Recall that a graph $G$ is arc transitive if the natural action of $\mathrm{Aut}(G)$ on $A(G) = \{ (u,v) | \{u,v\} \in E(G)\}$ is transitive.
In other words, given $(u,v),(u'.v') \in A(G)$ one finds a $g \in \mathrm{Aut}(G)$ such that $g (u,v) := (g(u),g(v)) = (u',v').$
Our lecturer said it is not completely straightforward to show that $K_n$ is an arc transitive graph. As far as I can see since $\mathrm{Aut}(K_n) = S_n$ one can always find a suitable permutation that maps a pair of vertices into another pair of vertices thus showing that $K_n$ is indeed arc transitive.
Am I missing something? I recall the lecturer talked about considering some cases (i.e if the arcs are not disjoint) but I don't see how this could create any real obstacle in finding the suitable permutation.
Am I missing something crucial here?