I know that the symmetric group $S_n$ is generated by $(12)$ and $(2345\dots n)$.
Let $G$ be a transitive subgroup of $S_n$ (transitive with respect to the natural action of $S_n$ on $\{1,2,\dots,n\}$) that contains a transposition and an $(n-1)$-cycle. Prove that $G=S_n$.