Let H be a connected wild hereditary path algebra. We prove that if Z is a quasi-simple regular brick, and [r]Z indecomposable regular of quasi-length r and with quasi-top Z, then $ra{d}^{r}En{d}_H\left(\left[r\right]Z\right)=0$.

Let $G$ be a finite group with a Sylow 2-subgroup $P$ which is either quaternion or semi-dihedral. Let $k$ be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial $kG$-modules, whose restrictions to $P$ are isomorphic to the direct sum of the known exotic endotrivial $kP$-modules and some projective modules. This provides a description of the group $T\left(G\right)$ of endotrivial $kG$-modules.

We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. We thereby correct and generalize a result of Farnsteiner [Math. Nachr. 202 (1999)]. Furthermore we show that modules with certain length of composition series are periodic. We apply these results to G-transitive blocks of the universal enveloping algebras of restricted p-Lie algebras and prove that G-transitive principal blocks only allow components...