On the Cohomology of Finite Groups of Lie Type

Guido Mislin

Introduction

Let IF_{q} denote the finite field with q elements and let G = XIF_{q} be a connected (not necessarily split) reductive group scheme over spec(IF_{q}) . We will be interested in the cohomology of the finite groups G(IF_{q^{n}}) of IF_{q^{n}}–rational points of G, with coefficients in ZZ/ℓ where ℓ is a prime different from p = char(IF_{q}) . These groups are closely related and present special cases of the groups referred to in the title. A finite group of Lie type is, by definition, a central quotient of a group of the form G(IF_{q^{n}}) . For instance, all finite Chevalley groups (as defined in [Gor] or [Car]) are of this kind. For simplicity, we will formulate our results for the groups G(IF_{q^{n}}) rather than for these more general central quotients; it should be clear to the reader how to apply the results, mutatis mutandis, to the general finite groups of Lie type. The basic references for reductive group schemes are [DeGr] or [Dem] (see also [Jan] for an account of the basic results).

It is well known that if a homomorphism ϕ : τ → π of finite groups induces an isomorphism in H^{∗}( ; ZZ/ℓ) , ℓ a prime, then the kernel of ϕ has order prime to ℓ and the image of ϕ has an index prime to ℓ in π (cf. [Jac] ); in particular, it will follow that τ and π will have isomorphic ℓ–Sylow subgroups. Obviously, the converse statement is in general false, as one can see by looking at the inclusion map of an ℓ–Sylow subgroup. Our main theorem shows, however, that for the natural inclusions of groups of Lie type a converse statement holds. The precise statement is as follows.

Theorem. Let G be a connected reductive IF_{q}–group scheme and let ℓ be a prime different from char(IF_{q}) = p. Then the following are equivalent:

(i) the inclusion G(IF_{q}) → G(IF_{q^{n}}) induces an H^{∗}( ; ZZ/ℓ)–isomorphism

(ii) the groups G(IF_{q}) and G(IF_{q^{n}}) have isomorphic ℓ–Sylow subgroups.

In Section 1 we will discuss trace formulas and prove the Theorem. We will also point out the relationship of the Theorem with conjugacy questions concerning the ℓ–subgroups of G(IF_{q}) and G(IF_{q^{n}}) . In Section 2 we show how to adapt the proof of the Theorem to cover the Suzuki and Ree groups (as defined in [Gor]) .

The notational conventions of the Introduction will be kept throughout this paper.

1. Trace Formulas

As in the Introduction, G = X_{IF_{q}} denotes always a connected reductive IF_{q}–group scheme. Let \overline{IF}_{q} be the algebraic closure of IF_{q} and put

\overline{G} = G ×_{spec(IF_{q})} spec(\overline{IF}_{q}) ,

the reductive \overline{IF}_{q}–group scheme obtained by base change from G. We will write ϕ : \overline{G} → \overline{G} for the Frobenius endomorphism associated with the IF_{q}–form G of \overline{G}. Similarly, ϕ^{n} will denote the Frobenius of the IF_{q^{n}}–form G ×_{spec(IF_{q})} spec(IF_{q^{n}}) of \overline{G}. Thus, if G = spec(A) and \overline{G} = spec(A ⊗ \overline{IF}_{q}) then ϕ is given by the IF_{q}-homomorphism which maps x ∈ A ⊗ \overline{IF}_{q} to x^{q}. One has therefore a natural diagram of Lang maps

\begin{aligned}G(IF_{q}) & → & \overline{G} & \underrightarrow{1/ϕ} & \overline{G} \\ ↓ & & ∥ & & ↓ & ψ_{n} & , \\ G(IF_{q^{n}}) & → & \overline{G} & \overrightarrow{1/ϕ^{n}} & \overline{G} \end{aligned}  (1)

where ψ_{n} is given on \overline{IF}_{q}–rational points by x ↦ x · ϕ(x) · ϕ^{2}(x) · . . . · ϕ^{n−1}(x) . We will write H_{et}^{∗}(\overline{G}; ZZ/ℓ^{j}) for the etale cohomology of \overline{G} with coefficients in the constant sheaf with stalks ZZ/ℓ^{j}, ℓ a prime different from p = char(IF_{q}) . As usual, the ℓ–adic cohomology H_{et}^{∗}(\overline{G}; I Q_{ℓ}) is defined as

\begin{matrix}(lim \\ ← \\ j \end{matrix} H_{et}^{∗}(\overline{G}; ZZ/ℓ^{j})) ⊗ I Q.

