Introduction to the subgroup structure of algebraic groups

Martin W. Liebeck

Imperial College, London SW7 2BZ, UK

Abstract This is an introduction to the subgroup structure of simple algebraic groups and of related finite groups of Lie type. Included are discussions of subgroups containing maximal tori (parabolic and subsystem subgroups), subgroups containing specified types of unipotent elements, reduction theorems for subgroups of classical groups, and recent material on subgroups of exceptional groups.

Introduction

The purpose of this article is to discuss various results concerning the subgroups of simple algebraic groups G and of the corresponding finite groups of Lie type G^{F} (where F is a Frobenius morphism). There are five sections. The first contains some background on simple groups, automorphisms and reductive subgroups. In the second section we present material on two important classes of subgroups which contain a maximal torus of G: the parabolic subgroups, and the reductive “subsystem” subgroups. Section 3 contains a discussion of unipotent classes, and of subgroups of G containing various particular types of such elements. In section 4 we concentrate on closed subgroups of classical groups G. We present a recent reduction theorem which shows that any such subgroup either lies in a member of a class of naturally defined “geometric” subgroups of G, or is essentially a quasisimple group acting irreducibly on the natural module for G. Use of this result, together with a standard process involving Lang’s theorem for linking finite and algebraic groups, yields a new proof of a well known reduction theorem of Aschbacher for finite classical groups, which we discuss. In the final section 5, we describe the picture for exceptional groups G. Again, there is a reduction theorem, reducing the study of subgroups H to the case where H is almost simple, and we sketch also the substantial body of recent results concerning the latter case.

1 Generalities

Let K be an algebraically closed field of characteristic p (allowing p = 0). The simple algebraic groups over K were classified by Chevalley [Ch], and fall into the following families:

classical types : A_{n}(K),  B_{n}(K),  C_{n}(K), D_{n}(K)
(examples : SL_{n+1}(K), SO_{2n+1}(K), Sp_{2n}(K), SO_{2n}(K))
exceptional types : G_{2}(K), F_{4}(K), E_{6}(K), E_{7}(K), E_{8}(K)

For each fixed type (e.g. A_{n}(K) with n fixed), there may be several pairwise non-isomorphic simple algebraic groups, which can usually (but not always) be distinguished by their centres. For example, when p = 0 or p does not divide n + 1, the groups of type A_{n}(K) are SL_{n+1}(K) and its quotients by subgroups of the group of scalar matrices; the group with the largest centre, SL_{n+1}(K), is the simply connected group of type A_{n}(K), while the group with trivial centre, P SL_{n+1}(K), is the adjoint group of this type. (When p divides n + 1, more subtle considerations are required to distinguish the different groups of type A_{n}(K) - see for instance [Ca3, 1.11].)

Let G be a simple algebraic group over K, and fix a maximal torus T of G. A root subgroup of G is a 1-dimensional T -invariant unipotent subgroup U_{α} = {U_{α}(c) : c ∈ K}, where α ∈ X(T) = Hom(T, K^{∗}), and for t ∈ T, c ∈ K,

U_{α}(c)^{t} = U_{α}(α(t)c).

The root subgroups U_{α} generate G, and the collection of roots α forms the root system Φ of G. The Weyl group W = N_{G}(T)/T is a finite group generated by reflections corresponding to roots in Φ. Associated with the root system Φ is the Dynkin diagram of G, which has nodes labelled by a set Δ of fundamental roots α_{1}, …, α_{n}, and edges determined by the inner products among these roots. If α_{0} is the highest root in Φ – that is, α_{0} = ∑ c_{i}α_{i} with ∑ c_{i} maximal – then the adjoining of −α_{0} to the Dynkin diagram (with appropriate edges) gives the extended Dynkin diagram of G. Full descriptions of root systems and diagrams (and many other related things) can be found in [Bou, p.250].

Next we describe the automorphism group of G as an abstract group. In order to do this, we first list various examples of automorphisms.

(1) Inner automorphisms.

(2) Graph automorphisms of types A_{n}, D_{n}, D_{4}, E_{6}. For these types, the Dynkin diagram possesses a symmetry of order 2,2,3,2 respectively, which extends linearly to a permutation ρ of the root system Φ. The graph automorphism corresponding to the symmetry ρ is given by

U_{α}(c) → U_{ρ(α)}(ϵ_{α}c) (α ∈ Σ(G), c ∈ K),

where each ϵ_{α} is ±1 (see [St1, p.156]).

(3) Field automorphisms. For each automorphism ϕ of the field K, there is a corresponding field automorphism of G given by

U_{α}(c) → U_{α}(c^{ϕ}) (α ∈ Φ, c ∈ K).

Note that the field automorphisms which are morphisms are those of the form U_{α}(c) → U_{α}(c^{q}), where q is a power of p. We shall write σ_{q} for this particular field automorphism.

(4) More graph automorphisms of types B_{2} (p = 2), F_{4} (p = 2), G_{2} (p = 3). For these types the Dynkin diagram has a symmetry of order 2, extending to a permutation ρ of Φ which interchanges long and short roots. This time the corresponding graph automorphism of G is given by

U_{α}(c) → U_{ρ(α)}(ϵ_{α}c^{p(α)}),

where p(α) = 1 if α is a long root, p(α) = p if α is a short root, and ϵ_{α} = ±1 (see [St1, p.156]).

By [St1, Theorem 30], the group of automorphisms of G as abstract group is generated by the automorphisms under (1)–(4) above. On the other hand, the automorphism group of G as algebraic group (i.e. the group of automorphisms α such that α and α^{−1} are both morphisms) is generated by the automorphisms just of types (1) and (2).

The Frobenius morphisms of G are the conjugates of σ_{q} and of τσ_{q}, where τ is a graph automorphism and q is a power of p. By [St2, 10.13], these are precisely those surjective morphisms F of G such that the fixed point group G^{F} is finite. When G is of adjoint type, O^{p′}(G^{F}) is usually simple, and is a simple group of Lie type over 𝔽_{q}.

