SUBGROUPS OF EXCEPTIONAL GROUPS

MARTIN W. LIEBECK

Imperial College London SW7 2BZ

Abstract.

In this article we present some recent results concerning the subgroups of the simple algebraic groups of exceptional type, and of the corresponding finite groups of Lie type. There are six sections. The first contains some general observations, while in the second we focus on connected subgroups. The third section contains results on infinite closed subgroups, and in the last three sections we discuss finite subgroups.

1. Generalities

Let K be an algebraically closed field of characteristic p (allowing p = 0), and let G be a simple algebraic group over K. Fix a maximal torus T of G, and let Φ denote the root system of G relative to T. For α ∈ Φ, let U_{α} = {U_{α}(c) : c ∈ K} be the corresponding root subgroup of G. Let α_{1}, …, α_{n} be a set of fundamental roots, and α_{0} the highest root in Φ - that is, α_{0} = ∑ c_{i}α_{i} with ∑ c_{i} maximal.

A well known result of Borel and Tits [2, 2.3] implies that any closed connected subgroup of G is either reductive, or lies in a canonically defined parabolic subgroup of G. As a consequence, many questions on subgroups can be resolved by studying the classes of reductive and parabolic subgroups of G.

We now describe a natural family of reductive subgroups of G which play a big role in the general theory of subgroups. These are the subsystem subgroups, which we define to be those connected semisimple subgroups of G which are normalized by a maximal torus.

Subsystem 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 [3] 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 [6].

For example, by removing appropriate nodes from the extended E_{8} diagram (see [5, 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 1.1 ([13, 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 [5, p.250], from 1.1(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 [31, 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.

2. Connected subgroups

For the remainder of the article, we assume that our simple algebraic group G is of one of the exceptional types G_{2}, F_{4}, E_{6}, E_{7} and E_{8}. The study of connected subgroups of G is based on the following fundamental result, obtained by Dynkin [12] for p = 0 and by Seitz [27] for p > 0.

Theorem 2.1 ([12, 27]) Let M be a maximal closed connected subgroup of the exceptional algebraic group 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 [27, 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.

Work is well under way on removing all the characteristic restrictions, by Seitz and the author.

Discussion of the proof of Theorem 2.1 Here is a very brief outline of some of the main ideas in Seitz’s proof of 2.1. The analysis involves heavy use of representation theory, via the action of M on L(G), the Lie algebra of G.

A first observation on this action is that we can assume that M has no fixed points on L(G) (apart from the centre Z(L(G)), which is 0 except when (G, p) = (E_{6}, 3) or (E_{7}, 2)): for otherwise M ≤ C_{G}(v) for some v ∈ L(G)\Z(L(G)), which can be shown to lie in a parabolic or subsystem subgroup (see [27, 1.3]).

By the Borel-Tits theorem mentioned in the previous section, we can assume that M is reductive. For c ∈ K^{∗}, define T (c) = ∏ h_{α}(c), the product being taken over all positive roots α in the root system of M, and let

T_{1} = {T (c) : c ∈ K^{∗}},

a 1-dimensional torus in M. Using the maximality of M, it is shown in [27, 2.3] that there is a system Π(G) of fundamental roots in Φ such that for all α ∈ Π(G), c ∈ K^{∗},

T (c)e_{α} = c^{l_{α}}e_{α}, l_{α} ∈ {0, 2}

where e_{α} is a root element of L(G) corresponding to α. Thus the 1-dimensional torus T_{1} ≤ M determines a labelling of the Dynkin diagram of G with 0 or 2 over each node. The labelled diagram determines the weights of T_{1} on the whole of L(G). Of course there are only finitely many possible such labelled diagrams.

The analysis now proceeds in several steps;

(1) Reduce to the case where M is simple (this is done in [27, §3]).

(2) For each type of simple group M, find all possible composition factors of L(G) ↓ M (these must have dimension at most dim G, and must also have highest T_{1}-weight bounded above by 2 ∑ c_{i}, where α_{0} = ∑ c_{i}α_{i}, as this is the highest possible T_{1}-weight on L(G) for any of the labelled diagrams).

(3) Find all T_{1}-weights of the possible composition factors.

