TITLE: Solvable transitive groups of prime degree
QUESTION [7 upvotes]: 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

REPLY [4 votes]: Happened to come across the following Satz in Huppert, Endliche Gruppen I, [S. 163].