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)

Let p be a prime number, G a finite group, P a p-subgroup of G and k an algebraically closed field of characteristic p. We study the relationship between the category F P (G) and the behavior of p-permutation kG-modules with vertex P under the Brauer construction. We give a sufficient condition for F P (G) to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we give a criterion for the categories arising from the data of (b, G)-Brauer pairs in the sense of Alperin-Broué and Broué-Puig to be saturated fusion systems on the underlying pgroup.

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 p-block A of a finite group G has an abelian defect group P , then A and its Brauer corresponding p-block B of N G (P ) are derived equivalent. We demonstrate in this paper that Broué's conjecture holds for non-principal 3-blocks A with elementary abelian defect group P of order 9 of the simple Held group and the sporadic simple Suzuki group.

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In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and (A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and (A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group Mn+1(p) of order p n+1 and exponent p n for n 2, and where A is not necessarily a full defect p-block but its defect group P = Mn+1(p) is normal in G. The proof is independent of the classification of finite simple groups. Mathematics Subject Classification (2000). Primary 20C20, Secondary 20C05.