We conclude this section with a brief discussion of the closed connected subgroups of G. Let M be such a subgroup. The largest connected unipotent normal subgroup of M, denoted R_{u}(M), is called the unipotent radical of M. The factor group M/R_{u}(M) is a connected reductive group, hence is equal to a commuting product DZ, where D is connected semisimple and Z is a torus.

Supose now that Q_{1} = R_{u}(M) ≠ 1. Define

Q_{2} = R_{u}(N_{G}(Q_{1})^{0}), Q_{3} = R_{u}(N_{G}(Q_{2})^{0}), …, Q_{i} = R_{u}(N_{G}(Q_{i−1})^{0}).

Evidently the chain Q_{1} ≤ Q_{2} ≤ … ≤ Q_{i} ≤ … of connected unipotent subgroups becomes stationary, at a subgroup Q_{r} with Q_{r} = R_{u}(N_{G}(Q_{r})^{0}). A basic result of Borel and Tits [BT, 2.3] says that for such a subgroup Q_{r}, N_{G}(Q_{r}) is necessarily a parabolic subgroup of G. Thus we have

Theorem 1.1 ([BT]) If M is a closed connected subgroup of G, then either M is reductive, or M lies in a (canonically defined) parabolic subgroup of G.

As a consequence of this result, many questions on subgroups can be resolved by studying the classes of reductive and parabolic subgroups of G.

2 Subgroups containing a maximal torus

Continue to assume that T is a maximal torus of the simple algebraic group G. We begin this section with a discussion of the reductive subgroups containing T.

Subsystem subgroups

A subsystem subgroup of G is defined to be a connected semisimple subgroup which is normalized by a maximal torus. Such subgroups are determined by the root system Φ, as follows. Let Σ be a closed subsystem of Φ (that is, Σ is closed under taking negatives, and if α, β ∈ Σ and α + β ∈ Φ, then α + β ∈ Σ). Then Σ is itself a root system, and the subgroup G(Σ) = ⟨U_{α} : α ∈ Σ⟩ is a T -invariant semisimple subgroup with root system Σ. Thus G(Σ) (and all its conjugates) is a subsystem subgroup. Indeed, except for some special cases which occur only when p = 2 or 3 and there is more than one root length, these are the only subsystem subgroups. Moreover, any connected reductive subgroup of G containing T is of the form DZ, where D is a subsystem subgroup and Z a torus commuting with D.

There is an elegant algorithm due to Borel and de Siebenthal [BS] which determines all closed subsystems of Φ, and proceeds as follows. Start with the Dynkin diagram of G, and adjoin −α_{0} to form the extended diagram. Remove any collection of nodes from this, and repeat the process with all connected components of the resulting graph. The diagrams obtained in this way are the Dynkin diagrams of subsystems, and there is one conjugacy class of subsystem subgroups for each such subsystem. In particular, G possesses only finitely many conjugacy classes of subsystem subgroups. Lists of subsystems for G of exceptional type can be found in [Ca1].

For example, by removing appropriate nodes from the extended E_{8} diagram (see [Bou, p. 250]), we can obtain subsystem subgroups of E_{8} of type A_{1}E_{7}, D_{8}, A_{8}, A_{2}E_{6}, A_{4}A_{4} and so on.

Important examples of reductive subgroups containing maximal tori are the centralizers of semisimple elements in G. Indeed, if t is a semisimple element of G, then t is conjugate to an element of T, so C_{G}(t)^{0} contains a conjugate of T; thus C_{G}(t)^{0} = DZ for some subsystem subgroup D and commuting torus Z. Which subsystem subgroups occur in this way ? Here is the answer in the important special case where t has prime order.

Theorem 2.1 ([GL, 14.1]) Let x ∈ G be a semisimple element of prime order r, and let C′ be the subsystem subgroup (C_{G}(x)^{0})′. If Σ is the Dynkin diagram of the root system of C′, then one of the following holds:

(i) Σ is obtained by deleting nodes from the Dynkin diagram of G;

(ii) Σ is obtained from the extended Dynkin diagram of G by deleting one node α_{i}, where r = c_{i}, the coefficient of α_{i} in the highest root α_{0}.

For example, when G = E_{8} we have

\begin{aligned}α_{0} = & 2 & 4 & 6 & 5 & 4 & 3 & 2 \\ & & & 3 \end{aligned}

where the node α_{i} of the E_{8} Dynkin diagram is labelled by its coefficient c_{i}. Referring to the extended Dynkin diagram [Bou, p.250], from 2.1(ii) we see that E_{8} possesses the following classes of semisimple elements:

for p ≠ 2, elements of order 2 with centralizers of type D_{8} and A_{1}E_{7}; for p ≠ 3, elements of order 3 with centralizers of type A_{8} and A_{2}E_{6}; for p ≠ 5, elements of order 5 with centralizer of type A_{4}A_{4}.

The primes dividing the coefficients c_{i} of the highest root α_{0} are called bad primes for G. The bad primes are as follows (see [SS, I, §4]):

G = A_{n} : none
G = B_{n}, C_{n}, D_{n} : only 2
G = G_{2}, F_{4}, E_{6}, E_{7} : 2 and 3
G = E_{8} : 2, 3 and 5.

We now discuss the subsystem subgroups of the finite groups G^{F}, where F is a Frobenius morphism. Let D = G(Σ) be a subsystem subgroup of G with root system Σ. By a subsystem subgroup of G^{F} we mean a subgroup of the form (D^{g})^{F}, where D^{g} is an F -stable G-conjugate of D. An application of Lang’s theorem shows that the G^{F} -classes of such subgroups are in 1-1 correspondence with the F -classes in N_{G}(D)/N_{G}(D)^{0}, which is isomorphic to N_{W} (Σ)/W (Σ)W (Σ^{⊥}), the group of symmetries induced by W on the Dynkin diagram of Σ (see [Ca2, §2]).

