Maximal PSL2 subgroups of exceptional groups of Lie type David A. Craven October 24, 2016 Abstract In this article we study embeddings of PSL2(q0) into exceptional groups G(q) for G = F4;E6;E7, and q0 and q powers of the same prime p. With a few possible exceptions, we prove that there are no maximal subgroups with socle such a simple group inside an almost simple group with socle G(q), except for those that arise as fixed points of a maximal positive-dimensional subgroup of the corresponding algebraic group. In the few remaining cases we provide considerable information about a potential maximal subgroup. Contents 1 Introduction 2 2 Notation and preliminaries 5 3 Maximal subgroups 9 4 Unipotent and semisimple elements 14 4.1 Actions of unipotent elements . 14 4.2 Blueprints and element orders . 16 4.3 Blueprints inside A1s......................................... 18 a 4.4 Traces for modules of PGL2(p )................................... 19 4.5 The graph automorphism of F4 ................................... 20 4.6 sl2-subalgebras of L(G)....................................... 21 a 5 Modules for SL2(p ) 25 a 5.1 Modules for SL2(2 )......................................... 25 a 5.2 Modules for SL2(3 )......................................... 29 5.3 Modules for SL2(p).......................................... 31 a 5.4 Modules for SL2(p ) for p > 5 and a > 1.............................. 35 6 Some PSL2s inside E6 in characteristic 3 39 7 Proof of the theorems: strategy 41 1 8 F4 43 8.1 Characteristic 2 . 43 8.2 Characteristic 3 . 46 8.3 Characteristic at least 5 . 49 9 E6 51 9.1 Characteristic 2 . 51 9.2 Characteristic 3 . 52 9.3 Characteristic at least 5 . 52 10 E7 in characteristic 2 57 11 E7 in odd characteristic: PSL2 embedding 74 11.1 Characteristic 3 . 74 11.2 Characteristic at least 5 . 75 12 E7 in odd characteristic: SL2 embedding 82 12.1 Characteristic 3 . 82 12.2 Characteristic at least 5 . 86 A Actions of maximal positive-dimensional subgroups on minimal and adjoint modules 101 1 Introduction Classifying the maximal subgroups of a finite group is one of the most fundamental problems in the field of finite group theory. Michael Aschbacher and Len Scott [5] reduced the problem for all finite groups to understanding H1(G; M) for all simple modules M for all finite simple groups G, and classifying all maximal subgroups of almost simple groups. This paper is a contribution towards the latter, ambitious goal. For alternating and classical groups there is in some sense no complete answer, since the dimensions of the classical groups (and degrees of the alternating groups) tend to infinity, although there is substantial work in this direction. However, for sporadic and exceptional groups there is a possibility of a complete answer being known. For sporadic groups, a complete answer is known for all groups but the Monster, and here we concentrate on exceptional groups of Lie type. There is a classification of maximal subgroups for exceptional groups 2 G = G(q) for G not of type F4, E6, E6, E7 and E8 already, and so we focus on these cases. What is known in the literature so far is summarized in Section 3, but broadly speaking, all maximal subgroups are known in these groups apart possibly from various almost simple maximal subgroups, and these are either a small list of simple groups that are not Lie type in defining characteristic, and if the potential maximal is Lie type in defining characteristic then what is left are groups of small rank and small field size, together with a large a collection of possible subgroups PSL2(p ), the focus of this paper. We prove the following theorems, for any almost simple group of the appropriate type. Theorem 1.1 Let p be a prime, a > 1 be an integer, and let q be a power of p. Let G be an almost simple ∗ a group with socle F4(q), and suppose that H is an almost simple group with F (H) = PSL2(p ). If H is maximal in G then one of the following holds: 2 (i) pa = 9; a (ii) p = 13, H = PSL2(13) and is a Serre embedding; a ∗ (iii) q = p , p > 13, F (H) = PSL2(q), and H is the intersection of G with a maximal algebraic A1 subgroup of the algebraic group F4. The definition of a Serre embedding is given formally in Definition 4.7, but informally it is a copy of PSL2(h + 1) where h is the Coxeter number of G and this subgroup contains a regular unipotent element. (This subgroup is named after Serre as he constructed copies of PSL2(h + 1) (if h + 1 is a prime) over all fields in [27].) In