Why the normalizer of the Sylow $p$-subgroups of the symmetric group of degree $p$ has order $p(p-1)$ and is known as Frobenius group $F_{p(p-1)}$? 
Why the normalizer of the Sylow $p$-subgroups of the symmetric group of degree $p$ has order $p(p-1)$ and is known as Frobenius group $F_{p(p-1)}$?


I am trying to understand the statements on Wikipedia about Sylow subgroups of the symmetric group, where the above statement has been made.
Of course, the Sylow $p$-subgroup $C_p$ of $S_p$ is a normal subgroup of a group of order $p(p-1)$ by Sylow theorems. But how to show that it is the maximal subgroup normalizing $C_p$ in $S_p$?
Moreover, what is the relation between this normalizer and the Frobenius group $F_{p(p-1)}$?
Thank you.

My approach is more combinatorial than algebraic. Any group isomorphic to $(\mathbb Z_p)^n$ can be written as a permutation group on $p^n$ points, with any Frobenius complement a group that fixes exactly one of these points, therefore, maximally, a group of order $p^n - 1$. It can be shown that this maximal group always exists. Of course, this is not the maximal normalizer of the $(\mathbb Z_p)^n$ group in the symmetric group on $n$ letters (though it is for $n = 1$); the other normalizers, however, fix more than one point and therefore do not serve as Frobenius complements.
