In _{''n''} or Alt(''n'').

_{''n''} is the commutator subgroup of the symmetric group S_{''n''} with _{''n''} is abelian _{5} is the smallest non-abelian _{4} has the Klein four-group V as a proper _{4} onto . We have the

_{''n''} that are conjugate by an element of A_{''n''} must have the same cycle decomposition (group theory), 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 .
Examples:
*The two permutations (123) and (132) are not conjugates in A_{3}, although they have the same cycle shape, and are therefore conjugate in S_{3}.
*The permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A_{8}, although the two permutations have the same cycle shape, so they are conjugate in S_{8}.

_{''n''} 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_{''n''} is simple for .

_{''n''} is the symmetric group S_{''n''}, with inner automorphism group A_{''n''} and outer automorphism group Z_{2}; the outer automorphism comes from conjugation by an odd permutation.
For and 2, the automorphism group is trivial. For the automorphism group is Z_{2}, with trivial inner automorphism group and outer automorphism group Z_{2}.
The outer automorphism group of A_{6} is Klein four-group, the Klein four-group , and is related to Symmetric group#Automorphism group, the outer automorphism of S_{6}. The extra outer automorphism in A_{6} swaps the 3-cycles (like (123)) with elements of shape 3^{2} (like (123)(456)).

_{4} is isomorphic to PSL_{2}(3)Robinson (1996), [ p. 78] and the symmetry group of chiral tetrahedral symmetry.
* A_{5} is isomorphic to PSL_{2}(4), PSL_{2}(5), and the symmetry group of chiral icosahedral symmetry. (See for an indirect isomorphism of using a classification of simple groups of order 60, and Projective linear group#Action on p points, here for a direct proof).
* A_{6} is isomorphic to PSL_{2}(9) and PSp_{4}(2)'.
* A_{8} is isomorphic to PSL_{4}(2).
More obviously, A_{3} is isomorphic to the cyclic group Z_{3}, and A_{0}, A_{1}, and A_{2} are isomorphic to the trivial group (which is also for any ''q'').

_{4} 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 , there does not necessarily exist a subgroup of ''G'' with order ''d'': the group , 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 , A_{''n''} has no nontrivial (that is, proper) _{''n''} is a _{5} is the smallest solvable group, non-solvable group.

_{''n''} (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. These were first computed in .
:$H\_2(\backslash mathrm\_n,\backslash mathrm)=0$ for $n\; =\; 1,2,3$;
:$H\_2(\backslash mathrm\_n,\backslash mathrm)=\backslash mathrm/2$ for $n\; =\; 4,5$;
:$H\_2(\backslash mathrm\_n,\backslash mathrm)=\backslash mathrm/6$ for $n\; =\; 6,7$;
:$H\_2(\backslash mathrm\_n,\backslash mathrm)=\backslash mathrm/2$ for $n\; \backslash geq\; 8$.

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, an alternating group is the group of even permutations of a finite set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. The alternating group on a set of ''n'' elements is called the alternating group of degree ''n'', or the alternating group on ''n'' letters and denoted by ABasic properties

For , the group Aindex
Index may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a halo megastructure in the ''Halo'' series of ...

2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group.
The group Aif and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondit ...

and simple if and only if or . Asimple group
Simple or SIMPLE may refer to:
*Simplicity
Simplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something i ...

, having order 60, and the smallest non- solvable group.
The group Anormal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group i ...

, namely the identity and the double transpositions that is the kernel of the surjection of Aexact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules) such that the image
An SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teid ...

. In Galois theory, this map, or rather the corresponding map , 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 ARelation with symmetric group

:''See Symmetric group#Relation with alternating group , Symmetric group''.Generators and relations

AAutomorphism group

For , except for , the automorphism group of AExceptional isomorphisms

There are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are: * A Examples ''S''_{4} and ''A''_{4}

Example A_{5} as a subgroup of 3-space rotations

Example: the 15 puzzle

It can be proved that the 15 puzzle, a famous example of the sliding puzzle, can be represented by the alternating group $A\_$, because the combinations of the 15 puzzle can be generated by Permutation#Definition, 3-cycles. In fact, any $2\; \backslash times\; k\; -\; 1$ sliding puzzle with square tiles of equal size can be represented by $A\_$.Subgroups

Anormal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group i ...

s. Thus, Asimple group
Simple or SIMPLE may refer to:
*Simplicity
Simplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something i ...

for all . AGroup 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_{1}: Abelianization

H_{2}: Schur multipliers

Notes

References

* * *External links

* * {{DEFAULTSORT:Alternating Group Finite groups Permutation groups