On the indecomposability of induced modules (Q1081687)

Let G be a finite group with normal subgroup N, and let H be a subgroup of G with normal subgroup U contained in N. Let (K,R,F) be a p-modular system which is ''large enough'', and let \(A\in \{K,R,F\}\). Let S be an AU- module such that \(S^ N\) is indecomposable and \(T_ G(S^ N)=T_ H(S)N\) for the inertial groups. The author proves that induction gives a bijection between isomorphism classes of indecomposable direct summands of \(S^ H\) and \(S^ G\). This generalizes the well-known case where \(N=U\). Some consequences of this result are also given. Moreover, the author offers the following converse to Green's Indecomposability Theorem: Let G be a finite group with normal subgroup N, and let S be an N-module with inertial group T. If \(S^ G\) is indecomposable then T/N is a p-group.