4 cosmetics; added 21 characters in body

Is the following true ?

*

Every solvable transitive subgroup $G\subset S_p$ 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$in$S_p$. \mathfrak{S}_p$. The quotient $G/C$ is cyclic of order dividing $p-1$. If $G$ is not cyclic, then it has exactly $p$ subgroups of index $p$.*p$. I need such a result for an arithmetic application. A reference or a short argument will be appreciated. Addendum. For those interested in the arithmetic application, see http://arxiv.org/abs/1005.2016 3 improved title Solvable transitive subgroupsgroups of thesymmetricgrouponplettersprimedegree 2 Addendum Is the following true ? *Every solvable transitive subgroup$G\subset 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$in$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

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