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

_{''n''} is the _{''n''} with _{''n''} is abelian _{5} is the smallest non-abelian _{4} has the _{4} onto . We have the

_{''n''} that are conjugate by an element of A_{''n''} 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 .
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 $n\; \backslash geq\; 8$.

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of even permutation
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s of a finite set
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of the symmetric group
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, having order 60, and the smallest non-solvable group
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, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent
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to be solved by radicals, as established by Lodovico Ferrari
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Conjugacy classes

As in thesymmetric group
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, 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
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s. Thus, Asimple group
In mathematics, a simple group is a nontrivial Group (mathematics), group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgrou ...

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