  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 $Ext_{KG}^1(M,M)
\neq 0$ implies that $Ext_{KG}^i(M,M) \neq 0$ for all $i \geq 1$. 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.