As an example, let G = SL_{6} (type A_{5}), and let D = (SL_{2})^{3}, a subsystem subgroup of type (A_{1})^{3}. Here N_{G}(D)/N_{G}(D)^{0} ≅ S_{3}, permuting the factors SL_{2}. Take F = σ_{q}, so G^{F} = SL_{6}(q). As F acts trivially on W, the F -classes in S_{3} are just the conjugacy classes, with representatives 1, (12) and (123). If D^{g} is an F -stable conjugate of D, then g^{F} g^{−1} ∈ N_{G}(D), hence maps to a conjugate w of one of the above representatives. The action of F on D^{g} is like that of wF on D; thus the three G^{F} -classes of subsystem subgroups of G^{F} arising from conjugates of D are as follows:

(1) w = 1 : D^{F} = SL_{2}(q)^{3};

(2) w = (12) : F -action on D^{g} is (g_{1}, g_{2}, g_{3}) → (g_{2}^{(q)}, g_{1}^{(q)}, g_{3}^{(q)}) for g_{i} ∈ SL_{2}, so

(D^{g})^{F} = SL_{2}(q^{2}) × SL_{2}(q);

(3) w = (123) : F -action is (g_{1}, g_{2}, g_{3}) → (g_{3}^{(q)}, g_{1}^{(q)}, g_{2}^{(q)}), and

(D^{g})^{F} = SL_{2}(q^{3}).

Parabolic subgroups

From some points of view (such as conjugacy classes, overgroups, intersections and so on), the parabolic subgroups of G are well understood. However, there are some interesting aspects of their internal structure which are perhaps less well known. We sketch one of these aspects.

Let Δ denote a system of fundamental roots in the root system Φ, and let J ⊆ Δ. Write Φ_{J} for the root system Φ ∩ ℤJ, and define the parabolic subgroup

P_{J} = ⟨T, U_{α}, U_{−β} : α ∈ Φ^{+}, β ∈ Φ_{J}^{+}⟩.

Then P_{J} = QL, where

Q = ⟨U_{α} : α ∈ Φ^{+}\Φ_{J}^{+}⟩, L = ⟨T, U_{β} : β ∈ Φ_{J}⟩.

The group Q is the unipotent radical of P_{J}, and L is a Levi subgroup. The structure of L is transparent - it is a commuting product of a torus and a semisimple group having Dynkin diagram J. The action of L on Q is much less transparent. Take an L-composition series

1 = Q_{0} ◁ Q_{1} ◁ … ◁ Q_{r} = Q.

Results in [ABS] establish that each factor Q_{i}/Q_{i−1} has the structure of an irreducible KL-module, of which the high weight can be calculated within the root system Φ. Moreover, by a theorem of Richardson [Ri], L has finitely many orbits on the vectors of each of these irreducible modules (and in particular, has an open dense orbit). Irreducible modules for algebraic groups with only finitely many orbits have been classified in [Kac, GLMS]; these modules are rather rare, and quite a large proportion of them occur within parabolics, as above.

It may be helpful to give an example. Let G = E_{8}, and let P_{J} be the parabolic subgroup with Levi subgroup A_{7}T_{1}. Then the above composition series is 1 < Q_{1} < Q_{2} < Q, and the irreducible quotient modules are V_{8}, Λ^{2}V_{8} and Λ^{3}V_{8}, where V_{8} is the natural 8-dimensional module for L. Consequently, GL_{8} has finitely many orbits on V_{8} and Λ^{2}V_{8} (which is obvious), and also on Λ^{3}V_{8} (which is far from obvious).

Representatives (or even just the numbers) of orbits for the modules occuring as above are not available in any systematic form. However, there is one special case in which there is an elegant description [RRS] - that in which Q is abelian (in which case Q is itself an irreducible module for L). A rather surprising phenomenon occurs in this case:

Theorem 2.2 ([RRS, 2.10]) If P = QL is a parabolic subgroup such that Q = R_{u}(P) is abelian, then the number of orbits of L on Q is equal to the number of P \G/P -double cosets.

Indeed, there is a natural correspondence between the orbits and the double cosets in this case.

The matters discussed above have been taken much further by Röhrle and others (see [Rö] for example).

3 Unipotent elements

Recall the Jordan decomposition for elements of the simple algebraic group G: each x ∈ G has a unique expression x = x_{s}x_{u}, where x_{s} is semisimple, x_{u} is unipotent, and x_{s}, x_{u} commute. We have already commented briefly on the classes and centralizers of semisimple elements. In this section we turn to unipotent elements, and discuss some results concerning their classes and overgroups. We begin with a theorem of Lusztig.

Theorem 3.1 ([Lu]) The number of conjugacy classes of unipotent elements in a simple algebraic group is finite.

For example, in SL_{n}(K), the unipotent classes are parametrized by Jordan canonical forms, so the number of classes is the number of partitions of n.

There are complete descriptions of the conjugacy classes of unipotent elements in all types of simple algebraic group; see [Ca3, Chapter 5] for discussion and references. Here we highlight two “extreme” classes:

(1) long root elements: these are non-identity elements of root subgroups U_{α} corresponding to a long root α ∈ Φ. (For example in SL_{n} or Sp_{n}, these are the transvections.) For a long root element u, we have C_{G}(u) = P′, where P is a parabolic subgroup (usually maximal).

(2) regular unipotent elements: these are conjugates of ∏_{α∈Δ} U_{α}(1), where Δ is a fundamental system in Φ. (In SL_{n}, they are the unipotent elements with a single Jordan block.) For a regular unipotent element u we have C_{G}(u) = U × Z(G), where U is an abelian unipotent group of dimension equal to the rank of G.

