On inclusion of permutation modules

Let H ₁, H ₂ be subgroups of a finite group G. Assume that G = ∪ _i=1 ^m H ₂ y _i H ₁ = ∪ _j=1 ⁿ H ₁ g _j H ₁ and that y ₁ = 1, g ₁ = 1. Let D _i be the set consisting of right cosets of H ₂ contained in H ₂ y _i H ₁ and let d _j (j = 1, ..., n) be the set consisting of right cosets contained in H ₁ g _j H{ia1}. We define the n×m matrix M _z (z = 1, ...,m) whose columns and rows are indexed by D _i and d _j respectively and the (d _k ,D _l ) entry is \D _z g _k ∩ D _l \. Let M = (M ₁, ..., M _m ). Assume that $1_{H_1 }^G$ and $1_{H_2 }^G$ are semisimple permutation modules of a finite group G. In this paper, by using the matrix M, we give some sufficient and necessary conditions such that $1_{H_1 }^G$ is isomorphic to a submodule of $1_{H_2 }^G$ . As an application, we prove Foulkes’ conjecture in special cases.