View source for Symmetric group:S3

{{perspectives}}
==Definition==

===Verbal definitions===

The symmetric group <math>S_3</math> can be defined in the following equivalent ways:

* It is the [[member of family::symmetric group]][[member of family::symmetric group on finite set| ]] on a set of three elements, viz., the group of all permutations of a three-element set. In particular, it is a [[member of family::symmetric group of prime degree]] and [[member of family::symmetric group of prime power degree]].
* It is the {{dihedral group}} of order six (degree three), viz., the group of (not necessarily orientation-preserving) symmetries of the equilateral triangle.
* It is the [[member of family::special linear group]] of [[member of family::special linear group of degree two|degree two]] <math>SL(2,2)</math> over the [[member of family::field:F2|field of two elements]]. It turns out that, because of the nature of the prime two, it is also the [[member of family::projective special linear group]] of [[member of family::projective special linear group of degree two|degree two]] <math>PSL(2,2)</math>, the [[member of family::general linear group]] of [[member of family::general linear group of degree two|degree two]] <math>GL(2,2)</math>, and the [[member of family::projective general linear group]] of [[member of family::projective general linear group of degree two|degree two]] <math>PGL(2,2)</math>.
* It is the [[member of family::general affine group]] of [[general affine group of degree one|degree one]] over the [[field:F3|field of three elements]], i.e., <math>GA(1,3)</math> (sometimes also written as <math>AGL(1,3)</math>).
* It is the [[member of family::general semilinear group]] of [[member of family::general semilinear group of degree one|degree one]] over the [[field:F4|field of four elements]], i.e., <math>\Gamma L(1,4)</math>.
* It is the [[member of family::von Dyck group]] with parameters <math>(2,2,3)</math>, and in particular, is a [[member of family::Coxeter group]]. In particular, it has the presentation (where <math>e</math> denotes the identity element):

<math>\langle a,b,c \mid a^2 = b^2 = c^3 = abc = e \rangle</math>.

In the Coxeter language, this is written as:

<math>\langle s_1, s_2 \mid s_1^2 = s_2^2 = (s_1s_2)^3 = e \rangle</math>.

===Multiplication table===

We portray elements as permutations on the set <math>\{ 1,2,3 \}</math> using the [[cycle decomposition]]. The ''row element is multiplied on the left and the column element on the right'', with the assumption of ''functions written on the left''. ''This means that the column element is applied first and the row element is applied next''.


{{#lst:element structure of symmetric group:S3|multiplication table}}


If we used the opposite convention (i.e., functions written on the right), the row element is to be multiplied on the right and the column element on the left.

Here is the multiplication table where we use the [[one-line notation]] for permutations, where, as in the previous multiplication table, the column permutation is applied first and then the row permutation. Thus, with the left action convention, the row element 
is multiplied on the left and the column element on the right:

{{#lst:element structure of symmetric group:S3|multiplication table in one-line notation}}

{{basicdef}}
{{particular group}}

==Families==

The symmetric group on three elements is part of some important families:

{| class="sortable" border="1"
! Generic name for family member !! Definition !!Parametrization of family !! Parameter value(s) for this member !! Other members !! Comments 
|-
|[[member of family::symmetric group on finite set]] <math>S_n</math> || group of all permutations on a finite set || by a nonnegative integer <math>n</math>, denoting size of set acted on || <math>n = 3</math>, so the group is <math>S_3</math> || {{#ask: [[member of family::symmetric group on finite set]]|limit = 0|searchlabel = click here for a list}} || 
|-
| [[member of family::Coxeter group]] || has a presentation of a particular form || Coxeter matrix describing the presentation || || {{#ask: [[member of family::Coxeter group]]|limit = 0|searchlabel = click here for a list}}|| [[symmetric groups on finite sets are Coxeter groups]]
|-
| [[member of family::dihedral group]] <math>D_{2n}</math> || semidirect product of a cyclic group and a two-element group acting via the inverse map || by a positive integer <math>n</math> that's half the order || <math>n = 3</math>, so the group is <math>D_6</math> || {{#ask: [[member of family::dihedral group]]|limit = 0|searchlabel = click here for a list}}|| 
|-
| [[member of family::general affine group]] <math>GA(n,F)</math>|| semidirect product of a vector space over a field with the general linear group acting on that vector space || name of field <math>F</math>, degree <math>n</math>(i.e., dimension of vector space). For a finite field, we may also write the group as <math>GA(n,q)</math> where <math>q</math> is the size of the field || [[field:F3]] (size <math>q = 3</math>), [[general affine group of degree one over a finite field|degree one]], so the group is <math>GA(1,\mathbb{F}_3)</math> or <math>GA(1,3)</math> || {{#ask: [[member of family::general affine group of degree one over a finite field]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::general linear group]] <math>GL(n,F)</math> || [[general linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field.  || [[field:F2]] (size <math>q = 2</math>), [[member of family::general linear group of degree two|degree two]], so the group is <math>GL(2,\mathbb{F}_2)</math> or <math>GL(2,2)</math> || {{#ask: [[member of family::general linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::projective general linear group]] <math>PGL(n,F)</math> || [[projective general linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field.   ||  [[field:F2]] (size <math>q = 2</math>), [[member of family::projective general linear group of degree two|degree two]], so the group is <math>PGL(2,\mathbb{F}_2)</math> or <math>PGL(2,2)</math> || {{#ask: [[member of family::projective general linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::special linear group]] <math>SL(n,F)</math> || [[special linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field.  || [[field:F2]] (size <math>q = 2</math>), [[member of family::special linear group of degree two|degree two]], so the group is <math>SL(2,\mathbb{F}_2)</math> or <math>SL(2,2)</math> || {{#ask: [[member of family::special linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::projective special linear group]] <math>PSL(n,F)</math> || [[projective special linear group]] of finite degree over a finite field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field.  || [[field:F2]] (size <math>q = 2</math>), [[member of family::projective special linear group of degree two|degree two]], so the group is <math>PSL(2,\mathbb{F}_2)</math> or <math>PSL(2,2)</math> || {{#ask: [[member of family::special linear group]]|limit = 0|searchlabel = click here for a list}} ||
|-
| [[member of family::general semilinear group]] <math>\Gamma L(n,F)</math> || semidirect product of general linear group and automorphism group of base field || name of field <math>F</math>, degree <math>n</math>. <math>F</math> may be replaced by its size <math>q</math> in case of a finite field. || [[field:F4]] (size <math>q = 4</math>), [[general semilinear group of degree one|degree one]], so the group is <math>\Gamma L(1,\mathbb{F}_4) = \Gamma L(1,4)</math> || {{#ask: [[member of family::general semilinear group]]|limit = 0|searchlabel = click here for a list}} ||
|}

==Elements==

{{further|[[Element structure of symmetric group:S3]]}}

===Conjugacy class structure===

As for any [[symmetric group]], [[cycle type determines conjugacy class]]. The cycle types, in turn, are parametrized by the unordered integer partitions of <math>3</math>. The conjugacy classes are described below.

{{#lst:element structure of symmetric group:S3|conjugacy class structure}}

This group is one of three finite groups with the property that any two elements of the same order are conjugate. The other two are the [[cyclic group of order two]] and the [[trivial group]].

For an interpretation of the conjugacy class structure based on the other equivalent definitions of the group, visit [[Element structure of symmetric group:S3#Conjugacy class structure]].
===Automorphism class structure===

The classification of elements upto automorphism is the same as that upto conjugation; this is because the symmetric group on three elements is a [[complete group]]: a [[centerless group]] where every automorphism is inner.

