Perhaps I will add the the question from number theory linked above - related to prime number theorem - is basically asking for elementary proof of $n>\frac{p_n}{\ln p_n}$, i.e. $p_n < n\ln p_n$. (Where $p_n$ denotes the $n$-th prime.)
Just to have short summary also here in chat. (So that users have basic idea what the question is aboiut without having to click on the link.)
The content of the theorem and its proof is still quite confusing to me, the comments contain some useful information, but what is written there is still confusing, for example the 2nd comment concludes that $N$ acts regularly, and then proceeds with the assumptions $N_{\alpha}\ne 1$, but regularity already implies $N_{\alpha} = 1$.
@StefanH You might add link as oneboxed, so that it is visually distinguished from other text. (BTW I have removed my comment on your question to reduce noise in comments.)
Let $p$ be a prime. Let $G$ be a solvable, non-regular, transitive permutation group such that some element fixes no point, and each element fixing some point fixes exactly $p$ points. Suppose that for $g \notin N_G(G_{\alpha})$ we have
$$
G_{\alpha}^g \cap G_{\alpha} = 1
$$
and that $p$ ...
Let $p$ be a prime. Let $G$ be a solvable, non-regular, transitive permutation group such that some element fixes no point, and each element fixing some point fixes exactly $p$ points. Suppose that for $g \notin N_G(G_{\alpha})$ we have
$$
G_{\alpha}^g \cap G_{\alpha} = 1
$$
and that $p$ ...
