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In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by A_{n} or Alt(n).## Basic properties

## Conjugacy classes

## Automorphism group

## Exceptional isomorphisms

## Subgroups

A_{4} is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d: the group G = A_{4, of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any additional element (except e) generates the whole group.
Group homology
The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large n, it is constant.
H1: Abelianization
The first homology group coincides with abelianization, and (since $A\_n$ is perfect, except for the cited exceptions) is thus:
$H\_1(A\_3,mathbf\{Z\})=A\_3^\{text\{ab\}\}\; =\; A\_3\; =\; mathbf\{Z\}/3$;
$H\_1(A\_4,mathbf\{Z\})=A\_4^\{text\{ab\}\}\; =\; mathbf\{Z\}/3$;
$H\_1(A\_n,mathbf\{Z\})=0$ for $n=1,2$ and $ngeq\; 5$.
H2: Schur multipliers The Schur multipliers of the alternating groups An (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 a triple cover. In these cases, then, the Schur multiplier is of order 6.$H\_2(A\_n,mathbf\{Z\})=0$ for $n\; =\; 1,2,3$;
$H\_2(A\_n,mathbf\{Z\})=mathbf\{Z\}/6$ for $n\; =\; 6,7$;
$H\_2(A\_n,mathbf\{Z\})=mathbf\{Z\}/2$ for $n\; =\; 4,5$ and $n\; geq\; 8$.
References
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For instance, the alternating group of degree 4 is A_{4} = {e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23)} (see cycle notation).

For n > 1, the group A_{n} is the commutator subgroup of the symmetric group S_{n} with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : S_{n} → {1, −1} explained under symmetric group.

The group A_{n} is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A_{5} is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.

As in the symmetric group, the conjugacy classes in A_{n} consist of elements with the same cycle shape. 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}.

$n$ | $mbox\{Aut\}(A\_n)$ | $mbox\{Out\}(A\_n)$ |

$ngeq\; 4,\; nneq\; 6$ | $S\_n,$ | $C\_2,$ |

$n=1,2,$ | $1,$ | $1,$ |

$n=3,$ | $C\_2,$ | $C\_2,$ |

$n=6,$ | $S\_6\; rtimes\; C\_2$ | $V=C\_2\; times\; C\_2$ |

For n > 3, except for n = 6, the automorphism group of A_{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 n = 1 and 2, the automorphism group is trivial. For n = 3 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 the Klein four-group V = Z_{2} × Z_{2}, and is related to 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)).

There are some isomorphisms between some of the small alternating groups and small groups of Lie type. These are:

- A
_{4}is isomorphic to PSL_{2}(3) 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. - 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_{1} and A_{2} are isomorphic to the trivial group (which is also SL_{1}(q)=PSL_{1}(q) for any q).

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Last updated on Monday August 18, 2008 at 12:48:02 PDT (GMT -0700)

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Last updated on Monday August 18, 2008 at 12:48:02 PDT (GMT -0700)

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