Sharply t-transitive groups. Suppose that $G$ acts sharply $t$-transitively on a set $\\{1,\cdots, n\\}$. Then I want to show that if $n = t + 2$ then $G = A_n$. I can indeed show this, but I feel it's unnecessarily messy. My way involves showing that $G$ must contain double transpositions, but never single ones. It would be nice to see a nice snappy argument though.

The conditions completely determine the order of the group to be $\frac{n!}{2!}$ (see for example my answer to Degree of a permutation group), so we know it will be a subgroup of $S_n$ of index $2$. The only such subgroup is $A_n$.