(4) For each combination of composition factors having total dimension dim G, see if the T_{1}-weights on this combination match the T_{1}-weights corresponding to a labelled diagram.

By carrying out these steps for each type of simple group, Seitz arrives at a rather short list of possibilities for the restriction L(G) ↓ M. For each possibility, he either obtains a contradiction to the maximality of M, or constructs a maximal subgroup M with the appropriate restriction and shows it is unique up to conjugacy.

For example, in the case where M = B_{2} and G = E_{8}, assuming that p ≠ 2, 3, 5 and M has no fixed points on L(G), just two possibilities for L(G) ↓ M arise (see [27, 6.5]):

(a) L(G) ↓ M = 12^{4} ⊕ 20^{2} ⊕ 02^{6} ⊕ 10^{4},

(b) L(G) ↓ M = 32 ⊕ 06 ⊕ 02, where the superscripts in (a) denote the multiplicities of the corresponding composition factors. With a great deal of effort, Seitz shows that a subgroup B_{2} in case (a) necessarily lies in a subsystem subgroup A_{4}A_{4} of G, while there is a unique class of maximal subgroups B_{2} satisfying (b) (see [27, 6.6, 6.7]).

We now turn to a discussion of arbitrary connected reductive subgroups of the exceptional algebraic group G. 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 [17]).

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

Theorem 2.2 ([24, 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”.

Remarks on the proof of Theorem 2.2 The general case of 2.2 can be deduced quickly from the case where H is simple and QL is minimal subject to being parabolic and containing H, so assume we are in the latter situation. Take an L-composition series

1 = Q_{0} ⊲ Q_{1} ⊲ … ⊲ Q_{r} = Q.

Results in [1] 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 (see the discussion in §1 of Seitz’s article in these proceedings).

By the minimality assumption, H lies in no parabolic subgroup of QL/Q ≅ L. Using 2.1 and some further argument, we can identify the possible embeddings of H in QL/Q, and hence obtain the composition factors of the restriction to H of each of the irreducible modules Q_{i}/Q_{i−1}. For example, when H = A_{2} the list of composition factors which can occur is as follows (up to taking duals and field twists):

\begin{matrix}10, 20, 30, 60, 11, 12, 22, 10 ⊗ 10^{(q)}, 10 ⊗ 01^{(q)}, 10 ⊗ 11^{(q)}, \\ 11 ⊗ 10^{(q)}, 10 ⊗ 20^{(q)}, 20 ⊗ 10^{(q)}, 10 ⊗ 02^{(q)}, 02 ⊗ 10^{(q)}, \end{matrix}

where the superscript (q) indicates a twist by a q-power field morphism.

At this point we require information about the first cohomology group H^{1}(H, V) for each possible H-composition factor V. We show that H^{1}(H, V) = 0 in all but one case, the exception being G = E_{8}, H = A_{1}, p = 11 or 17, QL a D_{7}-parabolic, and V = (p−2)⊗1^{(p)}. Excluding this exceptional case, we conclude that the semidirect product QH has just one class of closed complements to Q, giving the result. The exceptional case requires further argument (see [24, 3.7]).

As a consequence of Theorem 2.2, 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 2.1(ii) for p = 0 or p > 7).

Theorem 2.3 ([24, 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 [24, 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 2.3, the conjugacy classes and centralizers in G of all simple connected subgroups of rank at least 2 are explicitly listed in [24, §8]; the same is done for subgroups of type A_{1} in [21].

Using 2.3, 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.

Theorem 2.4 ([24, 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.

This result is not obvious, and indeed there are examples in small characteristics of reductive subgroups with non-reductive centralizers.

Theorem 2.5 ([24, Theorem 3; 21]) 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)).

3. Infinite closed subgroups

For the purpose of applications, particularly to the study of finite subgroups, as well as for intrinsic interest, it is important extend Theorem 2.1 by dropping the hypothesis of connectedness on the maximal subgroup M: that is, to determine the maximal closed subgroups of positive dimension in the exceptional group G. This is achieved by the next result. On the face of it, there seems to be no reason why there should not be many such maximal closed subgroups M such that M^{0} is not maximal connected. Surprisingly, it turns out that when M^{0} does not contain a maximal torus of G, there are just three classes of subgroups where this happens:

Theorem 3.1 ([22, Theorem 1]) Suppose that M is a maximal closed subgroup of G of positive dimension. 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 2.1;

(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).

