TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 9, September 1998, Pages 3409{3482 S 0002-9947(98)02121-7 ON THE SUBGROUP STRUCTURE OF EXCEPTIONAL GROUPS OF LIE TYPE MARTIN W. LIEBECK AND GARY M. SEITZ Abstract. We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q)ofLie type in the natural characteristic. Our approach is to show that for sufficiently large q (usually q>9 suffices), X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension. Introduction In this paper we establish results aimed at the study of the subgroup structure of the finite exceptional groups of Lie type, in particular of their maximal subgroups. Let G(r) be an exceptional group of Lie type over a finite field Fr of order r,where p0 ris a power of the prime p.WriteG(r)=O (Gσ), where G is a simple algebraic ¯ group over Fr and σ is a Frobenius morphism of G. The reduction theorem [LS1, Theorem 2] determines all maximal subgroups of G(r) which are not almost simple. Thus we concentrate on almost simple maximal subgroups of G(r); let M be such a subgroup. Of particular interest is the so-called “generic case”, in which F ∗(M)=X(q), a group of Lie type over a field Fq, also of characteristic p. Our approach to the generic case is to attempt to show that the embedding X(q) <G(r) lifts to an embedding X¯ G,whereX¯is a simple closed connected subgroup of G of the same type as X(q).≤ Once this is achieved, we are in a position to apply results from [Se2, LS1] on the subgroup structure of algebraic groups. The first lifting result for exceptional groups appeared in [ST1], and showed that the finite embedding can be lifted under certain hypotheses, one of the hypotheses being that the characteristic p be suitably large. In this paper we establish lifting results in which there are hypotheses on the size of the ground field Fq, but no assumptions on the characteristic. Consequently, the results hold for all but finitely many possibilities for the group X(q). For example, Theorem 6 below shows that for q sufficiently large, either X(q) is the group of fixed points of an automorphism of Received by the editors October 11, 1996. 1991 Mathematics Subject Classification. Primary 20G40, 20E28. The authors acknowledge the support of NATO Collaborative Research Grant CRG 931394. The second author also acknowledges the support of an NSF Grant. c 1998 American Mathematical Society 3409 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3410 MARTIN W. LIEBECK AND GARY M. SEITZ p0 ¯ ¯ G(r), or X(q)=O (Xσ) with X maximal among closed connected M σ -invariant subgroups of G. (In most cases, “sufficiently large” means that q>9.)h i Once the lifting has been achieved we can apply the results of [Se2, LS1], which determine those subgroups X¯ of G as in the last sentence of the previous paragraph, under some mild characteristic restrictions. These restrictions are much weaker than those required for the result in [ST1]; in particular, p =0orp>7covers all the restrictions. However, in view of the results obtained in this paper, it becomes a high priority issue to extend the analysis in [Se2, LS1] so as to remove the characteristic restrictions. Such an analysis is under way. The preceding discussion has been couched in terms of maximal subgroups, but our results apply much more generally. We establish lifting results for all subgroups X(q)ofG(r), with q sufficiently large. Namely, in Theorem 1 we show that any such subgroup is contained in a closed connected subgroup X¯ of G stabilizing precisely the same subspaces of the Lie algebra L(G)asX(q). We also show (Theorem 4 and Corollary 5) that this connected group X¯ is almost always proper p0 ¯ in G, and that, given mild characteristic restrictions, X(q)=O (Xδ)forsome Frobenius morphism δ (Theorem 10). At this point the results of [LS2] determine the conjugacy class of X(q)inG(r). Finally, we use our lifting results to clear up a nagging problem on classical groups. In Theorem 11, we show that an embedding X<Y <Cof finite groups of Lie type in the same characteristic, with C classical and X absolutely irreducible on the usual module for C, can usually be lifted to a