On Group Ring Automorphisms

For a finite group G, the group Outcent(ZpG) of outer central automorphisms of ZpG only depends on the Morita equivalence class of ZpG, which allows reduction to a basic order for its calculation. If the group ring is strongly related to a graduated order, it is often possible to give an explicit description of the basic order. In this paper, we show that Outcent(B)=1 for a block B of ZpG with cyclic defect group. We also prove that Outcent(B0(3)(A6))= 1 for the principal block B0(3)(A6) of Z3A6; this allows us to verify a conjecture of Zassenhaus for the perfect group of order 1080.