Some computations for the exceptional groups relevant to the classification of p-compact groups

Kasper K. S. Andersen Department of Mathematics University of Copenhagen

Universitetsparken 5 DK–2100 København Ø

Denmark kksa@math.ku.dk

February 8, 2001

Dear reader,

The present manuscript comprises my Ph.D. thesis as submitted to the Faculty of Science at the University of Copenhagen. The thesis is written under the supervision of Jesper Michael Møller, who I would like to thank for his constant optimism, interest and support.

The thesis consists of three parts. The first part forms the core of a forthcoming joint paper with Jesper Grodal, Jesper Møller and Antonio Viruel. The last two parts are papers in preparation.

Kasper Klinkby Sonne Andersen Copenhagen, january 2001

Contents

I Elementary abelian subgroups of the exceptional simple groups 4

1 Elementary abelian subgroups of the exceptional simple groups 5 1.1 Introduction and notation . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The group E_{8}(ℂ), p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The group F_{4}(ℂ), p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 The groups E_{6}(ℂ) and 3E_{6}(ℂ), p = 3 . . . . . . . . . . . . . . . . . . 12 1.5 The group 2E_{7}(ℂ), p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 The group E_{8}(ℂ), p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7 Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.8 Corrections and misprints . . . . . . . . . . . . . . . . . . . . . . . . 44 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

II Fusion, Solvability and Swan’s theorem 48

2 Fusion, Solvability and Swan’s theorem 49 2.1 Solvability and fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2 Swan’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

CONTENTS  3

III On the Poincaré series of cohomology rings of finite groups 62

3 On the Poincaré series of cohomology rings of finite groups 63 3.1 Zeroes and poles of the Poincaré series . . . . . . . . . . . . . . . . . 63 3.2 Poincaré series of wreath products . . . . . . . . . . . . . . . . . . . . 71 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Part I

Chapter 1

Elementary abelian subgroups of the exceptional simple groups Elementary abelian subgroups of the exceptional simple groups

1.1 Introduction and notation

Let G be one of the exceptional simple complex Lie groups, i.e. one of type G_{2}, F_{4}, E_{6}, E_{7} or E_{8}. Let p be a prime number. For reasons explained below, we are interested in determining the conjugacy classes of elementary abelian p-subgroups E of G. These may conveniently be divided in two classes, the toral ones and the nontoral ones. Here, an abelian subgroup A is called toral if A is contained in a torus of G and nontoral if it is not. If H is a maximal torus of G, then any two subsets of H are conjugate in G if and only if they are conjugate in the normalizer of the maximal torus N_{G}(H) (see Lemma (1.1.3(4))). Since the action of the Weyl group W = N_{G}(H)/H of G on H is well known, this makes the determination of the conjugacy classes of toral subgroups routine. In fact, in most cases these are the only elementary abelian p-subgroups:

(1.1.1) Proposition. Assume that p ≥ 3 if G has type G_{2}, p ≥ 5 if G has type F_{4}, E_{6} or E_{7} and p ≥ 7 if G has type E_{8}. Then all elementary abelian p-subgroups of G are toral.

(1.1.2) Proof: By [18, Exercise 5.11, p. 209] the conclusion holds if p is not a torsion prime for G. Thus the result follows by examining the list [18, I.4.4, p. 178–179]. □

Thus we are reduced to describing the conjugacy classes of nontoral elementary abelian p-subgroups in the finitely many cases left by the proposition.

In the present paper we carry out the computations for all odd primes p. For each conjugacy class of nontoral elementary abelian p-subgroups, we give an explicit representative E ⊆ G and compute its Weyl group, which by definition is the group W (E) = N_{G}(E)/C_{G}(E). We also compute the class distribution of E, i.e. the number of elements in E from each conjugacy class of elements of order p in G. Moreover we determine the oversize of E, which by definition is the codimension of the largest toral subgroup of E. In almost all cases we also compute the centralizer C_{G}(E) and some auxiliary information which we need for our applications.

Briefly outlined, the strategy of our proof is as follows. The important work of Griess [13] gives the maximal nontoral elementary abelian p-subgroups and some further information as well. Unfortunately this paper is not very accessible or explicit and we have therefore tried to keep our treatment as independent as possible. We have collected some corrections to [13] in the last section.

Following Griess, we first find representatives for the maximal nontoral elementary abelian p-subgroups. We then get lower bounds for their Weyl groups by producing explicit elements in these. From this we are able to identify the nonmaximal nontoral elementary abelian p-subgroups and get lower bounds for their Weyl groups. Finally we get exact results on the Weyl groups by computing centralizers.

The reason for our interest in the problem is that the computations form the core of the classification of p-compact groups [10] for odd primes p. This work, which is joint with Jesper Grodal, Jesper Møller and Antonio Viruel, is the subject of the forthcoming paper [1]. I am happy to take this opportunity to thank all three of them deeply for their collaboration.

Our notation is fairly standard. For elements x, y in a group G we let y^{x} = x^{−1}yx denote the conjugate of y by x, and [x, y] = x^{−1}y^{−1}xy be the commutator of x and y.

Let ℂ be the field of complex numbers and 𝔽_{p} the finite field with p elements. Now let K be an arbitrary field. We let K^{×} denote the multiplicative group of K. The algebra of n × n-matrices over K is denoted M_{n}(K) and we let I_{n} ∈ M_{n}(K) be the identity matrix. The diagonal matrix with entries a_{1}, …, a_{n} ∈ K is denoted diag(a_{1}, …, a_{n}). We let GL_{n}(K) and SL_{n}(K) be the general linear group and the special linear group respectively. For an elementary abelian p-subgroup E ⊆ G of rank r, the Weyl group W (E) is a subgroup of GL(E). Fixing a basis of E, we may consider W (E) as a subgroup of GL_{r}(𝔽_{p}). We make the convention that all matrices acts on columns. For a permutation σ in the symmetric group Σ_{n} we also let σ denote the associated permutation matrix A = [a_{ij}], where

a_{ij} = δ_{i,σ(j)} = \{ \begin{matrix}1 & for i = σ(j) \\ 0 & otherwise. \end{matrix}

We shall on occasions also use the following special notation. For matrices A_{1} ∈ M_{n_{1}}(K), …, A_{m} ∈ M_{n_{m}}(K) we let

A_{1} ⊕ … ⊕ A_{m} = [ \begin{matrix}A_{1} & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & A_{m} \end{matrix} ]

denote the n × n-block matrix with the A_{i}’s in the diagonal (n = n_{1} + … + n_{m}). We also need the “blowup” homomorphism Δ_{n,m} : M_{n}(K) ⟶ M_{mn}(K) defined by

Δ_{n,m} ([ \begin{matrix}a_{11} & ⋯ & a_{1n} \\ ⋮ & ... & ⋮ \\ a_{n1} & ⋯ & a_{nn} \end{matrix} ]) = [ \begin{matrix}a_{11}I_{m} & ⋯ & a_{1n}I_{m} \\ ⋮ & ... & ⋮ \\ a_{n1}I_{m} & ⋯ & a_{nn}I_{m} \end{matrix} ] ,

i.e. the mn × mn-block matrix obtained by replacing each entry a_{ij} by a_{ij}I_{m}.

As we have p = 3 in almost all the cases we consider we use some special notation. We let ∗ denote an arbitrary element of 𝔽_{3} and ε denote an element of the multiplicative group 𝔽_{3}^{×}. Moreover we let ω = e^{2πi/3} and η = e^{2πi/9}. We also consider the elements

β = diag(1, ω, ω^{2}) = [ \begin{matrix}1 & 0 & 0 \\ 0 & ω & 0 \\ 0 & 0 & ω^{2} \end{matrix} ] ,  γ = (1, 2, 3) = [ \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ] ,

τ_{1} = \frac{e^{−πi/18}}{\sqrt{3}} [ \begin{matrix}1 & ω^{2} & 1 \\ 1 & 1 & ω^{2} \\ ω^{2} & 1 & 1 \end{matrix} ] ,  τ_{2} = diag(η, η^{−2}, η) = [ \begin{matrix}η & 0 & 0 \\ 0 & η^{−2} & 0 \\ 0 & 0 & η \end{matrix} ]

in SL_{3}(ℂ) and note that β^{τ_{1}} = βγ, γ^{τ_{1}} = γ, β^{τ_{2}} = β and γ^{τ_{2}} = βγ.

For complex Lie groups we use standard notation taken from the literature on algebraic groups. Basis references are [15] and [17]. For the root systems we use the conventions used in [3]. We shall need the following well-known result repeatedly.

(1.1.3) Lemma. Let G be an algebraic group over an algebraically closed field K.

(1) If A is an abelian subgroup of G and x ∈ A, then A is toral in G if and only if A is toral in C_{G}(x).

(2) If G is simply connected and reductive, then the centralizer of a semisimple element is connected.

(3) If G is simply connected and reductive and A is an abelian subgroup of G consisting of semisimple elements generated by at most two elements, then A is toral.

(4) If H is a maximal torus, then two subsets of H are conjugate in G if and only if they are conjugate in N_{G}(H). If A is toral in G, then W (A) = N_{G}(A)/C_{G}(A) is isomorphic to a subquotient of the Weyl group W = N_{G}(H)/H of G.

(5) Assume that G is reductive, let Z ⊆ G be a central subgroup, and let π : G → G/Z be the quotient homomorphism. Then A ⊆ G/Z is toral in G/Z if and only if π^{−1}(A) is toral in G.

(6) Assume char K = 0, let G be connected and let 𝔤 be the Lie algebra of G. If S ⊆ G is a finite subset of G, then the Lie algebra of C_{G}(S) is given by

𝔠_{𝔤}(S) = {x ∈ 𝔤| Ad(s)(x) = x for all s ∈ S}.

In particular, if S ⊆ G is a finite subgroup, then

dim C_{G}(S) = \frac{1}{|S|} ∑_{s∈S} tr Ad(s)|_{𝔤}.

(1.1.4) Proof: (1): Obviously, if A is toral in C_{G}(x) then A is toral in G. Conversely, if A is toral in G, then A ⊆ H for a torus H in G. Since x ∈ A we get H ⊆ C_{G}(x) and thus A is toral in C_{G}(x).

(2): Our assumptions imply that G is semisimple, and thus the result follows from [18, Theorem II.3.9, p. 197].

(3): This follows directly from part (2), cf. [18, II.5.1, p. 206].

(4): The first part follows by a Frattini argument. Let A, A^{g} ⊆ H be conjugate. Then H and H^{g^{−1}} are maximal tori of C_{G}(A). Thus we may write H = H^{g^{−1}c} with c ∈ C_{G}(A) and we conclude that n = g^{−1}c ∈ N_{G}(H). Then A^{n^{−1}} = A^{c^{−1}g} = A^{g}, which proves the first part. The second part follows similarly, cf. [16, Proposition 1.1(i)].

(5): By [15, Corollary 21.3.C] we know that if H is a maximal torus of G, then π(H) is a maximal torus of G/Z, and all maximal tori of G/Z are of this form. Since G is reductive we have Z ⊆ H for all maximal tori H of G by [15, Corollary 26.2.A(b)]. From the above we then see that if H′ is a maximal torus of G/Z, then π^{−1}(H′) is a maximal torus of G and all maximal tori of G are obtained in this way. From this relation between the maximal tori of G and G/Z, the claim follows immediately.

(6): This first part follows by combining [15, Theorem 13.4(a)] and [15, Theorem 12.5]. Now assume that S ⊆ G is a finite group, and let χ be the character of the adjoint representation of G restricted to S. Then the dimension of

𝔠_{𝔤}(S) = {x ∈ 𝔤| Ad(s)(x) = x for all s ∈ S}.

equals the multiplicity of the trivial character in χ. By the orthogonality relations this is given by

(χ|1) = \frac{1}{|S|} ∑_{s∈S} χ(s),

and we are done. □

For some of our computations we have used the computer algebra system MAGMA [2]. More precisely we use this for our computations in 3E_{6}(ℂ) and E_{8}(ℂ). For both of these, we represent the group by the action given by its smallest faithful complex representation. We also use the computer to make some computations of minimal overgroups in GL_{r}(𝔽_{p}) relevant to the exact determination of the Weyl groups of the elementary abelian p-group in consideration.

The remainder of this part of thesis is divided into 7 sections. The first two which are extremely short deals with E_{8}(ℂ) in the case p = 5 and F_{4}(ℂ) in the case p = 3 respectively. The next sections deals in succession with the groups 3E_{6}(ℂ) and E_{6}(ℂ), 2E_{7}(ℂ) and E_{8}(ℂ). In section 1.7 we compute some of the centralizers C_{G}(E). The final section collects some corrections to the papers [7; 8; 13].

