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{{short description|Group of even permutations of a finite set}}
{{More footnotes|date=January 2008}}
{{Group theory sidebar |Finite}}

In [[mathematics]], an '''alternating group''' is the [[Group (mathematics)|group]] of [[even permutation]]s of a [[finite set]]. The alternating group on a set of {{mvar|n}} elements is called the '''alternating group of degree {{mvar|n}}''', or the '''alternating group on {{mvar|n}} letters''' and denoted by {{math|A{{sub|''n''}}}} or {{math|Alt(''n'').}}

== Basic properties ==
For {{nowrap|''n'' > 1}}, the group A<sub>''n''</sub> is the [[commutator subgroup]] of the [[symmetric group]] S<sub>''n''</sub> with [[Index of a subgroup|index]] 2 and has therefore [[factorial|''n''!]]/2 elements. It is the [[kernel (algebra)|kernel]] of the signature [[group homomorphism]] {{nowrap|sgn : S<sub>''n''</sub> → {{mset|1, −1}}}} explained under [[symmetric group]].

The group A<sub>''n''</sub> is [[abelian group|abelian]] [[if and only if]] {{nowrap|''n'' ≤ 3}} and [[simple group|simple]] if and only if {{nowrap|1=''n'' = 3}} or {{nowrap|''n'' ≥ 5}}.<!-- Note A3 is in fact a simple group of order 3. A1 and A2 are groups of order 1, so not usually called simple, and A4 has a non-identity proper normal subgroup so is not simple. -->  A<sub>5</sub> is the smallest non-abelian [[simple group]], having [[order of a group|order]] 60, and thus the smallest non-[[solvable group]].

The group A<sub>4</sub> has the [[Klein four-group]] V as a proper [[normal subgroup]], namely the identity and the double transpositions {{nowrap|{{mset| (),  (12)(34), (13)(24), (14)(23) }}}}, that is the kernel of the [[surjection]] of A<sub>4</sub> onto {{nowrap|1=A<sub>3</sub> ≅ Z<sub>3</sub>}}. We have the [[exact sequence]] {{nowrap|1=V → A<sub>4</sub> → A<sub>3</sub> = Z<sub>3</sub>}}. In [[Galois theory]], this map, or rather the corresponding map {{nowrap|S<sub>4</sub> → S<sub>3</sub>}}, corresponds to associating the [[Lagrange resolvent]] cubic to a quartic, which allows the [[quartic polynomial]] to be solved by radicals, as established by [[Lodovico Ferrari]].

== Conjugacy classes ==
As in the [[symmetric group]], any two elements of A<sub>''n''</sub> that are conjugate by an element of  A<sub>''n''</sub> must have the same [[cycle shape]]. The converse is not necessarily true, however. If the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape {{harv|Scott|1987|loc=§11.1, p299}}.

Examples:
*The two [[permutation]]s (123) and (132) are not conjugates in A<sub>3</sub>, although they have the same cycle shape, and are therefore conjugate in S<sub>3</sub>.
*The permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A<sub>8</sub>, although the two permutations have the same cycle shape, so they are conjugate in S<sub>8</sub>.

== Relation with symmetric group ==
:''See [[Symmetric group#Relation with alternating group | Symmetric group]]''.
As finite symmetric groups are the groups of all permutations of a set with finite elements, and the alternating groups are groups of even permutations, alternating groups are subgroups of finite symmetric groups.

== Generators and relations ==
For ''n'' ≥ 3, A<sub>''n''</sub> is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating set is often used to prove that A<sub>''n''</sub> is simple for {{nowrap|''n'' ≥ 5}}.

== Automorphism group ==
{{details|Automorphisms of the symmetric and alternating groups}}
{| align="right" class=wikitable
|-
! ''n''
! Aut(A<sub>''n''</sub>)
! Out(A<sub>''n''</sub>)
|- align=center
|''n'' ≥ 4, ''n'' ≠ 6
| S<sub>''n''</sub>
| Z<sub>2</sub>
|- align=center
| ''n'' = 1, 2
| Z<sub>1</sub>
| Z<sub>1</sub>
|- align=center
| ''n'' = 3
| Z<sub>2</sub>
| Z<sub>2</sub>
|- align=center
| ''n'' = 6
| S<sub>6</sub> ⋊ Z<sub>2</sub>
| V = Z<sub>2</sub> × Z<sub>2</sub>
|}

For {{nowrap|''n'' > 3}}, except for {{nowrap|1=''n'' = 6}}, the  [[automorphism group]] of A<sub>''n''</sub> is the symmetric group S<sub>''n''</sub>, with [[inner automorphism group]] A<sub>''n''</sub> and [[outer automorphism group]] Z<sub>2</sub>; the outer automorphism comes from conjugation by an odd permutation.