For the convenience of the reader, we recall some basic facts on H_{et}^{∗}(\overline{G}; I Q_{ℓ}). By the classification of connected reductive group schemes over an algebraically closed field, there exists a unique reductive algebraic group L over IC with the same Root Data as \overline{G}. The associated Lie group of complex points L(IC)^{top}, with the strong topology, satisfies (cf. [FrPa])

H_{sing}^{∗}(L(IC)^{top}; I Q_{ℓ}) ≅ H_{et}^{∗}(\overline{G}; I Q_{ℓ}).

Moreover, the Lie group L(IC)^{top} has complex dimension N = dim(\overline{G}), and L(IC)^{top} is homeomorphic to K × IR^{N}, where K denotes a maximal compact subgroup of L(IC)^{top} and, of course,

H_{sing}^{∗}(L(IC)^{top}; I Q_{ℓ}) ≅ H_{sing}^{∗}(K; I Q_{ℓ}).

It is well known that H_{sing}^{∗}(K; I Q_{ℓ}) is an exterior algebra over I Q_{ℓ} on odd dimensional generators. Thus

H_{et}^{∗}(\overline{G}; I Q_{ℓ}) ≅ ⋀ W

with W a graded vector space over I Q_{ℓ}. In particular,

H_{et}^{N}(\overline{G}; I Q_{ℓ}) ≅ I Q_{ℓ} ,

where N = dim(\overline{G}), and therefore a morphism f : \overline{G} → \overline{G} of schemes over spec(\overline{IF_{q}}) has a well-defined degree

d(f) ∈ ZZ_{ℓ} ,

ZZ_{ℓ} ⊂ I Q_{ℓ} the ℓ–adic integers, satisfying

f^{∗}(x) = d(f) · x

for all x ∈ H_{et}^{N}(\overline{G}; I Q_{ℓ}) . Because the algebra H_{et}^{∗}(\overline{G}; ZZ/ℓ) satisfies Poincaré duality it is clear that

H_{et}^{∗}(f; ZZ/ℓ) : H_{et}^{∗}(\overline{G}; ZZ/ℓ) → H_{et}^{∗}(\overline{G}; ZZ/ℓ)

is an isomorphism if and only if d(f) ∈ ZZ_{ℓ} is a unit. It is also useful to observe that H_{et}^{∗}(\overline{G}; I Q_{ℓ}) has the structure of a Hopf-algebra, with coalgebra structure induced by the multiplication in \overline{G}. If we put

V = P H_{et}^{∗}(\overline{G}; I Q_{ℓ}) ,

the subspace of primitive elements, then we have a natural isomorphism

⋀ V = H_{et}^{∗}(\overline{G}; I Q_{ℓ}) .

It follows that if f : \overline{G} → \overline{G} is a morphism of schemes over \overline{IF}_{q}, such that the induced map f^{∗} : H_{et}^{∗}(\overline{G}; I Q_{ℓ}) → H_{et}^{∗}(\overline{G}; I Q_{ℓ}) maps V = P H_{et}^{∗}(\overline{G}; I Q_{ℓ}) to itself (e.g., if f is a morphism of group schemes), then

d(f) = det(f^{∗} | V) ,

because H_{et}^{N}(\overline{G}; I Q_{ℓ}) ≅ ⋀^{max} V, the largest non-vanishing exterior power of V. Moreover, if we put f^{∗} = 1 − g, then the linear map g maps V into itself and satisfies

det(f^{∗} | V) = det(1 − g | V) = grT r(g) ,

where we mean by grT r(g) the graded trace

∑(−1)^{i}T r(g : H_{et}^{i}(\overline{G}; I Q_{ℓ}) → H_{et}^{i}(\overline{G}; I Q_{ℓ})) .

