Connecting the Permutation Representations of a Group

Abstract

Suppose that a finite group $G$ acts on two sets $X$ and $Y$, and that $FX$ and $FY$ are the natural permutation modules for a field $F$. We examine conditions which imply that $FX$ can be embedded in $FY$, in other words that $(\ast )$: There is
an injective $G$-map $FX\to FY$. For primitive groups we show that $(\ast )$
holds if the stabilizer of a point in $Y$ has a `maximally overlapping’ orbit on $X$. For groups of rank three, we show that $(\ast )$ holds unless a specific divisibility condition on the eigenvalues of an orbital matrix of $G$ is satisfied. Both results are obtained by constructing suitable incidence geometries.