For {{nowrap|1=''n'' = 1}} and 2, the automorphism group is trivial. For {{nowrap|1=''n'' = 3}} the automorphism group is Z<sub>2</sub>, with trivial inner automorphism group and outer automorphism group Z<sub>2</sub>.

The outer automorphism group of A<sub>6</sub> is [[Klein four-group|the Klein four-group]] {{nowrap|1=V = Z<sub>2</sub> × Z<sub>2</sub>}}, and is related to [[Symmetric group#Automorphism group|the outer automorphism of S<sub>6</sub>]]. The extra outer automorphism in A<sub>6</sub> swaps the 3-cycles (like&nbsp;(123)) with elements of shape 3<sup>2</sup> (like&nbsp;{{nowrap|(123)(456)}}).

== Exceptional isomorphisms ==
There are some [[exceptional isomorphism]]s between some of the small alternating groups and small [[groups of Lie type]], particularly [[projective special linear group]]s. These are:
* A<sub>4</sub> is isomorphic to PSL<sub>2</sub>(3)<ref name="Robinson-p78">Robinson (1996), [{{Google books|plainurl=y|id=lqyCjUFY6WAC|page=78|text=PSL}} p. 78]</ref> and the [[symmetry group]] of chiral [[tetrahedral symmetry]].
* A<sub>5</sub> is isomorphic to PSL<sub>2</sub>(4), PSL<sub>2</sub>(5), and the symmetry group of chiral [[icosahedral symmetry]]. (See<ref name="Robinson-p78"/>  for an indirect isomorphism of {{nowrap|PSL<sub>2</sub>(F<sub>5</sub>) → A<sub>5</sub>}} using a classification of simple groups of order 60, and [[Projective linear group#Action on p points|here]] for a direct proof).
* A<sub>6</sub> is isomorphic to PSL<sub>2</sub>(9) and PSp<sub>4</sub>(2)'.
* A<sub>8</sub> is isomorphic to PSL<sub>4</sub>(2).

More obviously, A<sub>3</sub> is isomorphic to the [[cyclic group]] Z<sub>3</sub>, and A<sub>0</sub>, A<sub>1</sub>, and A<sub>2</sub> are isomorphic to the [[trivial group]] (which is also {{nowrap|1=SL<sub>1</sub>(''q'') = PSL<sub>1</sub>(''q'')}} for any ''q'').
<!-- This part has a few errors, comment out until they are fixed. A4 is not perfect, SL(4,2)=PSL(4,2)=A8 is not the Schur cover of A8 -->
<!--
The associated extensions {{nowrap|SL<sub>''n''(''q'') → PSL<sub>''n''</sub>(''q'')}} are [[universal perfect central extension]]s for A<sub>4</sub>, A<sub>5</sub>, A<sub>8</sub>, by uniqueness of the universal perfect central extension;
for PSL<sub>2</sub>(9) <math>\cong</math> A<sub>6</sub>, the associated extension is a perfect central extension, but not universal: there is a 3-fold [[Schur multiplier|covering group]].
-->

==Examples S<sub>4</sub> and A<sub>4</sub>==
{|
|-
|  style="vertical-align:top;"|[[File:Symmetric group 4; Cayley table; numbers.svg|thumb|350px|[[Cayley table]] of the [[symmetric group]] S<sub>4</sub><br><br>The [[Parity of a permutation|odd permutations]] are colored:<br>[[Transposition (mathematics)|Transpositions]] in green and [[Cycles and fixed points|4-cycles]] in orange]] || &nbsp;&nbsp;&nbsp; ||  style="vertical-align:top;"|[[File:Alternating group 4; Cayley table; numbers.svg|thumb|350px|Cayley table of the alternating group A<sub>4</sub><br>Elements: The even permutations (the identity, eight [[Cycles and fixed points|3-cycles]] and three <nowiki>double-</nowiki>[[Transposition (mathematics)|transpositions]] (double transpositions in boldface))<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px|Klein four-group]]<br>[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|60px|Cyclic group Z3]]  [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|60px|Cyclic group Z3]]  [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|60px|Cyclic group Z3]]  [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|60px|Cyclic group Z3]]]]
|}