1.2 The group E_{8}(ℂ), p = 5

(1.2.1) Theorem (E_{8}). Up to conjugacy, E_{8}(ℂ) contains a unique nontoral elementary abelian 5-subgroup E. This group has rank 3, is 1-oversize and has Weyl group W (E) = SL_{3}(𝔽_{5}).

(1.2.2) Proof: This follows from [13, Lemma 10.3], except for the result on the oversize. However since E is the only nontoral elementary abelian subgroup up to conjugacy, it must be exactly 1-oversize. □

1.3 The group F_{4}(ℂ), p = 3

(1.3.1) Theorem (F_{4}). Up to conjugacy, F_{4}(ℂ) contains a unique nontoral elementary abelian 3-subgroup E. This group has rank 3, is 1-oversize and has Weyl group W (E) = SL_{3}(𝔽_{3}).

(1.3.2) Proof: This follows from [13, Theorem 7.4], except for the result on the oversize. However since E is the only nontoral elementary abelian subgroup up to conjugacy, it must be exactly 1-oversize. □

1.4 The groups E_{6}(ℂ) and 3E_{6}(ℂ), p = 3

(1.4.1) Setup. In this section we consider the elementary abelian 3-subgroups of the groups of type E_{6} over ℂ. The adjoint group is denoted E_{6}(ℂ), and the simply connected group is denoted by 3E_{6}(ℂ). The group 3E_{6}(ℂ) has two nonisomorphic faithful irreducible 27-dimensional representations, these have highest weight λ_{1} and λ_{6} respectively and are dual to each other. An explicit construction of 3E_{6}(ℂ) based on one of these representations was originally given by Freudenthal [11]. This construction is described in more detail in [8, section 2] from which we take most of our notation. In particular we let 𝕂 be the 27-dimensional complex vector space consisting of triples m = (m_{1}, m_{2}, m_{3}) of complex 3×3-matrices m_{i}, 1 ≤ i ≤ 3 where addition and scalar multiplication is defined coordinatewise. We define a cubic form ⟨·⟩ on 𝕂 by

⟨m⟩ = det(m_{1}) + det(m_{2}) + det(m_{3}) − tr(m_{1}m_{2}m_{3}).

Then 3E_{6}(ℂ) is the subgroup of GL(𝕂) preserving the form ⟨·⟩. Moreover the stabilizer in 3E_{6}(ℂ) of the element (I_{3}, 0, 0) ∈ 𝕂 is the group F_{4}(ℂ). For g_{1}, g_{2}, g_{3} ∈ SL_{3}(ℂ) we have the element s_{g_{1},g_{2},g_{3}} of 3E_{6}(ℂ) given by

s_{g_{1},g_{2},g_{3}} (m_{1}, m_{2}, m_{3}) = (g_{1}m_{1}g_{2}^{−1}, g_{2}m_{2}g_{3}^{−1}, g_{3}m_{3}g_{1}^{−1})

for m = (m_{1}, m_{2}, m_{3}) ∈ 𝕂. This gives a representation of SL_{3}(ℂ)^{3} which has kernel C_{3} generated by (ωI_{3}, ωI_{3}, ωI_{3}), and we thus get an embedding of SL_{3}(ℂ)^{3}/C_{3} in 3E_{6}(ℂ). We will denote the element s_{g_{1},g_{2},g_{3}} by [g_{1}, g_{2}, g_{3}].

We let {e_{j,k}^{i}}, 1 ≤ i, j, k ≤ 3 be the natural basis of 𝕂 consisting of the elements e_{j,k}^{i} whose entries are all 0 except for the (j, k)-entry of the i’th matrix which equals 1. The elements of 3E_{6}(ℂ) which acts diagonally with respect to this basis of 𝕂 form a maximal torus H in 3E_{6}(ℂ). Let e_{j,k} be the 3 × 3-matrix whose only nonzero element is 1 occurring at the (j, k)-entry, and m_{i}^{j,k} be the (j, k)-entry of the matrix m_{i}. We then have H-invariant subgroups

u_{α_{1}}(t) = [I_{3}, I_{3} + te_{1,3}, I_{3}] , u_{−α_{1}}(t) = [I_{3}, I_{3} + te_{3,1}, I_{3}] ,
u_{α_{2}}(t) = [I_{3} + te_{2,1}, I_{3}, I_{3}] , u_{−α_{2}}(t) = [I_{3} + te_{1,2}, I_{3}, I_{3}] ,
u_{α_{3}}(t) = [I_{3}, I_{3} + te_{2,1}, I_{3}] , u_{−α_{3}}(t) = [I_{3}, I_{3} + te_{1,2}, I_{3}] ,

u_{α_{4}}(t) : (m_{i})_{i=1,2,3} ↦ (m_{i} + t · [ \begin{matrix}0 & −m_{i+2}^{2,3} & 0 \\ 0 & 0 & 0 \\ 0 & m_{i+2}^{2,1} & 0 \end{matrix} ])_{i=1,2,3}

u_{−α_{4}}(t) : (m_{i})_{i=1,2,3} ↦ (m_{i} + t · [ \begin{matrix}0 & 0 & 0 \\ m_{i+1}^{3,2} & 0 & −m_{i+1}^{1,2} \\ 0 & 0 & 0 \end{matrix} ])_{i=1,2,3}

u_{α_{5}}(t) = [I_{3}, I_{3}, I_{3} + te_{2,1}] , u_{−α_{5}}(t) = [I_{3}, I_{3}, I_{3} + te_{1,2}] ,
u_{α_{6}}(t) = [I_{3}, I_{3}, I_{3} + te_{1,3}] , u_{−α_{6}}(t) = [I_{3}, I_{3}, I_{3} + te_{3,1}] .

Here, in the description of u_{±α_{4}}(t), the m_{i}’s should be counted cyclicly mod 3, e.g. m_{i+2} = m_{1} for i = 2.

The associated roots α_{i}, 1 ≤ i ≤ 6, of these root subgroups form a base for the root system Φ(E_{6}) of 3E_{6}(ℂ) (our numbering agrees with [3, p. 260–262]). For this base of Φ(E_{6}), the highest weight of 𝕂 is λ_{1}. Furthermore the root subgroups u_{±α_{i}}, 1 ≤ i ≤ 6, have been chosen so that they satisfy [17, 8.1.1(i) and 8.1.4(i)], i.e. they form part of a realization ([17, p. 133]) of Φ(E_{6}) in 3E_{6}(ℂ). For α = ±α_{i}, 1 ≤ i ≤ 6, and t ∈ ℂ^{×}, we may then define the elements

n_{α}(t) = u_{α}(t)u_{−α}(−1/t)u_{α}(t), h_{α}(t) = n_{α}(t)n_{α}(1)^{−1}.

Then the maximal torus consists of the elements h(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = ∏_{i=1}^{6} h_{α_{i}}(t_{i}) and the normalizer N(H) of the maximal torus is generated by H and the elements n_{i} = n_{α_{i}}(1), 1 ≤ i ≤ 6. It should be noted that this notation differs from the one used in [8]. More precisely, the element h(α, β, γ, δ, ε, ζ) in [8] is h(δ, α^{−1}, γ^{−1}, β, ε^{−1}, ζ) in our notation, and the elements n_{1}, n_{2}, n_{3}, n_{4}, n_{5} and n_{6} in [8] equals respectively n_{1}h_{α_{1}}(−1)h_{α_{3}}(−1), n_{2}h(−1, 1, 1, −1, 1, −1), n_{3}h_{α_{1}}(−1), n_{4}, n_{5}h_{α_{6}}(−1) and n_{6}h_{α_{5}}(−1)h_{α_{6}}(−1) in our notation.

From the description of the root system of type E_{6} in [3, p. 260–262] we see that the center Z of 3E_{6}(ℂ) is cyclic of order 3 and is generated by the element z = [I_{3}, ω^{2}I_{3}, ωI_{3}]. We consider also the element a = [ωI_{3}, I_{3}, I_{3}]. A straightforward computation shows that the roots of the centralizer C_{3E_{6}(ℂ)}(a) are

{±α_{1}, ±α_{2}, ±α_{3}, ±α_{5}, ±α_{6}, ±α̃, ±(α_{1} + α_{3}), ±(α_{5} + α_{6}), ±(α_{2} − α̃)},

where α̃ is the longest root. The Dynkin diagram for this centralizer is the same as the extended Dynkin diagram for E_{6} with the node α_{4} removed. In particular it has type A_{2}A_{2}A_{2} and basis {α_{1}, α_{3}, α_{5}, α_{6}, α_{2}, −α̃}. Since 3E_{6}(ℂ) is simply connected, Lemma (1.1.3(2)) implies that the centralizer C_{3E_{6}(ℂ)}(a) is connected, and thus it is generated by the maximal torus H and the root subgroups u_{±α}(t) where α runs through the simple roots in the basis {α_{1}, α_{3}, α_{5}, α_{6}, α_{2}, −α̃} of the root system of centralizer. Now note that uα̃(t) = [I_{3} + te_{3,1}, I_{3}, I_{3}] and u_{−α̃}(t) = [I_{3} + te_{1,3}, I_{3}, I_{3}] are root subgroups with associated roots α̃ and −α̃ respectively. Since these along with H and the root subgroups u_{±α_{1}}, u_{±α_{2}}, u_{±α_{3}}, u_{±α_{5}} and u_{±α_{6}} generate the subgroup SL_{3}(ℂ)^{3}/C_{3} of 3E_{6}(ℂ) from above, we conclude that C_{3E_{6}(ℂ)}(a) = SL_{3}(ℂ)^{3}/C_{3}.

To describe the conjugacy classes of elementary abelian 3-subgroups we need to introduce some more elements. Consider the following elements in SL_{3}(ℂ)^{3}/C_{3} ⊆ 3E_{6}(ℂ):

x_{1} = [I_{3}, β, β] ,  x_{2} = [β, β, β] ,  y_{1} = [I_{3}, γ, γ^{2}] ,  y_{2} = [γ, γ, γ] .

We also need the following elements in N(H):

s_{1} = n_{1}n_{3}n_{4}n_{2}n_{5}n_{4}n_{3}n_{1}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}
s_{2} = n_{1}n_{2}n_{3}n_{1}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4} ·
n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}

The action of these elements are as follows:

s_{1}(m_{1}, m_{2}, m_{3}) = (m_{3}, m_{1}, m_{2}), s_{2}(m_{1}, m_{2}, m_{3}) = (m_{3}^{T}, m_{2}^{T}, m_{1}^{T}),

where m_{i}^{T} denotes the transpose of m_{i}. From the description of the subgroup SL_{3}(ℂ)^{3}/C_{3} we see that conjugation by these elements acts as follows

[g_{1}, g_{2}, g_{3}]^{s_{1}} = [g_{2}, g_{3}, g_{1}] , [g_{1}, g_{2}, g_{3}]^{s_{2}} = [(g_{1}^{−1})^{T} , (g_{3}^{−1})^{T} , (g_{2}^{−1})^{T}] .

It may be checked that the image of s_{2} in the Weyl group W = W (E_{6}) equals the longest element in W. Moreover the element s_{1}s_{2}^{−1} lies in the subgroup F_{4}(ℂ) and its image in W (F_{4}) equals the longest element in W (F_{4}). We will not need these facts however.

(1.4.2) Lemma. We have

z = h(ω, 1, ω^{2}, 1, ω, ω^{2}), a = h(ω, 1, ω^{2}, 1, ω^{2}, ω), x_{1} = h(ω, 1, ω, 1, ω, ω),
x_{2} = h(1, ω^{2}, ω^{2}, 1, ω^{2}, 1), y_{1} = n_{1}n_{3}n_{5}n_{6}h_{α_{5}}(−1),
y_{2} = n_{1}n_{2}n_{3}n_{4}n_{3}n_{1}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{6}n_{5}h_{α_{2}}(−1).

Moreover conjugation by the element

n_{1}n_{4}n_{2}n_{3}n_{1}n_{4}n_{5}n_{4}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4} · h_{α_{2}}(−1)h_{α_{4}}(−1)

acts as follows:

a ↦ x_{2}, x_{2} ↦ a, y_{1} ↦ s_{1}, y_{2} ↦ y_{2}^{2}, x_{2}x_{1}^{−1} ↦ h_{α_{4}}(ω) = [τ_{2}, τ_{2}, τ_{2}] .

