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.

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Let K be an algebraically closed field of characteristic zero, and let G be a connected reductive algebraic group over K. We address the problem of classifying triples (G, H, V ), where H is a proper connected subgroup of G, and V is a finite-dimensional irreducible G-module such that the restriction of V to H is multiplicity-free -that is, each of its composition factors appears with multiplicity 1. A great deal of classical work, going back to Weyl, Dynkin, Howe, Stembridge and others, and also more recent work of the authors, can be set in this context. In this paper we determine all such triples in the case where H and G are both simple algebraic groups of type A, and H is embedded irreducibly in G. While there are a number of interesting familes of such triples (G, H, V ), the possibilities for the highest weights of the representations defining the embeddings H < G and G < GL(V ) are very restricted. For example, apart from two exceptional cases, both weights can only have support on at most two fundamental weights; and in many of the examples, one or other of the weights corresponds to the alternating or symmetric square of the natural module for either G or H.is MF, where the sum is over all weights (b 1 , . . . , b n−1 ) for which there exist non-negative integers c 1 , . . . , c n such that RESULTS OF STEMBRIDGE AND CAVALLIN 13Proposition 6.3.5. Let W be the irreducible module for A n (n ≥ 2) with high weight δ = ω 1 + ω n . Then ∧ 3 (W ) is MF. ProofThis is easily checked by Magma for n ≤ 6. So assume n ≥ 7. The high weight of any composition factor of ∧ 3 (W ) ↓ X is subdominant to λ = 3ω 1 +3ω n = (30 • • • 03). However the dominant weights λ, λ − α 1 , and λ − α n clearly cannot occur in in ∧ 3 (W ). The remaining dominant weights that are subdominant to λ are listed below (where as usual α 1 , . . . , α n form a system of fundamental roots for X = A n

Chapter 1. Introduction Acknowledgments Chapter 2. Preliminaries 2.1. Notation 2.2. Weights and multiplicities 2.3. Some dimension calculations 2.4. Clifford theory 2.5. Subgroup structure Chapter 3. The C 1 , C 3 and C 6 collections 3.1. The main result 3.2. Proof of Proposition 3.1.1 Chapter 4. Imprimitive subgroups 4.1. The main result 4.2. Preliminaries 4.3. Proof of Proposition 4.1.1 Chapter 5. Tensor product subgroups, I 5.1. The main result 5.2. Proof of Proposition 5.1.1

IntroductionLet X and G be simple, connected, linear algebraic groups over an algebraically closed field k of characteristic p (possibly p = 0), and suppose that rank(A') > 5 rank (G). In this paper we determine all closed, connected subgroups of G which are isomorphic to X. We also prove a corresponding result for finite groups of Lie type. It turns out that there are just finitely many conjugacy classes of such subgroups X in G, except when X = D 4 , G = E-, and p = 2; in this case there are infinitely many classes.Many of our subgroups X contain long root subgroups of G. Arbitrary such simple subgroups of G are classified in [25, Theorem 2.1], and we make extensive use of the results and methods of [25] in our proofs. For finite groups, some very special cases of our results can be found in [22].Our result for connected subgroups X (Theorem 1) has some overlap with recent work [26] of the first author and G. Seitz, in which arbitrary simple connected subgroups of simple algebraic groups of exceptional type are determined, assuming some mild restrictions on the characteristic of the field k. In Theorem 1, we make no assumption on the characteristic, and we also cover classical types; moreover, Theorem 1 is required for the proof of our result on finite groups (Theorem 2). As a consequence of Theorem 2, we determine all almost simple maximal subgroups of finite exceptional groups satisfying the condition on ranks (Theorem 3).To state our results we require a definition taken from [25]. A subsystem subgroup of G is a simple, closed, connected subgroup Y which is normalized by a maximal torus of G. Let 2(G) be the root system of G, and for a e 2(G), let U Q denote the T-root subgroup corresponding to a, where T is a fixed maximal torus normalizing Y. Then Y = (U a \ a e 2 0 ), where either 2 0 is a closed subsystem of 2(G) or (G, p) is {B n , 2), (C n , 2), (F 4 , 2) or (G 2 , 3) and 2 0 lies in the dual of a closed subsystem. The subsystem subgroups of G are well known and can be listed by an application of the result in [6]. We can now state our result for algebraic groups. Throughout, by a simple algebraic group we shall mean a simple connected linear algebraic group. THEOREM 1. Let G be a simple algebraic group over an algebraically closed field k of characteristic p. Suppose that X is a simple, closed, connected subgroup of G such that rank(A') > 2 rank (G). Then there is a subsystem subgroup Y of G such that X *s Y and one of the following holds: where r is a graph automorphism of Y of order 2;(III) X, Y are as in Table 1; (IV) X = D 4 , p = 2 and Y = G = £ 7 ; there are infinitely many G-conjugacy classes of subgroups D 4 in G, all lying in an E^-parabolic. Each possibility for X in Table 1 corresponds to a unique conjugacy class of subgroups in G. Furthermore, no subgroup X in Table 1 and (F 4 , £ 6 ), and for p * 2, (D n , A^-x) and (C 4 , £ 6 j.(2) In (IV), the projection of X = D 4 in the Levi factor £ 6 lies in a subgroup F 4) and is generated by short root subgroups of this F 4 . An explicit parameter...