For personal interest, I'm working on an solution of Serge Lang's Algebra.
Then I came across a problem about a doubly transitive group, exercise 47(b) of Chapter 1. I know there's already a question on this site about the same issue, but I couldn't find the answer in that question, so I'm here to ask.
Question is below, and I have no idea to deal with. I've tried for some $G\ni t:(a,s)\mapsto(s,a)$ where $H$ is isotropy subgroup of $s$, but I cannot prove that $t$ is order 2.
Any help would be greatly welcomed.
$G$ acts transitively and faithfully over a set $S$ which has more than one element and $H$ is isotropy subgroup of $s\in S$. Prove that $G$ is doubly transitive if and only if $G=HTH$, where $|T|=2$, $T<G$.
I already know that $G$ acts doubly transitive if and only if $\forall s\in S$, $G_s$ acts transitively on $S\setminus\{s\}$.