This is a consequence of the fact that V has a bases consisting of elements of odd degrees, so that grT r(g), as defined above, is also equal to

∑(−1)^{j}T r(∧^{j}P g : ∧^{j}V → ∧^{j}V) ,

where P g denotes the restriction of g to V. Our next goal is to analyse the degree of the morphism ψ_{n} in the diagram (1). An easy spectral sequence argument, applied to the diagram (1), shows the following (see [FrMi] for the case of an IF_{q}–split G) .

Proposition 1.1. Let G = X_{IF_{q}} and ψ_{n} : \overline{G} → \overline{G} be as above. Then for every prime ℓ different from p = char(IF_{q}) the following are equivalent:

(i) G(IF_{q}) → G(IF_{q^{n}}) induces an H_{∗}( ; ZZ/ℓ)–isomorphism,

(ii) the degree d(ψ_{n}) ∈ ZZ_{ℓ} is an ℓ–adic unit.

To prove the theorem of the Introduction, we need to express the degree d(ψ_{n}) in terms of the orders of the groups G(IF_{q}) and G(IF_{q^{n}}) . This will be done by interpreting d(ψ_{n}) in terms of graded traces and by relating these to the orders of the groups G(IF_{q}) and G(IF_{q^{n}}) using the Lefschetz trace formula applied to the Frobenius endomorphism of \overline{G} .

Lemma 1.2. Let \overline{G} be a connected reductive \overline{IF}_{q}–group scheme and ϕ : \overline{G} → \overline{G} the Frobenius endomorphism associated with an IF_{q}–form of \overline{G}. Then the degree d(1/ϕ) of the Lang map 1/ϕ : \overline{G} → \overline{G} is given by

d(1/ϕ) = ∑(−1)^{i}T r(ϕ^{∗} : H_{et}^{i}(\overline{G}; I Q_{ℓ}) → H_{et}^{i}(\overline{G}; I Q_{ℓ})) .

Proof: As observed above, the Hopf–algebra H_{et}^{∗}(\overline{G}; I Q_{ℓ}) is an exterior algebra ∧V on V = P H_{et}^{∗}(\overline{G}; I Q_{ℓ}), the subspace of primitive elements. If we denote by gr T r(ϕ^{∗}) the graded trace ∑(−1)^{i}T r(ϕ^{∗} : H_{et}^{i}(\overline{G}; I Q_{ℓ}) → H_{et}^{i}(\overline{G}; I Q_{ℓ})) then, because all homogeneous elements of V have odd cohomological degree, we have

gr T r(ϕ^{∗}) = ∑(−1)^{j}T r(∧^{j}P ϕ^{∗} : ∧^{j}V → ∧^{j}V) ,  (2)

where P ϕ^{∗} : P H_{et}^{∗}(\overline{G}; I Q_{ℓ}) → P H_{et}^{∗}(\overline{G}; I Q_{ℓ}) is the map induced from the morphism of group schemes ϕ : \overline{G} → \overline{G}; as observed earlier, the right hand side of (2) is therefore just det(1 − P ϕ^{∗}) . On the other hand, 1/ϕ : \overline{G} → \overline{G} maps x ∈ P H_{et}^{∗}(\overline{G}; I Q_{ℓ}) to x − ϕ^{∗}x, which is equal to (1 − P ϕ^{∗})x. Thus det(1 − P ϕ^{∗}) = d(1/ϕ) which shows that d(1/ϕ) = grT r(ϕ^{∗}) as claimed.

From the diagram (1) above we see that ψ_{n} ∘ (1/ϕ) = 1/ϕ^{n}. The following corollary is then immediate.

