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{{For|the number of labeled trees in graph theory|Cayley's formula}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In [[group theory]], '''Cayley's theorem''', named in honor of [[Arthur Cayley]], states that every [[group (mathematics)|group]] ''G'' is [[group isomorphism|isomorphic]] to a [[subgroup]] of the [[symmetric group]] acting on ''G''.<ref>Jacobson (2009), p. 38.</ref> This can be understood as an example of the [[group action]] of ''G'' on the elements of ''G''.<ref>Jacobson (2009), p. 72, ex. 1.</ref>
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A [[permutation]] of a set ''G'' is any [[bijective]] [[function (mathematics)|function]] taking ''G'' onto ''G''; and the set of all such functions forms a group under [[function composition]], called ''the symmetric group on'' ''G'', and written as Sym(''G'').<ref>Jacobson (2009), p. 31.</ref>
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Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as ('''''R''''',+)) as a [[permutation group]] of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


== History ==
<!--'''PNG''' (currently default in production)
Although Burnside<ref>{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | location = Cambridge | year = 1911 | edition = 2 | isbn = 0-486-49575-2}}</ref>
:<math forcemathmode="png">E=mc^2</math>
attributes the theorem
to Jordan,<ref>{{Citation | last = Jordan | first = Camille | author-link = Camille Jordan | title = Traite des substitutions et des equations algebriques | publisher = Gauther-Villars | location = Paris | year = 1870}}</ref>
Eric Nummela<ref>{{Citation | last = Nummela | first = Eric | title = Cayley's Theorem for Topological Groups | journal = American Mathematical Monthly | volume = 87 | issue = 3 | year = 1980 | pages = 202–203 | doi = 10.2307/2321608 | jstor = 2321608 | publisher = Mathematical Association of America}}</ref>
nonetheless argues that the standard name&mdash;"Cayley's Theorem"&mdash;is in fact appropriate. Cayley, in his original 1854 paper,<ref>{{Citation  | last = Cayley | first = Arthur | author-link = Arthur Cayley | title = On the theory of groups as depending on the symbolic equation θ<sup>n</sup>=1  | journal = Phil. Mag. | volume = 7 | issue = 4 | pages = 40–47 | year = 1854}}</ref>  
showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an isomorphism).  However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.


== Proof of the theorem ==
'''source'''
Where ''g'' is any element of ''G'', consider the function ''f''<sub>''g''</sub> : ''G'' → ''G'', defined by ''f''<sub>''g''</sub>(''x'') = ''g''*''x''. By the existence of inverses, this function has a two-sided inverse, <math>f_{g^{-1}}</math>. So multiplication by ''g'' acts as a [[bijective]] function. Thus, ''f''<sub>''g''</sub> is a permutation of ''G'', and so is a member of Sym(''G'').
:<math forcemathmode="source">E=mc^2</math> -->


The set <math>K = \{f_g: g \in G\}</math> is a subgroup of Sym(''G'') which is isomorphic to ''G''. The fastest way to establish this is to consider the function ''T'' : ''G'' → Sym(''G'') with ''T''(''g'') = ''f''<sub>''g''</sub> for every ''g'' in ''G''. ''T'' is a [[group homomorphism]] because (using "•" for composition in Sym(''G'')):
<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> (f_g \cdot f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_{(g*h)}(x)\mbox{,}</math>
==Demos==
for all ''x'' in ''G'', and hence:
:<math> T(g) \cdot T(h) = f_g \cdot f_h = f_{(g*h)} = T(g*h).</math>
The homomorphism ''T'' is also [[injective]] since ''T''(''g'') = id<sub>''G''</sub> (the identity element of Sym(''G'')) implies that ''g*x'' = ''x'' for all ''x'' in ''G'', and taking ''x'' to be the identity element ''e'' of ''G'' yields ''g'' = ''g''*''e'' = ''e''. Alternatively, ''T'' is also [[injective]] since, if ''g''*''x''=''g'''*''x'' implies ''g''=''g' '' (by post-multiplying with the inverse of ''x'', which exists because ''G'' is a group).


Thus ''G'' is isomorphic to the image of ''T'', which is the subgroup ''K''.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


''T'' is sometimes called the ''regular representation of'' ''G''.


