View source for Alternating group:A4

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

The alternating group <math>A_4</math> is defined in the following equivalent ways:

* It is the group of even permutations (viz., the [[member of family::alternating group]]) on four elements.
* It is the [[member of family::von Dyck group]] (sometimes termed ''triangle group'', though [[triangle group]] has an alternative interpretation) with parameters <math>(2,3,3)</math> (sometimes written in reverse order as <math>(3,3,2)</math>).
* It is the group of orientation-preserving symmetries of a [[regular tetrahedron]]. When viewed in this light, it is called the '''tetrahedral group''', and its symbol as a [[point group]] is <math>T</math> or <math>332</math>. {{further|[[Classification of finite subgroups of SO(3,R)]], [[Linear representation theory of alternating group:A4]]}}
* It is the [[member of family::projective special linear group]] of [[member of family::projective special linear group of degree two|degree two]] over the field of three elements, viz., <math>PSL(2,3)</math>.
* It is the [[member of family::general affine group]] of degree <math>1</math> over the field of four elements, viz., <math>GA(1,4)</math> (also written as <math>AGL(1,4)</math>.

===Equivalence of definitions===

* [[von Dyck group with parameters (2,3,3) is isomorphic to A4]]
* [[Tetrahedral group is isomorphic to A4]]
* [[PSL(2,3) is isomorphic to A4]]
* [[GA(1,4) is isomorphic to A4]]

{{quotation|'''IMPORTANT NOTE''': This page concentrates on <math>A_4</math> as an abstract group in its own right. To learn more about this group as a [[subgroup of index two]] inside [[symmetric group:S4]], see [[A4 in S4]].}}

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 12}}

===Basic arithmetic functions===

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value order|12}} || As <math>A_n, n = 4</math>: <math>n!/2 = 4!/2 = 12</math><br>As <math>PSL(2,q), q = 3</math>: <math>(q^3 - q)/\operatorname{gcd}(2,q - 1) = (3^3 - 3)/2 = 12</math><br>As <math>GA(1,q), q = 4</math>: <math>q(q - 1) = 4(3) = 12</math><br>As [[von Dyck group]] with parameters <math>(p,q,r) = (2,3,3)</math>: <math>\frac{2}{1/p + 1/q + 1/r - 1} = \frac{2}{1/2 + 1/3 + 1/3 - 1} = \frac{2}{1/6} = 12</math><br>See [[element structure of alternating group:A4#Order computation]] for more information.
|-
| {{arithmetic function value given order|exponent of a group|6|12}} || Elements of order <math>2,3</math>.
|-
| {{arithmetic function value given order|derived length|2|12}} || Derived series goes through Klein four-group of double transpositions.
|-
| [[nilpotency class]] ||  -- || -- || not a nilpotent group.
|-
| {{arithmetic function value given order|Frattini length|1|12}} || [[Frattini-free group]]: intersection of maximal subgroups is trivial.
|-
| {{arithmetic function value given order|minimum size of generating set|2|12}} || <math>(1,2,3), (1,2)(3,4)</math>
|-
| {{arithmetic function value given order|subgroup rank of a group|2|12}} || 
|-
| {{arithmetic function value given order|max-length of a group|3|12}} || 
|-
| {{arithmetic function value given order|composition length|3|12}} ||
|-
| {{arithmetic function value given order|chief length|2|12}} || The unique minimal normal subgroup is also the unique maximal normal subgroup and has order <math>2^2 = 4</math> and index <math>3</math>.
|}

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

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value given order|number of conjugacy classes|4|12}} || As <math>A_n, n = 4</math>: (Number of pairs of non-self-conjugate partitions of <math>n</math>) + 2(Number of self-conjugate partitions of <math>n</math>) = 2 + 2(1) = 4<br>As <math>PSL(2,q), q = 3</math>: <math>(q + 5)/2 = (3 + 5)/2 = 4</math><br>As <math>GA(1,q), q = 4</math>, <math>q = 4</math><br>See [[element structure of alternating group:A4#Number of conjugacy classes]] for more information.
|-
| {{arithmetic function value given order|number of orbits under automorphism group|3|12}} || See [[element structure of alternating group:A4]]. The two conjugacy classes of elements of order three are fused under the action of the automorphism group.
|-
| {{arithmetic function value given order|number of equivalence classes under real conjugacy|3|12}} || See [[element structure of alternating group:A4]]. The two conjugacy classes of elements of order three are fused under real conjugacy because they are inverses of each other.
|-
|  {{arithmetic function value given order|number of equivalence classes under rational conjugacy|3|12}} ||See [[element structure of alternating group:A4]]. The two conjugacy classes of elements of order three are fused under the action of the automorphism group.
|-
| {{arithmetic function value given order|number of conjugacy classes of real elements|2|12}} ||See [[element structure of alternating group:A4]]. The identity element and the conjugacy class of double transpositions are the only such classes.
|-
| {{arithmetic function value given order|number of conjugacy classes of rational elements|2|12}} ||See [[element structure of alternating group:A4]]. The identity element and the conjugacy class of double transpositions are the only such classes.
|-
| {{arithmetic function value given order|number of conjugacy classes of subgroups|5|12}} || See [[subgroup structure of alternating group:A4]]
|-
| {{arithmetic function value given order|number of subgroups|10|12}} || See [[subgroup structure of alternating group:A4]]
|}