Corollary 1.3. The degree of the map ψ_{n} in the diagram (1) is given by

d(ψ_{n}) = \frac{gr T r(ϕ^{n})^{∗}}{gr T r(ϕ^{∗})} ∈ ZZ_{ℓ} .

The Lefschetz trace formula in H_{c}^{∗}, etale cohomology with compact supports, permits to relate the number of IF_{q^{n}}– rational points of certain schemes to a graded trace. The following proposition is well known (see [Mil], and also [DeLu] for a general discussion of the Lefschetz trace formula; the formula we use here is 1.9.4, page 174, of [Del]).

Proposition 1.4. Let X_{IF_{q}} be a quasi-projective IF_{q}–scheme, \overline{X} = X_{IF_{q}} ×_{spec(IF_{q})} spec(\overline{IF}_{q}) , and ϕ : \overline{X} → \overline{X} the associated Frobenius endomorphism. Then

|X_{IF_{q}}(IF_{q})| = ∑(−1)^{i}T r(ϕ_{c}^{∗} : H_{c}^{i}(\overline{X}; I Q_{ℓ}) → H_{c}^{i}(\overline{X}; I Q_{ℓ})) .  (3)

In accordance with the notation used above we will write gr T r(ϕ_{c}^{∗}) for the right hand side of the formula (3). It remains to relate gr T r(ϕ^{∗}) to gr T r(ϕ_{c}^{∗}) in case X_{IF_{q}} = G. Since \overline{G} is a smooth quasi-projective variety of dimension N over \overline{IF}_{q}, there is a natural Poincaré– Duality pairing (cf. [Mil])

< , >: H_{et}^{j}(\overline{G}; I Q_{ℓ}) × H_{c}^{2N−j}(\overline{G}; I Q_{ℓ}) →I Q_{ℓ}(−N)

satisfying < ϕ^{∗}x, ϕ_{c}^{∗}y >= q^{N} < x, y >. Therefore < x, ϕ_{c}^{∗}y >=< q^{N}(ϕ^{∗})^{−1}(x), y > which, since the pairing < , > is non-degenerate, implies that

T r(ϕ_{c}^{∗} | H_{c}^{2N−j}(\overline{G}; I Q_{ℓ})) = q^{N}T r((ϕ^{∗})^{−1} | H_{et}^{j}(\overline{G}; I Q_{ℓ})) .

Thus, taking graded traces, we infer

gr T r(ϕ_{c}^{∗}) = q^{N} · gr T r((ϕ^{∗})^{−1}) .  (4)

From the computations in the proof of the Lemma 1.2 we see that

gr T r((ϕ^{∗})^{−1}) \begin{matrix}= det(1 − (P ϕ^{∗})^{−1}) = (−1)^{r} det(P ϕ^{∗})^{−1} det(1 − P ϕ^{∗}) \\ = (−1)^{r} det(P ϕ^{∗})^{−1} · gr T r(ϕ^{∗}) \end{matrix}

where r = dim P H_{et}^{∗}(\overline{G}; I Q_{ℓ}) . Note that det P ϕ^{∗} = u ∈ ZZ_{ℓ} is a unit, because ϕ induces an isomorphism in H_{et}^{∗}(\overline{G}; ZZ/ℓ) for any ℓ prime to p. Applying the same reasoning to ϕ^{n}, we obtain

gr T r(((ϕ^{n})^{∗})^{−1}) = (−1)^{r}u^{−n} · gr T r(ϕ^{n})^{∗} .

Therefore, combining this with Corollary 1.3, Proposition 1.4 and formula (4), we obtain

d(ψ_{n}) \begin{matrix}= \frac{u^{n} · gr T r((ϕ^{n})^{∗})^{−1}}{u · gr T r(ϕ^{∗})^{−1}} = \frac{q^{−nN} · u^{n} · gr T r(ϕ_{c}^{∗})^{n}}{q^{−N} · u · gr T r(ϕ_{c}^{∗})} \\ = (\frac{u}{q^{N}} )^{n−1} · \frac{|G(IF_{q^{n}})|}{|G(IF_{q})|} , u ∈ ZZ_{ℓ}^{∗} . \end{matrix}

It is now plain that d(ψ_{n}) is an ℓ–adic unit if and only if G(IF_{q}) and G(IF_{q^{n}}) have isomorphic ℓ–Sylow subgroups and, in view of Proposition 1.1 the proof of the Theorem is therefore completed.

