View source for Symmetric group:S4

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{{particular group}}
[[Importance rank::2| ]]
==Definition==

The symmetric group <math>S_4</math> or <math>\operatorname{Sym}(4)</math>, also termed the '''symmetric group of degree four''', is defined in the following equivalent ways:

* The group of all permutations, i.e., the [[member of family::symmetric group]][[member of family::symmetric group on finite set| ]] on a set of size four. In particular, it is a [[member of family::symmetric group of prime power degree]].
* The [[member of family::triangle group]] (''not'' the [[von Dyck group]], but its double) <math>\Delta(2,3,3)</math>. In other words, it has the [[presentation]] (where <math>e</math> denotes the identity element):

<math>\langle s_1, s_2, s_3 \mid s_1^2 = s_2^2 = s_3^2 = e, (s_1s_2)^2 = (s_2s_3)^3 = (s_1s_3)^3 = e \rangle</math>.

In particular, it is a [[member of family::Coxeter group]].

* The '''full tetrahedral group''': The group of all (not necessarily orientation-preserving) symmetries of the [[regular tetrahedron]]. This is denoted as <math>T_h</math>.
* The [[member of family::von Dyck group]] with parameters <math>(2,3,4)</math> (sometimes written in reverse order as <math>(4,3,2)</math>). In other words, it has the presentation (with <math>e</math> denoting the identity element):

<math>\langle a,b,c \mid a^2 = b^3 = c^4 = abc = e \rangle</math>.

* The '''octahedral group''' or '''cube group''': group of orientation-preserving symmetries of the cube (or equivalently, the octahedron). This is denoted as <math>O</math>.
* The [[member of family::projective general linear group]] of [[member of family::projective general linear group of degree two|degree two]] over the [[field:F3|field of three elements]]: <math>PGL(2,3)</math>.
* The [[member of family::general affine group]] of [[member of family::general affine group of degree two|degree two]] over the [[field:F2|field of two elements]]: <math>GA(2,2)</math>.
* The [[member of family::projective special linear group]] of [[member of family::projective special linear group of degree two|degree two]] over [[ring:Z4]], the ring <math>\mathbb{Z}/4\mathbb{Z}</math> of the integers modulo 4: <math>PSL(2,\mathbb{Z}/4\mathbb{Z})</math>.
* The [[member of family::general semiaffine group]] of [[member of family::general semiaffine group of degree one|degree one]] over [[field:F4]], i.e., the group <math>\Gamma A(1,4)</math> or <math>A \Gamma L(1,4)</math>.
===Equivalence of definitions===

The following is a list of proofs of the equivalence of various definitions:

* [[Full tetrahedral group is isomorphic to S4]]
* [[Octahedral group is isomorphic to S4]]
* [[Triangle group with parameters (2,3,3) is isomorphic to S4]]
* [[von Dyck group with parameters (2,3,4) is isomorphic to S4]]
* [[PGL(2,3) is isomorphic to S4]]
* [[GA(2,2) is isomorphic to S4]]
* [[PSL(2,Z4) is isomorphic to S4]]

==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]] || group of all permutations on a finite set || by a nonnegative integer, denoting size of set acted on || 4 || {{#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::von Dyck group]] || <math>\langle a,b,c \mid a^p = b^q = c^r = abc = e \rangle</math> || <math>(p,q,r)</math>|| (2,3,4) || {{#ask: [[member of family::von Dyck group]]|limit = 0|searchlabel = click here for a list}}|| <math>S_4</math> is a spherical von Dyck group, i.e., it occurs as a finite subgroup of <math>SO(3,\R)</math>. In particular, this makes it a [[Coxeter group]]. {{further|[[Classification of finite subgroups of SO(3,R)]]}}
|-
| [[member of family::triangle group]] || <math>\langle s_1,s_2,s_3 \mid s_1^2 = s_2^2 = s_3^2 = (s_1s_2)^p = (s_2s_3)^q = (s_3s_1)^r = e \rangle</math> || <math>(p,q,r)</math>|| (2,3,3) || {{#ask: [[member of family::triangle group]]|limit = 0|searchlabel = click here for a list}} || <math>S_4</math> is a spherical triangle group
|-
| [[member of family::projective general linear group]] || [[projective general linear group]] of given degree over a given field || name of field, degree || [[field:F3]] (size three), [[projective general linear group of degree two|degree two]] || ||
|}

