Editing Cayley's theorem

{{short description|Representation of groups by permutations}}
{{For|the number of labeled trees in graph theory|Cayley's formula}}

In the mathematical discipline of [[group theory]], '''Cayley's theorem''', named in honour of [[Arthur Cayley]], states that every [[group (mathematics)|group]] {{mvar|G}} is [[group isomorphism|isomorphic]] to a [[subgroup]] of a [[symmetric group]].<ref>{{harvtxt|Jacobson|2009|p=38}}</ref> 
More specifically, {{mvar|G}} is isomorphic to a subgroup of the symmetric group <math>\operatorname{Sym}(G)</math> whose elements are the [[permutation]]s of the underlying set of {{mvar|G}}.
Explicitly,
* for each <math>g \in G</math>, the left-multiplication-by-{{mvar|g}} map <math>\ell_g \colon G \to G</math> sending each element {{mvar|x}} to {{math|''gx''}} is a [[permutation]] of {{mvar|G}}, and
* the map <math>G \to \operatorname{Sym}(G)</math> sending each element {{mvar|g}} to <math>\ell_g</math> is an [[injective]] [[homomorphism]], so it defines an isomorphism from {{mvar|G}} onto a subgroup of <math>\operatorname{Sym}(G)</math>.
The homomorphism <math>G \to \operatorname{Sym}(G)</math> can also be understood as arising from the left translation [[Group action (mathematics)|action]] of {{mvar|G}} on the underlying set {{mvar|G}}.<ref>{{harvtxt|Jacobson|2009|p=72, ex. 1}}</ref>

When {{mvar|G}} is finite, <math>\operatorname{Sym}(G)</math> is finite too.  The proof of Cayley's theorem in this case shows that if {{mvar|G}} is a finite group of order {{mvar|n}}, then {{mvar|G}} is isomorphic to a subgroup of the standard symmetric group <math>S_n</math>.  But {{mvar|G}} might also be isomorphic to a subgroup of a smaller symmetric group, <math>S_m</math> for some <math>m<n</math>; for instance, the order 6 group <math>G=S_3</math> is not only isomorphic to a subgroup of <math>S_6</math>, but also (trivially) isomorphic to a subgroup of <math>S_3</math>.<ref name="Cameron2008">{{cite book|author=Peter J. Cameron|title=Introduction to Algebra, Second Edition|url=https://archive.org/details/introductiontoal00came_088|url-access=limited|year=2008|publisher=Oxford University Press|isbn=978-0-19-852793-0|page=[https://archive.org/details/introductiontoal00came_088/page/n144 134]}}</ref>  The problem of finding the minimal-order symmetric group into which a given group {{mvar|G}} embeds is rather difficult.<ref>{{Cite journal | doi = 10.2307/2373739| jstor = 2373739| title = Minimal Permutation Representations of Finite Groups| journal = American Journal of Mathematics| volume = 93| issue = 4| pages = 857–866| year = 1971| last1 = Johnson | first1 = D. L.}}</ref><ref>{{Cite journal | doi = 10.1023/A:1023860730624| year = 2003| last1 = Grechkoseeva | first1 = M. A.| journal = Siberian Mathematical Journal|title=On Minimal Permutation Representations of Classical Simple Groups| volume = 44| issue = 3| pages = 443–462| s2cid = 126892470}}</ref>

[[Jonathan Lazare Alperin|Alperin]] and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups".<ref name="AlperinBell1995">{{cite book|author1=J. L. Alperin|author2=Rowen B. Bell|title=Groups and representations|url=https://archive.org/details/groupsrepresenta00alpe_213|url-access=limited|year=1995|publisher=Springer|isbn=978-0-387-94525-5|page=[https://archive.org/details/groupsrepresenta00alpe_213/page/n39 29]}}</ref>

When {{mvar|G}} is infinite, <math>\operatorname{Sym}(G)</math> is infinite, but Cayley's theorem still applies.

== History ==
When Cayley (1854) introduced what are now called ''groups'', the modern definitions did not exist, and it was not immediately clear that this was equivalent to what were then called groups, which are now called ''permutation groups''. Cayley's theorem unifies the two.

Although Burnside<ref>{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | page = 22  | location = Cambridge | year = 1911 | edition = 2 | url = https://babel.hathitrust.org/cgi/pt?id=uc1.b4062919;view=1up;seq=52;size=125 | isbn = 0-486-49575-2}}</ref> 
attributes the theorem 
to [[Camille Jordan|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's 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 = Philosophical Magazine | volume = 7 | issue = 42 | pages = 40–47 | year = 1854 | url = https://books.google.com/books?id=_LYConosISUC&pg=PA40 }}</ref> 
showed that the correspondence in the theorem is one-to-one, but he did not explicitly show it was a homomorphism (and thus an embedding).  However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.

