Editing Symmetric group (section)

{{Short description|Type of group in abstract algebra}}
{{Distinguish|Symmetry group}}
[[File:Symmetric group 4; Cayley graph 4,9.svg|thumb|320px|A [[Cayley graph]] of the symmetric group S<sub>4</sub> using the generators (red) a right [[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]], with [[Table (information)#:~:text=header|header]] omitted, of the symmetric group S<sub>3</sub>. The elements are represented as [[Permutation#Matrix representation|matrices]]. To the left of the matrices, are their [[Permutation#Two-line notation|two-line form]]. The black arrows indicate disjoint cycles and correspond to [[Permutation#Cycle notation|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>[[File:Symmetric group 3; Cayley table; positions.svg|310px]]<br>Some matrices are not arranged symmetrically to the main diagonal – thus the symmetric group is not abelian.]]
{{Group theory sidebar |Finite}}

In [[abstract algebra]], the '''symmetric group''' defined over any [[set (mathematics)|set]] is the [[group (mathematics)|group]] whose [[Element (mathematics)|elements]] are all the [[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]] of <math>n</math> symbols consists of the [[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]]) 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]]s, this article focuses on the finite symmetric groups: their applications, their elements, their [[conjugacy class]]es, a [[finitely presented group|finite presentation]], their [[subgroup]]s, their [[automorphism group]]s, 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]], the [[representation theory of Lie groups]], and [[combinatorics]]. [[Cayley's theorem]] states that every group <math>G</math> is [[group isomorphism|isomorphic]] to a [[subgroup]] of the symmetric group on (the [[underlying set]] of) <math>G</math>.