View source for Alternating group:A5

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

The alternating group <math>A_5</math>, also denoted <math>\operatorname{Alt}(5)</math>, and termed the '''alternating group of degree five''', is defined in the following ways:

# It is the group of even permutations (viz., the [[member of family::alternating group]]) on five elements.
# It is the [[member of family::von Dyck group]] (sometimes termed ''triangle group'', though the latter has a slightly different meaning) with parameters <math>(2,3,5)</math> (sometimes written in reverse order as <math>(5,3,2)</math>).
# It is the '''icosahedral group''', i.e., the group of orientation-preserving symmetries of the regular icosahedron (or equivalently, regular dodecahedron). Viewed this way, it is denoted <math>l</math> or <math>532</math>. {{further|[[Classification of finite subgroups of SO(3,R)]], [[Linear representation theory of alternating group:A5]]}}
# It is the [[member of family::projective special linear group]] of degree two over the field of four elements, viz., <math>PSL(2,4)</math>. It is also the [[member of family::special linear group]] of [[member of family::special linear group of degree two|degree two]] over the [[field:F4|field of four elements]], i.e., <math>SL(2,4)</math>. It is also the [[member of family::projective general linear group]] of [[member of family::projective general linear group of degree two|degree two]] over the [[field:F4|field of four elements]], i.e., <math>PGL(2,4)</math>.
# 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:F5|field of five elements]], viz., <math>PSL(2,5)</math>.

===Equivalence of definitions===

* The equivalence of the definitions ''within'' (4) is given by [[isomorphism between linear groups when degree power map is bijective]].
* [[PGL(2,4) is isomorphic to A5]]: equivalence of (1) and (4)
* [[PSL(2,5) is isomorphic to A5]]: equivalence of (1) and (5)

{{quotation|'''IMPORTANT NOTE''': This page concentrates on <math>A_5</math> as an abstract group in its own right. To learn more about this group as a [[subgroup of index two]] inside [[symmetric group:S5]], see [[A5 in S5]].}}

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 60}}

===Basic arithmetic functions===
{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value order|60}} || As <math>A_n, n = 5</math>, <math>n!/2 = 5!/2 = 60</math><br>As <math>SL(2,q), q = 4</math>, <math>q^3 - q = 4^3 - 4 = 60</math><br>As <math>PSL(2,q), q = 5</math>, <math>(q^3 - q)/\operatorname{gcd}(2,q-1) = (5^3 - 5)/2 = 60</math>.<br>As von Dyck group with parameters <math>(p,q,r) = (2,3,5)</math>: <math>\frac{2}{1/p + 1/q + 1/r - 1} = \frac{2}{1/2 + 1/3 + 1/5 - 1} = 60</math><br>See [[element structure of alternating group:A5#Order computation]] for more.
|-
| {{arithmetic function value given order|exponent of a group|30|60}} || As <math>A_n, n = 5</math>, <math>n</math> odd: <math>\operatorname{lcm} \{ 1,2,\dots,n - 2,n \}</math> (exclude <math>n - 1</math>): becomes <math>\operatorname{lcm}\{1,2,3,5 \} = 30</math><br>As <math>SL(2,q), q = 4</math> (power of 2): <math>2(q^2 - 1) = 2(4^2 - 1) = 30</math><br>As <math>PSL(2,q), q = 5, p = 5</math> (odd): <math>p(q^2 - 1)/4 = 5(5^2 - 1)/4 = 5(6) = 30</math>
|-
| [[derived length]] || -- || -- || not a solvable group.
|-
| [[nilpotency class]] || -- || -- || not a nilpotent group.
|-
| {{arithmetic function value given order|Frattini length|1|60}} || [[Frattini-free group]]: intersection of maximal subgroups is trivial.
|-
| {{arithmetic function value given order|minimum size of generating set|2|60}} || <math>(1,2,3), (1,2,3,4,5)</math>.
|-
| {{arithmetic function value given order|subgroup rank of a group|2|60}} || --
|-
| {{arithmetic function value given order|max-length of a group|4|60}} || --
|-
| {{arithmetic function value given order|composition length|1|60}} || --
|-
| {{arithmetic function value given order|chief length|1|60}} || --
|}