The following example illustrates the general result. Take G = Sℓ_{2} . It is easy to check the Theorem in this case directly because the cohomology rings H^{∗}(Sℓ_{2}(IF_{q}); ZZ/ℓ) , ℓ a prime different from the characteristic of IF_{q}, are completely known (cf. [FiPr]) . The cohomology is periodic of periode 4 ,and for ℓ an odd (prime prime to q) one has

H^{∗}(Sℓ_{2}(IF_{q}); ZZ/ℓ) ≅ \{ \begin{matrix}E(u) ⊗ P (v), u ∈ H^{3}and v ∈ H^{4} ( if ℓ divides q^{2}−1) \\ ZZ/ℓ (if ℓ does not divide q^{2}−1) \end{matrix} .

Here, E(u) denotes an exterior algebra on u, and P (v) a polynomial algebra on v (over ZZ/ℓ) . Since the order of Sℓ_{2}(IF_{q}) is q(q^{2} − 1) we see that the ℓ-Sylow subgroup of Sℓ_{2}(IF_{q}) is non-trivial if and only if ℓ divides q^{2} − 1 and thus, by the formula above, one has abstract isomorphisms of rings

H^{∗}(Sℓ_{2}(IF_{q}); ZZ/ℓ) ≅ H^{∗}(SL_{2}(IF_{q^{n}}); ZZ/ℓ)  (5)

if ℓ | q^{2} − 1.But Sℓ_{2}(IF_{q}) and Sℓ_{2}(IF_{q^{n}}) have isomorphic ℓ-Sylow subgroups if and only if q^{2} − 1 and q^{2n} − 1 contain the same power of ℓ, that is (assuming that ℓ divides q^{2} − 1), if and only if n is relatively prime to ℓ. Thus, in most cases, the isomorphism (5) is not induced by the restriction map

res : H^{∗}(Sℓ_{2}(IF_{q^{n}}); ZZ/ℓ) → H^{∗}(Sℓ_{2}(IF_{q}); ZZ/ℓ) .

Our general setting will take the following form in case G = Sℓ_{2} . One has

H_{et}^{∗}(\overline{Sℓ}_{2}; I Q_{ℓ}) = E(w) ,

an exterior algebra over I Q_{ℓ} in w ∈ H_{et}^{3} . (Recall that H_{et}^{∗}(\overline{Sℓ}_{2}; I Q_{ℓ}) ≅ H_{top}^{∗}(Sℓ_{2}(IC); I Q_{ℓ}), and Sℓ_{2}(IC) is homotopy equivalent to a three-dimensional sphere).

Clearly, P H_{et}^{∗} = H_{et}^{3} ≅ I Q_{ℓ} and one checks easily that ϕ^{∗}w = q^{2}w so that
d(1/ϕ) = det(1 − P ϕ^{∗}) = 1 − q^{2}

and thus

d(ψ_{n}) = \frac{q^{2n} − 1}{q^{2} − 1} = \frac{1}{q^{n−1}} · \frac{|Sℓ_{2}(IF_{q^{n}})|}{|Sℓ_{2}(IF_{q})|} .

According to our general formula, we must have

\frac{1}{q^{n−1}} = (\frac{u}{q^{N}})^{n−1}

where u = det P ϕ^{∗} and N = dim Sℓ_{2} ; this is indeed so, as det P ϕ^{∗} = q^{2} and dim Sℓ_{2} = 3 .

