The maximal subgroups of positive dimension in exceptional algebraic groups Martin W. Liebeck Gary M. Seitz Department of Mathematics Department of Mathematics Imperial College University of Oregon London SW7 2AZ Eugene, Oregon 97403 England USA 1 Contents Abstract 1. Introduction 1 2. Preliminaries 10 3. Maximal subgroups of type A1 34 4. Maximal subgroups of type A2 90 5. Maximal subgroups of type B2 150 6. Maximal subgroups of type G2 175 7. Maximal subgroups X with rank(X) 3 181 ≥ 8. Proofs of Corollaries 2 and 3 196 9. Restrictions of small G-modules to maximal subgroups 198 10. The tables for Theorem 1 and Corollary 2 212 11. Appendix: E8 structure constants 218 References 225 v Abstract In this paper we complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small. A number of consequences are obtained. It follows from the main theo- rem that a simple algebraic group over an algebraically closed field has only finitely many conjugacy classes of maximal subgroups of positive dimension. It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of Lie type are known. Received by the editor May 22, 2002. 2000 Mathematics Classification Numbers: 20G15, 20G05, 20F30. Key words and phrases: algebraic groups, exceptional groups, maximal subgroups, finite groups of Lie type. The first author acknowledges the hospitality of the University of Oregon, and thesecond author the support of an NSF Grant and an EPSRC Visiting Fellowship. vi 1 Introduction Let G be a simple algebraic group of exceptional type G2,F4,E6,E7 or E8 over an algebraically closed field K of characteristic p (where we set p = if K has characteristic zero). In this paper we determine the maximal closed∞ subgroups of positive dimension in G. Taken together with the results of [25, 30, 40] on classical groups, this provides a description of all maximal closed subgroups of positive dimension in simple algebraic groups. We obtain a variety of consequences, including a classification of maxi- mal subgroups of the associated finite groups of Lie type, apart from some subgroups of bounded order. The analysis of maximal subgroups of exceptional groups has a history stretching back to the fundamental work of Dynkin [11], who determined the maximal connected subgroups of G in the case where K has characteristic zero. The flavour of his result is that apart from parabolic subgroups and reductive subgroups of maximal rank, there are just a few further conjugacy classes of maximal connected subgroups, mostly of rather small dimension compared to dim G. In particular, G has only finitely many conjugacy classes of maximal connected subgroups. The case of positive characteristic was taken up by Seitz [31], who de- termined the maximal connected subgroups under some assumptions on p, obtaining conclusions similar to those of Dynkin. If p > 7 then all these as- sumptions are satisfied. This result was extended in [21], where all maximal closed subgroups of positive dimension in G were classified, under similar assumptions on p. In the years since [31, 21], the importance of removing the character- istic assumptions in these results has become increasingly clear, in view of applications to both finite and algebraic group theory. For example, [24, Theorem 1] shows that any finite quasisimple subgroup X(q) of G, with q a sufficiently large power of p, can be embedded in a closed subgroup of positive dimension in G; this is used to prove that maximal subgroups of finite exceptional groups Gσ (σ a Frobenius morphism) are, with a bounded number of exceptions, of the form Xσ with X a maximal closed subgroup of positive dimension in G (see [24, Corollaries 7,8]). Here we complete the solution of this problem. We determine all maximal closed subgroups of positive dimension in G in arbitrary characteristic. For the purposes of one of our applications to finite groups of Lie type, we in fact prove a slightly more general result, admitting the presence of field endomorphisms and graph automorphisms of G. Henceforth we simply refer 1 2 MARTIN W. LIEBECK AND GARY M. SEITZ to these as “morphisms of G”. Let G be of adjoint type, and define Aut (G) to be the abstract group generated by inner automorphisms of G, together with graph and field mor- phisms. In the statement below, by a subgroup of maximal