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?



Transitive permutation groups which all of their proper subgroups are intransitive
Consider the symmetric group $S$ on $n$ symbols in its action on the $k$-subsets of $\{1,\ldots,n\}$. If $k\ne2,4$ and $n=2k+1$, then the only transitive subgroups of $S$ are $S$ itself and the alternating group. Hence in this case the alternating group acts transitively but all its proper subgroups are intransitive.
See my "More Odd Graph Theory", Discrete Math. 32 (1980) 205-207. The basic idea
is that a group that is transitive on $k$-subsets must be $(k-1)$-transitive
and in most cases this leaves with only the symmetric and alternating groups.


