On the sequences induced from Auslander-Reiten sequences (Q581535)

Let F be a field of prime characteristic, let G be a finite group, let N be a normal subgroup of G, and let V be a non-projective indecomposable FN-module. Denote the Auslander-Reiten sequence terminating in V by S(V), and write \(V^ G=W_ 1\oplus...\oplus W_ n\) with indecomposable FG- modules \(W_ i\). Then the induced sequence \(S(V)^ G\) has a corresponding decomposition \(S(V)^ G=S_ 1\oplus...\oplus S_ n\). The author shows that \(S_ i\) is the Auslander-Reiten sequence terminating in \(W_ i\) if \(J(End_{FG}(W_ i))=\sum_{H}Im(Tr^ G_ H)\) where H ranges over the proper subgroups of a vertex of V.