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 24}}
===Basic arithmetic functions===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| {{arithmetic function value order|24}} || As <math>\! S_n, n = 4:</math> <math>\! n! = 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24</math> <br><br> as <math>\! PGL(2,q), q = 3:</math> <math>\!q^3 - q = 3^3 - 3 = 24</math><br><br>As <math>\! GA(2,q), q = 2:</math> <math>\! q^2 \cdot (q^2 - 1) \cdot (q^2 - q) = 2^2 \cdot (2^2 - 1)(2^2 - 2) = 4 \cdot 3 \cdot 2 = 24</math><br>As <math>\! PSL(2,\mathbb{Z}_4)</math>, where <math>\mathbb{Z}_4</math> is a length <math>l = 2</math> DVR over a field of size <math>q = 2</math>:<br><math>q^{3l - 2}(q - 1)(q +1)/m</math> where <math>m</math> is the number of square roots of unity in the ring. In this case, <math>m = 2</math> and we get <math>2^{3(2) - 2}(2 - 1)(2 + 1)/2 = 24</math><br>As <math>\Gamma A (1,q), q = 4, q = p^r, p = 2, r = 2</math>: <math>rq(q - 1) = 2(4)(4 - 1) = 24</math><br>As [[triangle group]] with parameters <math>(p,q,r) = (2,3,3)</math>: <math>\frac{4}{1/p + 1/q +1/r - 1} = \frac{4}{1/2 + 1/3 + 1/3 - 1} = \frac{4}{1/6} = 24</math><br>As [[von Dyck group]] with parameters <math>(p,q,r) = (2,3,4)</math>: <math>\frac{2}{1/p + 1/q + 1/r - 1} = \frac{2}{1/2 + 1/3 + 1/4 - 1} = \frac{2}{1/12} = 24</math><br><br>More information at [[element structure of symmetric group:S4#Order computation]]
|-
| {{arithmetic function value exponent given order|12|24}} || As <math>S_n, n = 4</math>: <math>\operatorname{lcm} \{1,2,\dots,n \} = \operatorname{lcm} \{ 1,2,3,4 \} = 12</math>
|-
| {{arithmetic function value given order|derived length|3|24}} || Derived series goes through [[alternating group:A4]] and Klein four-group of double transpositions.
|-
| [[nilpotency class]] || -- || || not a nilpotent group.  
|-
| {{arithmetic function value given order|Frattini length|1|24}} ||[[Frattini-free group]]: intersection of maximal subgroups is trivial.
|-
| {{arithmetic function value given order|minimum size of generating set|2|24}} ||<math>(1,2), (1,2,3,4)</math>; see also [[symmetric group on a finite set is 2-generated]]
|-
| {{arithmetic function value given order|subgroup rank of a group|2|24}} ||All proper subgroups are cyclic or have generating set of size two ([[dihedral group:D8|D8]], [[Klein four-group]], and [[alternating group:A4|A4]]) 
|-
| {{arithmetic function value given order|max-length of a group|4|24}} ||Series for Sylow 2-subgroup followed by whole group, OR derived series refined at end. 
|-
| {{arithmetic function value given order|composition length|4|24}}|| 
|-
| {{arithmetic function value given order|chief length|3|24}} || 
|}