The class of regular unipotent elements is dense in the set of all unipotent elements of G. In a (rather weak) sense, an arbitrary unipotent class in G lies somewhere “in between” the above two classes. To explain this, first extend the definition of regular by defining a unipotent element u to be semiregular if C_{G}(u) = Q×Z(G) with Q a unipotent subgroup. If u is any unipotent element of G, then there is a connected semisimple subsystem subgroup D = D_{1} … D_{r} of G (all D_{i} simple) such that u = u_{1} … u_{r}, with each u_{i} a semiregular element of D_{i}.

There is quite a large body of literature concerning the subgroups of G and G^{F} containing an element of some special type. Probably the most studied is the case where the element is a long root element. For the finite groups, complete lists of subgroups containing long root elements have been obtained by Kantor [Ka] for classical groups, and by Cooperstein [Co1, Co2, Co3] for exceptional groups. A more conceptual approach was found in [LS2]; this contains the following result for simple algebraic groups G, from which a corresponding result for the finite groups G^{F} is deduced.

Theorem 3.2 ([LS2]) Let X be a simple closed connected subgroup of G such that |U ∩ X| > 2 for some long root subgroup U of G. Then there is a simple subsystem subgroup Y of G such that X ≤ Y and one of the following holds:

(i) X = Y;

(ii) X = C_{Y} (τ) for some graph automorphism τ of Y;

(iii) (X, Y) = (G_{2}, B_{3}) or (C_{4}, E_{6}) (with p = 2 in the latter case).

Note that [LS2] also contains a result covering the case where |U ∩ X| = 2.

The connection with subsystem subgroups makes it easy to study conjugacy classes of, centralizers of, and restrictions of representations to, reductive subgroups X generated by long root elements.

We now move on to regular unipotent elements. Of course, every such element lies in a parabolic subgroup, but one would expect there to be very few reductive subgroups containing a regular unipotent element. In SL_{n}(K), provided p = 0 or p ≥ n, there is such a reductive subgroup: the image of SL_{2}(K) in an irreducible representation over K of high weight n − 1. And indeed, every simple algebraic group G contains a “regular” A_{1} subgroup in which each non-identity unipotent element is regular in G, provided p = 0 or p ≥ h, the Coxeter number of G. The next result shows that, at least among the maximal connected subgroups of G, these are usually the only reductive overgroups of regular unipotents.

Theorem 3.3 ([SaSe]) Suppose X is a maximal connected subgroup of G and X contains a regular unipotent element of G. Then one of the following holds:

(i) X is a parabolic subgroup;

(ii) X is a regular A_{1} subgroup (p = 0 or p ≥ h);

(iii) X = C_{G}(τ) with τ a graph automorphism, or (X, G) = (G_{2}, B_{3}).

For arbitrary unipotent elements, the following theorem of Testerman is the most general result available.

Theorem 3.4 ([Te]) Assume p > 0 and p is a good prime for G. Then each element of order p in G lies in a closed connected subgroup of type A_{1}.

4 Classical groups

In this section we shall describe reduction theorems concerning the subgroups of algebraic and finite classical groups. Let V be a vector space of finite dimension n over the algebraically closed field K, and write

G = Cl(V)

to indicate that G is one of the classical groups SL(V), Sp(V) or SO(V). The main results take the following form. We define a collection C of “natural” subgroups of G (or G^{F}). Then any closed subgroup of G (or of G^{F}) either lies in a member of C, or is essentially a quasisimple group acting irreducibly on V. We call such a result a “reduction theorem”, since for many purposes it reduces the study of subgroups to the study of irreducible representations of quasisimple groups. The reduction theorem for the finite groups G^{F} is deduced from the reduction theorem for G. This gives a new proof of a well known result of Aschbacher [As] on subgroups of finite classical groups.

We begin with an elementary reduction theorem for connected subgroups of G.

Proposition 4.1 Let H be a closed connected subgroup of G. Then one of the following holds:

(i) H ≤ G_{U}, the stabilizer in G of a proper nonzero subspace U of V;

(ii) V = V_{1} ⊗ V_{2} and H lies in a subgroup of G of the form Cl(V_{1}) ∘ Cl(V_{2}) acting naturally on the tensor product;

(iii) H is simple, and acts irreducibly and tensor-indecomposably on V.

Remarks 1. When G = Sp(V) or SO(V), the subspace U in (i) can be taken to be totally singular or non-degenerate, or, in the case where (G, p) = (SO(V), 2), nonsingular of dimension 1. Also, the precise tensor product subgroups occurring in (ii) are of the following types:

SL ⊗ SL < SL, Sp ⊗ SO < Sp, Sp ⊗ Sp < SO, SO ⊗ SO < SO.

2. The subgroups H in (iii) are usually maximal in one of the groups Cl(V), with an explicit list of exceptions (see [Se1, Theorem 1]).

Here is a sketch proof of 2.1. If H is non-reductive, then the Borel-Tits theorem 1.1 implies that H ≤ P for some parabolic subgroup P; and P ≤ G_{U} for some (totally singular) subspace U, giving conclusion (i). Thus we may assume that H is reductive; we may also take it that H is irreducible on V (otherwise (i) holds again). Then Z(H) acts as scalars on V, hence is finite. It follows that H is semisimple, say H = H_{1} … H_{t}, a commuting product of simple groups H_{i}. If t > 1 then H preserves a tensor decomposition V = V_{1} ⊗ … ⊗ V_{t}, and it is not hard to see that (ii) holds. And if t = 1 then H is simple and we obtain (iii).

Proposition 4.1 only covers connected subgroups, and hence is not applicable in some important situations - in particular it will not apply to the study of finite subgroups. What is required is the generalization given in Theorem 4.2 below, a reduction theorem for arbitrary closed subgroups of G.

In order to state the result, we need to define a number of classes of subgroups M of the classical group G = Cl(V):

Class C_{1}. Subspace stabilizers M = G_{U}, where U is a proper nonzero subspace of V; moreover, U is totally singular or non-degenerate, or, in the case where (G, p) = (SO(V), 2), nonsingular of dimension 1.

