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It is an open problem as to whether any bimodule inducing a Morita auto-equivalence of a block must have endopermutation source. We prove that, for blocks b with normal defect groups in odd characteristic, a stronger result holds, namely that all such bimodules have linear source. We also prove the analogous result in characteristic 2, provided that the defect group is of a specific, slightly restrictive, form. [ABSTRACT FROM AUTHOR]

We construct blocks of finite groups with arbitrarily large O -Morita Frobenius numbers. There are no known examples of two blocks defined over O , with isomorphic defect groups, that are not Morita equivalent but the corresponding blocks defined over k are. Therefore, the above strongly suggests that Morita Frobenius numbers are also unbounded, which would answer a question of Benson and Kessar. [ABSTRACT FROM AUTHOR]

We consider Donovan's conjecture in the context of blocks of groups G with defect group D and normal subgroups N ◁ G such that G = C D (D ∩ N) N , extending similar results for blocks with abelian defect groups. As an application we show that Donovan's conjecture holds for blocks with defect groups of the form Q 8 × C 2 n or Q 8 × Q 8 defined over a discrete valuation ring. [ABSTRACT FROM AUTHOR]

We prove that if b is a block of a finite group with normal abelian defect group and inertial quotient a direct product of elementary abelian groups, then Picent (b) is trivial. We also provide examples of blocks b of finite groups with non-trivial Picent (b). We even have examples with normal abelian defect group and abelian inertial quotient. [ABSTRACT FROM AUTHOR]

We calculate examples of Picard groups for 2-blocks with abelian defect groups with respect to a complete discrete valuation ring. These include all blocks with abelian 2-groups of 2-rank at most three with the exception of the principal block of J 1. In particular this shows directly that all such Picard groups are finite and Picent, the group of Morita auto-equivalences fixing the centre, is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases. [ABSTRACT FROM AUTHOR]

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian 2-groups. [ABSTRACT FROM AUTHOR]

Abstract We define a new invariant for a p -block of a finite group, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete the proof of Donovan's conjecture for 2-blocks with abelian defect groups of rank at most 4 and for 2-blocks with abelian defect groups of order at most 64. [ABSTRACT FROM AUTHOR]

We consider p-blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index p. Using this, we show that if B is a 2-block of a finite group with abelian defect group D ≅= C2a1 × ·· ·×C2ar × (C2)8, where ai > 1 for all i and r ≥ 0, then d < LL(B) ≤ 2a1 + · · · + 2ar + 2s - r + 1, where |D| = 2d. When s = 1 the upper bound can be improved to 2a1 +· · ·+ 2ar + 2 - r. Together these give sharp upper bounds for every isomorphism type of D. A consequence is that when D is an abelian 2-group the Loewy length is bounded above by |D| except when D is a Klein-four group and B is Morita equivalent to the principal block of A5. We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal 3-blocks. [ABSTRACT FROM AUTHOR]

Let q be a power of a prime, ℓ a prime not dividing q , d a positive integer coprime to both ℓ and the multiplicative order of q mod ℓ and n a positive integer. A. Watanabe proved that there is a perfect isometry between the principal ℓ -blocks of GL n ( q ) and GL n ( q d ) where the correspondence of characters is given by Shintani descent. In the same paper Watanabe also proved that if ℓ and q are odd and ℓ does not divide | GL n ( q 2 ) | / | U n ( q ) | then there is a perfect isometry between the principal ℓ -blocks of U n ( q ) and GL n ( q 2 ) with the correspondence of characters also given by Shintani descent. R. Kessar extended this first result to all unipotent blocks of GL n ( q ) and GL n ( q d ) . In this paper we extend this second result to all unipotent blocks of U n ( q ) and GL n ( q 2 ) . In particular this proves that any two unipotent blocks of U n ( q ) at unitary primes (for possibly different n ) with the same weight are perfectly isometric. We also prove that this perfect isometry commutes with Deligne–Lusztig induction at the level of characters. [ABSTRACT FROM AUTHOR]

W. Turner has proved that Broué's abelian defect group conjecture holds for certain unipotent blocks of the finite general linear group, the so-called Rouquier blocks. This together with theorems of A. Marcus and of J. Chuang and R. Rouquier proves that the conjecture holds for all blocks of such groups. We prove that other finite classical groups also possess analogues of Rouquier blocks at linear primes. [ABSTRACT FROM AUTHOR]

We show that the Green correspondence induces an injective group homomorphism from the linear source Picard group L (B) of a block B of a finite group algebra to the linear source Picard group L (C) , where C is the Brauer correspondent of B. This homomorphism maps the trivial source Picard group T (B) to the trivial source Picard group T (C). We show further that the endopermutation source Picard group E (B) is bounded in terms of the defect groups of B and that when B has a normal defect group E (B) = L (B). Finally we prove that the rank of any invertible B -bimodule is bounded by that of B. [ABSTRACT FROM AUTHOR]