(1.4.3) Proof: Both parts of the lemma may be checked by direct computation. The second part also follows from the first by using the following relations in N(H): The element n_{i} has image s_{α_{i}} in W ([17, 8.1.4(i)]), we have n_{i}^{2} = h_{α_{i}}(−1) ([17, 8.1.4(ii)]) and

n_{i}n_{j}n_{i} … = n_{j}n_{i}n_{j} …

for 1 ≤ i, j ≤ 6, where the number of factors on both sides equals the order of s_{α_{i}}s_{α_{j}} in W ([17, 9.3.2]). □

(1.4.4) Conjugacy classes. For our calculations, we need some information on the conjugacy classes of elements of order 3 in 3E_{6}(ℂ). These are given in [8, Table 2]: There are 7 such conjugacy classes, which we label 3A, 3B, 3B′, 3C, 3D, 3E and 3E′, where 3B′ and 3E′ denotes the inverses of the classes 3B and 3E. This notation is almost identical to the notation in [8], but differs from [13]. We will need the following, which follows quickly from [8, Table 2] using the action of W on H: We have z ∈ 3E, a, x_{2}, y_{2} ∈ 3C, x_{1}, y_{1} ∈ 3D and x_{2}x_{1}^{−1} ∈ 3A. Multiplication by z acts as follows on the conjugacy classes:

3A ↦ 3B, 3B ↦ 3B′, 3B′ ↦ 3A, 3C ↦ 3C, 3D ↦ 3D, 3E ↦ 3E′, 3E′ ↦ 1,

where 1 denotes the conjugacy class consisting of the identity element.

(1.4.5) Theorem (3E_{6}). The conjugacy classes of nontoral elementary abelian 3-subgroups of 3E_{6}(ℂ) are given by the following table.

rank name

ordered basis 3E_{6}(ℂ)-class distribution dim C_{3E_{6}(ℂ)}(E)
3 E_{3E_{6}}^{3}  ⟨a, x_{2}, y_{2}⟩  3C^{26}  0
4 E_{3E_{6}}^{4}  ⟨z, a, x_{2}, y_{2}⟩  3C^{78}3E^{1}3E′^{1}  0

Their Weyl groups with respect to the given ordered bases satisfies the following:

W (E_{3E_{6}}^{3}) ⊇ SL_{3}(𝔽_{3}), W (E_{3E_{6}}^{4}) ⊇ [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] .

(1.4.6) Remark. In section 1.7 we will prove that the Weyl groups of E_{3E_{6}}^{3} and E_{3E_{6}}^{4} are exactly the lower bounds given above.

(1.4.7) Proof:

NONTORAL SUBGROUPS: By [13, Theorem (11.13)], there are two conjugacy classes of nontoral elementary abelian 3-subgroups in 3E_{6}(ℂ), one nonmaximal of rank 3 and one maximal of rank 4. We may concretely realize these as follows. Consider the subgroups

E_{3E_{6}}^{3} = ⟨a, x_{2}, y_{2}⟩ and E_{3E_{6}}^{4} = ⟨z, a, x_{2}, y_{2}⟩ ,

which are readily seen to be elementary abelian 3-subgroups of rank 3 and 4 respectively. In particular both groups are subsets of C_{3E_{6}(ℂ)}(a) = SL_{3}(ℂ)^{3}/C_{3}, and since β, γ ∈ SL_{3}(ℂ) does not commute, we see that the preimages of E_{3E_{6}}^{3} and E_{3E_{6}}^{4} under the projection SL_{3}(ℂ)^{3} → SL_{3}(ℂ)^{3}/C_{3} are nonabelian. Thus by Lemma (1.1.3(5)) E_{3E_{6}}^{3} and E_{3E_{6}}^{4} are nontoral in SL_{3}(ℂ)^{3}/C_{3} = C_{3E_{6}(ℂ)}(a) and hence also nontoral in 3E_{6}(ℂ) by Lemma (1.1.3(1)). Thus by the above these two groups represent the conjugacy classes of nontoral elementary abelian 3-subgroups in 3E_{6}(ℂ).

WEYL GROUPS: Note that we have an inclusion F_{4}(ℂ) ⊆ 3E_{6}(ℂ). Thus the unique nontoral subgroup E of F_{4}(ℂ) of rank 3 from Theorem (1.3.1) may also be considered as a subgroup of 3E_{6}(ℂ). As its Weyl group in F_{4}(ℂ) is SL_{3}(𝔽_{3}), its Weyl group in 3E_{6}(ℂ) must contain SL_{3}(𝔽_{3}). In particular it has order divisible by 13 and since the order of W (E_{6}) is 2^{7} · 3^{4} · 5 which is not divisible by 13, we conclude by Lemma (1.1.3(4)) that E is nontoral in 3E_{6}(ℂ) as well. Thus by the above E must be conjugate to E_{3E_{6}}^{3}, and we get that W (E_{3E_{6}}^{3}) contains SL_{3}(𝔽_{3}). From this we immediately see that W (E_{3E_{6}}^{4}) contains the group

[ \begin{matrix}1 & 0 0 0 \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] .

Note that the element [I_{3}, β, β^{2}] commutes with z, a and x_{2} and conjugates y_{2} to y_{2}z. Thus it normalizes E_{3E_{6}}^{4} and produces the element

[ \begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ]

in W (E_{3E_{6}}^{4}). As a result we see that W (E_{3E_{6}}^{4}) contains

[ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] .

CLASS DISTRIBUTIONS: Since a ∈ 3C by (1.4.4) and W (E_{3E_{6}}^{3}) contains SL_{3}(𝔽_{3}) which acts transitively on E_{3E_{6}}^{3} \ {1}, the class distribution of E_{3E_{6}}^{3} follows immediately. Using this and the information given in (1.4.4) about the action of z on the conjugacy classes, the class distribution of E_{3E_{6}}^{4} is easily found.

CENTRALIZER DIMENSIONS: By Lemma (1.1.3(6)) we see that the dimension of C_{3E_{6}(ℂ)}(E) may be computed from the class distribution of E and the action of E on the Lie algebra 𝔢_{6}. The eigenvalue multiplicities on 𝔢_{6} for the different conjugacy classes in 3E_{6}(ℂ) are given in [8, Table 2], and using this we easily get the dimensions given above. As an example consider E_{3E_{6}}^{3}. We find:

dim C_{3E_{6}(ℂ)}(E_{3E_{6}}^{3}) = \frac{1}{|E_{3E_{6}}^{3}|} ∑_{x∈E_{3E_{6}}^{3}} tr x|_{𝔢_{6}} = \frac{1}{27} · (78 + 26 · (24 + 27ω + 27ω^{2})) = 0.

A similar computation for E_{3E_{6}}^{4} finishes the proof. □

(1.4.8) Notation. We now turn to the group E_{6}(ℂ). As above we let Z be the center of 3E_{6}(ℂ) and we let π : 3E_{6}(ℂ) → E_{6}(ℂ) = 3E_{6}(ℂ)/Z denote the projection. For g ∈ 3E_{6}(ℂ) we write simply \overline{g} instead of π(g) and similarly we let \overline{S} = π(S) for a subset S ⊆ 3E_{6}(ℂ).

(1.4.9) Lemma. Let E be a rank 2 nontoral elementary abelian 3-subgroup of E_{6}(ℂ). Then the Weyl group W (E) is a subgroup of SL_{2}(𝔽_{3}).

(1.4.10) Proof: Let E = ⟨\overline{g_{1}}, \overline{g_{2}}⟩. By Lemma (1.1.3) part (5) and (3) the group ⟨g_{1}, g_{2}⟩ ⊆ 3E_{6}(ℂ) is nonabelian. Thus setting z′ = [g_{1}, g_{2}] ∈ Z we have z′ ≠ 1. As-

sume that σ ∈ W (E) is represented by the matrix [ \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ], i.e. we have

σ(\overline{g_{1}}) = (\overline{g_{1}})^{a_{11}}(\overline{g_{2}})^{a_{21}} and σ(\overline{g_{2}}) = (\overline{g_{1}})^{a_{12}}(\overline{g_{2}})^{a_{22}}. Since σ is given by a conjugation in E_{6}(ℂ), it lifts to a conjugation in 3E_{6}(ℂ). Now the relation [g_{1}, g_{2}] = z′ ∈ Z shows (z′)^{a_{11}·a_{22}−a_{12}·a_{21}} = z′, so since z′ ≠ 1 we have σ ∈ SL_{2}(𝔽_{3}). □

(1.4.11) Theorem (E_{6}). The conjugacy classes of nontoral elementary abelian 3-subgroups of E_{6}(ℂ) are given by the following table,

rank name

ordered basis  3E_{6}(ℂ)-class distribution  dim C_{E_{6}(ℂ)}(E)
2 E_{E_{6}}^{2a}  ⟨\overline{y_{1}}, \overline{x_{2}}⟩  3C^{18}3D^{6}3E^{1}3E′^{1}  8
2 E_{E_{6}}^{2b}  ⟨\overline{y_{1}}, \overline{x_{1}}⟩  3D^{24}3E^{1}3E′^{1}  14
3 E_{E_{6}}^{3a}  ⟨\overline{a}, \overline{y_{1}}, \overline{x_{2}}⟩  3C^{60}3D^{18}3E^{1}3E′^{1}  2
3 E_{E_{6}}^{3b}  ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩  3C^{78}3E^{1}3E′^{1}  0
3 E_{E_{6}}^{3c}  ⟨\overline{a}, \overline{y_{1}}, \overline{x_{1}}⟩  3C^{6}3D^{72}3E^{1}3E′^{1}  8
3 E_{E_{6}}^{3d}  ⟨\overline{x_{2}x_{1}^{−1}}, \overline{y_{1}}, \overline{x_{1}}⟩ 3A^{2}3B^{2}3B′^{2}3C^{48}3D^{24}3E^{1}3E′^{1}  4
4 E_{E_{6}}^{4a}  ⟨\overline{a}, \overline{y_{2}}, \overline{y_{1}}, \overline{x_{2}}⟩  3C^{186}3D^{54}3E^{1}3E′^{1}  0
4 E_{E_{6}}^{4b} ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}, \overline{y_{1}}, \overline{x_{1}}⟩ 3A^{6}3B^{6}3B′^{6}3C^{150}3D^{72}3E^{1}3E′^{1}  2

where the last entry in the row corresponding to E ⊆ E_{6}(ℂ) is the class distribution of π^{−1}(E) ⊆ 3E_{6}(ℂ).

These groups are all 1-oversize and their Weyl groups with respect to the given ordered bases satisfies the following:

W (E_{E_{6}}^{2a}) = [ \begin{matrix}ε & ∗ \\ 0 & ε \end{matrix} ] , W (E_{E_{6}}^{2b}) = SL_{2}(𝔽_{3}), W (E_{E_{6}}^{3a}) = [ \begin{matrix}ε_{1} & ∗ & ∗ \\ 0 & ε_{2} & ∗ \\ 0 & 0 & ε_{2} \end{matrix} ] ,

W (E_{E_{6}}^{3b}) ⊇ SL_{3}(𝔽_{3}), W (E_{E_{6}}^{3c}) = [ \begin{matrix}ε & ∗ ∗ \\ {0 \atop 0} & SL_{2}(𝔽_{3}) \end{matrix} ] , W (E_{E_{6}}^{3d}) = [ \begin{matrix}ε & 0 0 \\ {0 \atop 0} & SL_{2}(𝔽_{3}) \end{matrix} ] ,

W (E_{E_{6}}^{4a}) ⊇ [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det & ∗ \\ 0 0 & 0 & det \end{matrix} ] , W (E_{E_{6}}^{4b}) = [ \begin{matrix}ε_{1} & ∗ & ∗ ∗ \\ 0 & ε_{2} & 0 0 \\ {0 \atop 0} & {0 \atop 0} & SL_{2}(𝔽_{3}) \end{matrix} ] ,

where det denotes the determinant of the matrix from GL_{2}(𝔽_{3}) in the description of W (E_{E_{6}}^{4a}).

(1.4.12) Remark. In section 1.7 we will prove that the Weyl groups of E_{E_{6}}^{3b} and E_{E_{6}}^{4a} are

exactly the lower bound given above.

(1.4.13) Proof:

MAXIMAL NONTORAL SUBGROUPS: By [13, Theorem (11.14)], there are two conjugacy classes of maximal nontoral elementary abelian 3-subgroups in E_{6}(ℂ), both of which have rank 4. We may concretely realize these as follows. Consider the subgroups

E_{a} = ⟨z, a, y_{1}, y_{2}, x_{2}⟩ and E_{b} = ⟨z, a, x_{2}x_{1}^{−1}, y_{1}, x_{1}⟩

of C_{3E_{6}(ℂ)}(a) = SL_{3}(ℂ)^{3}/C_{3}. Since the commutator subgroup of both of these is Z, we see that E_{E_{6}}^{4a} = π(E_{a}) and E_{E_{6}}^{4b} = π(E_{b}) are elementary abelian subgroups of rank 4 in E_{6}(ℂ). It follows from Lemma (1.1.3(5)) that both E_{E_{6}}^{4a} and E_{E_{6}}^{4b} are nontoral in E_{6}(ℂ). We will see below that their class distributions are as given in the table. From this it follows that they are not conjugate and thus represents the two conjugacy classes of maximal elementary abelian 3-subgroups in E_{6}(ℂ).

