On inclusion of permutation modules

Let H 1, H 2 be subgroups of a finite group G. Assume that G = . mi=1 H 2 y i H 1 = . nj=1 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 1GH1 and 1GH2 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 1GH1 is isomorphic to a submodule of 1GH2. As an application, we prove Foulkes. conjecture in special cases.