Alternating group
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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 An or Alt(n).
For instance: {1234, 1342, 1423, 2143, 2314, 2431, 3124, 3241, 3412, 4132, 4213, 4321} is the alternating group of degree 4.
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Basic properties
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, having order 60.
Conjugacy classes
As in the symmetric group, the conjugacy classes in An consist of elements with the same cycle shape. However, if the cycle shape consists of cycles of odd length with no two cycles the same length, then there are exactly two conjugacy classes for this cycle shape. For instance, the permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A8, although the two permutations have the same cycle shape, so they are conjugate in S8.
Automorphism group
The automorphism group of An is the symmetric group Sn, except when n=6. The automorphism group of A6 is A6.22. The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 32 (like (123)(456)).
Exceptional isomorphisms
There are some isomorphisms between some of the small alternating groups and small groups of Lie type. These are:
- A5 is isomorphic to PSL2(4) and PSL2(5)
- A6 is isomorphic to PSL2(9) and PSp4(2)'
- A8 is isomorphic to PSL4(2)
More obviously, A3 is isomorphic to the cyclic group C3, and A1 and A2 are isomorphic to the trivial group.