Alternating Group - Automorphism Group

Automorphism Group

For more details on this topic, see Automorphisms of the symmetric and alternating groups.

For n > 3, except for n = 6, the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2; 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 Z2, with trivial inner automorphism group and outer automorphism group Z2.

The outer automorphism group of A6 is the Klein four-group V = Z2 × Z2, and is related to the outer automorphism of S6. The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 32 (like (123)(456)).

Read more about this topic:  Alternating Group

Famous quotes containing the word group:

    “It’s important to remember that feminism is no longer a group of organizations or leaders. It’s the expectations that parents have for their daughters, and their sons, too. It’s the way we talk about and treat one another. It’s who makes the money and who makes the compromises and who makes the dinner. It’s a state of mind. It’s the way we live now.”
    —Anna Quindlen (20th century)