Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

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.

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes ℓ such that a Sylow ℓ-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely that they only lie in very particular Harish-Chandra series then allow us to deduce from it the McKay conjecture for the prime 2, hence for characters of odd degree.

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The authors determine all the absolutely irreducible representations of degree up to 250 of quasi-simple finite groups, excluding groups that are of Lie type in their defining characteristic. Additional information is also given on the Frobenius-Schur indicators and the Brauer character fields of the representations.

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The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p -degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL 2 (q) for all prime powers q ≥ 4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q = 2 e or q = 3 e respectively, and e > 1 is odd. Since our conditions are also satisfied by the sporadic simple group J 1 , it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.

We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite non-abelian simple groups, there exists a triple of conjugate elements with product 1 which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).