{| class=wikitable
|+ [[Cycle graph (algebra)|Cycle graphs]]
|- align=center valign=top 
| [[File:GroupDiagramMiniC3.svg|200px]]<BR>A<sub>3</sub> = Z<sub>3</sub> (order 3)
| [[File:GroupDiagramMiniA4.svg|200px]]<BR>A<sub>4</sub> (order 12)
| [[File:GroupDiagramMiniA4xC2.png|200px]]<BR>A<sub>4</sub> × Z<sub>2</sub> (order 24)
|- align=center valign=top 
|[[File:GroupDiagramMiniD6.svg|200px]]<BR>S<sub>3</sub> = Dih<sub>3</sub> (order 6)
|[[File:Symmetric group 4; cycle graph.svg|200px]]<BR>S<sub>4</sub> (order 24)
|[[File:Alternating group 4; cycle graph; subgroup of S4.svg|200px]]<BR>A<sub>4</sub> in S<sub>4</sub> on the left
|}

== Example A<sub>5</sub> as a subgroup of 3-space rotations ==
[[File:A5_in_SO(3).gif|thumb|A<sub>5</sub> < SO<sub>3</sub>('''R''')
{{legend|gray|[[ball (mathematics)|ball]] – radius {{pi}} – [[principal homogeneous space]] of SO(3)}}
 {{legend|yellow|[[icosidodecahedron]] – radius {{pi}} – conjugacy class of 2-2-cycles}}
 {{legend|purple|[[icosahedron]] – radius 4{{pi}}/5 – half of the [https://groupprops.subwiki.org/wiki/Splitting_criterion_for_conjugacy_classes_in_the_alternating_group split] conjugacy class of 5-cycles}}
 {{legend|green|[[dodecahedron]] – radius 2{{pi}}/3 – conjugacy class of 3-cycles}}
 {{legend|red|icosahedron – radius 2{{pi}}/5 – second half of split 5-cycles}}
]]
[[File:Compound of five tetrahedra.png|thumb|Compound of five tetrahedra. A<sub>5</sub> acts on the dodecahedron by permuting the 5 inscribed tetrahedra. Even permutations of these tetrahedra are exactly the symmetric rotations of the dodecahedron and characterizes the {{nowrap|A<sub>5</sub> < SO<sub>3</sub>('''R''')}} correspondence.]]

A<sub>5</sub> is the group of isometries of a dodecahedron in 3-space, so there is a representation {{nowrap|A<sub>5</sub> → SO<sub>3</sub>('''R''')}}.

In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element. Each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians. Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A<sub>5</sub> is {{nowrap|1=1 + 12 + 12 + 15 + 20 = 60}}, we obtain four distinct (nontrivial) polyhedra.

The vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of (2,2)-cycles, which is represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by {{pi}} radians, and so can be represented by a vector of length {{pi}} in either of two directions. Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices.

The two conjugacy classes of twelve 5-cycles in A<sub>5</sub> are represented by two icosahedra, of radii 2{{pi}}/5 and 4{{pi}}/5, respectively. The nontrivial outer automorphism in {{nowrap|Out(A<sub>5</sub>) ≃ Z<sub>2</sub>}} interchanges these two classes and the corresponding icosahedra.

==Example: the 15 puzzle==
[[File:15-puzzle magical.svg|thumb|150px|A [[15 puzzle]].]]
It can be proved that the [[15 puzzle]], a famous example of the [[sliding puzzle]], can be represented by the alternating group A<sub>15</sub>,<ref>{{cite web |last1=Beeler |first1=Robert |title=The Fifteen Puzzle: A Motivating Example for the Alternating Group |url=https://faculty.etsu.edu/beelerr/fifteen-supp.pdf |website=faculty.etsu.edu/ |publisher=East Tennessee State University |access-date=2020-12-26 |archive-date=2021-01-07 |archive-url=https://web.archive.org/web/20210107214840/https://faculty.etsu.edu/beelerr/fifteen-supp.pdf |url-status=dead }}</ref> because the combinations of the 15 puzzle can be generated by [[Permutation#Definition|3-cycles]]. In fact, any {{nowrap|2''k'' − 1}} sliding puzzle with square tiles of equal size can be represented by A<sub>2''k''−1</sub>.

==Subgroups==
A<sub>4</sub> is the smallest group demonstrating that the converse of [[Lagrange's theorem (group theory)|Lagrange's theorem]] is not true in general: given a finite group ''G'' and a divisor ''d'' of {{abs|''G''}}, there does not necessarily exist a subgroup of ''G'' with order ''d'': the group {{nowrap|1=''G'' = A<sub>4</sub>}}, of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any distinct nontrivial element generates the whole group.

For all {{nowrap|''n'' > 4}}, A<sub>''n''</sub> has no nontrivial (that is, proper) [[normal subgroup]]s. Thus, A<sub>''n''</sub> is a [[simple group]] for all {{nowrap|''n'' > 4}}. A<sub>5</sub> is the smallest [[solvable group|non-solvable group]].

