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Alternating group
In mathematics an alternating group is the group of even permutations of a 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 An. For instance:
{1234,
1342,
1423,
2143,
2314,
2431,
3124,
3241,
3412,
4132,
4213,
4321}
is the alternating group of degree 4.
For n > 1, the group An is a normal subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : Sn → {1, -1} explained under symmetric group.
The group An is abelian iff n ≤ 3 and simple iff n = 3 or n ≥ 5.
A5 is the smallest non-abelian simple group.
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