View source for Symmetric group

{{Short description|Type of group in abstract algebra}}
[[File:Symmetric group 4; Cayley graph 4,9.svg|thumb|320px|A [[Cayley graph|Cayley graph]] of the symmetric group S<sub>4</sub> using the generators (red) a right [[Circular shift|circular shift]] of all four set elements, and (blue) a left circular shift of the first three set elements.]]
[[File:Symmetric group 3; Cayley table; matrices.svg|thumb|320px|[[Cayley table|Cayley table]], with header omitted, of the symmetric group S<sub>3</sub>. The elements are represented as matrices. To the left of the matrices, are their two-line form. The black arrows indicate disjoint cycles and correspond to cycle notation. Green circle is an odd permutation, white is an even permutation and black is the identity. <br><br>These are the positions of the six matrices<br>310px<br>Some matrices are not arranged symmetrically to the main diagonal – thus the symmetric group is not abelian.]]
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In [[Abstract algebra|abstract algebra]], the '''symmetric group''' defined over any [[Set (mathematics)|set]] is the [[Group (mathematics)|group]] whose [[Element (mathematics)|elements]] are all the [[Bijection|bijection]]s from the set to itself, and whose group operation is the [[Function composition|composition of functions]].  In particular, the finite symmetric group <math>\mathrm{S}_n</math> defined over a [[Finite set|finite set]] of <math>n</math> symbols consists of the [[Permutation|permutation]]s that can be performed on the <math>n</math> symbols.<ref name=Jacobson-def>{{harvnb|Jacobson|2009|p=31}}</ref> Since there are <math>n!</math> (<math>n</math> [[Factorial|factorial]]) such permutation operations, the [[Order (group theory)|order]] (number of elements) of the symmetric group <math>\mathrm{S}_n</math> is <math>n!</math>.

Although symmetric groups can be defined on [[Infinite set|infinite set]]s, this article focuses on the finite symmetric groups: their applications, their elements, their [[Conjugacy class|conjugacy class]]es, a finite presentation, their [[Subgroup|subgroup]]s, their automorphism groups, and their [[Group representation|representation]] theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.

The symmetric group is important to diverse areas of mathematics such as [[Galois theory]], [[Invariant theory|invariant theory]], the representation theory of Lie groups, and [[Combinatorics|combinatorics]]. [[Cayley's theorem]] states that every group <math>G</math> is [[Group isomorphism|isomorphic]] to a [[Subgroup|subgroup]] of the symmetric group on (the underlying set of) <math>G</math>.

== Definition and first properties ==
The symmetric group on a finite set <math>X</math> is the group whose elements are all bijective functions from <math>X</math> to <math>X</math> and whose group operation is that of [[Function composition|function composition]].<ref name=Jacobson-def /> For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of '''degree''' <math>n</math> is the symmetric group on the set <math>X = \{1, 2, \ldots, n\}</math>.

The symmetric group on a set <math>X</math> is denoted in various ways, including <math>\mathrm{S}_X</math>, <math>\mathfrak{S}_X</math>, <math>\Sigma_X</math>, <math>X!</math>, and <math>\operatorname{Sym}(X)</math>.<ref name=Jacobson-def />  If <math>X</math> is the set <math>\{1, 2, \ldots, n\}</math> then the name may be abbreviated to <math>\mathrm{S}_n</math>, <math>\mathfrak{S}_n</math>, <math>\Sigma_n</math>, or <math>\operatorname{Sym}(n)</math>.<ref name=Jacobson-def />

Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in {{harv|Scott|1987|loc=Ch. 11}}, {{harv|Dixon|Mortimer|1996|loc=Ch. 8}}, and {{harv|Cameron|1999}}.

The symmetric group on a set of <math>n</math> elements has [[Order (group theory)|order]] <math>n!</math> (the [[Factorial|factorial]] of <math>n</math>).<ref>{{harvnb|Jacobson|2009|p=32 Theorem 1.1}}</ref> It is [[Abelian group|abelian]] if and only if <math>n</math> is less than or equal to 2.<ref>{{cite web|title=Symmetric Group is not Abelian/Proof 1|url=https://proofwiki.org/wiki/Symmetric_Group_is_not_Abelian/Proof_1}}</ref> For <math>n=0</math> and <math>n=1</math> (the [[Empty set|empty set]] and the singleton set), the symmetric groups are [[Trivial group|trivial]] (they have order <math>0! = 1! = 1</math>). The group S<sub>''n''</sub> is [[Solvable group|solvable]] if and only if <math>n \leq 4</math>. This is an essential part of the proof of the [[Abel–Ruffini theorem]] that shows that for every <math>n > 4</math> there are [[Polynomial|polynomial]]s of degree <math>n</math> which are not solvable by radicals, that is, the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients.

== Applications ==
The symmetric group on a set of size ''n'' is the [[Galois group]] of the general [[Polynomial|polynomial]] of degree ''n'' and plays an important role in [[Galois theory]].  In [[Invariant theory|invariant theory]], the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called [[Symmetric function|symmetric function]]s.  In the representation theory of Lie groups, the [[Representation theory of the symmetric group|representation theory of the symmetric group]] plays a fundamental role through the ideas of [[Schur functor]]s.  