rank we mean a subgroup containing a maximal torus of G, and Symk denotes the sym- ˉ metric group of degree k. Also Fp denotes the algebraic closure of the prime field Fp. Recall also that a Frobenius morphism of G is an endomorphism σ whose fixed point group Gσ is finite. Here is our main result. Theorem 1 Let G1 be a group satisfying G G1 Aut (G); in the case ≤ ≤ ˉ where G1 contains a Frobenius morphism of G, assume that K = Fp. Let X be a proper closed connected subgroup of G which is maximal among proper closed connected NG1 (X)-invariant subgroups of G. Then one of the following holds: (a) X is either parabolic or reductive of maximal rank; 2 (b) G = E7, p = 2 and NG(X) = (2 D4).Sym3; 6 × (c) G = E8, p = 2, 3, 5 and NG(X) = A1 Sym5; 6 × (d) X is as in Table 1 below. The subgroups X in (b), (c) and (d) exist, are unique up to conjugacy in Aut(G), and are maximal among closed, connected NG(X)-invariant sub- groups of G. Table 1 GX simple X not simple G2 A1 (p 7) ≥ F4 A1 (p 13),G2 (p = 7),A1G2 (p = 2) ≥ 6 E6 A2 (p = 2, 3),G2 (p = 7),A2G2 6 6 C4 (p = 2),F4 6 E7 A1 (2 classes, p 17, 19 resp.),A1A1 (p = 2, 3),A1G2 (p = 2), ≥ 6 6 A2 (p 5) A1F4,G2C3 ≥ E8 A1 (3 classes, p 23, 29, 31 resp.),A1A2 (p = 2, 3),A1G2G2 (p = 2), ≥ 6 6 B2 (p 5) G2F4 ≥ MAXIMAL SUBGROUPS OF EXCEPTIONAL ALGEBRAIC GROUPS 3 Theorem 1 determines the maximal subgroups M of any such group G1 such that M G is closed and has positive dimension: namely, either M ∩ contains G, or M = NG1 (X) for X as in (a)-(d) and MG = G1. Below we shall give applications with G1 = G and with G1 = G σ , where σ is a Frobenius morphism of G. h i In Tables 10.1 and 10.2 at the end of the paper we present further infor- mation concerning the subgroups X in Table 1: (1) We give the precise action (as a sum of explicit indecomposable modules) of X on L(G), and also, in the cases G = F4,E6,E7, on the module V , where V is the restricted irreducible G-module of high weight λ4, λ1, λ7 respectively (of dimension 26 δp,3, 27, 56). These actions are recorded in Tables 10.1 and 10.2, and proofs− can be found in Section 9. (2) We give the values of NG(X): X ; this is always at most 2. In all | | cases where X has a factor A2, NG(X) induces a graph automorphism on this factor, and the only other case where NG(X): X = 2 is that in which | | G = E8 and X = A1G2G2, where NG(X) has an element interchanging the two G2 factors. These facts follow from the constructions of the maximal subgroups A1A2,A1G2G2 < E8 in [31, p.46, 39], of A2G2 < E6 in [31, 3.15], and from [24, 8.1] for A2 < E7 and A2 < E6. The subgroups of G of type (a) in Theorem 1 are well understood. Max- imal parabolic subgroups correspond to removing a node of the Dynkin diagram (possibly two nodes if G1 contains an element involving a graph or graph-field morphism). Subgroups which are reductive of maximal rank are easily determined. They correspond to various subsystems of the root system of G, and we give a complete list (with a proof in Section 8) of those whose normalizers are maximal in G, in Table 10.3 in Section 10. Likewise, Table 10.4 lists the maximal connected subgroups of maximal rank (again with a proof in Section 8). Application of Theorem 1 with G1 = G gives a complete determina- tion of the maximal closed subgroups of positive dimension in G - they are just the subgroups NG(X) for X as in (a)-(d). We state this formally for completeness: Corollary 2 (i) The maximal closed subgroups of positive dimension in G are as follows: maximal parabolics; normalizers of reductive subgroups of 2 maximal rank, as listed in Table 10.3; the subgroups (2 D4).Sym3 < E7 × and A1 Sym5 < E8 in Theorem 1(b,c); and subgroups NG(X) with X as × 4 MARTIN W. LIEBECK AND GARY M. SEITZ in Table 1. (ii) The maximal closed connected subgroups of G are as follows: max- imal parabolics; maximal closed connected subgroups of maximal rank, as listed in Table 10.4; and all subgroups X in Table 1, omitting the subgroup A1G2G2 < E8. The subgroup A1G2G2 < E8 in Table 1 lies in a subgroup F4G2 so is not maximal connected; however its normalizer in E8 interchanges the two G2 factors, and indeed NE8 (X) is maximal in E8. On glancing at the main results of [21, 31] and comparing them with our Theorem 1, the reader will notice that the conclusions are very similar.