Outer Automorphism Group - The Outer Automorphisms of The Symmetric and Alternating Groups

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.

\begin{align} n\neq 6: \mathrm{Out}(S_n) & = 1 \\ n\geq 3,\ n\neq 6: \mathrm{Out}(A_n) & = C_2 \\ \mathrm{Out}(S_6) & = C_2 \\ \mathrm{Out}(A_6) & = C_2 \times C_2 \end{align}

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