4. Finite subgroups: reduction theorem

Let H be a finite subgroup of the exceptional algebraic group G. We begin by describing a reduction theorem for such subgroups; this result reduces the study to the case where H is almost simple.

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 3.1 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 [2]. Some rather interesting local subgroups emerge in this situation:

Theorem 4.1 ([4, 8]) 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 4.1 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 1.1 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 [4].

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

Theorem 4.2 ([4, 22]) 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 3.1 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 [22, Theorem 2]; this result determines all maximal subgroups of G^{F} which are not almost simple.

In view of the above reduction theorem, attention now focuses on the finite (almost) simple subgroups of G. This is the subject of the next two sections.

5. Finite simple subgroups: generic case

Let S be a finite simple subgroup of the exceptional algebraic group G. Recall that G is defined in characteristic p. If p > 0, write Lie(p) to denote the set of all finite simple groups of Lie type over fields of characteristic p. Also write Lie(p′) for the set of finite simple groups of Lie type over fields of characteristic not equal to p.

The analysis breaks naturally into two very different cases:

(1) the generic case, in which S ∈ Lie(p), and

(2) the non-generic case, in which S is alternating, sporadic, or in Lie(p′). In this section we discuss the generic case. Write S = S(q), a simple group of Lie type over 𝔽_{q}, where q = p^{e}. Here the aim is to lift the embedding S(q) < 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 2.1, 2.3 and 3.1 to identify the connected subgroup S̄, and hence also S.

The first such lifting result was proved by Seitz and Testerman [28], under the assumption that the characteristic p is large. More recently, Seitz and the author [25] have obtained a lifting result, under assumptions only on the field size q:

Theorem 5.1 ([25]) Let S = S(q) < G, q = p^{e}, and assume

q > t(G)  if S = L_{2}(q), ^{2}B_{2}(q) or ^{2}G_{2}(q),
q > 9 and S ≠ A_{2}^{ϵ}(16) otherwise

(where t(G) is a constant defined in terms of the root system of G; see below for the precise definition). Then either S is of the same type as G, or S lies in a proper connected subgroup D of G, such that S and D stabilize precisely the same subspaces of L(G).

The point about this result is that there are no assumptions on the characteristic p. Therefore, since there are only finitely many non-generic possibilities (see the next section), Theorem 5.1 covers all but finitely many simple subgroups S. The subgroups S of the same type as G in the conclusion, are identified by [23, 5.1].

The constant t(G) has been calculated, using a computer, by R. Lawther (private communication), in all cases except G = E_{8}: we have t(G) = u(G).(2, p−1), where u(G) = 12, 68, 124 or 388, according as G = G_{2}, F_{4}, E_{6} or E_{7}, respectively. The definition of u(G) in terms of the root system is given in the proof of Proposition 5.2 below.

Discussion of proof of Theorem 5.1 Before discussing the consequences of 5.1, we discuss the proof, at least when q is large. In this case the proof is based on the following proposition, which in fact applies to all simple algebraic groups G.

Proposition 5.2 There is a constant u(G), such that if x is a semisimple element of G, with |x| > u(G), then x lies in a subgroup X of G satisfying

(1) X is an infinite closed subgroup, and

(2) every x-invariant subspace of L(G) is also X-invariant.

Proof Choose a maximal torus T of G containing x, and let Φ be the root system of G relative to T. Define L ≤ ℤΦ to be the ℤ-span of the set

{α − β : α, β ∈ Φ, α(x) = β(x)}.

Set X = Ann_{T} (L) := {t ∈ T : λ(t) = 1 for all λ ∈ L}. Clearly X is a closed subgroup of G containing x.

We claim that x and X stabilize the same subspaces of L(G). For L(G) = L(T) ⊕ ∑_{α∈Φ} L_{α}, where L_{α} is the T -weight space of L corresponding to α. By definition of X, whenever two roots agree on x, they agree on X; thus if x fixes a sum of L_{α}’s, so does X, and the claim follows.