corresponding embedding of simple algebraic groups; such embeddings of algebraic groups are known [Se1]. See [Se3] for further discussion of this problem. We now state our results in detail, beginning with our main lifting results. As indicated above, these require no assumption on the characteristic p, but do need an assumption on q,whereX=X(q) is as above. In order to specify this assumption we need the following definition. Definition. Let G be a simple adjoint algebraic group, and let Σ = Σ(G)bethe root system of G. For a subgroup L of the lattice ZΣ, let t(L) be the exponent of the torsion subgroup of ZΣ/L.Forα, β Σ, call the element α β of ZΣaroot difference. Define ∈ − t(Σ(G)) = max t(L):La subgroup of ZΣ generated by root differences . { } Theorem 1. Let X = X (q) be a quasisimple group of Lie type in characteristic p, and suppose that X<G,whereGis a simple adjoint algebraic group of exceptional type, also in characteristic p. Assume that 2 2 q>t(Σ(G)).(2,p 1) if X = A1(q), B2(q) or G2(q), q>9and X = A−(16) otherwise. 6 2 Then the following hold: (i) there is a closed connected subgroup X¯ of G containing X, such that every X-invariant subspace of the Lie algebra L(G) is also X¯ -invariant; (ii) if also X Gσ where σ is a Frobenius morphism, then there is a σ-stable ≤ ¯ and NG(X)-stable subgroup X containing X such that each X-invariant subspace of each G-composition factor of L(G) is also X¯ -invariant. Theorem 4 below will show that the subgroup X¯ in the conclusion is proper in G, unless X has the same type as G. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE SUBGROUP STRUCTURE OF EXCEPTIONAL GROUPS OF LIE TYPE 3411 Remark. If p is a good prime for G,thenL(G) is irreducible as a G-module, so the subspace invariance conditions in conclusions (i) and (ii) are equivalent; however, for bad primes L(G) can be reducible (see Proposition 1.10 below), in which case (i) is stronger than (ii). In view of Theorem 1, it is of interest to know the values of t(Σ(G)), particularly for groups G of exceptional type. Dr Ross Lawther has informed us that he has verified (using a computer) that t(G2) = 12, t(F4) = 68, t(E6) = 124 and t(E7)= 388, while t(E8) is unknown at present. 2 2 The bounds in Theorem 1 for groups of type A1, B2, G2 are relatively large, and follow from Corollary 3 below, which we shall deduce from the following general proposition. It should be possible to improve these bounds with additional work. Proposition 2. Let G be a simple adjoint algebraic group, and let x be a semisim- ple element of G of finite order greater than mt(Σ(G)), for some positive integer m. Then there is an infinite closed subgroup S of G such that the following both hold: (i) x S, and some nontrivial power of x, of order greater than m,liesinS0; (ii) every∈ x-invariant subspace of L(G) is also S-invariant. Corollary 3. Let X = X (q) be a quasisimple group of Lie type in characteristic p, and suppose that X<G,whereGis a simple adjoint algebraic group, also in characteristic p. Assume that q>t(Σ(G))m(X),wherem(X)is the order of the Schur multiplier of X/Z(X). Then conclusions (i) and (ii) of Theorem 1 hold. Note that Proposition 2 and Corollary 3 apply to all types of simple algebraic group, not just exceptional types. The proofs of Proposition 2 and Corollary 3 are rather short, and are given in 2. Reducing the bounds on q to those in Theorem 1 requires a great deal more effort.§ After some preliminary results in 3, this is carried out for p =2in 4,5 and in 6forp= 2. With some further effort,§ our proof could no doubt6 be extended§§ to improve§ the q>9 bound for subgroups X of rank larger than 2; indeed, when X has rank greater than half the rank of G, [LST, Theorem 2] gives a stronger conclusion assuming only that q>2. In order to make use of Theorem 1 (or Corollary 3), we need to know that the subgroup X¯ is proper in G. For this it suffices to show that X¯ acts reducibly on some G-composition factor of L(G). This information is provided by the next result. ( ) Definition. We define X and G to be of the same type if X ∼= Gσ∞ for some Frobenius morphism σ;wealsosaythatXis a group of type G.