Connecting the Permutation Representations of a Group

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 \( FX \rightarrow FY\). 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.