Symmetric Group - Automorphism Group

Automorphism Group

For more details on this topic, see Automorphisms of the symmetric and alternating groups.
n
1 1
1 1
1

For, is a complete group: its center and outer automorphism group are both trivial.

For n = 2, the automorphism group is trivial, but S2 is not trivial: it is isomorphic to, which is abelian, and hence the center is the whole group.

For n = 6, it has an outer automorphism of order 2:, and the automorphism group is a semidirect product

In fact, for any set X of cardinality other than 6, every automorphism of the symmetric group on X is inner, a result first due to (Schreier & Ulam 1937) according to (Dixon & Mortimer 1996, p. 259).

Read more about this topic:  Symmetric Group

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