:Binary operations]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

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Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

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