Find a generating set for Sym(p)

Let $p$ be a prime. Show that $S_p = \langle \sigma, \tau \rangle$ where $\sigma$ is any transposition and $\tau$ any $p$-cycle.

Let $\sigma = (a_1\ a_2)$ and $\tau = (a_1\ b_2\ \ldots\ b_p)$. (We have $a_2 = b_i$ for some $i$.) By a previous exercise, $\tau^k(a_1) = a_2$ for some $k$. Because $\tau$ has prime order, $\langle \sigma, \tau \rangle = \langle \sigma, \tau^k \rangle$. Relabeling the elements $\{ 1, \ldots, n \}$, by the previous exercise we have $S_p = \langle \sigma, \tau^k \rangle = \langle \sigma, \tau \rangle$.