It remains to define the constant u(G) in such a way as to force X to be infinite, provided |x| > u(G). Let M denote the set of all sublattices of ℤΦ which are generated by root differences (i.e. by elements of the form α − β for α, β ∈ Φ). Then M is a finite set. Now define

u(G) = max {|Ann_{T} (M)/Ann_{T} (M)^{0}| : M ∈ M}.

Now L ∈ M, so |X/X^{0}| ≤ u(G). Say |X/X^{0}| = m. Then provided |x| > u(G), we have 1 ≠ x^{m} ∈ X^{0}, and hence X is infinite. This completes the proof of the proposition.

We now indicate how this proposition is used in the proof of 5.1. Thus let S = S(q) < G, and suppose q = p^{e} > t(G) = u(G).(2, p − 1). Now S contains a subgroup isomorphic to SL_{2}(q), L_{2}(q) or ^{2}B_{2}(q), and hence contains a semisimple element x of order at least (q + 1)/(2, p − 1). Then |x| > u(G), so by 5.2, there is an infinite closed subgroup X, containing x, and fixing the same subspaces of L(G) as x. Define

Y = ⟨S, X⟩.

Then Y is infinite, closed, and fixes the same subspaces of L(G) as S. Moreover, S ∩ X^{0} ≠ 1 as it contains a non-identity power of x, so S ≤ Y^{0}. Taking D = Y^{0}, we have the conclusion of Theorem 5.1, provided we show that either D is proper in G or S is of the same type as G. If D were equal to G, then D, hence S, would be irreducible on every G-composition factor of L(G). Thus the proof of 5.1 (for q > t(G)) is completed by the following result.

Theorem 5.3 ([25, Theorem 4]) Let S = S(p^{e}) < G, an exceptional algebraic group in characteristic p, and suppose S is irreducible on each Gcomposition factor of L(G). Then S is of the same type as G.

Thus the proof of Theorem 5.1 for q large is fairly elementary. However, reducing the bound to q > 9 takes a great deal more effort, and accounts for much of the length of [25].

A number of consequences of Theorem 5.1, particularly for the finite groups G^{F} (where F is a Frobenius morphism of G), are obtained in [25]. We conclude the section by stating a couple of these.

Theorem 5.4 ([25, Corollary 8]) 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 of the same type as G, or

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

The subgroups under (i) are determined up to conjugacy by [23, 5.1]; and the subgroups under (ii) are given by 3.1, provided p = 0 or p > 7. As remarked after 2.1, work is currently under way on removing the characteristic restrictions.

Theorem 5.5 ([25, Theorem 10]) Let S = S(q) < G, with q = p^{e} as in Theorem 5.1, and suppose p = 0 or p > 7. There S lies in a connected simple subgroup of G of the same type as S.

6. Finite simple subgroups: non-generic case

In the non-generic case, G is an exceptional algebraic group over K of characteristic p, and S is a finite simple subgroup of G which is alternating, sporadic, or in Lie(p′).

We begin by discussing the case where K = ℂ, so that 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, [20] 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, Ryba, 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 a very small number of open cases remaining (see [7, 9, 10, 14, 15, 16, 29]). Here is a summary of their results for G of type E_{8} (which of course contains all the other exceptional groups):

Theorem 6.1 The complete list of finite simple groups, some cover of which embeds in E_{8}(ℂ), is as follows:

A_{n}, 5 ≤ n ≤ 17,
L_{2}(q), q ≤ 61, q ≠ 23, 43, 47, 53, 59,
L_{3}(q), q ≤ 5,
U_{3}(3), U_{3}(8), U_{4}(2), U_{4}(3), P Sp_{4}(5), Sp_{6}(2), Ω_{8}^{+}(2),
G_{2}(3),^{3}D_{4}(2),^{2}F_{4}(2)′,
M_{11}, M_{12}, J_{2},
2B_{2}(8)^{?}.

The question mark next to the group ^{2}B_{2}(8) indicates that its existence in E_{8}(ℂ) has not yet been established, although it is strongly believed to be there. All the other groups have been shown to exist in E_{8}(ℂ).