=== Alternative setting of proof ===
* accessibility:
An alternative setting uses the language of [[group action]]s. We consider the group <math>G</math> as a G-set, which can be shown to have permutation representation, say <math>\phi</math>.  
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** 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.


Firstly, suppose <math>G=G/H</math> with <math>H=\{e\}</math>. Then the group action is <math>g.e</math> by [[group action|classification of G-orbits]] (also known as the orbit-stabilizer theorem).
==Test pages ==


Now, the representation is faithful if <math>\phi</math> is injective, that is, if the kernel of <math>\phi</math> is trivial. Suppose <math>g\in\ker\phi</math> Then, <math>g=g.e=\phi(g).e</math> by the equivalence of the permutation representation and the group action. But since <math>g \in \ker\phi</math>, <math>\phi(g)=e</math> and thus <math>\ker\phi</math> is trivial. Then <math>\mathrm{Im} \phi < G</math> and thus the result follows by use of the [[first isomorphism theorem]].
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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==Remarks on the regular group representation==
*[[Inputtypes|Inputtypes (private Wikis only)]]
The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left [[coset]] of the subgroup generated by the element.
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
==Examples of the regular group representation==
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 .
Z<sub>2</sub> = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12).
 
Z<sub>3</sub> = {0,1,2} with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E.g. 1 + 1 = 2 corresponds to (123)(123)=(132).
 
Z<sub>4</sub> = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).
 
The elements of [[Klein four-group]] {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).
 
S<sub>3</sub> ([[dihedral group of order 6]]) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements:
 
<!-- Looks ugly if it's left-aligned/non-square cells, so a bit of customization is good here -->
{| class="wikitable" style="text-align: center;"
! style="width: 1.5em; height: 1.5em;" | *
! style="width: 1.5em;" | ''e''
! style="width: 1.5em;" | ''a''
! style="width: 1.5em;" | ''b''
! style="width: 1.5em;" | ''c''
! style="width: 1.5em;" | ''d''
! style="width: 1.5em;" | ''f''
! permutation
|-
! style="height: 1.5em;" | ''e''
| ''e'' || ''a'' || ''b'' || ''c'' || ''d'' || ''f'' || ''e''
|-
! style="height: 1.5em;" | ''a''
| ''a'' || ''e'' || ''d'' || ''f'' || ''b'' || ''c'' || (12)(35)(46)
|-
! style="height: 1.5em;" | ''b''
| ''b'' || ''f'' || ''e'' || ''d'' || ''c'' || ''a'' || (13)(26)(45)
|-
! style="height: 1.5em;" | ''c''
| ''c'' || ''d'' || ''f'' || ''e'' || ''a'' || ''b'' || (14)(25)(36)
|-
! style="height: 1.5em;" | ''d''
| ''d'' || ''c'' || ''a'' || ''b'' || ''f'' || ''e'' || (156)(243)
|-
! style="height: 1.5em;" | ''f''
| ''f'' || ''b'' || ''c'' || ''a'' || ''e'' || ''d'' || (165)(234)
|}
 
== See also ==
* [[Containment order]], a similar result in order theory
* [[Frucht's theorem]], every group is the automorphism group of a graph
* [[Yoneda lemma]], an analogue of Cayley's theorem in category theory
* [[representation theorem]]
 
== Notes ==
{{Reflist}}
 
== References ==
* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| series= | publisher=Dover| isbn = 978-0-486-47189-1}}.
 
[[Category:Permutations]]
[[Category:Theorems in group theory]]
[[Category:Articles containing proofs]]
 
[[de:Satz von Cayley]]
[[es:Teorema de Cayley]]
[[fr:Théorème de Cayley]]
[[it:Teorema di Cayley]]
[[he:משפט קיילי]]
[[hu:Cayley-tétel]]
[[nl:Stelling van Cayley]]
[[pl:Twierdzenie Cayleya]]
[[ru:Теорема Кэли (теория групп)]]
[[fi:Cayleyn lause]]
[[sv:Cayleys sats]]
[[uk:Теорема Келі (теорія груп)]]
[[zh:凱萊定理]]

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]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

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

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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