Selfextensions of modules over group algebras

Let KG be a group algebra with G a finite group and K a field and M an indecomposable KG-module. We pose the question, whether ExtKG1(M,M)≠0 implies that ExtKGi(M,M)≠0 for all i≥1. We give a positive answer in several important special cases such as for periodic groups and give a positive answer also for all Nakayama algebras, which allows us to improve a classical result of Gustafson. We then specialise the question to the case where the module M is simple, where we obtain a positive answer also for all tame blocks of group algebras. For simple modules M, the appendix provides a Magma program that gives strong evidence for a positive answer to this question for groups of small order.