In the theory of [[Coxeter group]]s, the symmetric group is the Coxeter group of type A<sub>''n''</sub> and occurs as the [[Weyl group]] of the [[General linear group|general linear group]].  In [[Combinatorics|combinatorics]], the symmetric groups, their elements ([[Permutation|permutation]]s), and their [[Group representation|representations]] provide a rich source of problems involving Young tableaux, [[Plactic monoid|plactic monoid]]s, and the [[Bruhat order]].  [[Subgroup]]s of symmetric groups are called [[Permutation group|permutation group]]s and are widely studied because of their importance in understanding [[Group action (mathematics)|group action]]s, [[Homogeneous space|homogeneous space]]s, and automorphism groups of [[Graph (discrete mathematics)|graph]]s, such as the [[Higman–Sims group]] and the [[Higman–Sims graph]].

== Group properties and special elements ==
The elements of the symmetric group on a set ''X'' are the [[Permutation|permutation]]s of ''X''.

=== Multiplication ===
The group operation in a symmetric group is function composition, denoted by the symbol ∘ or simply by just a composition of the permutations. The composition {{nowrap|''f'' ∘ ''g''}} of permutations ''f'' and ''g'', pronounced "''f'' of ''g''", maps any element ''x'' of ''X'' to ''f''(''g''(''x'')). Concretely, let (see [[Permutation|permutation]] for an explanation of notation):

: <math> f = (1\ 3)(4\ 5)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{pmatrix} </math>
: <math> g = (1\ 2\ 5)(3\ 4)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}.</math>

Applying ''f'' after ''g'' maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing ''f'' and ''g'' gives
: <math> fg = f\circ g = (1\ 2\ 4)(3\ 5)=\begin{pmatrix} 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end{pmatrix}.</math>

A [[Cyclic permutation|cycle]] of length {{nowrap|1=''L'' = ''k'' · ''m''}}, taken to the ''k''th power, will decompose into ''k'' cycles of length ''m'': For example, ({{nowrap|1=''k'' = 2}}, {{nowrap|1=''m'' = 3}}),
: <math> (1~2~3~4~5~6)^2 = (1~3~5) (2~4~6).</math>

=== Verification of group axioms ===
To check that the symmetric group on a set ''X'' is indeed a [[Group (mathematics)|group]], it is necessary to verify the group axioms of closure, associativity, identity, and inverses.<ref>{{cite book |last1=Vasishtha |first1=A.R. |last2=Vasishtha |first2=A.K. |chapter=2. Groups S3 Group Definition |chapter-url={{GBurl|45eCTUS6YnQC|p=49}} |title=Modern Algebra |publisher=Krishna Prakashan Media |date=2008 |isbn=9788182830561 |pages=49 }}</ref>
# The operation of function composition is closed in the set of permutations of the given set ''X''.
# Function composition is always associative.
# The trivial bijection that assigns each element of ''X'' to itself serves as an identity for the group.
# Every bijection has an [[Inverse function|inverse function]] that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too.

=== Transpositions, sign, and the alternating group ===
{{main article|Transposition (mathematics)}}
A '''transposition''' is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation ''g'' from above can be written as ''g'' = (1 2)(2 5)(3 4). Since ''g'' can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas ''f'' is an even permutation.

The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation.

The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd.  Thus we can define the '''sign''' of a permutation:

:<math>\operatorname{sgn}f = \begin{cases} +1, & \text{if }f\mbox { is even} \\ -1, & \text{if }f \text{ is odd}. \end{cases}</math>

With this definition,
:<math>\operatorname{sgn}\colon \mathrm{S}_n \rightarrow \{+1, -1\}\ </math>
is a [[Group homomorphism|group homomorphism]] ({+1, −1} is a group under multiplication, where +1 is e, the neutral element). The [[Kernel (algebra)|kernel]] of this homomorphism, that is, the set of all even permutations, is called the '''[[Alternating group|alternating group]]''' A<sub>''n''</sub>. It is a [[Normal subgroup|normal subgroup]] of S<sub>''n''</sub>, and for {{nowrap|''n'' ≥ 2}} it has {{nowrap|''n''!/2}} elements. The group S<sub>''n''</sub> is the [[Semidirect product|semidirect product]] of A<sub>''n''</sub> and any subgroup generated by a single transposition.

Furthermore, every permutation can be written as a product of ''adjacent transpositions'', that is, transpositions of the form {{nowrap|(''a'' ''a''+1)}}.  For instance, the permutation ''g'' from above can also be written as {{nowrap|1=''g'' = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5)}}. The sorting algorithm [[Bubble sort|bubble sort]] is an application of this fact.  The representation of a permutation as a product of adjacent transpositions is also not unique.

=== Cycles ===
A [[Cyclic permutation|cycle]] of ''length'' ''k'' is a permutation ''f'' for which there exists an element ''x'' in {1, ..., ''n''} such that ''x'', ''f''(''x''), ''f''<sup>2</sup>(''x''), ..., ''f''<sup>''k''</sup>(''x'') = ''x'' are the only elements moved by ''f''; it conventionally is required that {{nowrap|''k'' ≥ 2}} since with {{nowrap|1=''k'' = 1}} the element ''x'' itself would not be moved either. The permutation ''h'' defined by
: <math>h = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 1 & 3 & 5\end{pmatrix}</math>
is a cycle of length three, since {{nowrap|1=''h''(1) = 4}}, {{nowrap|1=''h''(4) = 3}} and {{nowrap|1=''h''(3) = 1}}, leaving 2 and 5 untouched. We denote such a cycle by {{nowrap|(1 4 3)}}, but it could equally well be written {{nowrap|(4 3 1)}} or {{nowrap|(3 1 4)}} by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are ''disjoint'' if they have disjoint subsets of elements. Disjoint cycles [[Commutative property|commute]]: for example, in S<sub>6</sub> there is the equality {{nowrap|1=(4 1 3)(2 5 6) = (2 5 6)(4 1 3)}}. Every element of S<sub>''n''</sub> can be written as a product of disjoint cycles; this representation is unique [[Up to|up to]] the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.

