I have to proof, that a symmetric group $S_n$, with $n \geq 2$ with support $\{a_1, a_2, ... , a_p\}$ such that $σ(a_k)= a_k+1$ (when $k < p$) and $σ(a_p)=a_1$, is generated by transposition $(1,2)$ and the cycle $σ=(1,2,...,n)$.
I started by verifying that $σ(1)(1,2)σ(2) = (2,3)$. Then by iteration I show that the next transposition $(3,4) = σ(1)(2,3) σ(2)...$ Iterating until transposition $(n,1)$. After this how can I then proof that $S_n$ is generated by this transposition?
Thanks!