Existence proofs for the above subgroups vary from the trivial to the very difficult. As an example of the trivial, observe that A_{17} embeds in SO_{16} via the natural permutation representation, and hence some cover of A_{17} lies in the subsystem subgroup D_{8} of E_{8}. On the other hand, the existence of most of the L_{2}(q) subgroups comes into the very difficult category (see [29] for example). The next result, taken from [10], is an illustration of an elegant proof which is somewhere in between.

Proposition 6.2 ([10, 6.1]) The group ^{2}F_{4}(2) embeds in E_{6}(ℂ).

Proof Write S = ^{2}F_{4}(2). Observe first that S has a complex irreducible character χ of degree 78, with values given in [11]. Let V be a 78-dimensional space affording χ. Write χ^{2^{+}}, χ^{3^{−}}, χ^{4^{−}} to denote the symmetric square, alternating cube, and alternating fourth power of χ, respectively. Calculation shows that

(χ^{2^{+}}, 1) = (χ^{3^{−}}, 1) = 1, (χ^{4^{−}}, 1) = 0.

Therefore, up to scalars, S fixes a unique nonzero symmetric bilinear form ( , ) on V, a unique nonzero alternating trilinear form ( , ,) on V, but no nonzero alternating 4-linear form.

Now define a multiplication [ .] on V by

([xy], z) = (x, y, z)

for all x, y, z ∈ V. Then [xy] = −[yx]. Further,

(x, y, z, u) = ([xy], [zu]) + ([yz], [xu]) + ([zx], [yu])

defines an S-invariant alternating 4-linear form on V, which must therefore be the zero form. It follows that

([[xy]z] + [[yz]x] + [[zx]y], u) = 0

for all x, y, z, u ∈ V, and hence the multiplication [ .] satisfies the Jacobi identity. Consequently V becomes a Lie algebra under this multiplication. Because S is irreducible on V, this Lie algebra is simple. There are only three simple complex Lie algebras of dimension 78, namely B_{6}, C_{6} and E_{6}, and so S embeds in one of the corresponding Lie groups. However, S does not embed in B_{6}(ℂ) or C_{6}(ℂ) as it has no nontrivial character of degree 12 or 13. Therefore S < E_{6}(ℂ), completing the proof.

Thus the question of the isomorphism types of finite simple subgroups of complex exceptional groups is just about settled. 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 [33], and is also proved in [30].

Theorem 6.3 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 conclude by discussing the non-generic case when the characteristic p is nonzero. An observation of Serre (see [29, §5]) assures us that if S is a finite simple subgroup of G(ℂ), a complex Lie group, then also S < G(K) for any algebraically closed field K of any positive characteristic. On the other hand, there are certainly finite simple subgroups which occur only in positive characteristic: as an obvious example, observe that A_{18} embeds in SO_{16}(2) and in SO_{16}(3) via the natural permutation representation, and hence some cover of A_{18} lies in E_{8}(K) when K has characteristic 2 or 3, but not in characteristic zero. The following recent result of Seitz and the author classifies all such instances.

Theorem 6.4 ([26]) Suppose S is a finite simple group, some cover of which lies in E_{8}(K), where K is an algebraically closed field of characteristic p, and S ∉ Lie(p). Then one of the following holds:

(i) some cover of S lies in E_{8}(ℂ);

(ii) S, p are as in Table 1 below. Each subgroup in Table 1 exists, apart possibly from the two cases marked with a question mark.

\begin{matrix}S & p \\ A_{18} & 2, 3 \\ L_{4}(3), L_{4}(5), Ω_{7}(3), J_{3}, F i_{22} & 2 \\ T h & 3 \\ Ru, HS & 5 \\ M_{22} & 2, 5 \\ ^{2}B_{2}(32)^{?} & 5 \\ ^{2}B_{2}(8)^{?} & any \end{matrix}

Table 1

The sporadic case of this result was established in [18]. The existence of the subgroups in Table 1 is given by the following embeddings:

L_{4}(3) < Ω_{7}(3) < F i_{22} < ^{2}E_{6}(2) (see [11]),
M_{22} < F i_{22} < ^{2}E_{6}(2)  (see [11]),
L_{4}(5) < E_{8}(4)  (see [8]),
J_{3} < U_{9}(2) < E_{8}(2)  (see [11]),
T h < E_{8}(3)  (see [32]),
Ru, HS, M_{22} < E_{7}(5)  (see [14, 19]).

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