A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

Abstract. This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups. Main resultsBrauer's famous height zero conjecture [9] from 1955 states that a p-block B of a finite group has an abelian defect group if and only if every ordinary irreducible character in B has height zero.Here we are concerned with one direction of this conjecture: (HZC1) If a p-block B of a finite group has abelian defect groups, then every ordinary irreducible character of B has height zero. One of the main aims of this paper is the proof of the following result: Theorem 1.1. The 'if part' (HZC1) of Brauer's height zero conjecture holds for all finite groups.Our proof relies on the crucial paper of Berger and Knörr [3] where they show that this direction of the conjecture holds for all groups, provided that it holds for all quasi-simple groups. An alternative proof of this reduction was later given by Murai [41].Many particular cases of (HZC1) had been considered before. Olsson [44] showed the claim for the covering groups of alternating groups.

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This is the accepted version of the paper.This version of the publication may differ from the final published version. We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form C 2 m × C 2 m for m ≥ 2 has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita equivalence types of 2-blocks with abelian defect groups of rank 2, from which we conclude that Donovan's conjecture holds for such 2-groups. A further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16. The proof uses the classification of finite simple groups. Permanent

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