LOWER BOUNDS FOR WEYL GROUPS OF MAXIMAL NONTORAL SUB-GROUPS: We can find lower bounds for the Weyl groups of the maximal nontoral elementary abelian 3-subgroups by conjugating with elements coming from the centralizer C_{3E_{6}(ℂ)}(a) = SL_{3}(ℂ)^{3}/C_{3} and the normalizer N(H) of the maximal torus.

The elements \overline{[β^{2}, I_{3}, I_{3}]}, \overline{[I_{3}, τ_{1}, τ_{1}^{2}]}, \overline{s_{1}} and \overline{s_{2}} normalize E_{E_{6}}^{4a} and conjugation by these elements induce the automorphisms given by the matrices

[ \begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{matrix} ]

on E_{E_{6}}^{4a}. Moreover, by Lemma (1.4.2) we may conjugate the ordered basis of E_{E_{6}}^{4a}

into the ordered basis ⟨\overline{x_{2}}, \overline{y_{2}^{2}}, \overline{s_{1}}, \overline{a}⟩ . Noting that the element \overline{[τ_{1}, τ_{1}, τ_{1}]} commutes with \overline{y_{2}}, \overline{s_{1}} and \overline{a} and conjugates \overline{x_{2}} into \overline{x_{2}y_{2}}, we see that W (E_{E_{6}}^{4a}) contains the automorphism given by the matrix

[ \begin{matrix}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] .

The above matrices are easily seen to generate the group

W′(E_{E_{6}}^{4a}) = [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det & ∗ \\ 0 0 & 0 & det \end{matrix} ] ,

and thus W_{E_{6}}^{4a} contains this group.

Now consider E_{E_{6}}^{4b} and let σ = −(2, 3) ∈ SL_{3}(ℂ). We then see that the elements \overline{[I_{3}, τ_{1}, τ_{1}^{2}]}, \overline{[I_{3}, τ_{2}β, τ_{2}^{2}]}, \overline{[σ, I_{3}, I_{3}]}, \overline{[γ, I_{3}, I_{3}]}, \overline{[I_{3}, β^{2}, I_{3}]} and \overline{s_{2}} normalize E_{E_{6}}^{4b}, and conjugation by these elements induce the automorphisms given by the matrices

[ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] ,

[ \begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{matrix} ]

on E_{E_{6}}^{4b}. We see that these matrices generate the group

W′(E_{E_{6}}^{4b}) = [ \begin{matrix}ε_{1} & ∗ & ∗ ∗ \\ 0 & ε_{2} & 0 0 \\ {0 \atop 0} & {0 \atop 0} & SL_{2}(𝔽_{3}) \end{matrix} ]

and thus W_{E_{6}}^{4b} contains this group.

ORBIT COMPUTATION: Any elementary abelian 3-subgroup of rank 1 is toral since E_{6}(ℂ) is connected. As we already know that E_{E_{6}}^{4a} and E_{E_{6}}^{4b} are representatives of the maximal nontoral elementary abelian 3-subgroups, we may find the conjugacy classes of nontoral elementary abelian 3-subgroups of rank 2 and 3 by studying subgroups of these.

Under the action of W′(E_{E_{6}}^{4a}), the set of rank 2 subgroups of E_{E_{6}}^{4a} has orbit representatives

E_{E_{6}}^{2a} = ⟨\overline{y_{1}}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{y_{1}}⟩ and ⟨\overline{a}, \overline{y_{2}}⟩ ,

and under the action of W′(E_{E_{6}}^{4b}), the set of rank 2 subgroups of E_{E_{6}}^{4b} has orbit representatives

E_{E_{6}}^{2a} = ⟨\overline{y_{1}}, \overline{x_{2}}⟩ , E_{E_{6}}^{2b} = ⟨\overline{y_{1}}, \overline{x_{1}}⟩ , ⟨\overline{a}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{y_{1}}⟩ , ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}⟩ and ⟨\overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ .

Similarly we find that under the action of W′(E_{E_{6}}^{4a}), the set of rank 3 subgroups of E_{E_{6}}^{4a} has orbit representatives

E_{E_{6}}^{3a} = ⟨\overline{a}, \overline{y_{1}}, \overline{x_{2}}⟩ , E_{E_{6}}^{3b} = ⟨\overline{a}, \overline{y_{2}}, \overline{x_{2}}⟩ and ⟨\overline{a}, \overline{y_{1}}, \overline{y_{2}}⟩ ,

and that under the action of W′(E_{E_{6}}^{4b}), the set of rank 3 subgroups of E_{E_{6}}^{4b} has orbit representatives

E_{E_{6}}^{3a} = ⟨\overline{a}, \overline{y_{1}}, \overline{x_{2}}⟩ , E_{E_{6}}^{3c} = ⟨\overline{a}, \overline{y_{1}}, \overline{x_{1}}⟩ , E_{E_{6}}^{3d} = ⟨\overline{x_{2}x_{1}^{−1}}, \overline{y_{1}}, \overline{x_{1}}⟩ and ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ .

OTHER NONTORAL SUBGROUPS: We see directly that the subgroups ⟨\overline{a}, \overline{x_{2}}⟩, ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}⟩, ⟨\overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ and ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ are toral. Noting that the elements β and γ are conjugate in SL_{3}(ℂ) we see that the group ⟨\overline{a}, \overline{y_{1}}, \overline{y_{2}}⟩ is conjugate to the group ⟨\overline{a}, \overline{[I_{3}, β, β^{2}]}, \overline{x_{2}}⟩ which is obviously toral. Thus we see that the groups ⟨\overline{a}, \overline{y_{1}}, \overline{y_{2}}⟩, ⟨\overline{a}, \overline{y_{1}}⟩ and ⟨\overline{a}, \overline{y_{2}}⟩ are also toral. Using the fact that [y_{1}, x_{1}] = [y_{1}, x_{2}] = z we see from Lemma (1.1.3(5)) that both E_{E_{6}}^{2a} and E_{E_{6}}^{2b} are nontoral in E_{6}(ℂ). Since the groups E_{E_{6}}^{3a}, E_{E_{6}}^{3c} and E_{E_{6}}^{3d} all contain either E_{E_{6}}^{2a} or E_{E_{6}}^{2b} they are also nontoral. Using Lemma (1.1.3(5)) we see that the group E_{E_{6}}^{3b} is nontoral in E_{6}(ℂ), since we know that π^{−1}(E_{E_{6}}^{3b}) = E_{3E_{6}}^{4} is nontoral in 3E_{6}(ℂ) by Theorem (1.4.5).

OVERSIZE: Since the groups ⟨\overline{a}, \overline{y_{1}}, \overline{y_{2}}⟩ and ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ are toral and are codimension 1 subgroups of E_{E_{6}}^{4a} and E_{E_{6}}^{4b} respectively, we see that E_{E_{6}}^{4a} and E_{E_{6}}^{4b} are at most 1-oversize. Since they are nontoral they are both exactly 1-oversize. Since the other nontoral elementary abelian 3-subgroups are contained in one of these, they are also 1-oversize.

CLASS DISTRIBUTIONS: Using (1.4.4) and the action of the groups W′(E_{E_{6}}^{4a}) and W′(E_{E_{6}}^{4b}) it is not hard to verify the class distributions in the table. As an example consider the group E_{E_{6}}^{4b}. From the action of W′(E_{E_{6}}^{4b}) we see that E_{E_{6}}^{4b} \ {1} contains 2 elements conjugate to \overline{a}, 6 elements conjugate to \overline{x_{2}x_{1}^{−1}}, 24 elements conjugate to \overline{x_{1}} and 48 elements conjugate to \overline{x_{2}}. Thus by (1.4.4), the set π^{−1}(E_{E_{6}}^{4b} \\{1\}) contains 6 elements from each of the classes 3A, 3B and 3B′, 3·(2+48) = 150 elements from the class 3C and 3 · 24 = 72 elements from the class 3D. Including the elements z and z^{2} from the classes 3E and 3E′ respectively, we get the class distribution of π^{−1}(E_{E_{6}}^{4b})\{1} given in the table. Similar computations give the remaining entries in the table. Since these distributions are different we see that the groups in the table are not conjugate and thus they provide a set of representatives for the conjugacy classes of nontoral elementary abelian 3-subgroups of E_{6}(ℂ).

CENTRALIZER DIMENSIONS: Noting that Z acts trivially on the Lie algebra 𝔢_{6}, we see from Lemma (1.1.3(6)) that for any finite subgroup E ⊆ E_{6}(ℂ) we have dim C_{E_{6}(ℂ)}(E) = dim C_{3E_{6}(ℂ)}(π^{−1}(E)). Thus we can use the class distributions above and [8, Table 2] to find the dimensions of the centralizers. As an example consider E_{E_{6}}^{4b}:

dim C_{E_{6}(ℂ)}(E_{E_{6}}^{4b}) = dim C_{3E_{6}(ℂ)}(π^{−1}(E_{E_{6}}^{4b})) = \frac{1}{|π^{−1}(E_{E_{6}}^{4b})|} ∑_{x∈π^{−1}(E_{E_{6}}^{4b})} tr x|_{𝔢_{6}} =

\frac{1}{3^{5}} · (3 · 78 + 18 · (36 + 21ω + 21ω^{2}) + 150 · (24 + 27ω + 27ω^{2})+

72 · (30 + 24ω + 24ω^{2})) = 2.

Similar computations give the results for the other groups in the table.

LOWER BOUNDS FOR OTHER WEYL GROUPS: We now show that the other matrix groups in the table are all lower bounds for the remaining Weyl groups. To do this consider one of the nonmaximal groups E from the table. We then have E ⊆ E_{E_{6}}^{4a} or E ⊆ E_{E_{6}}^{4b}, and we get a lower bound on W (E) by considering the action on E of the subgroup of W′(E_{E_{6}}^{4a}) or W′(E_{E_{6}}^{4b}) fixing E. As an example we see that E_{E_{6}}^{2a} ⊆ E_{E_{6}}^{4a} and that the stabilizer of E_{E_{6}}^{2a} inside W′(E_{E_{6}}^{4a}) is

[ \begin{matrix}GL_{2}(𝔽_{3}) & {0 \atop 0} & {0 \atop 0} \\ 0 0 & det & x \\ 0 0 & 0 & det \end{matrix} ] ,

where det is the determinant of the matrix from GL_{2}(𝔽_{3}). The action of such a matrix on E_{E_{6}}^{2a} is given by

\overline{y_{1}} ↦ (\overline{y_{1}})det, \overline{x_{2}} ↦ (\overline{y_{1}})^{x}(\overline{x_{2}})det.

Thus W (E_{E_{6}}^{2a}) contains the group

W′(E_{E_{6}}^{2a}) = [ \begin{matrix}ε & ∗ \\ 0 & ε \end{matrix} ]

as claimed. Similar computations show that for the groups E = E_{E_{6}}^{2b}, E_{E_{6}}^{3a}, E_{E_{6}}^{3c} and E_{E_{6}}^{3d}, the group W′(E) occurring in the theorem is a lower bound for the Weyl group W (E).

For the group E_{E_{6}}^{3b} = ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩ we note that by Theorem (1.4.5) the group E_{3E_{6}}^{3} = ⟨a, x_{2}, y_{2}⟩ has Weyl group at least SL_{3}(𝔽_{3}). Thus we see that W (E_{E_{6}}^{3b}) also contains SL_{3}(𝔽_{3}).

EXACT WEYL GROUPS: By now we know that the groups in the table is a set of representatives for the conjugacy classes of nontoral elementary abelian 3-subgroups of E_{6}(ℂ), that they are all 1-oversize and that the matrix groups given in the theorem are lower bounds for the Weyl groups. We thus only need to verify that for all the groups except E_{E_{6}}^{3b} and E_{E_{6}}^{4a} the Weyl group is the one given above.