Cycles admit the following conjugation property with any permutation <math>\sigma</math>, this property is often used to obtain its [[Symmetric group#Generators and relations|generators and relations]].

:<math>\sigma\begin{pmatrix} a & b & c & \ldots \end{pmatrix}\sigma^{-1}=\begin{pmatrix}\sigma(a) & \sigma(b) & \sigma(c) & \ldots\end{pmatrix}</math>

=== Special elements ===
Certain elements of the symmetric group of {1, 2, ..., ''n''} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set).

The '''{{visible anchor|order reversing permutation}}''' is the one given by:
:<math>\begin{pmatrix} 1 & 2 & \cdots & n\\
n & n-1 & \cdots & 1\end{pmatrix}.</math>
This is the unique maximal element with respect to the [[Bruhat order]] and the
[[Longest element of a Coxeter group|longest element]] in the symmetric group with respect to generating set consisting of the adjacent transpositions {{nowrap|(''i'' ''i''+1)}}, {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}.

This is an involution, and consists of <math>\lfloor n/2 \rfloor</math> (non-adjacent) transpositions
:<math>(1\,n)(2\,n-1)\cdots,\text{ or }\sum_{k=1}^{n-1} k = \frac{n(n-1)}{2}\text{ adjacent transpositions: }</math>
:: <math>(n\,n-1)(n-1\,n-2)\cdots(2\,1)(n-1\,n-2)(n-2\,n-3)\cdots,</math>

so it thus has sign:

:<math>\mathrm{sgn}(\rho_n) = (-1)^{\lfloor n/2 \rfloor} =(-1)^{n(n-1)/2} = \begin{cases}
+1 & n \equiv 0,1 \pmod{4}\\
-1 & n \equiv 2,3 \pmod{4}
\end{cases}</math>
which is 4-periodic in ''n''.

In S<sub>2''n''</sub>, the ''[[Faro shuffle|perfect shuffle]]'' is the permutation that splits the set into 2 piles and interleaves them. Its sign is also <math>(-1)^{\lfloor n/2 \rfloor}.</math>

Note that the reverse on ''n'' elements and perfect shuffle on 2''n'' elements have the same sign; these are important to the classification of [[Clifford algebra]]s, which are 8-periodic.

== Conjugacy classes ==

The [[Conjugacy class|conjugacy class]]es of S<sub>''n''</sub> correspond to the [[Permutation#Cycle type|cycle types]] of permutations; that is, two elements of S<sub>''n''</sub> are conjugate in S<sub>''n''</sub> if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S<sub>5</sub>, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S<sub>''n''</sub> can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another.  Continuing the previous example,
<math display="block">k = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5\end{pmatrix},</math>
which can be written as the product of cycles as (2 4).
This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is,
<math display="block">(2~4)\circ(1~2~3)(4~5)\circ(2~4)=(1~4~3)(2~5).</math>
It is clear that such a permutation is not unique.

Conjugacy classes of S<sub>''n''</sub> correspond to integer partitions of ''n'': to the partition {{nowrap|1=''μ'' = (''μ''<sub>1</sub>, ''μ''<sub>2</sub>, ..., ''μ''<sub>''k''</sub>)}} with <math display="inline">n=\sum_{i=1}^k \mu_i</math> and {{nowrap|''μ''<sub>1</sub> ≥ ''μ''<sub>2</sub> ≥ ... ≥ ''μ''<sub>''k''</sub>}}, is associated the set ''C''<sub>''μ''</sub> of permutations with cycles of lengths {{nowrap|''μ''<sub>1</sub>, ''μ''<sub>2</sub>, ..., ''μ''<sub>''k''</sub>}}. Then ''C''<sub>''μ''</sub>  is a conjugacy class of S<sub>''n''</sub>, whose elements are said to be of cycle-type <math>\mu</math>.

== Low degree groups ==
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.

;S<sub>0</sub> and S<sub>1</sub>: The symmetric groups on the [[Empty set|empty set]] and the singleton set are trivial, which corresponds to {{math|1=0! = 1! = 1}}.  In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S<sub>0</sub>, its only member is the empty function.

;S<sub>2</sub>: This group consists of exactly two elements: the identity and the permutation swapping the two points.  It is a [[Cyclic group|cyclic group]] and is thus [[Abelian group|abelian]].  In [[Galois theory]], this corresponds to the fact that the [[Quadratic formula|quadratic formula]] gives a direct solution to the general quadratic polynomial after extracting only a single root.  In [[Invariant theory|invariant theory]], the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting {{math|1=''f''<sub>s</sub>(''x'', ''y'') = ''f''(''x'', ''y'') + ''f''(''y'', ''x'')}}, and {{math|1=''f''<sub>a</sub>(''x'', ''y'') = ''f''(''x'', ''y'') − ''f''(''y'', ''x'')}}, one gets that {{math|1=2⋅''f'' = ''f''<sub>s</sub> + ''f''<sub>a</sub>}}.  This process is known as [[Symmetrization|symmetrization]].