Class C′_{1}. Subspace-pair stabilizers: here G = SL(V) and M = G_{U,W}, where U, W are proper subspaces of V such that dim U +dim W = dim V and either U ⊆ W or U ∩ W = 0; also, dim U ≠ dim W.

Class C_{2}. Stabilizers of orthogonal decompositions: here V = ⨁_{1}^{t} V_{i}, where t > 1 and the subspaces V_{i} are mutually orthogonal and isometric, and M = G_{{V_{1},... ,V_{t}}}; in general, M ≅ Cl(V_{1}) wr Sym_{t} (where Sym_{t} denotes the symmetric group of degree t).

Class C_{3}. Stabilizers of totally singular decompositions: here G = Sp(V) or SO(V), V = W ⊕ W′ where W, W′ are maximal totally singular subspaces, and M = G_{{W,W′}}; if dim V = 2m, then M ≅ GL_{m}.2 or GL_{m} (the latter only when G = SO(V) and m is odd).

Class C_{4}. Tensor product subgroups: either (i) V = V_{1} ⊗ V_{2} and M = Cl(V_{1}) ∘ Cl(V_{2}) as in 4.1(ii), or (ii) V = ⨂_{1}^{t} V_{i} with t > 1, the V_{i} mutually isometric, and we have

M = N_{G}(∏ Cl(V_{i})), where ∏ Cl(V_{i}) acts naturally on the tensor product; in general, we have M = (∏ Cl(V_{i})).Sym_{t}.

Class C_{5}. Finite local subgroups: let r be a prime different from p, and let R be an extraspecial r-group of order r^{1+2m}, or, when r = 2, a central product of such a group with a cyclic group of order 4. Every faithful irreducible representation of R has degree r^{m}. Take n = dim V to be r^{m}, and embed R as a subgroup of G = Cl(V) via one of these representations, where Cl(V) is SL(V) if r is odd or R = C_{4} ∘ 2^{1+2m}, and Cl(V) is Sp(V) or SO(V) otherwise. Then the members of the class C_{5} are the subgroups N_{G}(R) with R as above. If G_{1} = N_{SL(V)}(G), Z = Z(G) and M = N_{G_{1}}(R), then M̄ = M/Z ≅ C_{Aut(R)}(Z(R)). Thus the subgroups in the class are as follows:

\begin{matrix}G = SL_{r^{m}}, & M̄ = r^{2m}.Sp_{2m}(r) \\ G = Sp_{2^{m}}, & M̄ = 2^{2m}.O_{2m}^{−}(2) \\ G = SO_{2^{m}}, & M̄ = 2^{2m}.O_{2m}^{+}(2). \end{matrix}

Definition We write C(G) for the collection C_{1} ∪ C′_{1} ∪ … ∪ C_{5} of subgroups of G defined above.

We can now state the reduction theorem for G. If p > 0, let F be a Frobenius morphism of G such that G^{F} is a finite classical group.

Theorem 4.2 ([LS5]) Let G = Cl(V) be a classical algebraic group, and let H be a closed subgroup of G. Then one of the following holds:

(i) H is contained in a member of C(G), which is F -stable if H is F -stable;

(ii) modulo scalars, H is almost simple, and E(H) (the unique quasisimple normal subgroup of H) acts irreducibly on V.

We remark that in the theorem, H may of course be finite or infinite; if it is infinite, the conclusion of (ii) can be strengthened to say that E(H) also acts tensor-indecomposably on V.

The proof of 4.2 is naturally somewhat more complicated than that of 4.1, but uses only elementary linear algebra, together with a small amount of Lie algebra theory.

We now outline the use of Theorem 4.2 in proving a corresponding reduction theorem for finite classical groups; details can be found in [LS5]. Continue to assume that G = Cl(V) = SL(V), Sp(V) or SO(V), and let F be a Frobenius morphism of G such that G^{F} = SL_{n}(q), SU_{n}(q), Sp_{n}(q) or SO_{n}^{±}(q); write G^{F} = Cl_{n}(q). The natural (finite) module for G^{F} is V^{F^{u}} = V_{n}(q^{u}), where u = 2 if G^{F} is unitary and u = 1 otherwise.

Let H be a subgroup of G^{F}. Then 4.2(i) or (ii) holds for H. If 4.2(i) holds then H ≤ M^{F} for some M ∈ C(G); thus to obtain a reduction theorem for G^{F}, we need to analyse the G^{F} -classes of subgroups M^{F}. This can be done using Lang’s theorem and some linear algebra. We sketch some of the arguments - for details, see [LS5].

If M lies in C_{1}, then M = G_{U} for some subspace U of V and we easily see that H lies in a member of the following class of subgroups of G^{F}:

Class C_{1}^{F}. Stabilizers in G_{F} of subspaces of V_{n}(q^{u}).

Now let M ∈ C_{2}, so M = G_{{V_{1},... ,V_{t}}} where V = ⨁ V_{i}, and F permutes the factors Cl(V_{i}). Moreover, the group of permutations induced by M^{F} on the factors is a subgroup of C_{Sym_{t}}(F). If ⟨F ⟩ induces a nontrivial imprimitive subgroup of Sym_{t}, then M^{F} permutes the sums of subspaces corresponding to each block of imprimitivity. Hence, replacing M by another member of C_{2} if necessary, we may assume that either ⟨F ⟩ induces a trivial group, or t is prime and ⟨F ⟩ induces a transitive group. In the first case, in general M^{F} ≅ Cl_{m}(q) wr Sym_{t} (where n = mt); in the second, roughly speaking, M^{F} ≅ Cl_{m}(q^{t}).t. Thus we obtain the following classes of subgroups of G^{F}:

Class C_{2}^{F}. Stabilizers in G^{F} of orthogonal decompositions of V_{n}(q^{u}) (type Cl_{m}(q) wr Sym_{t}, where n = mt).