==Group homology==
{{see also|Symmetric group#Homology}}
The [[group homology]] of the alternating groups exhibits stabilization, as in [[stable homotopy theory]]: for sufficiently large ''n'', it is constant. However, there are some low-dimensional exceptional homology. Note that the [[Symmetric group#Homology|homology of the symmetric group]] exhibits similar stabilization, but without the low-dimensional exceptions (additional homology elements).

=== ''H''<sub>1</sub>: Abelianization ===
The first [[homology group]] coincides with [[abelianization]], and (since A<sub>''n''</sub> is [[perfect group|perfect]], except for the cited exceptions) is thus:
:''H''<sub>1</sub>(A<sub>''n''</sub>, Z) = Z<sub>1</sub> for ''n'' = 0, 1, 2;
:''H''<sub>1</sub>(A<sub>3</sub>, Z) = A{{su|b=3|p=ab|lh=1em}} = A<sub>3</sub> = Z<sub>3</sub>;
:''H''<sub>1</sub>(A<sub>4</sub>, Z) = A{{su|b=4|p=ab|lh=1em}} = Z<sub>3</sub>;
:''H''<sub>1</sub>(A<sub>''n''</sub>, Z) = Z<sub>1</sub> for ''n'' ≥ 5.

This is easily seen directly, as follows. A<sub>''n''</sub> is generated by 3-cycles – so the only non-trivial abelianization maps are {{nowrap|A<sub>''n''</sub> → Z<sub>3</sub>,}} since order-3 elements must map to order-3 elements – and for {{nowrap|''n'' ≥ 5}} all 3-cycles are conjugate, so they must map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like (123) must map to the same element as its inverse (321), but thus must map to the identity, as it must then have order dividing 2 and 3, so the abelianization is trivial.

For {{nowrap|''n'' < 3}}, A<sub>''n''</sub> is trivial, and thus has trivial abelianization. For A<sub>3</sub> and A<sub>4</sub> one can compute the abelianization directly, noting that the 3-cycles form two conjugacy classes (rather than all being conjugate) and there are non-trivial maps {{nowrap|A<sub>3</sub> ↠ Z<sub>3</sub>}} (in fact an isomorphism) and {{nowrap|A<sub>4</sub> ↠ Z<sub>3</sub>}}.

=== ''H''<sub>2</sub>: Schur multipliers ===
{{main|Covering groups of the alternating and symmetric groups}}
The [[Schur multiplier]]s of the alternating groups A<sub>''n''</sub> (in the case where ''n'' is at least 5) are the cyclic groups of order 2, except in the case where ''n'' is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6.<ref name="raw">{{citation|first=Robert |last=Wilson |author-link=Robert Arnott Wilson |date=October 31, 2006 |url=http://www.maths.qmul.ac.uk/~raw/fsgs.html |title=The finite simple groups, 2006 versions |chapter=Chapter 2.7: Covering groups |chapter-url=http://www.maths.qmul.ac.uk/~raw/fsgs_files/alt.ps |url-status=dead |archive-url=https://web.archive.org/web/20110522121819/http://www.maths.qmul.ac.uk/~raw/fsgs.html |archive-date=May 22, 2011 }}</ref> These were first computed in {{Harv|Schur|1911}}.

:''H''<sub>2</sub>(A<sub>''n''</sub>, Z) = Z<sub>1</sub> for ''n'' = 1, 2, 3;
:''H''<sub>2</sub>(A<sub>''n''</sub>, Z) = Z<sub>2</sub> for ''n'' = 4, 5;
:''H''<sub>2</sub>(A<sub>''n''</sub>, Z) = Z<sub>6</sub> for ''n'' = 6, 7;
:''H''<sub>2</sub>(A<sub>''n''</sub>, Z) = Z<sub>2</sub> for ''n'' ≥ 8.

== Notes ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation |last1=Robinson |first1=Derek John Scott |title=A course in the theory of groups |edition=2 |series=Graduate texts in mathematics |volume=80 |year=1996 |publisher=Springer |isbn=978-0-387-94461-6 }}
*{{citation
|first=Issai
|last=Schur
|title=Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen
|journal=[[Journal für die reine und angewandte Mathematik]]
|volume=1911
|year=1911
|issue=139
|pages=155–250
|author-link=Issai Schur
|doi=10.1515/crll.1911.139.155
|s2cid=122809608
}}
*{{Citation | last1=Scott | first1=W.R. | title=Group Theory | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-65377-8 | year=1987 }}

{{Refend}}

==External links==
* {{mathworld | urlname = AlternatingGroup  | title = Alternating group }}
* {{mathworld | urlname = AlternatingGroupGraph  | title = Alternating group graph}}

{{DEFAULTSORT:Alternating Group}}
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[[Category:Permutation groups]]