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Suppose that H is a subgroup of $S_p$ of order p, then show that $N_G(H) / C_G(H) \simeq Aut(H)$. I showed that $N_G(H)$ is of order p(p - 1), but I have no idea what Aut(H) look like or $C_G(H)$.

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Hint: Consider the induced map $\varphi_x:h\mapsto xhx^{-1}$ for $x\in N_G(H)$. Is this map an automorphism of $H$?

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