;S<sub>3</sub>: S<sub>3</sub> is the first nonabelian symmetric group.  This group is isomorphic to the [[Dihedral group of order 6|dihedral group of order 6]], the group of reflection and rotation symmetries of an [[Equilateral triangle|equilateral triangle]], since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations.  In Galois theory, the sign map from S<sub>3</sub> to S<sub>2</sub> corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the A<sub>3</sub> kernel corresponds to the use of the [[Discrete Fourier transform|discrete Fourier transform]] of order 3 in the solution, in the form of Lagrange resolvents.{{citation needed|date=September 2009}}

;S<sub>4</sub>: The group S<sub>4</sub> is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, [[Rencontres numbers|9, 8 and 6]] permutations, of the [[Cube|cube]].<ref>{{cite thesis |first=J. |last=Neubüser |title=Die Untergruppenverbände der Gruppen der Ordnungen ̤100 mit Ausnahme der Ordnungen 64 und 96 |date=1967 |type=PhD |publisher=Universität Kiel |url=}}</ref> Beyond the group [[Alternating group|A<sub>4</sub>]], S<sub>4</sub> has a [[Klein four-group]] V as a proper [[Normal subgroup|normal subgroup]], namely the even transpositions {{math|{(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)},}} with quotient S<sub>3</sub>. In [[Galois theory]], this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by [[Biography:Lodovico Ferrari|Lodovico Ferrari]]. The Klein group can be understood in terms of the Lagrange resolvents of the quartic.  The map from S<sub>4</sub> to S<sub>3</sub> also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree ''n'' of dimension below {{math|''n'' − 1}}, which only occurs for {{math|1=''n'' = 4}}.

;S<sub>5</sub>: S<sub>5</sub> is the first non-solvable symmetric group.  Along with the [[Special linear group|special linear group]] {{math|SL(2, 5)}} and the icosahedral group {{math|A<sub>5</sub> × S<sub>2</sub>}}, S<sub>5</sub> is one of the three non-solvable groups of order 120, up to isomorphism.  S<sub>5</sub> is the [[Galois group]] of the general quintic equation, and the fact that S<sub>5</sub> is not a [[Solvable group|solvable group]] translates into the non-existence of a general formula to solve quintic polynomials by radicals.  There is an exotic inclusion map {{math|S<sub>5</sub> → S<sub>6</sub>}} as a transitive subgroup; the obvious inclusion map {{math|S<sub>''n''</sub> → S<sub>''n''+1</sub>}} fixes a point and thus is not transitive. This yields the outer automorphism of S<sub>6</sub>, discussed below, and corresponds to the resolvent sextic of a quintic.

;S<sub>6</sub>: Unlike all other symmetric groups, S<sub>6</sub>, has an outer automorphism.  Using the language of [[Galois theory]], this can also be understood in terms of Lagrange resolvents. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map {{math|S<sub>5</sub> → S<sub>6</sub>}} as a transitive subgroup (the obvious inclusion map {{math|S<sub>''n''</sub> → S<sub>''n''+1</sub>}} fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S<sub>6</sub>—see ''[[Automorphisms of the symmetric and alternating groups]]'' for details.

:Note that while A<sub>6</sub> and A<sub>7</sub> have an exceptional [[Schur multiplier]] (a triple cover) and that these extend to triple covers of S<sub>6</sub> and S<sub>7</sub>, these do not correspond to exceptional Schur multipliers of the symmetric group.

=== Maps between symmetric groups ===

Other than the trivial map {{math|1=S<sub>''n''</sub> → C<sub>1</sub> ≅ S<sub>0</sub> ≅ S<sub>1</sub>}} and the sign map {{math|S<sub>''n''</sub> → S<sub>2</sub>}}, the most notable homomorphisms between symmetric groups, in order of [[Relative dimension|relative dimension]], are:
* {{math|S<sub>4</sub> → S<sub>3</sub>}} corresponding to the exceptional normal subgroup {{math|V < A<sub>4</sub> < S<sub>4</sub>}};
* {{math|S<sub>6</sub> → S<sub>6</sub>}} (or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of S<sub>6</sub>.
* {{math|S<sub>5</sub> → S<sub>6</sub>}} as a transitive subgroup, yielding the outer automorphism of S<sub>6</sub> as discussed above.
There are also a host of other homomorphisms {{math|S<sub>''m''</sub> → S<sub>''n''</sub>}} where {{math|''m'' < ''n''}}.

== Relation with alternating group ==
For {{nowrap|''n'' ≥ 5}}, the [[Alternating group|alternating group]] A<sub>''n''</sub> is [[Simple group|simple]], and the induced quotient is the sign map: {{nowrap|A<sub>''n''</sub> → S<sub>''n''</sub> → S<sub>2</sub>}} which is split by taking a transposition of two elements. Thus S<sub>''n''</sub> is the semidirect product {{nowrap|A<sub>''n''</sub> ⋊ S<sub>2</sub>}}, and has no other proper normal subgroups, as they would intersect A<sub>''n''</sub> in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in A<sub>''n''</sub> (and thus themselves be A<sub>''n''</sub> or S<sub>''n''</sub>).

S<sub>''n''</sub> acts on its subgroup A<sub>''n''</sub> by conjugation, and for {{nowrap|''n'' ≠ 6}}, S<sub>''n''</sub> is the full automorphism group of A<sub>''n''</sub>: Aut(A<sub>''n''</sub>) ≅ S<sub>''n''</sub>. Conjugation by even elements are [[Inner automorphism|inner automorphism]]s of A<sub>''n''</sub> while the outer automorphism of A<sub>''n''</sub> of order 2 corresponds to conjugation by an odd element. For {{nowrap|1=''n'' = 6}}, there is an [[Automorphisms of the symmetric and alternating groups#exceptional outer automorphism|exceptional outer automorphism]] of A<sub>''n''</sub> so S<sub>''n''</sub> is not the full automorphism group of A<sub>''n''</sub>.

