Show that the symmetric group $S_p=\left<\sigma, \tau\right>$, where $\sigma$ is any transposition and $\tau$ is any p-cycle and p is a prime number.

Question

Let $\sigma = (a_1\ a_2) \ \ and \ \ \tau = (a_1\ b_2\ \ldots\ b_p)$. (We have $a_2 = b_i$ for some $i$.) We know that $S_p$ is generated by $(a_1\ a_2)$ and $(a_1\ a_2\ \ldots\ a_p)$. I want to show that $\tau^k(a_1) = a_2$ for some $k$. Please help me.

Thank you

Answer 1

As Michael stated before, you have included that information in the question itself, if $b_{i}=a_{2}$ for some $i$, then $\tau^{i-1}(a_{1})=b_{i}=a_{2}$ or $\tau^{k}(a_{1})=a_{2}$ for $k=i-1$.