Automorphism of a symmetric group, except fot $n = 6$

Question

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...

https://math.stackexchange.com/questions/123413/outer-automorphisms-of-s-6 — Angina Seng, Jul 13 '20 at 16:05
The key is to prove that the conjugates of transpositions are transpositions. Unless $n=6$ no other conjugacy classes of involutions have the same number of elements as the set of transpositions. — Angina Seng, Jul 13 '20 at 16:07

score 0 · Answer 1 · answered Jul 13 '20 at 16:37

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.

Automorphism of a symmetric group, except fot $n = 6$

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