An application on Nagao's lemma (Q1063682)

Let G be a finite group and F a field of characteristic \(p>0\). Let V be a right FG-module. For a subgroup H of G the fixed point set of H in V is denoted by \(V^ H\). The trace map \(Tr^ G_ H: V^ H\to V^ G\) is defined by \(Tr^ G_ H(v)=\sum_{g\in H\setminus G}vg\), where \(H\setminus G\) denotes a complete set of coset representatives of H in G. Using Nagao's lemma the authors give two proofs for their main theorem: Let V be an indecomposable FG-module in a p-block B, and let P be a p- subgroup of G. Then each composition factor of the \(FN_ G(P)\)-module \(V(P)=V^ P/\sum_{A\setminus P}Tr^ P_ A(V^ A)\), where A runs over the set of proper subgroups of P, belongs to a block b of \(N_ G(P)\) such that \(b^ G=B\).