recent work of Tim Burness and Donna Testerman, this case has been solved, and proved to be a subcase of (iii) above, so pa = 9 is the only outstanding case. It seems difficult to remove the first possibility, although it might be possible using more advanced geometric techniques than employed here; of course no such maximal subgroup is known. In Section 8.2 we give more information about the case pa = 9, where we give the action of a potential maximal subgroup on the minimal module; such a subgroup does exist, but is contained inside a positive-dimensional subgroup, and representation-theoretic techniques do not seem able to prove uniqueness. Kay Magaard [24] proved Theorem 1.1 for p > 5 in his PhD thesis, with the extra condition that q = 13 in (ii). For E6 we have a complete theorem, as the Serre embedding can be shown to lie in F4. Theorem 1.2 Let p be a prime, a > 1 be an integer, let q be a power of p, and let G be an almost simple 2 group with socle either E6(q) or E6(q). There does not exist an almost simple maximal subgroup H of G ∗ a with F (H) = PSL2(p ). Almost all of this theorem was obtained by Aschbacher [4] using geometric techniques, where only the a case q = p = 11 and H contains a semiregular unipotent element, from class E6(a1), is left open; here we remove it using the Lie algebra structure of the adjoint module L(G). a For E7, here we again have some potential exceptions, but not the case p = 9, which was completed in [9]. This time the difficult cases are the Serre embedding and pa = 7; 8; 25. Theorem 1.3 Let p be a prime, a > 1 be an integer and let q be a power of p. Let G be an almost simple ∗ a group with socle E7(q), and suppose that H is an almost simple subgroup with F (H) = PSL2(p ). If H is maximal in G then one of the following holds: (i) pa = 7, pa = 8 or pa = 25; a a (ii) p = 19, H = PSL2(p ) and is a Serre embedding; a ∗ (iii) q = p , p > 17, F (H) = PSL2(q), and H is the intersection of G with a maximal algebraic A1 subgroup of the algebraic group E7. Again, Burness and Testerman have showed that (ii) is a subcase of (iii). In the case (i) where pa = 8, we can give the composition factors of H on the minimal module, and can give the precise module structure a as well whenever 8 j q. For p = 7, there are unresolved cases of potential copies of PSL2(7) where the preimage of the subgroup in the simply connected version of E7 is both 2 × PSL2(7) and SL2(7). In both cases the module structures on the minimal module can be given precisely, but it seems difficult to progress 3 a further using these techniques. In the case of p = 25, this is a copy of SL2(25) inside the simply connected version of E7 with centres coinciding, and we have complete information about the module structures on both the minimal and adjoint modules. If it exists then it is a maximal subgroup of E7(q) for the smallest q into which the group embeds. We do not deal with maximal subgroups of E8 here, and only consider it for certain lemmas, which will be useful in a later treatment of this case. A rough estimate is that, with current techniques, attempting E8 here would result in many unresolved cases and double the length of this work. For exceptional groups other than E8, the minimal module has dimension much smaller than the dimension of the group (as an algebraic group) and we can use representation theory to analyse this module. We can still do things with the Lie algebra for E8, as we did in [9], The strategy for the proofs of these theorems is given in Section 7, and relies heavily on computer calculations in three ways: (i) The first is to compute the traces of semisimple elements of large order on various modules for excep- tional groups. Tables of these traces are available for elements of small order, but we need them for very large orders, sometimes in the hundreds. For this we can use the program that Litterick produced in his PhD thesis [22], or construct the normalizer of a torus explicitly in Magma and take the con- jugacy classes, then compute their eigenvalues. (Litterick has produced a much faster algorithm for computing traces of elements on fundamental modules, but we do not need this for our cases.) (ii) The second is to do large linear algebra problems. To find all sets of composition factors that could arise as the composition factors of the restriction of a G-module to a subgroup H involves checking many possible combinations against these large lists of possible sets of composition factors.