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We classify all 2‐blocks with abelian defect groups of rank 4 up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application is that Broué's abelian defect group conjecture holds for all blocks under consideration here. [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]

Let b be a block with normal abelian defect group and abelian inertial quotient. We prove that every Morita auto-equivalence of b has linear source. We note that this improves upon results of Zhou and also Boltje, Kessar and Linckelmann. [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 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]

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]

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]