Show that $S_p=\langle \tau,\sigma \mid \tau^2,\sigma ^p\rangle$

Question

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.

Answer 1

This is false as stated. The group $$ G=\langle \tau,\sigma\mid \tau^2=1=\sigma^p\rangle $$ is always infinite (if $p>1$). It does have $S_p$ as its quotient. There is a surjective homomorphism $f:G\to S_p, \sigma\mapsto (123\cdots p), \tau\mapsto (12)$.