Here $\displaystyle S_p$ is the symmetric group of order $\displaystyle p!$. We must prove that an arbitrary transposition (that is, a cycle of length 2) and cycle of length $\displaystyle p$ generate $\displaystyle S_p$.Show that if $\displaystyle p$ is prime, $\displaystyle S_p=\langle\sigma,\tau\rangle$ where $\displaystyle \sigma$ is any transposition and $\displaystyle \tau$ is any $\displaystyle p$-cycle.

I can more or less guess intuitively how these work together, but it would be an enormous task to show it. I suspect there is an easier way, using homomorphisms or some such. Any hints would be much appreciated!