The theorem was later published by [[Walther Dyck]] in 1882<ref>{{Citation | last=von Dyck | year=1882 | first=Walther | author-link=Walther Dyck | title=Gruppentheoretische Studien |trans-title=Group-theoretical Studies | url=https://archive.org/stream/mathematischean54behngoog#page/n38/mode/1up | doi=10.1007/BF01443322 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=20 | issue=1 | page=30| hdl=2027/njp.32101075301422 | s2cid=179178038 | hdl-access=free }}. {{in lang|de}}</ref> and is attributed to Dyck in the first edition of Burnside's book.<ref>{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | page = 22  | location = Cambridge | year = 1897 | edition = 1 | url = https://archive.org/stream/cu31924086163726#page/n43/mode/2up }}</ref>

== Background ==

A ''permutation'' of a set {{mvar|A}} is a [[bijective]] [[function (mathematics)|function]] from {{mvar|A}} to {{mvar|A}}. The set of all permutations of {{mvar|A}} forms a group under [[function composition]], called ''the symmetric group on'' {{mvar|A}}, and written as <math>\operatorname{Sym}(A)</math>.<ref>{{harvtxt|Jacobson|2009|p=31}}</ref>
In particular, taking {{mvar|A}} to be the underlying set of a group {{mvar|G}} produces a symmetric group denoted <math>\operatorname{Sym}(G)</math>.

== Proof of the theorem ==
If ''g'' is any element of a group ''G'' with operation ∗, consider the function {{nowrap|''f''<sub>''g''</sub> : ''G'' → ''G''}}, defined by {{nowrap|1=''f''<sub>''g''</sub>(''x'') = ''g'' ∗ ''x''}}. By the existence of inverses, this function has also an 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'').

The set {{nowrap|1=''K'' = {''f''<sub>''g''</sub> : ''g'' ∈ ''G''} }} is a subgroup of Sym(''G'') that is isomorphic to ''G''. The fastest way to establish this is to consider the function {{nowrap|''T'' : ''G'' → Sym(''G'')}} with {{nowrap|1=''T''(''g'') = ''f''<sub>''g''</sub>}} for every ''g'' in ''G''. ''T'' is a [[group homomorphism]] because (using · to denote composition in Sym(''G'')):

:<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) ,</math>
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 [[injective]] since {{nowrap|1=''T''(''g'') = id<sub>''G''</sub>}} (the identity element of Sym(''G'')) implies that {{nowrap|1=''g'' ∗ ''x'' = ''x''}} for all ''x'' in ''G'', and taking ''x'' to be the identity element ''e'' of ''G'' yields {{nowrap|1=''g'' = ''g'' ∗ ''e'' = ''e''}}, i.e. the kernel is trivial. Alternatively, ''T'' is also [[injective]] since {{nowrap|1=''g'' ∗ ''x'' = ''g''′ ∗ ''x''}} implies that {{nowrap|1=''g'' = ''g''′}} (because every group is [[cancellative]]).

Thus ''G'' is isomorphic to the image of ''T'', which is the subgroup ''K''.

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

=== Alternative setting of proof ===
An alternative setting uses the language of [[Group action (mathematics)|group action]]s. We consider the group <math>G</math> as acting on itself by left multiplication, i.e. <math>g \cdot x = gx</math>, which has a permutation representation, say <math>\phi : G \to \mathrm{Sym}(G)</math>.

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 = ge = g\cdot e = e</math>. Thus, <math>\ker\phi</math> is trivial. The result follows by use of the [[first isomorphism theorem]], from which we get <math>\mathrm{Im}\, \phi \cong G</math>.

==Remarks on the regular group representation==
The identity element of the group corresponds to the identity permutation. All other group elements correspond to [[derangements]]: permutations that do 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 that consists of cycles all of the same length: this length is the order of that element. The elements in each cycle form a right [[coset]] of the subgroup generated by the element.

==Examples of the regular group representation==
<math> \mathbb Z_2 = \{0,1\} </math> with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12) (see [[Permutation#Cycle_notation|cycle notation]]). E.g. 0 +1 = 1 and 1+1 = 0, so <math display=inline>1\mapsto0</math> and <math display=inline>0\mapsto1,</math> as they would under a permutation.

<math> \mathbb Z_3 = \{0,1,2\} </math> 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&nbsp;+&nbsp;1&nbsp;=&nbsp;2 corresponds to (123)(123)&nbsp;=&nbsp;(132).

<math> \mathbb Z_4 = \{0,1,2,3\} </math> 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, and the latter is how it is realized by its regular representation.

<!-- 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)
|}

== More general statement==

'''Theorem:''' 
Let {{mvar|G}} be a group, and let {{mvar|H}} be a subgroup.
Let <math>G/H</math> be the set of left cosets of {{mvar|H}} in {{mvar|G}}.
Let {{mvar|N}} be the [[Core (group)|normal core]] of {{mvar|H}} in {{mvar|G}}, defined to be the intersection of the conjugates of {{mvar|H}} in {{mvar|G}}. 
Then the quotient group <math>G/N</math> is isomorphic to a subgroup of <math>\operatorname{Sym}(G/H)</math>.

The special case <math>H=1</math> is Cayley's original theorem.

== See also ==
* [[Wagner–Preston theorem]] is the analogue for inverse semigroups.
* [[Birkhoff's representation theorem]], a similar result in order theory
* [[Frucht's theorem]], every finite group is the automorphism group of a graph
* [[Yoneda lemma]], a generalization 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| publisher=Dover| isbn = 978-0-486-47189-1}}.

[[Category:Permutations]]
[[Category:Theorems about finite groups]]
[[Category:Articles containing proofs]]