===Arithmetic functions of a counting nature===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| {{arithmetic function value given order|number of subgroups|30|24}} ||
|-
| {{arithmetic function value given order|number of conjugacy classes|5|24}} || For more information, see [[element structure of symmetric group:S4#Number of conjugacy classes]]<br>As <math>S_n, n =4:</math> the number of conjugacy classes is <math>\! p(n) = p(4) = 5</math>, where <math>p</math> is the [[number of unordered integer partitions]]; see [[cycle type determines conjugacy class]] <br><br>As <math>PGL(2,q), q = 3:</math> the number of conjugacy classes is <math>q + 2 = 3 +  2 = 5</math>; see [[element structure of projective general linear group of degree two over a finite field]] and [[element structure of symmetric group:S4#Interpretation as projective general linear group of degree two]]<br><br>as <math>GA(2,q), q = 2:</math> <math>\! q^2 + q - 1 = 2^2 + 2 - 1 = 5</math>; see [[element structure of general affine group of degree two over a finite field]] and [[element structure of symmetric group:S4#Interpretation as general affine group of degree two]]<br><br>As <math>\Gamma A(1,p^2), p = 2</math>: <math>p(p + 3)/2 = 2(2 + 3)/2 = 5</math>; see [[element structure of general semiaffine group of degree one over a finite field]] and [[element structure of symmetric group:S4#Interpretation as general semiaffine group of degree one]]
|-
| {{arithmetic function value given order|number of conjugacy classes of subgroups|11|24}} ||
|}

===Lists of numerical invariants===

{| class="sortable" border="1"
! List !! Value !! Explanation/comment
|-
| [[conjugacy class size set|conjugacy class sizes]] || <math>1,3,6,6,8</math> || See [[element structure of symmetric group:S4]], [[element structure of symmetric groups]], [[cycle type determines conjugacy class]]
|-
| [[order statistics]] || <math>1 \mapsto 1, 2 \mapsto 9, 3 \mapsto 8, 4 \mapsto 6</math> ||
|-
| [[degrees of irreducible representations]] || <math>1,1,2,3,3</math> || See [[linear representation theory of symmetric group:S4]], [[linear representation theory of symmetric groups]]
|}

==Group properties==

{{quotation|'''COMPARE AND CONTRAST''': ''Want to know more about how this group compares with symmetric groups of other degrees? Read [[contrasting symmetric groups of various degrees]]''.}}
{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
|[[Dissatisfies property::Abelian group]] || No || <math>(1,2)</math>, <math>(1,3)</math> don't commute || <math>S_n</math> is non-abelian, <math>n \ge 3</math>.
|-
|[[Dissatisfies property::Nilpotent group]] || No || [[Centerless group|Centerless]]: The [[center]] is trivial || <math>S_n</math> is non-nilpotent, <math>n \ge 3</math>.
|-
|[[Dissatisfies property::Metacyclic group]] || No || No [[cyclic normal subgroup]] || <math>S_n</math> is not metacyclic, <math>n \ge 4</math>.
|-
|[[Dissatisfies property::Supersolvable group]] || No || No [[cyclic normal subgroup]] || <math>S_n</math> is not supersolvable, <math>n \ge 4</math>.
|-
|[[Satisfies property::Solvable group]] || Yes || Length three, [[derived subgroup]] is [[A4 in S4]], its derived subgroup is [[Klein four-group]] || Largest <math>n</math> for which <math>S_n</math> is solvable.
|-
|[[Dissatisfies property::T-group]] || No || Double transposition generates non-normal [[2-subnormal subgroup]] || Only <math>n</math> for which <math>S_n</math> isn't a T-group.
|-
|[[Dissatisfies property::HN-group]] || No || Double transposition generates subnormal non-[[hypernormalized subgroup]] ||Only <math>n</math> for which <math>S_n</math> isn't hypernormalized.
|-
|[[Satisfies property::Complete group]] || Yes || Centerless and every automorphism's inner || [[Symmetric groups are complete]] except the ones of degree <math>2,6</math>.
|-
|[[Satisfies property::Monolithic group]] || Yes || Monolith is the Klein four-group of double transpositions || All symmetric groups are monolithic; <math>n=4</math> is the only case the monolith is not the alternating group.
|-
|[[Satisfies property::One-headed group]] || Yes || The alternating group is the unique [[maximal normal subgroup]] || True for all <math>n > 1</math>.
|-
|[[Satisfies property::Group having subgroups of all orders dividing the group order]] || Yes || (See subgroup list) || Largest <math>n</math> for which this is true.
|-
|[[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]] ||
|-
|}

