How can I prove that all automorphisms of a symmetric group $S_n = \{1, 2, ..., n\}$ are inner automorphism except for $n = 6$? I saw some related questions in the forum, but couldn't understand them yet.
And is this exception only applicable to $6$? Is it possible to construct an outer automorphism for it?
I first tried to prove it in a general way, hoping that somewhere along I would have to make an exception for $n = 6$, but that didn't work...
For finding the outer automorphism understand that there are 6 conjugates of the nontrivial $S_5$ living in $S_6$. Use the Elliott Configuration for example https://cp4space.wordpress.com/2012/11/24/outer-automorphism-of-s6/
And then consider the action of $S_6$ on those conjugates, this gives the outer automorphism.