The fact that $$\langle\tau,\sigma \mid \tau^2,\sigma ^p\rangle\ \subset\ S_p$$ is obvious. But how can I show the other inclusion? $p$ is a prime number. The initial question is to show that $S_p$ is generated by a transposition and a $p-$cycle. It looks to be different than $\langle\tau,\sigma \mid \tau^2,\sigma ^p\rangle$ but I don't understand why.