How to show that the normalizer N of a Sylow $p$-subgroup P of $S_p$ in $S_p$ is the affine linear group AGL(1,$p$)?

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I want to prove this as a special case of another fact that the normalizer of G in the group of set-bijections of G is isomorphic to the semi-direct product of Aut(G) and G itself. How do I link up the two things and prove them?