==Group properties==

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[Dissatisfies property::Abelian group]] || No || <math>(1,2,3)</math>, <math>(1,2)(3,4)</math> don't commute || <math>A_n</math> is non-abelian, <math>n \ge 4</math>.
|-
|[[Dissatisfies property::Nilpotent group]] || No || [[satisfies property::Centerless group|Centerless]]: The [[center]] is trivial || <math>A_n</math> is non-nilpotent, <math>n \ge 4</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 two, derived subgroup is Klein four-group || Largest <math>n</math> for which <math>A_n</math> is solvable.
|-
|[[Dissatisfies property::T-group]] || No || Double transposition generates non-normal [[2-subnormal subgroup]] || Only <math>n</math> for which <math>A_n</math> isn't a T-group.
|-
|[[Dissatisfies property::Group having subgroups of all orders dividing the group order]] || No || No subgroup of order six || 
|-
|[[Dissatisfies property::Ambivalent group]] || No || ||[[Classification of ambivalent alternating groups]]
|-
|[[Dissatisfies property::Rational group]] || No || ||
|-
|[[Dissatisfies property::Rational-representation group]] || No || ||
|-
|[[Satisfies property::Group in which every element is automorphic to its inverse]] || Yes || [[Alternating group implies every element is automorphic to its inverse]] ||
|-
|[[Satisfies property::Group in which any two elements generating the same cyclic subgroup are automorphic]] || Yes || [[Alternating group implies any two elements generating the same cyclic subgroup are automorphic]] ||
|-
|[[Satisfies property::Frobenius group]] || Yes || Frobenius kernel is Klein four-subgroup, complement is generated by 3-cycle ||
|-
|[[Satisfies property::Camina group]] || Yes || [[Derived subgroup]] is Klein four-subgroup, other two cosets are conjugacy classes.||
|}

==Endomorphisms==

===Automorphisms===

The automorphism group of the alternating group of degree four is isomorphic to the symmetric group of degree four. Since the alternating group of degree four is centerless, it embeds as a subgroup inside its automorphism group.

Another way of thinking of this is as follows: in the symmetric group of degree four, the alternating group of degree four is a subgroup of index two, and every automorphism of this subgroup is realized as the restriction to this subgroup of an inner automorphism of the symmetric group.

===Endomorphisms===

The endomorphisms of the alternating group of degree four are given by the following (i.e., equivalent to one of these up to composition with an automorphism):

* The trivial map.
* The identity map.
* The retraction to a subgroup of order three, with kernel being the [[Klein four-group]] comprising the identity and the double transpositions. (All such retractions are equivalent).

==Elements==

{{further|[[element structure of alternating group:A4]]}}

===Summary===

{{#lst:element structure of alternating group:A4|summary}}

===Up to conjugacy===

{{#lst:element structure of alternating group:A4|conjugacy class structure}}

===Up to automorphism===

The conjugacy classes of the identity element and double transpositions are invariant under all automorphisms.

An outer automorphism interchanges the conjugacy classes of elements of order three (each of size four). This can be realized, for instance, by viewing the alternating group as a subgroup of the [[symmetric group:S4|symmetric group of degree four]]. Any [[transposition]] or <math>4</math>-cycle in the symmetric group interchanges those two conjugacy classes

==Subgroups==

{{further|[[Subgroup structure of alternating group:A4]]}}
{{#lst:subgroup structure of alternating group:A4|summary}}

==Supergroups==

{{further|[[supergroups of alternating group:A4]]}}
{{#lst:supergroups of alternating group:A4|minimalist}}

==Endomorphisms==

{{further|[[endomorphism structure of alternating group:A4]]}}
{{#lst:endomorphism structure of alternating group:A4|summary}}

==Extensions==

''These are groups having the alternating group as a [[quotient group]]''
Perhaps the most important of these is <math>SL(2,3)</math>, which is the [[universal central extension]] of <math>PSL(2,3)</math>. The kernel of the projection mapping is a two-element subgroup, namely the identity matrix and the negative identity matrix.

==GAP implementation==

{{GAP ID|12|3}}

===Other definitions===
The alternating group can be constructed in many equivalent ways:

* As the alternating group of degree four, using GAP's [[GAP:AlternatingGroup|AlternatingGroup]] function:

<tt>AlternatingGroup(4)</tt>

* Using the von Dyck presentation. Here is a sequence of steps:
<pre>F := FreeGroup(3);
G := F/[F.1^3, F.2^3, F.3^2, F.1*F.2*F.3]</pre>

The output <math>G</math> is the alternating group.
* As the [[projective special linear group]], using GAP's [[GAP:ProjectiveSpecialLinearGroup|ProjectiveSpecialLinearGroup]] function:

<tt>PSL(2,3)</tt>