Class C′_{2}^{F}. Extension field subgroups (type Cl_{m}(q^{t}).t with n = mt and t prime).

Next, suppose that M ∈ C_{3}, so M is the stabilizer of a totally singular decomposition V = W ⊕ W′. Here M = GL\frac{n}{2} .2 (or GL\frac{n}{2} ), and M^{F} is GL\frac{n}{2} (q).2 or GU\frac{n}{2} (q).2 (or possibly a subgroup of small index in one of these); thus H lies in a member of the following class.

Class C_{3}^{F}. Subgroups of type GL\frac{n}{2} (q).2 and GU\frac{n}{2} (q).2, where G^{F} is symplectic or orthogonal.

Now consider M ∈ C_{4}, a tensor product subgroup. If M = N_{G}(∏ Cl(V_{i})), where V = ⨂ V_{i} and the V_{i} are isometric, we argue as in the case where M ∈ C_{2} that either F acts trivially on the factors, or t is prime and ⟨F ⟩ acts transitively. Thus H lies in one of the following subgroups.

Class C_{4}^{F}. Tensor product subgroups of type Cl_{n_{1}}(q) ∘ Cl_{n_{2}}(q) (n = n_{1}n_{2}), or of type N_{G^{F}} (∏_{1}^{t} Cl_{m}(q)) (n = m^{t}).

Class C′_{4}^{F}. Subgroups of type Cl_{m}(q^{t}).t with n = m^{t} and t prime.

Finally, suppose that M ∈ C_{5}, so M is a finite local subgroup N_{G}(R). In this case we argue that if M^{F} < M then M lies in a member of C_{1} ∪ … ∪ C_{4}. Hence we may take it that M^{F} = M, and define the class

Class C_{5}^{F}. Local subgroups N_{G^{F}} (R), as in C_{5}.

We have now outlined the proof of the reduction theorem for finite classical groups:

Theorem 4.3 ([As]) If G^{F} is a finite classical group as above, and H is a subgroup of G^{F}, then either

(i) H lies in a member of C_{1}^{F} ∪ … ∪ C_{5}^{F}, or

(ii) modulo scalars, H is almost simple, and E(H) is absolutely irreducible on the natural module for G^{F}.

Remarks 1. In fact, Aschbacher’s result [As] is more detailed than that stated in 4.3. His result involves two further classes of subgroups: C_{6}^{F}, a class of subgroups whose representations on V_{n}(q^{u}) are realised over a proper subfield of F_{q^{u}}; and C_{7}^{F}, the class of classical groups on V_{n}(q^{u}) which lie in G^{F}. His conclusion in (ii) is correspondingly more refined - for example, the representation of E(H) can be taken not to be realised over a proper subfield. In [LS5] similar methods to those outlined above are used to obtain the more refined result.

2. The subgroups in C′_{4}^{F} do not occur in [As], and indeed satisfy both conclusions (i) and (ii) of 4.3. These subgroups play a major role in the extension of Aschbacher’s theorem proved in [Se2, Corollary 6].

5 Exceptional groups

Througout this section, let G be a simple algebraic group of exceptional type G_{2}, F_{4}, E_{6}, E_{7} or E_{8} over the algebraically closed field K. We begin by discussing some results concerning connected reductive subgroups of G, for which there is now quite a satisfactory theory, at least under some mild assumptions on the characteristic p of K. We then move on to the study of arbitrary closed subgroups, which divides naturally into the cases of finite and infinite subgroups.

Connected subgroups

We begin with the following result of Seitz [Se3] on maximal connected subgroups, which underlies the entire body of work presented in this section.

Theorem 5.1 ([Se3]) Let M be a maximal closed connected subgroup of G. If M is simple, assume that either p = 0 or p > 7. Then either

(i) M is a parabolic subgroup or a subsystem subgroup; or

(ii) G and M are as follows (one Aut(G)-class for each M listed, except when otherwise stated):

G = G_{2} : M = A_{1}
G = F_{4} : M = A_{1}, G_{2} or A_{1}G_{2}
G = E_{6} : M = A_{2}, G_{2}, F_{4}, C_{4} or A_{2}G_{2}
G = E_{7} : M = A_{1} (two classes), A_{2}, A_{1}A_{1}, A_{1}G_{2}, A_{1}F_{4} or G_{2}C_{3}
G = E_{8} : M = A_{1} (three classes), B_{2}, A_{1}A_{2} or G_{2}F_{4}.

Remark The assumptions on p in [Se3, Theorem 1] are in fact much weaker than the assumption p = 0 or p > 7 in 5.1. Roughly speaking they are as follows:

if M = A_{1}, assume p = 0 or p > 7;

if M has rank 2, assume p = 0 or p > 5;

if M has rank 3 or (G, M) = (E_{8}, B_{4}), assume p ≠ 2.

In all other cases, no assumption on p is made. This remark applies to all the other results in this section in which characteristic assumptions are made.

We now turn to arbitrary connected reductive subgroups. We begin with a trivial observation on subgroups of GL_{n}(K) = GL(V): a subgroup H of GL(V) is completely reducible on V if and only if, whenever H lies in a parabolic subgroup QL of GL(V) (with unipotent radical Q, Levi subgroup L), then H lies in a Q-conjugate of L. If p = 0, or p ≥ dim V, then all connected reductive subgroups of GL(V) are completely reducible (see [Ja]).

Is there a similar result for subgroups of exceptional groups ? The answer is yes:

Theorem 5.2 ([LS3, Theorem 1]) Let H be a connected reductive subgroups of the exceptional algebraic group G, and assume that p = 0 or p > 7. If H lies in a parabolic subgroup QL of G, then H is contained in a Q-conjugate of L.

In other words, excluding some small characteristics, connected reductive subgroups are “completely reducible”.

As a consequence of this result, all connected reductive subgroups of G can be determined. To state the result, we need a definition.

