How to show that any solvable transitive subgroup of S$_p$ where $p$ is a prime has a conjugate contained in Aff($\mathbf F_p$)?

Question

Here Aff ($\mathbf F_p$) denotes the group of affine transformations $x\rightarrow ax+b,$ with $ a\neq 0, b\in \mathbf F_p$. What I've done is to show that the penultimate group in the solvable series has a $p-$cycle. How do I complete the proof? I think that somehow I've to link everything with the normalizer in G in of the set of bijections of G, but can't figure out how to do that and how that proves anything. Any solution to link to one available online or in some book will be deeply appreciated.

Answer 1

The group $A=\operatorname{Aff}(\mathbf{F}_p)$ is the normalizer of a Sylow subgroup of $S_p$. What holds in all finite groups $G$ is that if $P$ is a Sylow subgroup and $N=N_G(P)$ is its normalizer, then $N_G(N)=N$. In particular $A=N_{S_p}(A)$.

This doesn't immediately imply the claim for a solvable series might go by a route other than the full normalizer. We need the following extra (on top of the bit that the penultimate group $P$ is generated by a $p$-cycle). Assume that $N_1\le S_P$ is a group that contains $P$ as a normal subgroup. This means that $N_1\le A$. This implies that $P$ is a characteristic subgroup of $N_1$. So if $N_2\le S_p$ is any group with the property $N_1\unlhd N_2$, then we also have $P\unlhd N_2$ and, consequently, $N_2\le A$.

This, of course, implies that the solvable series never extends outside of $A$.