The Outer Automorphisms of The Symmetric and Alternating Groups
For more details on this topic, see Automorphisms of the symmetric and alternating groups.The outer automorphism group of a finite simple group in some infinite family of finite simple groups can almost always be given by a uniform formula that works for all elements of the family. There is just one exception to this: the alternating group A6 has outer automorphism group of order 4, rather than 2 as do the other simple alternating groups (given by conjugation by an odd permutation). Equivalently the symmetric group S6 is the only symmetric group with a non-trivial outer automorphism group.
Read more about this topic: Outer Automorphism Group
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