On modules with cyclic vertices in the Auslander-Reiten quiver (Q1081686)

Let M be an indecomposable, non-projective module belonging to a block B of non-cyclic defect of a modular group algebra in odd characteristic. If \(\to \tau M\to E\to M\to 0\) is the Auslander-Reiten sequence, then E is indecomposable and non-projective. In the parlance of \textit{C. M. Ringel} [Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)], M is ''at the end of a tube'' in the Auslander- Reiten quiver of B. The result holds in characteristic 2 as well, provided that the vertex of M is a normal subgroup. There is a nice corollary: if \(p>2\) and if the p-block B has modules \(M_ 1\), \(M_ 2\) with cyclic vertices such that there is an irreducible map \(M_ 1\to M_ 2\), then the defect group of B is cyclic.