Generators of symmetric group $S_n$ [duplicate]
Hint:
- show (or be convinced of the fact) that $S_n$ is generated by permutations of $i$ and $j$; so you should prove that they are in your generated subgroup of $S_n$
- $(i\,j)$ can be obtained from combining permutations $(i \, i+1)$
- as per the comment above, look at what happens if you apply $s = (1\,2\ldots n)$ $m$ times (i.e. $s^m$), then $t = (1 \, 2)$, and then $s^{-1}$ $m$ times (i.e. $s^{-m}$).
You should get there.