Conversely, for {{nowrap|''n'' ≠ 6}}, S<sub>''n''</sub> has no outer automorphisms, and for {{nowrap|''n'' ≠ 2}} it has no center, so for {{nowrap|''n'' ≠ 2, 6}} it is a [[Complete group|complete group]], as discussed in automorphism group, below.

For {{nowrap|''n'' ≥ 5}}, S<sub>''n''</sub> is an [[Almost simple group|almost simple group]], as it lies between the simple group A<sub>''n''</sub> and its group of automorphisms.

S<sub>''n''</sub> can be embedded into A<sub>''n''+2</sub> by appending the transposition {{nowrap|(''n'' + 1, ''n'' + 2)}} to all odd permutations, while embedding into A<sub>''n''+1</sub> is impossible for {{nowrap|''n'' > 1}}.

== Generators and relations ==
The symmetric group on {{mvar|n}} letters is generated by the adjacent transpositions <math> \sigma_i = (i, i + 1)</math> that swap {{mvar|i}} and {{math|''i'' + 1}}.<ref>{{citation|title = The Symmetric Group| edition = 2 | first = Bruce E. | last = Sagan |  publisher = Springer | year = 2001 | page = [{{GBurl|Jm-HBaMdt8sC|p=4}} 4] |isbn=978-0-387-95067-9}}</ref>  The collection <math>\sigma_1, \ldots, \sigma_{n-1}</math> generates {{math|S<sub>''n''</sub>}} subject to the following relations:<ref>{{citation|title = Combinatorics of Coxeter groups|last1 = Björner|first1 = Anders|last2 = Brenti|first2 = Francesco|publisher = Springer|year = 2005 |page = [{{GBurl|1TBPz5sd8m0C|p=4}} 4. Example 1.2.3] |isbn=978-3-540-27596-1}}</ref>
*<math>\sigma_i^2 = 1,</math>
*<math>\sigma_i\sigma_j = \sigma_j\sigma_i</math> for <math>|i-j| > 1</math>, and
*<math>(\sigma_i\sigma_{i+1})^3 =1,</math>
where 1 represents the identity permutation.  This representation endows the symmetric group with the structure of a [[Coxeter group]] (and so also a [[Reflection group|reflection group]]).

Other possible generating sets include the set of transpositions that swap {{math|1}} and {{mvar|i}} for {{math|2 ≤ ''i'' ≤ ''n''}},{{cn|date=May 2020}} and a set containing any {{mvar|n}}-cycle and a {{math|2}}-cycle of adjacent elements in the {{mvar|n}}-cycle.<ref>{{citation|title = Algebra | first = Michael | last = Artin |  publisher = Pearson | year = 1991 | at = Exercise 6.6.16 |isbn=978-0-13-004763-2}}</ref>

== Subgroup structure ==
A [[Subgroup|subgroup]] of a symmetric group is called a [[Permutation group|permutation group]].

=== Normal subgroups ===
The [[Normal subgroup|normal subgroup]]s of the finite symmetric groups are well understood. If {{nowrap|''n'' ≤ 2}}, S<sub>''n''</sub> has at most 2 elements, and so has no nontrivial proper subgroups. The [[Alternating group|alternating group]] of degree ''n'' is always a normal subgroup, a proper one for {{nowrap|''n'' ≥ 2}} and nontrivial for {{nowrap|''n'' ≥ 3}}; for {{nowrap|''n'' ≥ 3}} it is in fact the only nontrivial proper normal subgroup of S<sub>''n''</sub>, except when {{nowrap|1=''n'' = 4}} where there is one additional such normal subgroup, which is isomorphic to the Klein four group.

The symmetric group on an infinite set does not have a subgroup of index 2, as [[Biography:Giuseppe Vitali|Vitali]] (1915<ref>{{cite journal |first=G. |last=Vitali |title=Sostituzioni sopra una infinità numerabile di elementi |journal=Bollettino Mathesis |volume=7 |pages=29–31 |date=1915 |doi= |url=}}</ref>) proved that each permutation can be written as a product of three squares. However it contains the normal subgroup ''S'' of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of ''S'' that are products of an even number of transpositions form a subgroup of index 2 in ''S'', called the alternating subgroup ''A''. Since ''A'' is even a [[Characteristic subgroup|characteristic subgroup]] of ''S'', it is also a normal subgroup of the full symmetric group of the infinite set. The groups ''A'' and ''S'' are the only nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by Onofri (1929<ref>§141, p.124 in {{cite journal |first=L. |last=Onofri |title=Teoria delle sostituzioni che operano su una infinità numerabile di elementi |journal=Annali di Matematica |volume=7 |issue=1 |pages=103–130 |date=1929 |doi=10.1007/BF02409971 |s2cid=186219904 |url=|doi-access=free }}</ref>) and independently Schreier–[[Biography:Stanislaw Ulam|Ulam]] (1934<ref>{{cite journal |last1=Schreier |first1=J. |last2=Ulam |first2=S. |title=Über die Permutationsgruppe der natürlichen Zahlenfolge |journal=Studia Math |volume=4 |issue=1 |pages=134–141 |date=1933 |doi= 10.4064/sm-4-1-134-141|url=http://matwbn.icm.edu.pl/ksiazki/sm/sm4/sm4120.pdf}}</ref>). For more details see {{harv|Scott|1987|loc=Ch. 11.3}} or {{harv|Dixon|Mortimer|1996|loc=Ch. 8.1}}.

