Let $G$ be a doubly transitive subgroup of $S_n$ which contains an $n$-cycle, and let $G_{12}$ be the subgroup of $G$ consisting of all elements $g\in G$ for which $g(1)=1$ and $g(2)=2$. Then $G_{12}$ has an orbit on $\{3,4,...,n\}$ which is preserved by $N_{12}$, where $N$ is the normalizer of $G$ in $S_n$.

I can prove this by using the classification of doubly transitive groups with an $n$-cycle, which depends on the classification of finite simple groups. But I think there should be a 5-line proof which uses nothing more than the definition of double transitivity. Can anyone find such a proof?

The reason I believe that such a proof should exist is as follows. The main result of a 1954 paper by Uchiyama asserts that if $f(x)$ is a degree-$n$ polynomial in $\mathbf{F}_q[x]$ for which $\displaystyle\frac{f(x)-f(y)}{x-y}$ is absolutely irreducible, and if in addition the characteristic of $\mathbf{F}_q$ is sufficiently large compared to $n$, then the image set $f(\mathbf{F}_q)$ contains at least $q/2$ elements. In fact Uchiyama gives only part of the proof, and says he will give the full proof elsewhere; but his published papers do not contain the full proof. I can deduce Uchiyama's result from the group-theoretic assertion in a few lines, using tools Uchiyama knew. So I think this might be the proof he had in mind. Since he did not indicate that he used any nontrivial group theory, I suspect that if he had a proof of the group-theoretic result then it must have been very simple. The group-theoretic result is stronger than what Uchiyama needed, but still it seems like the most natural statement he might have tried to prove. I note that nowadays one can prove much more precise results than Uchiyama's by using tools developed in the past 60 years; but still it would be interesting to know whether there is indeed a simple proof along the lines I suggested, in case such a proof might provide further insights.