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

{| class="sortable" border="1"
! Function !! Value !! Explanation
|-
| [[number of subgroups]] || [[arithmetic function value::number of subgroups;59|59]] || [[subgroup structure of alternating group:A5]]
|-
| [[number of conjugacy classes]] || [[arithmetic function value::number of conjugacy classes;5|5]] || As <math>A_n, n = 5</math>: (2 * (number of self-conjugate partitions of 5)) + (number of conjugate pairs of non-self-conjugate partitions of 5) = <math>2(1) + 3 = 5</math> (more [[element structure of alternating groups|here]])<br>As <math>PSL(2,q)</math>, <math>q = 5</math>: <math>(q + 5)/2 = (5 + 5)/2 = 5</math> (more [[element structure of projective special linear group of degree two over a finite field#Number of conjugacy classes|here]])<br>As <math>PGL(2,q)</math>, <math>q = 4</math> (even): <math>q + 1 = 4 + 1 = 5</math> (more [[element structure of projective general linear group of degree two over a finite field#Number of conjugacy classes|here]])<br>See [[element structure of alternating group:A5#Number of conjugacy classes]] for more
|-
| [[number of conjugacy classes of subgroups]] || [[arithmetic function value::number of conjugacy classes of subgroups;9|9]] || [[subgroup structure of alternating group:A5]]
|}

==Group properties==

===Basic properties===

{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
|[[Dissatisfies property::abelian group]] || No || <math>(1,2,3)</math>, <math>(1,2,3,4,5)</math> don't commute || <math>A_n</math> is non-abelian, <math>n \ge 4</math>.
|-
|[[Dissatisfies property::nilpotent group]] || No || [[Centerless group|Centerless]]: The [[center]] is trivial || <math>A_n</math> is non-nilpotent, <math>n \ge 4</math>.
|-
|[[Dissatisfies property::metacyclic group]] || No || Simple and non-abelian || <math>A_n</math> is not metacyclic, <math>n \ge 4</math>.
|-
|[[Dissatisfies property::supersolvable group]] || No || Simple and non-abelian || <math>A_n</math> is not supersolvable, <math>n \ge 4</math>.
|-
|[[Dissatisfies property::solvable group]] || No ||  || <math>A_n</math> is not solvable, <math>n \ge 5</math>.
|-
|[[Satisfies property::simple group]] || Yes || Smallest [[satisfies property::simple non-abelian group]] || 
|}

===Other properties===

{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
|[[Satisfies property::T-group]] || Yes || Simple and non-abelian || 
|-
|[[Dissatisfies property::rational-representation group]] || No ||  ||
|-
|[[Dissatisfies property::rational group]] || No || ||
|-
|[[Satisfies property::ambivalent group]] || Yes ||  || Also see [[classification of ambivalent alternating groups]] 
|-
|[[Dissatisfies property::complete group]] || No || Conjugation by odd permutations of <math>S_5</math> gives outer automorphisms ||
|-
|[[Satisfies property::perfect group]] || Yes || Follows from it being a [[simple non-abelian group]]. || 
|-
|[[Dissatisfies property::Schur-trivial group]] || No || The [[Schur multiplier]] is [[cyclic group:Z2]]. ||
|-
|[[satisfies property::N-group]] || Yes || See [[classification of alternating groups that are N-groups]] || <matH>A_n</math> is a N-group only for <math>n \le 7</math>.
|}

==Endomorphisms==

===Automorphisms===

The automorphism group of <math>A_5</math> is the [[symmetric group:S5|symmetric group on five letters]] <math>S_5</math>, with <math>A_5</math> embedded in it as inner automorphisms.