=== Maximal subgroups ===

The [[Maximal subgroup|maximal subgroup]]s of S<sub>''n''</sub> fall into three classes: the intransitive, the imprimitive, and the primitive.  The intransitive maximal subgroups are exactly those of the form {{nowrap|S<sub>''k''</sub> × S<sub>''n''–''k''</sub>}} for {{nowrap|1 ≤ ''k'' < ''n''/2}}.  The imprimitive maximal subgroups are exactly those of the form {{nowrap|S<sub>''k''</sub> wr S<sub>''n''/''k''</sub>}}, where {{nowrap|2 ≤ ''k'' ≤ ''n''/2}} is a proper divisor of ''n'' and "wr" denotes the [[Wreath product|wreath product]].  The primitive maximal subgroups are more difficult to identify, but with the assistance of the [[O'Nan–Scott theorem]] and the [[Classification of finite simple groups|classification of finite simple groups]], {{harv|Liebeck|Praeger|Saxl|1988}} gave a fairly satisfactory description of the maximal subgroups of this type<!-- though beware of typos in the low degrees-->, according to {{harv|Dixon|Mortimer|1996|p=268}}.

=== Sylow subgroups ===
The [[Sylow subgroup]]s of the symmetric groups are important examples of [[P-group|''p''-groups]].  They are more easily described in special cases first:

The Sylow ''p''-subgroups of the symmetric group of degree ''p'' are just the cyclic subgroups generated by ''p''-cycles.  There are {{math|1=(''p'' − 1)!/(''p'' − 1) = (''p'' − 2)!}} such subgroups simply by counting generators.  The normalizer therefore has order {{math|''p''⋅(''p'' − 1)}} and is known as a [[Frobenius group]] {{math|''F''<sub>''p''(''p''−1)</sub>}} (especially for {{math|1=''p'' = 5}}), and is the affine general linear group, {{math|AGL(1, ''p'')}}.

The Sylow ''p''-subgroups of the symmetric group of degree ''p''<sup>2</sup> are the [[Wreath product|wreath product]] of two cyclic groups of order ''p''.  For instance, when {{math|1=''p'' = 3}}, a Sylow 3-subgroup of Sym(9) is generated by {{math|1=''a'' = (1 4 7)(2 5 8)(3 6 9)}} and the elements {{math|1=''x'' = (1 2 3), ''y'' = (4 5 6), ''z'' = (7 8 9)}}<!-- or just use (1,2,3) -->, and every element of the Sylow 3-subgroup has the form {{math|1=''a''<sup>''i''</sup>''x''<sup>''j''</sup>''y''<sup>''k''</sup>''z''<sup>''l''</sup>}} for {{tmath|1=0 \le i,j,k,l  \le 2}}.

The Sylow ''p''-subgroups of the symmetric group of degree ''p''<sup>''n''</sup> are sometimes denoted W<sub>''p''</sub>(''n''), and using this notation one has that {{math|W<sub>''p''</sub>(''n'' + 1)}} is the wreath product of W<sub>''p''</sub>(''n'') and W<sub>''p''</sub>(1).

In general, the Sylow ''p''-subgroups of the symmetric group of degree ''n'' are a direct product of ''a''<sub>''i''</sub> copies of W<sub>''p''</sub>(''i''), where {{math|1= 0 ≤ ''a<sub>i</sub>'' ≤ ''p'' − 1}} and {{math|1=''n'' = ''a''<sub>0</sub>&nbsp;+&nbsp;''p''⋅''a''<sub>1</sub>&nbsp;+&nbsp;...&nbsp;+&nbsp;''p''<sup>''k''</sup>⋅''a''<sub>''k''</sub>}} (the base ''p'' expansion of ''n'').

For instance, {{math|1=W<sub>2</sub>(1) =&nbsp;C<sub>2</sub> and W<sub>2</sub>(2) =&nbsp;D<sub>8</sub>}}, the dihedral group of order 8, and so a Sylow 2-subgroup of the [[Symmetric group|symmetric group]] of degree 7 is generated by {{math|{ (1,3)(2,4), (1,2), (3,4), (5,6) } }} and is isomorphic to {{math|D<sub>8</sub> × C<sub>2</sub>}}.

These calculations are attributed to {{harv|Kaloujnine|1948}} and described in more detail in {{harv|Rotman|1995|p=176}}.  Note however that {{harv|Kerber|1971|p=26}} attributes the result to an 1844 work of [[Biography:Augustin-Louis Cauchy|Cauchy]], and mentions that it is even covered in textbook form in {{harv|Netto|1882|loc=§39–40}}.

===<span id="Transitive subgroup anchor"></span> Transitive subgroups ===<!-- Transitive subgroup redirects to this anchor-->
A '''transitive subgroup''' of S<sub>''n''</sub> is a subgroup whose action on {1,&nbsp;2,&nbsp;,...,&nbsp;''n''} is transitive. For example, the Galois group of a (finite) [[Galois extension]] is a transitive subgroup of S<sub>''n''</sub>, for some ''n''.

