Let $\sigma\in S_n$ be an $n$-cycle, and let $\tau \in S_n$ be a $2$-cycle. Show by constructing a counterexample that $\sigma$ and $\tau$ need not generate $S_n$.
I know that $S_n$ is in fact generated by its cycles and its transpositions as I have proven this previously, so I feel like maybe I am just misunderstanding what is being asked. I initially approached this by trying to find an element of $S_n$ that cannot be written as a product of an $n$-cycle and a $2$-cycle, but this has not gotten me very far. If someone could nudge me in the right direction I would much appreciate it.
Consider $\sigma=(1234)$ and $\tau=(13)$ in $S_4$. Viewed geometrically, we have four objects in a square where $\sigma$ rotates the square by $90^\circ$ and $\tau$ reflects the square on a fixed axis. But the symmetry group of the square $D_8=\langle\sigma,\tau\rangle$ is a proper subgroup of $S_4$, so permutations such as $(12)$ cannot be generated.