Symmetric group

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, the symmetric group defined over any <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> is the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> whose <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a> are all the <a href="/facts/Bijection/j4gTuTmW">bijections</a> from the set to itself, and whose <a href="/facts/Group_operation/IQTYqINt">group operation</a> is the <a href="/facts/Function_composition/r5Pvnmth">composition of functions</a>. In particular, the finite symmetric group 
 
 
 
 
 
 S
 
 
 n
 
 
 
 
 {\displaystyle \mathrm {S} _{n}}
 
 defined over a <a href="/facts/Finite_set/za1newlP">finite set</a> of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 symbols consists of the <a href="/facts/Permutation/JSw9ukmv">permutations</a> that can be performed on the 
 
 
 
 n
 
 
 {\displaystyle n}
 
 symbols. Since there are 
 
 
 
 n
 !
 
 
 {\displaystyle n!}
 
 (
 
 
 
 n
 
 
 {\displaystyle n}
 
 <a href="/facts/Factorial/kkfy1Bwe">factorial</a>) such permutation operations, the <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a> (number of elements) of the symmetric group 
 
 
 
 
 
 S
 
 
 n
 
 
 
 
 {\displaystyle \mathrm {S} _{n}}
 
 is 
 
 
 
 n
 !
 
 
 {\displaystyle n!}
 
.
Although symmetric groups can be defined on <a href="/facts/Infinite_set/9rTN0xGH">infinite sets</a>, this article focuses on the finite symmetric groups: their applications, their elements, their <a href="/facts/Conjugacy_class/AFuPIFNW">conjugacy classes</a>, a <a href="/facts/Finitely_presented_group/f9emPUoI">finite presentation</a>, their <a href="/facts/Subgroup/r91mBgpa">subgroups</a>, their <a href="/facts/Automorphism_group/65vImKwe">automorphism groups</a>, and their <a href="/facts/Group_representation/FsuuDagQ">representation</a> 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 <a href="/facts/Galois_theory/54wscJfj">Galois theory</a>, <a href="/facts/Invariant_theory/5fIUh9h3">invariant theory</a>, the <a href="/facts/Representation_theory_of_Lie_groups/29R96ar2">representation theory of Lie groups</a>, and <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>. <a href="/facts/Cayley%2527s_theorem/a0M460K8">Cayley's theorem</a> states that every group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of the symmetric group on (the <a href="/facts/Underlying_set/TkuXca24">underlying set</a> of) 
 
 
 
 G
 
 
 {\displaystyle G}
 
.

Symmetric group open-in-new

Symmetric group