The Theorem of the Introduction can also be viewed as a result concerning the conjugacy relationship between the various ℓ– subgroups of G(IF_{q}) and G(IF_{q^{n}}) . For this purpose, denote by F rob_{ℓ}(F) the ”Frobenius category” of finite ℓ–subgroups of the group F ; its objects are the finite ℓ–subgroups of F , and morphisms are induced by inner automorphisms of F . It is a classical result that for a finite group F , the restriction maps fit together to give rise to an isomorphism

H^{∗}(F; ZZ/ℓ) → \begin{matrix}lim \\ ← \\ π ∈ F rob_{ℓ}(F)^{op} \end{matrix}H^{∗}(π; ZZ/ℓ) .

Thus, any homomorphism of finite groups φ : F_{1} → F_{2} inducing an equivalence of categories F rob_{ℓ}(F_{1}) → F rob_{ℓ}(F_{2}) will induce an isomorphism H^{∗}(F_{2}; ZZ/ℓ) → H^{∗}(F_{1}; ZZ/ℓ) . The converse of this statement is also true according to [Mis]. Thus, our Theorem implies the following corollary, which in the case of G = Gℓ_{n} is a well-known fact on the representation theory of finite ℓ-groups in characteristic p different from ℓ.

Corollary 1.5. Let G be a connected reductive IF_{q}–group scheme and let ℓ be a prime different from p = char(IF_{q}) . Suppose that G(IF_{q}) and G(IF_{q^{n}}) have isomorphic ℓ–Sylow subgroups. Then the inclusion G(IF_{q}) → G(IF_{q^{n}}) induces an equivalence of Frobenius categories of finite ℓ–subgroups

F rob_{ℓ}(G(IF_{q})) → F rob_{ℓ}(G(IF_{q^{n}})) .

2. Suzuki and Ree Groups

We will be considering the three families of groups, which in [Gor] are denoted by

2B_{2}(2^{n}) , ^{2}G_{2}(3^{n}) , ^{2}F_{4}(2^{n})

with n odd, say n = 2m + 1 ; ^{2}B_{2}(2^{n}) is isomorphic to the Suzuki group Sz(2^{n}) , which is simple for n > 1 , and the groups ^{2}G_{2}(3^{n}) , ^{2}F_{4}(2^{n}) are the simple groups of Ree type. The orders of these groups are (cf. [Gor]):

|^{2}B_{2}(q)| = q^{2}(q^{2} + 1)(q − 1) where q = 2^{2m+1}
|^{2}G_{2}(q)| = q^{3}(q^{3} + 1)(q − 1) where q = 3^{2m+1}
|^{2}F_{4}(q)| = q^{12}(q^{6} + 1)(q^{4} − 1)(q^{3} + 1)(q − 1) where q = 2^{2m+1}

Let G = G_{IF_{q}} denote the split reductive group scheme over spec(IF_{q}) of type B_{2} , G_{2} respectively F_{4} . Then \overline{G} = G ×_{spec IF_{q}} spec \overline{IF}_{q} admits an exceptional isogeny

ψ : \overline{G} → \overline{G}

such that ψ^{2} = ϕ, the Frobenius of the IF_{q}–form G of \overline{G}. Moreover

2G(q) = {x ∈ G(IF_{q}) | ψx = x}

agrees with ^{2}B_{2}(q) ,^{2}G_{2}(q) respectively ^{2}F_{4}(q) for G of type B_{2}, G_{2} respectively F_{4} (we always assume q = 2^{2m+1}, 3^{2m+1} respectively 2^{2m+1} , according to the case one considers). As in [FrMi], one gets then a commutative diagram of finite etale maps arising from Lang’s construction

2G(q) → \overline{G} \underrightarrow{1/ψ} \overline{G}
↓ ∥ ↓ θ
2G(q^{d}) → \overline{G} \overrightarrow{1/ψ^{d}} \overline{G}

where d is an odd natural number; θ is given on a point x ∈ \overline{G}(IF_{q}) by

θ(x) = x · ψ(x) · ψ^{2}(x) … ψ^{d−1}(x)

As before, it follows that the inclusion ^{2}G(q) → ^{2}G(q^{d}) is an H_{∗}( ; ZZ/ℓ) isomorphism (ℓ as always prime to q) if and only if θ^{∗} : H_{et}^{∗}(\overline{G}; I Q_{ℓ}) → H_{et}^{∗}(\overline{G}; I Q_{ℓ}) has a degree prime to ℓ (the degree of θ^{∗} is an ℓ–adic integer). There are now three cases to consider. For the computation of the relevant degrees of (Bψ)^{∗} we refer the reader to [AdMa].