==Arithmetic functions==

===Basic arithmetic functions===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value order|6}} || as [[symmetric group]] <math>\! S_n, n = 3:</math>, <math>\! n! = 3! = 3 \cdot 2 \cdot 1 = 6</math><br>as [[general linear group of degree two]] <math>\! GL(2,q), q = 2:</math> <math>\! (q^2 - 1)(q^2 - q) = (2^2 - 1)(2^2 - 2) = 6</math><br>as [[general affine group of degree one]] <math>GA(1,q)</math>, <math>q = 3</math>: <math>q(q - 1) = 3 \cdot 2 = 6</math><br>as [[dihedral group]] <math>D_{2n}, n = 3</math>: <math>2 \cdot 3 = 6</math><br>as [[general semilinear group of degree one]] <math>\Gamma L(1,q), q = 4, q = p^r, p = 2, r = 2</math>: <math>r(q - 1) = 2(4 - 1) = 2(3) = 6</math><br><br>For more information, see [[element structure of symmetric group:S3#Order computation]]
|-
| {{arithmetic function value given order|exponent of a group|6|6}} || Elements of order <math>2</math> and <math>3</math>.<br>as [[symmetric group]] <math>\! S_n, n = 3</math>: <math>\! \operatorname{lcm} \{ 1,2, \dots, n \} = \operatorname{lcm} \{ 1,2,3 \} = 6</math><br>As [[general linear group of degree two]] <math>GL(2,q), q = 2</math>, underlying prime <math>p = 2</math>: <math>p(q^2 - 1) = 2(2^2 - 1) = 2(3) = 6</math><br>As [[general affine group of degree one]] <math>GA(1,q)</math>, <math>q = 3</math>, underlying prime <math>p = 3</math>: <math>p(q - 1) = 3(3 - 1) = 3(2) = 6</math><br>as [[dihedral group]] <math>D_{2n}, n = 3</math>: <math>\operatorname{lcm} \{ 2,n \} = \operatorname{lcm} \{ 2, 3 \} = 6</math>
|-
| {{arithmetic function value given order|derived length|2|6}} || Cyclic subgroup of order three is abelian, has abelian quotient.
|-
| {{arithmetic function value given order|minimum size of generating set|2|6}} || <math>(1,2), (1,2,3)</math><br>As [[symmetric group on a finite set]]: 2 (see [[symmetric group on a finite set is 2-generated]])
|-
| {{arithmetic function value given order|subgroup rank of a group|2|6}} || All proper subgroups are cyclic.
|-
| {{arithmetic function value given order|max-length of a group|2|6}} || Subgroup series going through subgroup of order two or three.
|}

===Arithmetic functions of an element-counting nature===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value given order|number of conjugacy classes|3|6}} || The three classes are the identity element, the transpositions, and the 3-cycles.<br>As <matH>S_n, n =3 </math>: [[number of unordered integer partitions]] of 3, equals 3<br>As <math>GL(2,q), q = 2</math>: <matH>q^2 - 1 = 2^2 - 1 = 3</math><br>As <math>D_{2n}, n = 3</math>: <math>(n + 3)/2 = (3 + 3)/2 = 3</math><br>As <math>GA(1,q), q = 3</math>: <math>q = 3</math><br>As <math>\Gamma L(1,p^2), p = 2</math>: <math>(p^2 + 3p - 4)/2 = (2^2 + 3 \cdot 2 - 4)/2 = 3</math><br><br>For a more elaborate explanation of these formulas, see [[element structure of symmetric group:S3#Number of conjugacy classes]]
|-
| {{arithmetic function value given order|number of equivalence classes under real conjugacy|3|6}} || Same as the number of conjugacy classes, because the group is an [[ambivalent group]].
|-
|{{arithmetic function value given order|number of conjugacy classes of real elements|3|6}} || Same as the number of conjugacy classes, because the group is an [[ambivalent group]].
|-
|{{arithmetic function value given order|number of equivalence classes under rational conjugacy|3|6}} || Same as the number of conjugacy classes, because the group is a [[rational group]].
|-
|{{arithmetic function value given order|number of conjugacy classes of rational elements|3|6}} || Same as the number of conjugacy classes, because the group is a [[rational group]].
|}

===Arithmetic functions of a subgroup-counting nature===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value|number of subgroups|6}} || || See [[subgroup structure of symmetric group:S3]]
|-
| {{arithmetic function value|number of conjugacy classes of subgroups|4}} || || 
|-
| {{arithmetic function value given order|number of normal subgroups|3|6}} ||
|-
| {{arithmetic function value|number of automorphism classes of subgroups|4}} || ||
|}

===Lists of numerical invariants===

{| class="sortable" border="1"
! List !! Value !! Explanation/comment
|-
| [[conjugacy class size set|conjugacy class sizes]] || 1,2,3 || See [[cycle type determines conjugacy class]], [[element structure of symmetric group:S3]], [[element structure of symmetric groups]]
|-
| [[order statistics]] || <math>1 \mapsto 1, 2 \mapsto 3, 3 \mapsto 2</math> ||
|-
| [[degrees of irreducible representations]] || 1,1,2 || See [[linear representation theory of symmetric group:S3]], [[linear representation theory of symmetric groups]]
|-
| orders of subgroups || 1,2,2,2,3,6 || See [[subgroup structure of symmetric group:S3]]
|-
|}

