TITLE: Transitive permutation groups which all of their proper subgroups are intransitive
QUESTION [7 upvotes]: Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear
that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is
there any other class of groups with this property? I mean that which transitive groups has no proper transitive subgroup?

REPLY [3 votes]: A very repeatable construction to yield such permutation groups is to let a group act on the cosets of a small subgroup.
The idea is that if $H$ is a subgroup of $G$, a subgroup $T \leq G$ acts transitively on the cosets of $H$ if and only if $HT = G$. If $H$ is small, $T$ is very restricted (beginning, for example, with $[G:T] \leq |H|$) and it's easy, in many cases, to prove $T$ must be the whole $G$. If $G$ is regular, this corresponds to choosing $H = 1$, which forces $T=G$ without further discussion.