Automorphisms of the symmetric and alternating groups

In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements. , and thus . Formally, is complete and the natural map is an isomorphism. , and the outer automorphism is conjugation by an odd permutation. Indeed, the natural maps are isomorphisms. trivial: , and is a semidirect product. , and Among symmetric groups, only S6 has a non-trivial outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S6) = C2. This was discovered by Otto Hölder in 1895. The specific nature of the outer automorphism is as follows: the sole identity permutation maps to itself; a 2-cycle such as (1 2) maps to the product of three 2-cycles such as (1 2)(3 4)(5 6) and vice versa, there being 15 permutations each way; a 3-cycle such as (1 2 3) maps to the product of two 3-cycles such as (1 4 5)(2 6 3) and vice versa, accounting for 40 permutations each way; a 4-cycle such as (1 2 3 4) maps to another 4-cycle such as (1 6 2 4) accounting for 90 permutations; the product of two 2-cycles such as (1 2)(3 4) maps to another product of two 2-cycles such as (3 5)(4 6), accounting for 45 permutations; a 5-cycle such as (1 2 3 4 5) maps to other 5-cycles such as (1 3 6 5 2) accounting for 144 permutations; the product of a 2-cycle and a 3-cycle such as (1 2 3)(4 5) maps to a 6-cycle such as (1 2 5 3 4 6) and vice versa, accounting for 120 permutations each way; the product of a 2-cycle and a 4-cycle such as (1 2 3 4)(5 6) maps to another such permutation such as (1 4 2 6)(3 5) accounting for the 90 remaining permutations. Thus, all 720 permutations on 6 elements are accounted for. The outer automorphism does not preserve cycle structure in general, mapping single cycles to the product of two cycles and vice versa.