We verify the inductive blockwise Alperin weight (BAW) and the inductive Alperin-McKay (AM) conditions introduced by the second author for blocks of finite quasisimple groups with cyclic defect groups. Furthermore we establish a criterion that describes conditions under which the inductive AM condition for blocks with abelian defect groups implies the inductive BAW condition for those blocks.

Abstract. Using the classification of finite simple groups, we prove Alperin's weight conjecture and the character theoretic version of Broué's abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a finite group G has an abelian Sylow p-subgroup P, then the derived categories of the principal p-blocks of G and of the normalizer N G P of P in G are equivalent. We prove in this paper that Broué's conjecture holds for the principal 3-block of an arbitrary finite group G with an elementary abelian Sylow 3-subgroup P of order 9, by using initiated works for the case where G is simple, which are due to Puig, Okuyama, Waki, Miyachi, and the authors. The result depends on the classification of finite simple groups.  2002 Elsevier Science (USA)

We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p = 2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.

Given a dihedral 2-group P of order at least 8, we classify the splendid Morita equivalence classes of principal 2-blocks with defect groups isomorphic to P . To this end we construct explicit stable equivalences of Morita type induced by specific Scott modules using Brauer indecomposability and gluing methods; we then determine when these stable equivalences are actually Morita equivalences, and hence automatically splendid Morita equivalences. Finally, we compute the generalised decomposition numbers in each case.