By Lemma (1.4.9) the Weyl groups W (E_{E_{6}}^{2a}) and W (E_{E_{6}}^{2b}) are subgroups of SL_{2}(𝔽_{3}). From this we see that W (E_{E_{6}}^{2b}) = SL_{2}(𝔽_{3}) and that W (E_{E_{6}}^{2a}) is equal to either W′(E_{E_{6}}^{2a}) or SL_{2}(𝔽_{3}), since these are the only subgroups of SL_{2}(𝔽_{3}) containing W′(E_{E_{6}}^{2a}). We have E_{E_{6}}^{2a} = ⟨\overline{y_{1}}, \overline{x_{2}}⟩, and by (1.4.4) we see that the elements \overline{y_{1}} and \overline{x_{2}} are not conjugate in E_{6}(ℂ). In particular we see that W (E_{E_{6}}^{2a}) cannot act transitive on the nontrivial elements of E_{E_{6}}^{2a}, and we conclude that W (E_{E_{6}}^{2a}) = W′(E_{E_{6}}^{2a}) is the group from above.

Note that the groups E_{E_{6}}^{3a}, E_{E_{6}}^{3d}, and E_{E_{6}}^{4b} all contain E_{E_{6}}^{2a}. By using for example the computer program mentioned in the introduction, we find that the minimal overgroups of W′(E) in GL(E) are:

E_{E_{6}}^{3a} : [ \begin{matrix}ε & ∗ ∗ \\ {0 \atop 0} & SL_{2}(𝔽_{3}) \end{matrix} ] , [ \begin{matrix}∗ & ∗ & ∗ \\ 0 & ∗ & ∗ \\ 0 & 0 & ∗ \end{matrix} ]

E_{E_{6}}^{3d} : [ \begin{matrix}ε & 0 0 \\ \begin{matrix}∗ \\ ∗ \end{matrix} & SL_{2}(𝔽_{3}) \end{matrix} ] , [ \begin{matrix}ε & ∗ ∗ \\ {0 \atop 0} & SL_{2}(𝔽_{3}) \end{matrix} ] , [ \begin{matrix}ε & 0 0 \\ {0 \atop 0} & GL_{2}(𝔽_{3}) \end{matrix} ]

E_{E_{6}}^{4b} : [ \begin{matrix}ε_{1} & ∗ & ∗ ∗ \\ 0 & ε_{2} & 0 0 \\ {0 \atop 0} & {0 \atop 0} & GL_{2}(𝔽_{3}) \end{matrix} ] , [ \begin{matrix}ε_{1} & ∗ & ∗ ∗ \\ 0 & ε_{2} & ∗ ∗ \\ {0 \atop 0} & {0 \atop 0} & SL_{2}(𝔽_{3}) \end{matrix} ] , [ \begin{matrix}ε_{1} & ∗ & ∗ ∗ \\ 0 & ε_{2} & 0 0 \\ {0 \atop 0} & \begin{matrix}∗ \\ ∗ \end{matrix} & SL_{2}(𝔽_{3}) \end{matrix} ] .

From this we see that if W (E_{E_{6}}^{3a}) ≠ W′(E_{E_{6}}^{3a}) then W (E_{E_{6}}^{3a}) will contain one of the matrices

[ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ]

both fixing the subgroup E_{E_{6}}^{2a} of E_{E_{6}}^{3a}. However the automorphisms of E_{E_{6}}^{2a} induced by these elements are given by the matrices

[ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] .

Since none of these are contained in W (E_{E_{6}}^{2a}) we get W (E_{E_{6}}^{3a}) = W′(E_{E_{6}}^{3a}) as claimed. Similar computations using the above list of minimal overgroups show that W (E) = W′(E) for E = E_{E_{6}}^{3d} and E_{E_{6}}^{4b}. For the group E_{E_{6}}^{3c} we get that W′(E_{E_{6}}^{3c}) has only one minimal overgroup, namely:

[ \begin{matrix}ε & ∗ ∗ \\ {0 \atop 0} & GL_{2}(𝔽_{3}) \end{matrix} ]

If W (E_{E_{6}}^{3c}) contained this group, then it would contain the matrix

[ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ]

fixing the subgroup E_{E_{6}}^{2b}. However the action induced on E_{E_{6}}^{2b} is given by the matrix

[ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ]

which is not contained in W (E_{E_{6}}^{2b}). Thus we get W (E_{E_{6}}^{3c}) = W′(E_{E_{6}}^{3c}) as well. □

1.5 The group 2E_{7}(ℂ), p = 3

(1.5.1) Setup. In this section we consider the elementary abelian 3-subgroups of 2E_{7}(ℂ), the simply connected group of type E_{7} over ℂ. We let H be a maximal torus of 2E_{7}(ℂ), Φ(E_{7}) be the root system relative to H, and choose a realization ([17, p. 133]) (u_{α})_{α∈Φ(E_{7})} of Φ(E_{7}) in 2E_{7}(ℂ).

Note that we have an inclusion Φ(E_{6}) ⊆ Φ(E_{7}) of root systems (we use the notation of [3, p. 260–266] for the root systems Φ(E_{6}) and Φ(E_{7})). The subgroup G of 2E_{7}(ℂ) generated by the root subgroups (u_{α})_{α∈Φ(E_{6})} is isomorphic to 3E_{6}(ℂ) and these root subgroups form a realization of Φ(E_{6}) in G. Fixing an isomorphism G ≅ 3E_{6}(ℂ) and using the choice of root subgroups for 3E_{6}(ℂ) from section 1.4 we get the root subgroups u′_{±α_{i}}, 1 ≤ i ≤ 6 in G ⊆ 2E_{7}(ℂ). Now using [17, 8.1.4(iv)] we may actually suppose that u_{±α_{i}} = u′_{±α_{i}} for 1 ≤ i ≤ 6. For α = α_{i}, 1 ≤ i ≤ 7, and t ∈ ℂ^{×} we define the elements

n_{α}(t) = u_{α}(t)u_{−α}(−1/t)u_{α}(t), h_{α}(t) = n_{α}(t)n_{α}(1)^{−1}.

Then the maximal torus consists of the elements ∏_{i=1}^{7} h_{α_{i}}(t_{i}) and the normalizer N(H) of the maximal torus is generated by H and the elements n_{i} = n_{α_{i}}(1), 1 ≤ i ≤ 7.

As in section 1.4 we define the following elements in 2E_{7}(ℂ):

z = h_{α_{1}}(ω)h_{α_{3}}(ω^{2})h_{α_{5}}(ω)h_{α_{6}}(ω^{2}),  a = h_{α_{1}}(ω)h_{α_{3}}(ω^{2})h_{α_{5}}(ω^{2})h_{α_{6}}(ω),
x_{2} = h_{α_{2}}(ω^{2})h_{α_{3}}(ω^{2})h_{α_{5}}(ω^{2}),
y_{2} = n_{1}n_{2}n_{3}n_{4}n_{3}n_{1}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{6}n_{5}h_{α_{2}}(−1).

(1.5.2) Conjugacy classes. The conjugacy classes of elements of order 3 in 2E_{7}(ℂ) are given in [13, Table VI] and [7, Table 6] from which we take our notation. In particular, there are 5 such conjugacy classes, which we label 3A, 3B, 3C, 3D and 3E. Moreover these classes may be distinguished by their traces on 𝔢_{7}, except for the classes 3A and 3D which have the same trace. Since the trace of the element h ∈ H is given by 7 + ∑_{α∈Φ(E_{7})} α(h) we easily obtain the inclusions

3C[3E_{6}(ℂ)] ⊆ 3C[2E_{7}(ℂ)], 3E[3E_{6}(ℂ)] ⊆ 3B[2E_{7}(ℂ)], 3E′[3E_{6}(ℂ)] ⊆ 3B[2E_{7}(ℂ)],

corresponding to the inclusion 3E_{6}(ℂ) ≅ G ⊆ 2E_{7}(ℂ).

(1.5.3) Theorem (2E_{7}). The conjugacy classes of nontoral elementary abelian 3-subgroups of 2E_{7}(ℂ) are given by the following table.

rank name

ordered basis 2E_{7}(ℂ)-class distribution dim C_{2E_{7}(ℂ)}(E)

3 E_{2E_{7}}^{3}  ⟨a, x_{2}, y_{2}⟩  3C^{26}  3
4 E_{2E_{7}}^{4}  ⟨z, a, x_{2}, y_{2}⟩  3B^{2}3C^{78}  1

These groups are all 1-oversize and their Weyl groups with respect to the given ordered bases satisfies the following:

W (E_{2E_{7}}^{3}) ⊇ SL_{3}(𝔽_{3}), W (E_{2E_{7}}^{4}) ⊇ [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] .

(1.5.4) Remark. In section 1.7 we will prove that the Weyl groups of E_{2E_{7}}^{3} and E_{2E_{7}}^{4} are exactly the lower bounds given above.

(1.5.5) Proof:

NONTORAL SUBGROUPS: From the way the realization (u_{α})_{α∈Φ(E_{7})} is chosen above, it follows from Theorem (1.4.5) that E_{2E_{7}}^{3} and E_{2E_{7}}^{4} are elementary abelian 3-subgroups of 2E_{7}(ℂ) and that we have

W (E_{2E_{7}}^{3}) ⊇ SL_{3}(𝔽_{3}), W (E_{2E_{7}}^{4}) ⊇ [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] .

In particular we see that both W (E_{2E_{7}}^{3}) and W (E_{2E_{7}}^{4}) have orders divisible by 13 and since the order of W (E_{7}) is 2^{10} ·3^{4} ·5·7 which is not divisible by 13, we conclude by Lemma (1.1.3(4)) that E_{2E_{7}}^{3} and E_{2E_{7}}^{4} are nontoral in 2E_{7}(ℂ). However by [13, Theorem (11.16)] we know that there are precisely two conjugacy classes of nontoral elementary abelian 3-subgroups in 2E_{7}(ℂ), and thus E_{2E_{7}}^{3} and E_{2E_{7}}^{4} represent these two conjugacy classes.

CLASS DISTRIBUTIONS: The class distributions follows directly from the class distributions of the groups E_{3E_{6}}^{3} and E_{3E_{6}}^{4} given in Theorem (1.4.5) and the information in (1.5.2) about the behavior of conjugacy classes in 3E_{6}(ℂ) under the inclusion 3E_{6}(ℂ) ≅ G ⊆ 2E_{7}(ℂ).

CENTRALIZER DIMENSIONS: By using Lemma (1.1.3(6)), the class distributions just found and the traces given in [13, Table VI], the dimensions of the centralizers are verified easily.

OVERSIZE: Since the group ⟨\overline{z}, \overline{a}, \overline{x_{2}}⟩ is toral and has codimension 1 in E_{2E_{7}}^{4} we see that E_{2E_{7}}^{4} is at most 1-oversize. Since E_{2E_{7}}^{3} is contained in E_{2E_{7}}^{4} it is also at most 1-oversize. Now both E_{2E_{7}}^{3} and E_{2E_{7}}^{4} are nontoral, so they are both exactly 1-oversize.

LOWER BOUNDS FOR WEYL GROUPS: Using our realization (u_{α})_{α∈Φ(E_{7})} we may define a canonical map φ : W → N(H) as follows ([17, 9.3.3]): If w = s_{α_{i_{1}}} … s_{α_{ir}} is a reduced expression for w ∈ W we let φ(w) = n_{i_{1}} … n_{i_{r}} (by [17, 8.3.3 and 9.3.2] this does not depend on the reduced expression for w). Note that the element φ(w) is a representative in N(H) for w ∈ W. Now let w_{0} ∈ W be the longest element in W, and let n_{0} = φ(w_{0}). From [3, p. 264–266] it follows that w_{0} equals the scalar transformation −1, and so conjugation by n_{0} acts as inversion on H. Now let w ∈ W and define w′ by ww′ = w_{0}. Since w_{0} is central in our case, we have (ww′) w^{−1} = w^{−1} (ww′) = w′ so we conclude that w′w = ww′ = w_{0}. Now let ℓ be the length function on W. By [14, p. 16] we ℓ(w) + ℓ(w′) = ℓ(w_{0}). In general the map φ is not a homomorphism, but we do have φ(w_{1}w_{2}) = φ(w_{1})φ(w_{2}) if ℓ(w_{1}w_{2}) = ℓ(w_{1}) + ℓ(w_{2}) by [17, 9.3.4(i)]. From this it follows that φ(w)φ(w′) = φ(w′)φ(w) = φ(w_{0}) = n_{0}, and we conclude that n_{0} commutes with φ(w) for all w ∈ W.

Now consider the element

w = s_{1}s_{2}s_{3}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}.

Using the fact that the length of an element is given by the number of positive roots it sends to negative roots ([14, Corollary 1.7]), we see that the above product is a reduced expression for w. Thus we have y_{2} = φ(w)h_{α_{2}}(−1). From the above we then conclude that conjugation by n_{0} acts as follows:

z ↦ z^{2}, a ↦ a^{2}, x_{2} ↦ x_{2}^{2}, y_{2} ↦ y_{2}.

