Is the following true ?

> Every solvable transitive subgroup
> $G\subset\mathfrak{S}_p$ (the symmetric group on
> $p$ letters, where $p$ is a prime)
> contains a unique subgroup $C$ of
> order $p$ and is contained in the
> normaliser $N$ of $C$ in $\mathfrak{S}_p$.  The
> quotient $G/C$ is cyclic of order
> dividing $p-1$.  If $G$ is not cyclic,
> then it has exactly $p$ subgroups of
> index $p$.

I need such a result for an arithmetic application.  A reference or a short argument will be appreciated.

**Addendum.** For those interested in the arithmetic application, see http://arxiv.org/abs/1005.2016