==Endomorphisms==

{{further|[[Endomorphism structure of symmetric group:S3]]}}

===Automorphisms===

Since <math>S_3</math> is a complete group, it is isomorphic to its automorphism group, where each element of <math>S_3</math> acts on <math>S_3</math> by conjugation. In fact, for <math>n \ne 2,6</math>, the symmetric group <math>S_n</math> is a complete group. {{further|[[Symmetric groups on finite sets are complete]]}}

==Group properties==

===Important properties===

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[Dissatisfies property::Abelian group]] || No || <math>(1,2)</math> and <math>(2,3)</math> don't commute || Smallest non-abelian group
|-
|[[Dissatisfies property::Nilpotent group]] || No || [[Centerless group|Centerless]]: The [[center]] is trivial || Smallest non-nilpotent group
|-
|[[Satisfies property::Metacyclic group]] || Yes || Cyclic normal subgroup of order three, cyclic quotient of order two ||
|-
|[[Satisfies property::Supersolvable group]] || Yes || [[Metacyclic implies supersolvable]] ||
|-
|[[Satisfies property::Solvable group]] || Yes || Metacyclic implies solvable ||
|}

===Other properties===

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[Satisfies property::T-group]] || Yes || ||
|-
|[[Satisfies property::Monolithic group]] || Yes|| Unique minimal normal subgroup of order three || 
|-
|[[Satisfies property::One-headed group]] || Yes|| Unique maximal normal subgroup of order three ||
|-
|[[Satisfies property::Jordan-unique group]] || Yes || There is a unique composition series ||
|-
|[[Satisfies property::SC-group]] || Yes || Every subgroup of it is a [[C-group]] || [[C-group]] means that every subgroup is permutably complemented
|-
|[[Satisfies property::Rational-representation group]] || Yes || [[Symmetric groups are rational-representation]] ||
|-
|[[Satisfies property::Rational group]] || Yes || [[Symmetric groups are rational]] || Also see [[classification of rational dihedral groups]]
|-
|[[Satisfies property::Ambivalent group]] || Yes || [[Symmetric groups are ambivalent]] ||
|-
|[[Satisfies property::Complete group]] || Yes || [[Symmetric groups are complete]], except degrees <math>2,6</math> ||
|-
|[[Satisfies property::Frobenius group]] || Yes || Frobenius kernel is alternating group, complement is any subgroup of order two. || Frobenius group on account of being <math>GA(1,3)</math>.
|-
|[[Satisfies property::Camina group]] || Yes || ||
|-
|[[Satisfies property::Z-group]] || Yes || Both the 2-Sylow subgroup ([[S2 in S3]]) and the 3-Sylow subgroup ([[A3 in S3]]) are cyclic. ||
|-
|[[Satisfies property::Schur-trivial group]] || Yes || [[Schur multiplier of Z-group is trivial]] ||
|}

==Subgroups==

[[Image:S3latticeofsubgroups.png|500px]]

{{further|[[Subgroup structure of symmetric group:S3]]}}

{{#lst:subgroup structure of symmetric group:S3|summary}}

===Subgroup-defining functions and associated quotient-defining functions===

{{#lst:subgroup structure of symmetric group:S3|sdf summary}}

==Linear representation theory==

{{further|[[Linear representation theory of symmetric group:S3]]}}

===Summary===

{{#lst:linear representation theory of symmetric group:S3|summary}}

===Character table===

{{#lst:linear representation theory of symmetric group:S3|character table}}

==Distinguishing features==

===Smallest of its kind===

* This is the unique non-abelian group of smallest order. All groups of order up to <math>5</math>, and all other groups of order <math>6</math>, are abelian.
* This is the unique non-nilpotent group of smallest order. All groups of order up to <math>5</math>, and all other groups of order <math>6</math>, are nilpotent.
* This is the unique smallest nontrivial [[complete group]].

==GAP implementation==

{{access GAP implementation online using SAGE|id=5012}}

{{GAP ID|6|1}}

===Other descriptions===

{| class="sortable" border="1"
! Description !! Functions used
|-
| <tt>SymmetricGroup(3)</tt> || [[GAP:SymmetricGroup|SymmetricGroup]]
|-
| <tt>DihedralGroup(6)</tt> || [[GAP:DihedralGroup|DihedralGroup]]
|-
| <tt>GL(2,2)</tt> || [[GAP:GL|GL]]
|}