Thus n_{0} normalizes E_{2E_{7}}^{4} and induces the element

[ \begin{matrix}2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ]

in W (E_{2E_{7}}^{4}). Combined with the above we conclude that

W (E_{2E_{7}}^{4}) ⊇ [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ]

as claimed. □

1.6 The group E_{8}(ℂ), p = 3

(1.6.1) Setup. In this section we consider the elementary abelian 3-subgroups of the group E_{8}(ℂ), which is both the simply connected and adjoint group of type E_{8} over ℂ. By using [4, Table 2, p. 214] we see that the smallest faithful representation of E_{8}(ℂ) is the adjoint representation, i.e. the representation given by the action of E_{8}(ℂ) on its Lie algebra 𝔢_{8}, which has dimension 248. We may explicitly construct this representation by following the recipe in [6, Chapter 4]. Letting Φ(E_{8}) denote the root system of type E_{8} (we use the notation of [3, p. 268–270]), we have in particular a maximal torus H generated by the elements h_{α_{i}}(t), 1 ≤ i ≤ 8, t ∈ ℂ^{×} ([6, p. 92, p. 97]) and root subgroups u_{α}(t), α ∈ Φ(E_{8}), t ∈ ℂ. The normalizer N(H) of the maximal torus, is generated by H and the elements n_{i} = n_{α_{i}}, 1 ≤ i ≤ 8 ([6, p. 93, p. 101]). We let

h(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}, t_{7}, t_{8}) = ∏_{i=1}^{8} h_{α_{i}}(t_{i}).

Note that by [6, p. 100 and Lemma 6.4.4] the root subgroups u_{α} form a realization ([17, p. 133]) of Φ(E_{8}) in E_{8}(ℂ). In particular we have the following relations. The element n_{i} has image s_{α_{i}} in W = W (E_{8}) ([17, 8.1.4(i)]), we have n_{i}^{2} = h_{α_{i}}(−1) ([17, 8.1.4(ii)]) and

n_{i}n_{j}n_{i} … = n_{j}n_{i}n_{j} …

for 1 ≤ i, j ≤ 8, where the number of factors on both sides equals the order of s_{α_{i}}s_{α_{j}} in W ([17, 9.3.2]).

Now let \overline{a} = h_{α_{1}}(ω)h_{α_{2}}(ω)h_{α_{3}}(ω^{2}) ∈ E_{8}(ℂ). Direct computation shows that for any root α ∈ Φ(E_{8}) we have α(\overline{a}) = ω^{2⟨α,λ_{2}⟩}. From this we see that Dynkin diagram of the centralizer C_{E_{8}(ℂ)}(\overline{a}) is the same as the extended Dynkin diagram of E_{8} with the node α_{2} removed. Thus it has type A_{8} and basis

{α_{1}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}, α_{8}, −α̃},

where α̃ is the longest root. As in [3, p. 250–251] we identify Φ(A_{8}) with the set of elements in ℝ^{9} of the form e_{i} − e_{j} with i ≠ j and 1 ≤ i, j ≤ 9, where e_{i} denotes the i’th canonical basis vector in ℝ^{9}. We now consider SL_{9}(ℂ), which is the simply connected group of type E_{8} over ℂ. For any 1 ≤ i, j ≤ 9 we let e_{i,j} be the 9 × 9matrix over ℂ whose only nonzero element is 1 occurring at the (i, j)-entry. Given a root α′ = e_{i} − e_{j} ∈ Φ(A_{8}) we let u′_{α′}(t) = I_{9} + te_{i,j} for t ∈ ℂ. With respect to the maximal torus consisting of the diagonal matrices, this is a root subgroup of SL_{9}(ℂ) corresponding to the root α′. The roots α′_{i} = e_{i} − e_{i+1}, 1 ≤ i ≤ 8, form a basis of Φ(A_{8}). From the above we then see that

u′_{α′_{1}}(t) ↦ u_{α_{1}}(t), u′_{α′_{2}}(t) ↦ u_{α_{3}}(t), u′_{α′_{3}}(t) ↦ u_{α_{4}}(t), u′_{α′_{4}}(t) ↦ u_{α_{5}}(t),
u′_{α′_{5}}(t) ↦ u_{α_{6}}(t), u′_{α′_{6}}(t) ↦ u_{α_{7}}(t), u′_{α′_{7}}(t) ↦ u_{α_{8}}(t), u′_{α′_{8}}(t) ↦ u_{−α̃}(t)

defines a homomorphism SL_{9}(ℂ) → E_{8}(ℂ) onto the centralizer C_{E_{8}(ℂ)}(\overline{a}). It is easy to check that this map has kernel C_{3} = ⟨ωI_{9}⟩ and thus we may make the identification C_{E_{8}(ℂ)}(\overline{a}) = SL_{9}(ℂ)/C_{3}. For any g ∈ SL_{9}(ℂ) we denote by \overline{g} its image in SL_{9}(ℂ)/C_{3} = C_{E_{8}(ℂ)}(\overline{a}) ⊆ E_{8}(ℂ). In particular we see that a = ηI_{9} corresponds to the element \overline{a} from above. We also define the following elements in SL_{9}(ℂ) :

x_{1} = diag(1, ω, ω^{2}, 1, ω, ω^{2}, 1, ω, ω^{2}), x_{2} = diag(1, 1, 1, ω, ω, ω, ω^{2}, ω^{2}, ω^{2}),
x_{3} = diag(1, 1, 1, 1, 1, 1, ω, ω, ω),  y_{1} = (1, 2, 3)(4, 5, 6)(7, 8, 9),
y_{2} = (1, 4, 7)(2, 5, 8)(3, 6, 9).

From the explicit homomorphism above we easily find

\overline{a} = h_{α_{1}}(ω)h_{α_{2}}(ω)h_{α_{3}}(ω^{2}),  \overline{x_{1}} = h_{α_{1}}(ω)h_{α_{5}}(ω)h_{α_{8}}(ω),
\overline{x_{2}} = h_{α_{1}}(ω)h_{α_{3}}(ω^{2})h_{α_{5}}(ω^{2})h_{α_{6}}(ω), \overline{x_{3}} = h_{α_{1}}(ω^{2})h_{α_{3}}(ω)h_{α_{5}}(ω^{2})h_{α_{6}}(ω),

and a direct computation in E_{8}(ℂ) shows that

\begin{aligned}n_{−α̃} = & n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8} · \\ & n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8}. \end{aligned}

From this and the explicit homomorphism above we find, either by direct computation or by using the relations in N(H), that

y_{1} = n_{1}n_{3}n_{5}n_{6}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6}n_{5} ·
n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3} ·
n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8} · h_{α_{1}}(−1)h_{α_{2}}(−1)h_{α_{7}}(−1),
y_{2} = n_{2}n_{3}n_{1}n_{4}n_{2}n_{3}n_{4}n_{5}n_{4}n_{2}n_{3}n_{4}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{7}n_{6}n_{5}n_{4} ·
n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{6}n_{7}n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5} ·
n_{4}n_{3}n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{8}n_{7}n_{6} · h_{α_{2}}(−1)h_{α_{5}}(−1).

(1.6.2) Conjugacy classes. To distinguish subgroups of E_{8}(ℂ), we need some information on the conjugacy classes of elements of order 3. These are given in [13, Table VI] (which is taken from [7, Table 4]): There are 4 such conjugacy classes, which we label 3A, 3B, 3C and 3D. Moreover these classes may be distinguished by their traces on 𝔢_{8}. Since the trace of the element h ∈ H is given by 8 + ∑_{α∈Φ(E_{8})} α(h) we easily obtain that \overline{a} ∈ 3A, \overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}} ∈ 3B and \overline{x_{3}a^{−1}} ∈ 3D.

(1.6.3) Notation. If K is a field and n is a natural number, we define the general symplectic group GSp_{2n}(K) as the group of 2n × 2n matrices over K which leaves the skewsymmetric scalar product given by the matrix

B = \underbrace{[ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] ⊕ … ⊕ [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ]}_{n copies},

invariant up to a nonzero scalar. Thus we have

GSp_{2n}(K) = {X ∈ GL_{2n}(K)|X^{t}BX = cB, c ∈ K^{×}}.

We define χ : GSp_{2n}(K) → K^{×} as follows. If X ∈ GSp_{2n}(K) with X^{t}BX = cB, c ∈ K^{×}, we let χ(X) = c. This is easily seen to be a group homomorphism. The kernel of χ is the symplectic group Sp_{2n}(K) which is the subgroup leaving the above skew-symmetric scalar product invariant, i.e.

Sp_{2n}(K) = {X ∈ GL_{2n}(K)|X^{t}BX = B}.

These groups were first investigated by Jordan and Dickson. In [9, §110] the groups GSp_{2n}(K) and Sp_{2n}(K) are called the general Abelian linear group and the special Abelian linear group and denoted by respectively GA(2n, K) and SA(2n, K). It should be noted that our notation GSp_{2n}(K) is not standard. Indeed no standard notation seems to be available since this group does not occur as often in applications as the symplectic group Sp_{2n}(K).

(1.6.4) Theorem (E_{8}). The conjugacy classes of nontoral elementary abelian 3-subgroups of E_{8}(ℂ) are given by the following table.

rank name

ordered basis E_{8}(ℂ)-class distribution dim C_{E_{8}(ℂ)}(E)
3 E_{E_{8}}^{3a}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩  3A^{18}3B^{8}  8
3 E_{E_{8}}^{3b}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩  3B^{26}  14
4 E_{E_{8}}^{4a}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩  3A^{52}3B^{26}3D^{2}  4
4 E_{E_{8}}^{4b}  ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩  3A^{54}3B^{26}  2
4 E_{E_{8}}^{4c}  ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩  3B^{80}  8
5 E_{E_{8}}^{5a} ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩  3A^{156}3B^{80}3D^{6}  2
5 E_{E_{8}}^{5b}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩  3A^{162}3B^{80}  0

These groups are all 1-oversize and their Weyl groups with respect to the given ordered bases satisfies the following:

W (E_{E_{8}}^{3a}) = [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det \end{matrix} ] , W (E_{E_{8}}^{3b}) ⊇ SL_{3}(𝔽_{3}), W (E_{E_{8}}^{4a}) = [ \begin{matrix}SL_{3}(𝔽_{3}) & 0 0 0 \\ 0 0 0 & ε \end{matrix} ] ,

W (E_{E_{8}}^{4b}) = [ \begin{matrix}ε & ∗ ∗ & ∗ \\ {0 \atop 0} & GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 & 0 0 & det \end{matrix} ] , W (E_{E_{8}}^{4c}) ⊇ [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] ,

W (E_{E_{8}}^{5a}) = [ \begin{matrix}ε_{1} & ∗ ∗ ∗ & ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) & 0 0 0 \\ 0 & 0 0 0 & ε_{2} \end{matrix} ] , W (E_{E_{8}}^{5b}) = [ \begin{matrix}GSp_{4}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \\ ∗ \\ ∗ \end{matrix} \\ 0 0 0 0 & χ \end{matrix} ]

where det is the determinant of the matrix from GL_{2}(𝔽_{3}) in the description of W (E_{E_{8}}^{3a}) and W (E_{E_{8}}^{4b}). In the description of W (E_{E_{8}}^{5b}), χ denotes the value of the homomorphism χ : GSp_{4}(𝔽_{3}) → 𝔽_{3}^{×} defined in (1.6.3) evaluated on the matrix from GSp_{4}(𝔽_{3}).

(1.6.5) Remark. In section 1.7 we will prove that the Weyl groups of E_{E_{8}}^{3b} and E_{E_{8}}^{4c} are exactly the lower bounds given above.

(1.6.6) Proof:

MAXIMAL NONTORAL SUBGROUPS: By [13, Lemma (11.7) and Lemma (11.9)], any maximal nontoral elementary abelian 3-subgroup of E_{8}(ℂ) contains an element of type 3A. We may thus find representatives in C_{E_{8}(ℂ)}(\overline{a}) = SL_{9}(ℂ)/C_{3}. From [13, Corollary 11.10], it follows that there are two conjugacy classes of these maximal nontoral elementary abelian 3-subgroups: E_{E_{8}}^{5a} and E_{E_{8}}^{5b}, both of which have rank 5. Moreover, by [13, Lemma 11.5], their preimages in SL_{9}(ℂ) may be chosen to have the shape 3^{1+2}∘9×3×3 and 3^{1+4}∘9, where ∘ denotes a central product over C_{3} = ⟨ωI_{3}⟩. Using the representation theory of extraspecial 3-groups ([7, Chapter 5.5]) we find that E_{E_{8}}^{5a} is represented by ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩ and that E_{E_{8}}^{5b} is represented by ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩.

