On projective summands of induced modules (Q1121984)

Let F be an algebraically closed field of prime characteristic p, let G be a finite group, and let H be a subgroup of G. Using the Reynolds ideal, the author proves the following reciprocity theorem: If S is a simple FG-module with projective cover P, and if T is a simple FH-module with projective cover Q then the multiplicity of Q as a direct summand of \(Res^ G_ H(S)\) coincides with the multiplicity of P as a direct summand of \(Ind^ G_ H(T)\). This result is then applied to bound Cartan invariants and dimensions of projective modules.