Representations of the symmetric group are reducible over singly transitive
subgroups

A. S. Kleschev and J. K. Sheth

Let $F$ be an algebraically closed field of characteristic $p > 0$
and $\Sigma_n = {\rm Sym}(\Omega)$ be the symmetric group on an
$n$-element set $\Omega$.

Main Theorem. Let $p > 2$, $n > 4$, $G \le \Sigma_n$, and $D$ be a simple
$F\Sigma_n$-module with $\dim D > 1$. If the restriction $D_G$ is
irreducible then either $G \le \Sigma_{n-1}$ or $G$ is $2$-transitive
on $\Omega$.