Definition Let Y = Y_{1} … Y_{t} be a semisimple connected algebraic group, with each Y_{i} simple. We say that a closed connected semisimple subgroup X of Y is essentially embedded in Y if the following hold for all i:

(i) if Y_{i} is a classical group with natural module V_{i}, then either the projection of X in Y_{i} is irreducible on V_{i}, or Y_{i} = D_{n} and the projection lies in a natural subgroup B_{r}B_{n−r−1} of Y_{i}, irreducible in each factor with inequivalent representations;

(ii) if Y_{i} is of exceptional type, then the projection of X in Y_{i} is either Y_{i} or a maximal connected subgroup of Y_{i} not containing a maximal torus (hence given in 5.1(ii) for p = 0 or p > 7).

Theorem 5.3 ([LS3, Theorem 5]) Let X be a closed connected semisimple subgroup of the exceptional group G, and suppose that p = 0 or p > 7. Assume that X has no factor of type A_{1}. Then there is a subsystem subgroup Y of G such that X is essentially embedded in Y.

When X has a factor A_{1}, there is a similar result which is somewhat more complicated to state; we refer the reader to [LS3, Theorem 7]. As a consequence, when p = 0 there are only finitely many conjugacy classes of closed connected semisimple subgroups in G, whereas there are infinitely many when p > 0.

The classical factors of subsystem groups are of small rank (at most 8), so their essentially embedded closed connected semisimple subgroups can be determined, using some representation theory. Using 5.3, the conjugacy classes and centralizers in G of all simple connected subgroups of rank at least 2 are explicitly listed in [LS3, §8]; the same is done for subgroups of type A_{1} in [LT].

Using 5.3, one can work out centralizers of reductive subgroups of G, and restrictions of the Lie algebra L(G) to reductive subgroups. Several interesting consequences emerge, such as the following two results.

Corollary 5.4 ([LS3, Theorem 2]) Let X be a closed connected reductive subgroup of G, and assume that p = 0 or p > 7. Then C_{G}(X)^{0} is reductive.

Corollary 5.5 ([LS3 Theorem 3; LT]) Suppose that X is a closed connected simple subgroup of G, and p = 0 or p > 7. Then C_{L(G)}(X) = L(C_{G}(X)).

Closed subgroups: the infinite case

As in §4, for applications one needs to strengthen 5.1 in two ways: firstly, by dropping the assumption of connectedness on M; and secondly, by keeping track of F -stability, where F is a Frobenius morphism of G. This is achieved in the next result, which determines the maximal closed subgroups M of G of positive dimension. Surprisingly, it turns out that when M^{0} does not contain a maximal torus of G, there are just three classes for which M is maximal closed but M^{0} is not maximal connected:

Theorem 5.6 ([LS1, Theorem 1]) Suppose that M is a maximal closed (F - stable) subgroup of G with M^{0} ≠ 1. If M^{0} is simple, assume that p = 0 or p > 7. Then one of the following holds:

(i) M^{0} contains a maximal torus of G;

(ii) M^{0} is as in (ii) of 5.1;

(iii) G = E_{7} and M = (2^{2} × D_{4}).Sym_{3} (where M^{0} = D_{4} < A_{7} subsystem);

(iv) G = E_{8} and M = A_{1} × Sym_{5} (where M^{0} = A_{1} < A_{4}A_{4} subsystem);

(v) G = E_{8} and M = (A_{1}G_{2}G_{2}).2 (where M^{0} = A_{1}G_{2}G_{2} < F_{4}G_{2} < G).

As will be seen below, this result is fundamental also to the study of finite subgroups (both of G and of G^{F}), and it would be very desirable to remove the assumptions on the characteristic p. Work on this is in progress, by Seitz and the author.

Closed subgroups: the finite case

Let H be a finite subgroup of the exceptional algebraic group G. We begin by describing a reduction theorem for such subgroups.

Assume that H normalizes no connected subgroup D such that 1 < D < G. (Otherwise HD (hence H) lies in a maximal closed subgroup of positive dimension, given by 5.6 for p = 0 or p > 7.)

Consider first the case where H is local, so that H ≤ N_{G}(E) for some elementary abelian r-subgroup E of G (where r is prime). We may assume r ≠ p, by [BT]. Some rather interesting local subgroups emerge in this situation:

Theorem 5.7 ([Bor, CLSS]) Under the above hypotheses, one of the following holds:

G = G_{2} : E = 2^{3}, H = N_{G}(E) = 2^{3}.SL_{3}(2)
G = F_{4} or E_{6} : E = 3^{3}, H = N_{G}(E) = 3^{3}.SL_{3}(3) or 3^{3+3}.SL_{3}(3)
G = E_{8} : E = 2^{5} or 5^{3}, H = N_{G}(E) = 2^{5+10}.SL_{5}(2) or 5^{3}.SL_{3}(5).

Definition We call the subgroups N_{G}(E) in the conclusion of 5.3 exotic local subgroups.

Now assume that our finite subgroup H of G is non-local. Thus F^{∗}(H) = F_{1} × … × F_{t}, where the F_{i} are non-abelian simple groups. When t = 1, H is almost simple, which is our desired conclusion. So assume that t > 1.

At this point we apply a double centralizer argument due to Borovik. For each i, let CC(F_{i}) be the double centralizer C_{G}(C_{G}(F_{i})) of F_{i} in G. Then ∏ CC(F_{i}) is a commuting product of subgroups of G which is proper, nontrivial, and normalized by H. Therefore by assumption, CC(F_{i}) is finite for all i. If we choose x ∈ F_{i} of prime order at least 5, then CC(F_{i}) contains Z(C_{G}(x)). By the theory of centralizers in G (see 2.1 for the case where x is semisimple), Z(C_{G}(x)) can only be finite if the order of x is a bad prime for G. Hence G = E_{8} and each F_{i} is a simple {2, 3, 5}-group. The only such simple groups are Alt_{5}, Alt_{6} and U_{4}(2) (where Alt_{k} denotes the alternating group of degree k). Further argument now forces

G = E_{8}, F^{∗}(H) = Alt_{5} × Alt_{6}.

