[Dummit p132, Q34]

Prove that if p is a prime and P is a subgroup of $\displaystyle S_p$ of order p, then $\displaystyle |N_{S_p}(P)|=p(p-1)$.

[Argue that every conjugate of P contains exactly p-1 p-cycles and use the formula for the number of p-cycles to compute the index of $\displaystyle N_{S_p}(P)$ in $\displaystyle S_p$].