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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 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]

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]

Abstract In this paper we classify all blocks with defect group C 2 n × C 2 × C 2 up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of 2-blocks with abelian defect groups of rank at most 3. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index 2, so requires novel reduction techniques which we hope will be of wider interest. We note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for all blocks mentioned above. [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]