===Cayley's theorem===
[[Cayley's theorem]] states that every group ''G'' is isomorphic to a subgroup of some symmetric group.  In particular, one may take a subgroup of the symmetric group on the elements of ''G'', since every group acts on itself faithfully by (left or right) multiplication.

== Cyclic subgroups ==

[[Cyclic group]]s are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the [[Least common multiple|least common multiple]] of the lengths of its cycles.  For example, in S{{sub|5}}, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S{{sub|5}} are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size.{{cn|date=October 2022}} For example, S{{sub|5}} has no subgroup of order 15 (a divisor of the order of S{{sub|5}}), because the only group of order 15 is the cyclic group.  The largest possible order of a cyclic subgroup (equivalently, the largest possible order of an element in S{{sub|''n''}}) is given by [[Landau's function]].

== Automorphism group ==
{{details|Automorphisms of the symmetric and alternating groups}}
{| class="wikitable" style="float:right;"  cellspacing="2"
|- style="background:#a0e0a0;"
| ''n''
| Aut(S<sub>''n''</sub>)
| Out(S<sub>''n''</sub>)
| Z(S<sub>''n''</sub>)
|-
| ''n'' ≠ 2, 6
| S<sub>''n''</sub>
| C<sub>1</sub>
| C<sub>1</sub>
|-
| ''n'' = 2
| C<sub>1</sub>
| C<sub>1</sub>
| S<sub>2</sub>
|-
| ''n'' = 6
| S<sub>6</sub> ⋊ C<sub>2</sub>
| C<sub>2</sub>
| C<sub>1</sub>
|}
For {{nowrap|''n'' ≠ 2, 6}}, S<sub>''n''</sub> is a [[Complete group|complete group]]: its [[Center (group theory)|center]] and [[Outer automorphism group|outer automorphism group]] are both trivial.

For {{nowrap|1=''n'' = 2}}, the automorphism group is trivial, but S<sub>2</sub> is not trivial: it is isomorphic to C<sub>2</sub>, which is abelian, and hence the center is the whole group.

For {{nowrap|1=''n'' = 6}}, it has an outer automorphism of order 2: {{nowrap|1=Out(S<sub>6</sub>) = C<sub>2</sub>}}, and the automorphism group is a semidirect product {{nowrap|1=Aut(S<sub>6</sub>) = S<sub>6</sub> ⋊ C<sub>2</sub>}}.

In fact, for any set ''X'' of cardinality other than 6, every automorphism of the symmetric group on ''X'' is inner, a result first due to {{harv|Schreier|Ulam|1936}} according to {{harv|Dixon|Mortimer|1996|p=259}}.

== Homology ==
The group homology of S<sub>''n''</sub> is quite regular and stabilizes: the first homology (concretely, the abelianization) is:

:<math>H_1(\mathrm{S}_n,\mathbf{Z}) = \begin{cases} 0 & n < 2\\
\mathbf{Z}/2 & n \geq 2.\end{cases}</math>

The first homology group is the abelianization, and corresponds to the sign map S<sub>''n''</sub> → S<sub>2</sub> which is the abelianization for ''n'' ≥ 2; for ''n'' < 2 the symmetric group is trivial. This homology is easily computed as follows: S<sub>''n''</sub> is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps {{nowrap|S<sub>''n''</sub> → C<sub>''p''</sub>}} are to S<sub>2</sub> and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps {{nowrap|S<sub>''n''</sub> → S<sub>2</sub> ≅ {±1} }} send an involution to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of S<sub>''n''</sub>.

The second homology (concretely, the [[Schur multiplier]]) is:
:<math>H_2(\mathrm{S}_n,\mathbf{Z}) = \begin{cases} 0 & n < 4\\
\mathbf{Z}/2 & n \geq 4.\end{cases}</math>
This was computed in {{Harv|Schur|1911}}, and corresponds to the double cover of the symmetric group, 2 · S<sub>''n''</sub>.

Note that the [[Exceptional object|exceptional]] low-dimensional homology of the alternating group (<math>H_1(\mathrm{A}_3)\cong H_1(\mathrm{A}_4) \cong \mathrm{C}_3,</math> corresponding to non-trivial abelianization, and <math>H_2(\mathrm{A}_6)\cong H_2(\mathrm{A}_7) \cong \mathrm{C}_6,</math> due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map <math>\mathrm{A}_4 \twoheadrightarrow \mathrm{C}_3</math> extends to <math>\mathrm{S}_4 \twoheadrightarrow \mathrm{S}_3,</math> and the triple covers of A<sub>6</sub> and A<sub>7</sub> extend to triple covers of S<sub>6</sub> and S<sub>7</sub> – but these are not ''homological'' – the map <math>\mathrm{S}_4 \twoheadrightarrow \mathrm{S}_3</math> does not change the abelianization of S<sub>4</sub>, and the triple covers do not correspond to homology either.

The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map {{nowrap|S<sub>''n''</sub> → S<sub>''n''+1</sub>}}, and for fixed ''k'', the induced map on homology {{nowrap|''H''<sub>''k''</sub>(S<sub>''n''</sub>) → ''H''<sub>''k''</sub>(S<sub>''n''+1</sub>)}} is an isomorphism for sufficiently high ''n''. This is analogous to the homology of families Lie groups stabilizing.

The homology of the infinite symmetric group is computed in {{Harv|Nakaoka|1961}}, with the cohomology algebra forming a [[Hopf algebra]].