LOWER BOUNDS FOR WEYL GROUPS OF MAXIMAL NONTORAL SUB-GROUPS: We can find lower bounds for the Weyl groups of E_{E_{8}}^{5a} and E_{E_{8}}^{5b} by conjugating with elements in the centralizer C_{E_{8}(ℂ)}(\overline{a}) = SL_{9}(ℂ)/C_{3} and the normalizer N(H) of the maximal torus.

Note that

a = ηI_{3} ⊕ ηI_{3} ⊕ ηI_{3},  x_{1} = β ⊕ β ⊕ β,  x_{2} = I_{3} ⊕ ωI_{3} ⊕ ω^{2}I_{3},
x_{3} = I_{3} ⊕ I_{3} ⊕ ωI_{3},  y_{1} = γ ⊕ γ ⊕ γ

and (A ⊕ B ⊕ C)^{y_{2}} = B ⊕ C ⊕ A. Conjugation by τ_{1} ⊕ τ_{1} ⊕ τ_{1}, τ_{2} ⊕ τ_{2} ⊕ τ_{2} and I_{3} ⊕ β^{2} ⊕ β gives

τ_{1} ⊕ τ_{1} ⊕ τ_{1} : a ↦ a, x_{1} ↦ x_{1}y_{1}, x_{2} ↦ x_{2}, x_{3} ↦ x_{3}, y_{1} ↦ y_{1}, y_{2} ↦ y_{2}. (i)
τ_{2} ⊕ τ_{2} ⊕ τ_{2} : a ↦ a, x_{1} ↦ x_{1}, x_{2} ↦ x_{2}, x_{3} ↦ x_{3}, y_{1} ↦ x_{1}y_{1}, y_{2} ↦ y_{2}. (ii)
I_{3} ⊕ β^{2} ⊕ β : a ↦ a, x_{1} ↦ x_{1}, x_{2} ↦ x_{2}, x_{3} ↦ x_{3}, y_{1} ↦ x_{2}y_{1}, y_{2} ↦ x_{1}y_{2}. (iii)

Now consider the group E_{E_{8}}^{5a}. From (i)—(iii) we see that the elements \overline{τ_{1} ⊕ τ_{1} ⊕ τ_{1}}, \overline{τ_{2} ⊕ τ_{2} ⊕ τ_{2}} and \overline{I_{3} ⊕ β^{2} ⊕ β} normalize E_{E_{8}}^{5a} and that conjugation by these elements induces the automorphisms on E_{E_{8}}^{5a} given by the matrices

[ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] .

Letting σ = −(1, 4)(2, 5)(3, 6) ∈ SL_{9}(ℂ) we see that (A ⊕ B ⊕ C)^{σ} = B ⊕ A ⊕ C. Using this and the above we obtain that \overline{σ}, \overline{y_{2}} and \overline{I_{3} ⊕ I_{3} ⊕ β^{2}} normalize E_{E_{8}}^{5a} and that conjugation by these elements induces the automorphisms on E_{E_{8}}^{5a} given by the matrices

[ \begin{matrix}2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] .

By using the relations in N(H) given above or by direct computation, it may be checked that conjugation by the element

\begin{aligned}n = & n_{1}n_{2}n_{4}n_{2}n_{3}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{7}n_{6}n_{5}n_{4}n_{8}n_{7}n_{6} · \\ & n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{7} · h(1, 1, −1, −1, −1, 1, −1, −1) \end{aligned}

induces the automorphism on E_{E_{8}}^{5a} represented by the matrix

[ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix} ] .

It is easy to see that the above matrices generate the group

W′(E_{E_{8}}^{5a}) = [ \begin{matrix}ε_{1} & ∗ ∗ ∗ & ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) & 0 0 0 \\ 0 & 0 0 0 & ε_{2} \end{matrix} ] ,

and thus W (E_{E_{8}}^{5a}) contains this group.

Next consider the group E_{E_{8}}^{5b}. From (i)—(iii) we see that the elements \overline{τ_{1} ⊕ τ_{1} ⊕ τ_{1}}, \overline{τ_{2} ⊕ τ_{2} ⊕ τ_{2}} and \overline{I_{3} ⊕ β^{2} ⊕ β} normalize E_{E_{8}}^{5b} and that conjugation by these elements induces the automorphisms on E_{E_{8}}^{5b} given by the matrices

[ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] .

Now note that a = Δ_{3,3}(ηI_{3}), x_{2} = Δ_{3,3}(β) and y_{2} = Δ_{3,3}(γ). Noting also that Δ_{3,3}(M_{1}) commutes with M_{2} ⊕ M_{2} ⊕ M_{2} for any M_{1}, M_{2} ∈ M_{3}(ℂ) we see that the elements \overline{Δ_{3,3}(τ_{1})} and \overline{Δ_{3,3}(τ_{2})} normalize E_{E_{8}}^{5b}. The automorphisms induced on E_{E_{8}}^{5b} by conjugation with these elements have the matrices

[ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] .

The upper left 4 × 4-corner of these matrices is easily seen to generate the group

⟨[ \begin{matrix}SL_{2}(𝔽_{3}) & 0 0 0 0 \\ 0 0 0 0 & SL_{2}(𝔽_{3}) \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ]⟩ .

Using the coordinates (η_{1}, ξ_{1}, η_{2}, ξ_{2}) in (𝔽_{3})^{4} we see that this is exactly the same group as in [9, §114]. By comparing with [9, §110] and (1.6.3) we see that this is the symplectic group Sp_{4}(𝔽_{3}) from above.

By using the relations in N(H) given above or by direct computation, we get that conjugation by the element

\begin{aligned}n = & n_{2}n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6} · \\ & n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8} · h(1, −1, −1, −1, −1, 1, 1, 1) \end{aligned}

of N(H) induces the automorphism on E_{E_{8}}^{5b} represented by the matrix

[ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix} ] .

It now follows that W (E_{E_{8}}^{5b}) contains the group

W′(E_{E_{8}}^{5b}) = [ \begin{matrix}GSp_{4}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \\ ∗ \\ ∗ \end{matrix} \\ 0 0 0 0 & χ \end{matrix} ] .

LOWER BOUNDS FOR OTHER WEYL GROUPS: We now show that the other Weyl groups in the table are all lower bounds. To do this consider one of the nonmaximal groups E from the table. We then have E ⊆ E_{E_{8}}^{5a}, and we get a lower bound on W (E) by considering the action on E of the subgroup of W′(E_{E_{8}}^{5a}) fixing E. As an example we find that the stabilizer of E_{E_{8}}^{3a} inside W′(E_{E_{8}}^{5a}) is

[ \begin{matrix}ε_{1} & 0 & 0 & x & x \\ 0 & a_{11} & a_{12} & a_{13} & 0 \\ 0 & a_{21} & a_{22} & a_{23} & 0 \\ 0 & 0 & 0 & det & 0 \\ 0 & 0 & 0 & 0 & det \end{matrix} ] ,

where det = a_{11} · a_{22} − a_{12} · a_{21} ≠ 0. The action of such a matrix on E_{E_{8}}^{3a} is given by

\overline{x_{1}} ↦ (\overline{x_{1}})^{a_{11}}(\overline{y_{1}})^{a_{21}},  \overline{y_{1}} ↦ (\overline{x_{1}})^{a_{12}}(\overline{y_{1}})^{a_{22}},  \overline{a} ↦ (\overline{x_{1}})^{a_{13}}(\overline{y_{1}})^{a_{23}}(\overline{a})det.

Thus W (E_{E_{8}}^{3a}) contains the group

W′(E_{E_{8}}^{3a}) = [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det \end{matrix} ]

as claimed. Similar computations show that for the remaining groups E = E_{E_{8}}^{3b}, E_{E_{8}}^{4a}, E_{E_{8}}^{4b} and E_{E_{8}}^{4c}, the group W′(E) occurring in the theorem is a lower bound for the Weyl group W (E).

ORBIT COMPUTATION: Note first that all elementary abelian subgroups of rank at most two are toral by Lemma (1.1.3(3)). By using the lower bounds on the Weyl groups of E_{E_{8}}^{5a} and E_{E_{8}}^{5b} established above, we may find a set of representatives for the conjugacy classes of subgroups of E_{E_{8}}^{5a} and E_{E_{8}}^{5b} of rank 3 and 4.

Under the action of W′(E_{E_{8}}^{5a}), the set of rank 3 subgroups of E_{E_{8}}^{5a} has orbit representatives

\begin{matrix}E_{E_{8}}^{3a} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ , E_{E_{8}}^{3b} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩ , ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}⟩ , \\ ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{x_{1}}, \overline{x_{3}}⟩ and ⟨\overline{a}, \overline{x_{2}}, \overline{x_{3}}⟩ , \end{matrix}

and under the action of W′(E_{E_{8}}^{5b}), the set of rank 3 subgroups of E_{E_{8}}^{5b} has orbit representatives

E_{E_{8}}^{3a} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ , ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}⟩ and ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}⟩ .

Similarly we find that under the action of W′(E_{E_{8}}^{5a}), the set of rank 4 subgroups of E_{E_{8}}^{5a} has orbit representatives

\begin{matrix}E_{E_{8}}^{4a} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩ , E_{E_{8}}^{4b} = ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ , \\ E_{E_{8}}^{4c} = ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩ and ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}⟩ , \end{matrix}

and that under the action of W′(E_{E_{8}}^{5b}), the set of rank 4 subgroups of E_{E_{8}}^{5b} has orbit representatives

E_{E_{8}}^{4b} = ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ and E_{0} = ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}, \overline{y_{2}}⟩ .

CLASS DISTRIBUTIONS: Recall that by (1.6.2), \overline{a} is in the conjugacy class 3A, \overline{x_{1}} and \overline{x_{2}} are in the class 3B and \overline{x_{3}a^{−1}} belongs to the class 3D. Using the actions of W′(E_{E_{8}}^{5a}) and W′(E_{E_{8}}^{5b}) it is then straightforward to verify the class distributions given in the table. As an example consider the group E_{E_{8}}^{5a}. Under the action of W′(E_{E_{8}}^{5a}) it contains 156 elements conjugate to \overline{a}, 78 elements conjugate to \overline{x_{1}}, 2 elements conjugate to \overline{x_{2}} and 6 elements conjugate to \overline{x_{3}a^{−}1}, which gives the class distribution in the table. Similar computations give the results for the remaining groups.

We also see that the group E_{0} = ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}, \overline{y_{2}}⟩ has class distribution 3B^{80} and from the class distribution of E_{E_{8}}^{5b} we get E_{0} = (E_{E_{8}}^{5b} ∩ 3B) ∪ {1}. It then follows from [13, Lemma (11.5)] that E_{0} is toral.

CENTRALIZER DIMENSIONS: By using Lemma (1.1.3(6)), the class distributions found above and the traces given in [7, Table 4], the dimensions given in the last column of the table in the theorem are easily checked.

OTHER NONTORAL SUBGROUPS: We see directly that the groups

⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}⟩ , ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{x_{1}}, \overline{x_{3}}⟩ and ⟨\overline{a}, \overline{x_{2}}, \overline{x_{3}}⟩

are toral. Since the group ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}⟩ is a subgroup of E_{0} it is also toral. Alternatively, from the action of W′(E_{E_{8}}^{5a}) we see that it is conjugate to the group ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}⟩, which is visibly toral. Thus any nontoral elementary abelian 3-subgroup of E_{8}(ℂ) is conjugate to a group in the table. Moreover, since their class distributions differ, none of the groups occurring in the table are conjugate.

To see that the groups in the table are actually nontoral we may proceed as follows. The group E_{E_{8}}^{3a} contains the element \overline{a}, so by Lemma (1.1.3(1)) it is toral if and only if it is toral in C_{E_{8}(ℂ)}(\overline{a}) = SL_{9}(ℂ)/C_{3}. However this is not the case by Lemma (1.1.3(5)), since its lift to SL_{9}(ℂ) is nonabelian. The groups E_{E_{8}}^{4a} and E_{E_{8}}^{4b} are thus also nontoral since they contain E_{E_{8}}^{3a}. We saw above that the Weyl group of E_{E_{8}}^{3b} contains SL_{3}(𝔽_{3}), which has order 2^{4} · 3^{3} · 13. Since the Weyl group of E_{8}(ℂ) has order 2^{14} · 3^{5} · 5^{2} · 7 which is not divisible by 13 it follows from Lemma (1.1.3(4)) that E_{E_{8}}^{3b} is nontoral. Since E_{E_{8}}^{4c} contains E_{E_{8}}^{3b} it is also nontoral.