==Endomorphisms==

{{further|[[Endomorphism structure of symmetric group:S4]]}}
===Automorphisms===

Since <math>S_4</math> is a complete group, it is isomorphic to its automorphism group, where each element of <math>S_4</math> acts on <math>S_4</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]]}}

===Endomorphisms===

<math>S_4</math> admits five kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these four):

* The endomorphism to the trivial group
* The identity map
* Endomorphisms with kernel [[A4 in S4]]:
**The retraction to a group of order two, given by the sign homomorphism
*** The endomorphism with kernel [[A4 in S4]] and image the subgroup of order two generated by a double transposition
* The retraction to a symmetric group on three of the elements, with kernel being the Klein four-group comprising the identity element and the double transpositions. (Note that all such retractions are equivalent, and there are other equivalent endomorphisms obtained by composing such a retraction with an automorphism)

==Elements==

{{further|[[element structure of symmetric group:S4]]}}

===Conjugacy class structure===

There are five conjugacy classes, corresponding to the cycle types, because [[cycle type determines conjugacy class]]. Further, each cycle type corresponds to a partition of <math>4</math>.

{{#lst:element structure of symmetric group:S4|conjugacy class structure}}

===Automorphism class structure===

Since <math>S_4</math> is a [[complete group]], all its automorphisms are [[inner automorphism]]s, and, in particular, the classification of elements up to conjugacy is the same as the classification up to automorphisms.

==Subgroups==

{{further|[[Subgroup structure of symmetric group:S4]]}}

{{#lst:subgroup structure of symmetric group:S4|summary}}

==Subgroup-defining functions==

{| class="soritable" border="1"
! Subgroup-defining function !! Subgroup type in list !! Isomorphism class !! Comment
|-
| [[Center]] || trivial subgroup || [[trivial group]] || The group is [[centerless group|centerless]]
|-
| [[Derived subgroup]] || [[A4 in S4]] || [[alternating group:A4]] ||
|-
| [[Frattini subgroup]] || trivial subgroup|| [[trivial group]] || The normal cores of the maximal subgroups of orders <math>6</math> and <math>8</math> intersect trivially.
|-
| [[Socle]] || [[normal Klein four-subgroup of symmetric group:S4]] || [[Klein four-group]] || This subgroup is the unique [[minimal normal subgroup]], i.e., the [[monolith]], and the group is [[monolithic group|monolithic]].
|}

==Linear representation theory==

{{further|[[Linear representation theory of symmetric group:S4]]}}

===Summary===

{{#lst:linear representation theory of symmetric group:S4|summary}}

===Character table===

{{#lst:linear representation theory of symmetric group:S4|character table}}

==Supergroups==

{{further|[[supergroups of symmetric group:S4]]}}

The symmetric group <math>S_4</math> is contained in higher symmetric groups, most notably the [[symmetric group:S5|symmetric group on five elements]] <math>S_5</math>.

===Extensions===

These include <math>GL(2,3)</math> whose [[inner automorphism group]] is <math>S_4</math> (specifically <math>S_4</math> is the quotient of <math>GL(2,3)</math> by its scalar matrices).

==GAP implementation==

{{access GAP implementation online using SAGE|id=5011}}

{{GAP ID|24|12}}

===Short descriptions===

{| class="sortable" border="1"
! Description !! Functions used !! Group storage format (verification command) !! Memory usage
|-
| <tt>SymmetricGroup(4)</tt> || [[GAP:SymmetricGroup|SymmetricGroup]] || permutation group (<tt>IsPermGroup</tt>) || 189
|-
| <tt>PGL(2,3)</tt> || [[GAP:PGL|PGL]] || permutation group (<tt>IsPermGroup</tt>) || 2023
|}