== Representation theory ==
{{main article|Representation theory of the symmetric group}}

The [[Representation theory of the symmetric group|representation theory of the symmetric group]] is a particular case of the [[Representation theory of finite groups|representation theory of finite groups]], for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from [[Symmetric function|symmetric function]] theory to problems of [[Physics:Quantum mechanics|quantum mechanics]] for a number of [[Identical particles|identical particles]].

The symmetric group S<sub>''n''</sub> has order ''n''<nowiki>!</nowiki>. Its [[Conjugacy class|conjugacy class]]es are labeled by partitions of&nbsp;''n''. Therefore, according to the representation theory of a finite group, the number of inequivalent [[Irreducible representation|irreducible representation]]s, over the [[Complex number|complex number]]s, is equal to the number of partitions of&nbsp;''n''. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of ''n'' or equivalently Young diagrams of size&nbsp;''n''.

Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the [[Young symmetrizer]]s acting on a space generated by the [[Young tableau]]x of shape given by the Young diagram.

Over other [[Field (mathematics)|field]]s the situation can become much more complicated.  If the field ''K'' has [[Characteristic (algebra)|characteristic]] equal to zero or greater than ''n'' then by [[Maschke's theorem]] the [[Group ring|group algebra]] ''K''S<sub>''n''</sub> is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary).

However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of [[Module (mathematics)|module]]s rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called ''Specht modules'', and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their [[Dimension (vector space)|dimension]]s are not known in general.

The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory.

== See also ==
* [[Braid group]]
* [[History of group theory]]
* Signed symmetric group and [[Generalized symmetric group]]
* {{slink|Symmetry in quantum mechanics|Exchange symmetry or permutation symmetry}}
* [[Symmetric inverse semigroup]]
* [[Symmetric power]]

== Notes ==
{{reflist}}

== References ==
{{refbegin}}
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* {{Citation | last1=Dixon | first1=John D. | last2=Mortimer | first2=Brian | title=Permutation groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | series=Graduate Texts in Mathematics | isbn=978-0-387-94599-6 | mr=1409812 | year=1996 | volume=163 | url-access=registration | url=https://archive.org/details/permutationgroup0000dixo }}
* {{Citation| last=Jacobson| first=Nathan|  year=2009| title=Basic algebra| edition=2nd| volume = 1 | publisher=Dover| isbn = 978-0-486-47189-1}}.
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*{{Citation | last=Kerber | first=Adalbert | title=Representations of permutation groups. I | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | series=Lecture Notes in Mathematics, Vol. 240 | doi=10.1007/BFb0067943 | mr=0325752 | year=1971 | volume=240 | isbn=978-3-540-05693-5}}
* {{Citation | first1=M.W. | last1=Liebeck | first2=C.E. | last2=Praeger | first3=J. | last3=Saxl | title=On the O'Nan–Scott theorem for finite primitive permutation groups | journal=Journal of the Australian Mathematical Society | volume=44 | year=1988 | pages=389–396 | doi=10.1017/S144678870003216X | issue=3 | doi-access=free }}
* {{Citation | title=Homology of the Infinite Symmetric Group | first=Minoru | last=Nakaoka | journal=[[Annals of Mathematics]] | series=2 | volume=73 | number=2 |date=March 1961 | pages=229–257 | jstor=1970333 | doi=10.2307/1970333 }}
* {{Citation | last=Netto | first=Eugen |  title=Substitutionentheorie und ihre Anwendungen auf die Algebra | publisher=Leipzig. Teubner | language=de | jfm=14.0090.01 | year=1882}}
* {{Citation | last = Rotman | first = Joseph J. | date = 1995 | chapter = Extensions and Cohomology | title = An Introduction to the Theory of Groups | series = Graduate Texts in Mathematics | volume = 148 | pages = 154–216 | publisher = Springer | doi = 10.1007/978-1-4612-4176-8_7 | isbn = 978-1-4612-8686-8 | chapter-url = https://link.springer.com/content/pdf/10.1007%2F978-1-4612-4176-8_7.pdf }}
* {{Citation | last=Scott | first=W.R. | title=Group Theory | publisher=[[Social:Dover Publications|Dover Publications]] | isbn=978-0-486-65377-8 | year=1987 | pages=45–46}}
*{{citation | first=Issai | last=Schur |  title=Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen | journal=Journal für die reine und angewandte Mathematik | volume=1911 | year=1911 | issue=139 | pages=155–250 |doi=10.1515/crll.1911.139.155 | s2cid=122809608 }}
* {{Citation | last1=Schreier | first1=Józef | last2=Ulam | first2=Stanislaw |  title=Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm28/fm28128.pdf | language=de | zbl=0016.20301 | year=1936 | journal=[[Fundamenta Mathematicae]] | volume=28 | pages=258–260| doi=10.4064/fm-28-1-258-260 }}
{{refend}}

== External links==
* {{springer|title=Symmetric group|id=p/s091670}}
* {{mathworld | urlname = SymmetricGroup  | title = Symmetric group }}
* {{mathworld | urlname = SymmetricGroupGraph  | title = Symmetric group graph }}
*[http://www.ted.com/talks/marcus_du_sautoy_symmetry_reality_s_riddle.html Marcus du Sautoy: Symmetry, reality's riddle] (video of a talk)
* OEIS [http://oeis.org/search?q=Symmetric+Group Entries dealing with the Symmetric Group]

{{DEFAULTSORT:Symmetric Group}}
[[Category:Permutation groups]]
[[Category:Symmetry]]

{{Sourceattribution|Symmetric group}}