I am learning about automorphisms, and came across the fact that:
$$\text{Out} (S_n) = \{e \} \hspace{15pt} \text{when} \hspace{5pt} n \neq 6$$
I see why this is true for n in general, since $\text{Aut} (S_n) \cong S_n$ and I can convince myself that $\text{Inn}(S_n) \cong S_n$, meaning that:
$$\text{Out} (S_n) \cong \text{Aut}(S_n) / \text{Inn}(S_n) \cong \{e \}$$
What fails when n = 6?
For all $n \ne 6$, single transpositions have larger centralizers in $S_n$ than elements of order $2$ in any other conjugacy classes. So any automorphism of $S_n$ must map transpositions to transpositions, and from that it is not too difficult to deduce that the automorphism must be inner.
But in $S_6$, the elements $(1,2)$ and $(1,2)(3,4)(5,6)$ both have centralizers of order $48$, and it turns out that there is an automorphism of $S_6$ that maps one to the other.