Concretely, we can think of <math>A_5</math> as embedded in <math>S_5</math>, and <math>S_5</math> acting on <math>A_5</math> by conjugation. The automorphisms obtained this way are all the automorphisms of <math>A_5</math>.

===Other endomorphisms===

Since <math>A_5</math> is a finite simple group, it is a [[group in which every endomorphism is trivial or an automorphism]]. In particular, the endomorphisms of <math>A_5</math> are: the trivial map, and the automorphisms described above.

==Elements==

{{further|[[element structure of alternating group:A5]]}}

===Summary===

{{#lst:element structure of alternating group:A5|summary}}

===Conjugacy class structure===

{{#lst:element structure of alternating group:A5|conjugacy class structure}}

===Up to automorphism===

Under outer automorphisms, the fourth and fifth conjugacy classes get merged. Thus, the classes under automorphism are of size <math>1,15,20,24</math>.

==Subgroups==

{{further|[[Subgroup structure of alternating group:A5]]}}

{{#lst:subgroup structure of alternating group:A5|summary}}

==Linear representation theory==

{{further|[[linear representation theory of alternating group:A5]]}}

===Summary===

{{#lst:linear representation theory of alternating group:A5|summary}}

===Character table===

{{#lst:linear representation theory of alternating group:A5|character table}}

==Supergroups==

{{further|[[supergroups of alternating group:A5]]}}
{{#lst:supergroups of alternating group:A5|minimalist}}

==External links==

* {{atlas|A5}}

==GAP implementation==

{{GAP ID|60|5}}

'''Equivalent command''': Running this command constructs the group as <tt>AlternatingGroup(5)</tt>.

'''Memory usage''': The memory usage for the SmallGroup construction is 1452.

===Other descriptions===

{| class="sortable" border="1"
! Description !! Functions used !! Output storage format (verification command) !! Memory usage (can be computed by invoking the <tt>MemoryUsage</tt> function) !! Groups containing this as a subgroup
|-
| <tt>AlternatingGroup(5)</tt> || [[GAP:AlternatingGroup|AlternatingGroup]] || permutation group (<tt>IsPermGroup</tt>) || 194 || <tt>AlternatingGroup</tt> or <tt>SymmetricGroup</tt> of degree at least five.
|-
| <tt>PSL(2,4)</tt> || [[GAP:PSL|PSL]] || permutation group (<tt>IsPermGroup</tt>) || 2289 || itself
|-
| <tt>SL(2,4)</tt> || [[GAP:SL|SL]] || matrix group (<tt>IsMatrixGroup</tt>) || 1194 || <tt>GL(2,q)</tt>, for <math>q</matH> a power of 4.
|-
| <tt>PGL(2,4)</tt> || [[GAP:PGL|PGL]] || permutation group (<tt>IsPermGroup</tt>) || 2289 || itself
|-
| <tt>PSL(2,5)</tt> || [[GAP:PSL|PSL]] || permutation group (<tt>IsPermGroup</tt>) || 2125 || <tt>PGL(2,5)</tt>
|-
| <tt>PerfectGroup(60)</tt> or equivalently <tt>PerfectGroup(60,1)</tt> || [[GAP:PerfectGroup|PerfectGroup]] || finitely presented group (<tt>IsFpGroup</tt>) || 233 || --
|-
| <tt>SimpleGroup("Alt",5)</tt> || [[GAP:SimpleGroup|SimpleGroup]] || permutation group (<tt>IsPermGroup</tt>) || 229 || --
|-
| <tt>SmallSimpleGroup(60)</tt> || [[GAP:SmallSimpleGroup|SmallSimpleGroup]] || permutation group (<tt>IsPermGroup</tt>) || 229 || --
|-
| <tt>AllSmallNonabelianSimpleGroups([1..100])[1]</tt> || [[GAP:AllSmallNonabelianSimpleGroups|AllSmallNonabelianSimpleGroups]] || permutation group (<tt>IsPermGroup</tt>) || 229 || --
|}