Somewhat amazingly, there is such a subgroup H in G = E_{8} which normalizes no proper connected subgroup, as was first shown by Borovik [Bor].

We have now sketched the proof of the reduction theorem for finite subgroups of G:

Theorem 5.8 ([Bor, LS1]) Let H be a finite subgroup of the exceptional algebraic group G. Then one of the following holds:

(i) H is almost simple;

(ii) H is contained in a proper maximal closed subgroup of positive dimension in G (given by 5.6 if p = 0 or p > 7);

(iii) H is contained in an exotic local subgroup;

(iv) G = E_{8} and H = (Alt_{5} × Alt_{6}).2^{2}.

Remark A more complete reduction theorem for maximal subgroups of the finite groups G^{F} (F a Frobenius morphism), again with no characteristic assumptions, can be found in [LS1, Theorem 2].

In view of the above reduction theorem, attention now focuses on the finite (almost) simple subgroups of G. There is a considerable literature on this, which we now proceed to discuss.

Finite simple subgroups

We begin by discussing the case where K = ℂ (so G is a complex exceptional Lie group).

Let S be a finite simple subgroup of G = G(ℂ). As G has a nontrivial complex representation of degree at most 248, so does S. However, [LaSe] gives strong lower bounds for the degrees of nontrivial complex representations of finite groups of Lie type (and bounds are also available for alternating and sporadic groups). For example, if S = L_{n}(q) and χ is a nontrivial complex character of S, then χ(1) ≥ q^{n−1} − 1, and hence q^{n−1} − 1 ≤ 248; this gives a small list of possibilities for (n, q) with L_{n}(q) < G. In this way, one quickly reduces to a finite list of possibilities for S.

Cohen, Griess, Serre, Wales and others have taken this analysis much further, and there are now complete lists of isomorphism types of simple subgroups S of G, with just a few open cases remaining (see [CW1, CW2, CG, Ser]). However, the conjugacy classes of these subgroups remain largely undetermined. Could there even be infinitely many G-classes of simple subgroups of a given isomorphism type? The answer to this question, at least, is no, as the next result shows; this goes back to Weil [We], and is also proved in [Sl].

Theorem 5.9 Let H be a finite group, and X a linear algebraic group over an algebraically closed field of characteristic p, where either p = 0 or p does not divide |H|. Then, up to X-equivalence, there are only finitely many homomorphisms from H into X.

We mention one well known conjecture relating to the problem of this section. This is Kostant’s conjecture, that if h is the Coxeter number of the exceptional algebraic group G = G(ℂ), then G contains a copy of P SL_{2}(2h+1). (Note that h = 6, 12, 12, 18, 30 according as G = G_{2}, F_{4}, E_{6}, E_{7}, E_{8}, respectively, so 2h + 1 is a prime power.) This conjecture was verified case-by-case [CGL, KR] using a computer; and an elegant computer-free proof has been given by Serre [Ser].

We conclude with a discussion of finite simple subgroups S of exceptional algebraic groups G = G(K) when the algebraically closed field K has positive charateristic p. The analysis breaks up into two cases:

(1) the generic case, in which S = S(p^{e}) is of Lie type in characteristic p, and

(2) the non-generic case, in which S is alternating, sporadic or of Lie type in p′-characteristic.

For the non-generic case, similar methods to those used for the complex case K = ℂ are available, and work on this is in progress by Seitz and the author.

Finally we discuss the generic case. Here the aim is to lift the embedding S(p^{e}) < G to an embedding S̄ < G, where S̄ is a connected simple group of the same type as S. Once such a lifting is achieved, we can then use results such as 5.1, 5.3 and 5.6 to identify the connected subgroup S̄, and hence also S.

The first such lifting result was proved by Seitz and Testerman [ST], under the assumption that the characteristic p is large. More recently, Seitz and the author [LS4] have obtained a lifting result, under assumptions only on the field size p^{e}:

Theorem 5.10 ([LS4]) Let S = S(p^{e}) < G, and assume

p^{e} > 9  if S ≠ L_{2}(p^{e}), ^{2}B_{2}(p^{e}), ^{2}G_{2}(p^{e}),
p^{e} > t(G) otherwise

(where t(G) is a constant defined in terms of the root system of G). Then S lies in a proper connected N_{G}(S)-invariant subgroup D of G.

The constant t(G) has been calculated, using a computer, by R. Lawther (private communication), in all cases except G = E_{8}: t(G) = 12, 68, 124 or 388, according as G = G_{2}, F_{4}, E_{6} or E_{7}, respectively.

Theorem 5.10 covers all but finitely many simple subgroups S. If D is as in the conclusion, then of course DN_{G}(S) lies in a maximal closed subgroup of G of positive dimension; such subgroups are determined by 5.6 if p = 0 or p > 7.

A number of consequences of 5.10, particularly for the finite groups G^{F}, are obtained in [LS4]. We mention just one:

Corollary 5.11 ([LS4]) There is a constant c, such that if H is a maximal subgroup of the finite exceptional group G^{F} (F a Frobenius morphism), with |H| > c, then either

(i) H is the centralizer of a field, graph or graph-field automorphism of G^{F}, or

(ii) H = X̄^{F} for some maximal closed F -stable subgroup X̄ of G of positive dimension.

The subgroups under (i) are just subgroups of the same type as G^{F}, possibly twisted, over subfields of the defining finite field, and are determined up to conjugacy by [LS2, 5.1]; the subgroups under (ii) are given by 5.6, provided p = 0 or p > 7. As remarked after 5.6, work is currently under way on removing these characteristic restrictions.

Subgroup structure 19

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20 Liebeck

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Subgroup structure 21

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