(i) : G of type B_{2} In this case, H_{et}^{∗}(B\overline{G}; I Q_{ℓ}) ≅ I Q_{ℓ}[x_{4}, x_{8}] and (Bψ)^{∗}x_{4} = qx_{4} , (Bψ)^{∗}x_{8} = −q^{2}x_{8} . Therefore

\begin{aligned}deg θ & = (1 + q + q^{2} + … + q^{d−1})(1 − q^{2} + q^{4} − … + q^{2d−2}) \\ & = q^{2−2d}|^{2}B_{2}(q^{d})|/|^{2}B_{2}(q)| . \end{aligned}

We conclude that deg θ is prime to ℓ if and only if ^{2}B_{2}(q) and ^{2}B_{2}(q^{d}) have isomorphic ℓ–Sylow subgroups. Thus the following holds.

Proposition 2.1. Let q = 2^{2m+1} and d ≥ 1 an odd integer. Let ℓ be an odd prime. Then ^{2}B_{2}(q) → ^{2}B_{2}(q^{d}) induces an isomorphism in ZZ/ℓ–homology if and only if ^{2}B_{2}(q) and ^{2}B_{2}(q^{d}) have isomorphic ℓ–Sylow subgroups.

(ii) : G of type G_{2} We have H_{et}^{∗}(B\overline{G}; I Q_{ℓ}) ≅ I Q_{ℓ}[x_{4}, x_{12}] and (Bψ)^{∗}x_{4} = qx_{4} , (Bψ)^{∗}x_{12} = −q^{3}x_{12} which yields

deg θ \begin{matrix}= (1 + q + q^{2} + … + q^{d−1})(1 − q^{3} + q^{6} − … + q^{3d−3}) \\ = q^{3−3d}|^{2}G_{2}(q^{d})|/|^{2}G_{2}(q)| \end{matrix}

and the next result follows.

Proposition 2.2. Let q = 3^{2m+1} and d odd. Let ℓ be a prime different from 3 . Then ^{2}G_{2}(q) → ^{2}G_{2}(q^{d}) induces a ZZ/ℓ–homology isomorphism if and only if ^{2}G_{2}(q) and 2G(q^{d}) have isomorphic ℓ–Sylow subgroups.

(iii) : G of type F_{4} We proceed as in the other two cases and obtain:

H^{∗}(B\overline{G}; I Q_{ℓ}) ≅ I Q_{ℓ}[x_{4}, x_{12}, x_{16}, x_{24}] ,

(Bψ)^{∗}x_{4} = qx_{4} , (Bψ)^{∗}x_{12} = −q^{3}x_{12} , (Bψ)^{∗}x_{16} = q^{4}x_{16} , (Bψ)^{∗}x_{24} = −q^{6}x_{24} .

This implies that

deg θ = q^{12−12d}|^{2}F_{4}(q^{d})|/|^{2}F_{4}(q)|

and we get our final result.

Proposition 2.3. Let q = 2^{2m+1} , and d ≥ 1 odd. Let ℓ be an odd prime. Then 2F_{4}(q) → ^{2}F_{4}(q^{d}) induces a ZZ/ℓ–homology isomorphism if and only if ^{2}F_{4}(q) and ^{2}F_{4}(q^{d}) have isomorphic ℓ–Sylow subgroups.

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Mathematik ETH–Zentrum 8092 Zürich Switzerland  and  Dept. of Mathematics Ohio State University Columbus Ohio 43210 USA

May 90, revised Jan 91