OVERSIZE: Note that both E_{E_{8}}^{5a} and E_{E_{8}}^{5b} are 1-oversize in E_{8}(ℂ). For E_{E_{8}}^{5a} this follows from the fact that it is nontoral and contains the toral subgroup ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}⟩ of codimension 1. For E_{E_{8}}^{5b} it follows similarly using the toral subgroup E_{0} from above. Since the other groups in the table are contained in a maximal nontoral elementary abelian 3-subgroup, these are 1-oversize as well.

EXACT WEYL GROUPS: By now we know that the groups in the table is a set of representatives for the conjugacy classes of nontoral elementary abelian 3-subgroups of E_{8}(ℂ), that they are all 1-oversize and that the matrix groups given in the theorem are lower bounds for the Weyl groups. We thus only need to verify that for all the groups except E_{E_{8}}^{3b} and E_{E_{8}}^{4c} the Weyl group is the one given above.

First consider the group E_{E_{8}}^{3a} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩. Let x_{4} = τ_{2}^{−1} ⊕τ_{2}^{−1} ⊕τ_{2}^{−1} ∈ SL_{9}(ℂ). Note that conjugation by (2, 7, 3, 4)(5, 8, 9, 6) ∈ SL_{9}(ℂ) acts as follows:

a ↦ a, x_{1} ↦ x_{2}, x_{2} ↦ x_{1}^{2}, x_{3}a^{−1} ↦ x_{4}, y_{1} ↦ y_{2}, y_{2} ↦ y_{1}^{2}.

In particular we see that E_{E_{8}}^{3a} is conjugate to ⟨\overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩. If W (E_{E_{8}}^{3a}) was larger than the group W′(E_{E_{8}}^{3a}) from above, we then see (using for example the computer program mentioned in the introduction) that W (E_{E_{8}}^{3a}) would have to contain one of the groups

[ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & ε \end{matrix} ] or SL_{3}(𝔽_{3})

which are the minimal overgroups of W′(E_{E_{8}}^{3a}) inside GL_{3}(𝔽_{3}). Thus W (E_{E_{8}}^{3a}) would have to contain one of the matrices

[ \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} ] .

This would mean that inside C_{E_{8}(ℂ)}(\overline{x_{2}}, \overline{a}) there would be an element which conjugates \overline{y_{2}} into either \overline{y_{2}}^{2} or \overline{y_{2}a}. However, since C_{E_{8}(ℂ)}(\overline{a}) = SL_{9}(ℂ)/C_{3}, we get

C_{E_{8}(ℂ)}(\overline{x_{2}}, \overline{a}) = C_{SL_{9}(ℂ)/C_{3}}(\overline{x_{2}}) = \overline{⟨y_{2}, \{A ⊕ B ⊕ C | det ABC = 1\}⟩}.

From this it is easily seen that no such element exists, and thus we have W (E_{E_{8}}^{3a}) = W′(E_{E_{8}}^{3a}) as claimed.

Note that the groups E_{E_{8}}^{4a}, E_{E_{8}}^{4b}, E_{E_{8}}^{5a} and E_{E_{8}}^{5b} all contain E_{E_{8}}^{3a}. The minimal overgroups of W′(E) in GL(E) for these groups are found to be:

E_{E_{8}}^{4a} : [ \begin{matrix}GL_{3}(𝔽_{3}) & 0 0 0 \\ 0 0 0 & ε \end{matrix} ] , [ \begin{matrix}SL_{3}(𝔽_{3}) & 0 0 0 \\ ∗ ∗ ∗ & ε \end{matrix} ] , [ \begin{matrix}SL_{3}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \\ ∗ \end{matrix} \\ 0 0 0 & ε \end{matrix} ]

E_{E_{8}}^{4b} : [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] , [ \begin{matrix}ε_{1} & ∗ ∗ & ∗ \\ {0 \atop 0} & GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 & 0 0 & ε_{2} \end{matrix} ]

E_{E_{8}}^{5a} : [ \begin{matrix}ε_{1} & ∗ ∗ ∗ & ∗ \\ 0 0 0 & GL_{3}(𝔽_{3}) & 0 0 0 \\ 0 & 0 0 0 & ε_{2} \end{matrix} ] , [ \begin{matrix}ε_{1} & ∗ ∗ ∗ & ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) & 0 0 0 \\ 0 & ∗ ∗ ∗ & ε_{2} \end{matrix} ] , [ \begin{matrix}ε_{1} & ∗ ∗ ∗ & ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \\ ∗ \end{matrix} \\ 0 & 0 0 0 & ε_{2} \end{matrix} ]

E_{E_{8}}^{5b} : [ \begin{matrix}GSp_{4}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \\ ∗ \\ ∗ \end{matrix} \\ 0 0 0 0 & ε \end{matrix} ] .

From this we see that if W (E_{E_{8}}^{4a}) ≠ W′(E_{E_{8}}^{4a}) then W (E_{E_{8}}^{4a}) will contain one of the matrices

[ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ]

all fixing the subgroup E_{E_{8}}^{3a} of E_{E_{8}}^{4a}. However the automorphisms of E_{E_{8}}^{3a} induced by these elements are given by the matrices

[ \begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix} ] .

Since none of these are contained in W (E_{E_{8}}^{3a}) we get W (E_{E_{8}}^{4a}) = W′(E_{E_{8}}^{4a}) as claimed. Similar computations using the above list of minimal overgroups show that W (E) = W′(E) for E = E_{E_{8}}^{4b}, E_{E_{8}}^{5a} and E_{E_{8}}^{5b}. □

1.7 Centralizers

In this section we compute the centralizers of most of the elementary abelian subgroups given in the previous sections. Our calculations also enables us to finish the computation of Weyl groups begun earlier; we show that in the cases where only a lower bound was given, the Weyl group actually equals this lower bound. The notation for the various elements and subgroups is the same as before. For an abelian group A and a prime p, we let _{p}A denote the elementary abelian p-subgroup of A generated by the elements in A of order p.

The results of this section are used in [1] together with the Weyl group computations to show the vanishing of certain higher limits which occurs in the proof of the classification of p-compact groups at odd primes. The main technical result we need for this application is the following:

(1.7.1) Theorem. Let p be an odd prime, let G be one of the exceptional simple algebraic groups over ℂ considered in the previous sections and let E be a nontoral elementary abelian p-subgroup of G. Assume that p ≠ 3 or that (G, E) ≠ (3E_{6}(ℂ), E_{3E_{6}}^{3}), (E_{6}(ℂ), E_{E_{6}}^{2b}) and (E_{8}(ℂ), E_{E_{8}}^{3b}) for p = 3. Then we have _{p}Z(C_{G}(E)) = E. □

(1.7.2) Remark. Obviously, if E is an elementary abelian p-subgroup of G, then_{p}Z(C_{G}(E)) is an elementary abelian p-subgroup containing E. Thus if E is maximal, then the conclusion of Theorem (1.7.1) holds for E. However we are not going to need this since we can compute most of the centralizers and their centers directly.

We do not know if the conclusion of the theorem holds in the cases (G, E) = (E_{6}(ℂ), E_{E_{6}}^{2b}) and (G, E) = (E_{8}(ℂ), E_{E_{8}}^{3b}) with p = 3, since we do not know the exact structure of these centralizers.

For p = 3, G = 3E_{6}(ℂ) and E = E_{3E_{6}}^{3} we have _{p}Z(C_{G}(E)) = ⟨E, Z(G)⟩ due to the fact that Z(G) has p-torsion. However, in general we do not have _{p}Z(C_{G}(E)) = E for any nontoral elementary abelian p-subgroup E of a simple group G of adjoint type. The following example is inspired by [5, Section 3]. Let p ≥ 5 be a prime number and choose any m with 3 ≤ m ≤ p − 1. Let ξ = e^{2πi/p} and G = PSL_{mp}(ℂ). For an element g ∈ SL_{mp}(ℂ), the corresponding element in G is denoted by \overline{g}. Choose different integers i_{1}, …, i_{m} in {0, …, p − 1} satisfying i_{1} + … + i_{m} ≡ 0 (mod p) and consider the following elements in SL_{mp}(ℂ):

g_{1} = diag(\underbrace{1, ξ, …, ξ^{p−1}}, …, \underbrace{1, ξ, …, ξ^{p−1}}),
g_{2} = (1, 2, …, p)(p + 1, …, 2p) … ((m − 1)p + 1, …, mp) ,
g_{3} = diag(\underbrace{ξ^{i_{1}}, …, ξ^{i_{1}}}_{p}, \underbrace{ξ^{i_{2}}, …, ξ^{i_{2}}}_{p}, …, \underbrace{ξ^{i_{m}}, …, ξ^{i_{m}}}_{p}).

Note that E = ⟨\overline{g_{1}}, \overline{g_{2}}, \overline{g_{3}}⟩ is an elementary abelian p-subgroup of rank 3. Then we have

C_{PSL_{mp}(ℂ)}(\overline{g_{1}}) = \overline{⟨g_{2}, C_{SL_{mp}(ℂ)}(g_{1})⟩},
C_{PSL_{mp}(ℂ)}(\overline{g_{1}}, \overline{g_{2}}) = \overline{⟨g_{1}, g_{2}, C_{SL_{mp}(ℂ)}(g_{1}, g_{2})⟩}.

We directly get C_{SL_{mp}(ℂ)}(g_{1}, g_{2}) = Δ_{m,p} ({A ∈ GL_{m}(ℂ)| (det A)^{p} = 1}). Since m is coprime to p we actually have

C_{PSL_{mp}(ℂ)}(\overline{g_{1}}, \overline{g_{2}}) = \overline{⟨g_{1}, g_{2}, Δ_{m,p}(SL_{m}(ℂ))⟩} ≅ ⟨\overline{g_{1}}, \overline{g_{2}}⟩ × PSL_{m}(ℂ).

Here \overline{g_{3}} corresponds to the element x = \overline{diag(ξ^{i_{1}}, …, ξ^{i_{m}})} ∈ PSL_{m}(ℂ). Since i_{1}, …, i_{m} are different and m < p we get that C_{PSL_{m}(ℂ)}(x) = 𝕋_{m−1} is the maximal torus in PSL_{m}(ℂ). Therefore

C_{PSL_{mp}(ℂ)}(E) = \overline{⟨g_{1}, g_{2}⟩} × 𝕋_{m−1},

and hence _{p}Z(C_{PSL_{mp}(ℂ)}(E)) has rank 2 + (m − 1) = m + 1 > 3.

(1.7.3) Centralizers in 3E_{6}(ℂ) and E_{6}(ℂ). Let Θ : SL_{3}(ℂ) ⟶ SL_{3}(ℂ)^{3}/C_{3} ⊆ 3E_{6}(ℂ) denote the homomorphism given by Θ(g) = [g, g, g] for g ∈ SL_{3}(ℂ). We have already seen that C_{3E_{6}(ℂ)}(a) = SL_{3}(ℂ)^{3}/C_{3}. From this we get

C_{3E_{6}(ℂ)}(a, x_{2}) = C_{SL_{3}(ℂ)^{3}/C_{3}}(x_{2}) = ⟨y_{2}, (𝕋_{2} × 𝕋_{2} × 𝕋_{2}) /C_{3}⟩ ,
C_{3E_{6}(ℂ)}(a, x_{2}, y_{2}) = ⟨x_{2}, y_{2}, (⟨ωI_{3}⟩ × ⟨ωI_{3}⟩ × ⟨ωI_{3}⟩) /C_{3}⟩ = E_{3E_{6}}^{4}.

Thus C_{3E_{6}(ℂ)}(E_{3E_{6}}^{3}) = C_{3E_{6}(ℂ)}(E_{3E_{6}}^{4}) = E_{3E_{6}}^{4}. We now turn to the subgroups of E_{6}(ℂ). By Lemma (1.4.2) the group E_{E_{6}}^{2a} = ⟨\overline{x_{2}}, \overline{y_{1}}⟩ is conjugate to the group ⟨\overline{a}, \overline{s_{1}}⟩. Since a^{s_{1}} = az^{2} we obtain C_{E_{6}(ℂ)}(\overline{a}) = \overline{⟨s_{1}, SL_{3}(ℂ)^{3}/C_{3}⟩}, and from this we get

C_{E_{6}(ℂ)}(\overline{a}, \overline{s_{1}}) = \overline{⟨a, s_{1}, z, Θ(SL_{3}(ℂ))⟩} = \overline{⟨a, s_{1}, z⟩ × Θ(SL_{3}(ℂ))} ≅ ⟨\overline{a}, \overline{s_{1}}⟩ × PSL_{3}(ℂ).

We thus have C_{E_{6}(ℂ)}(E_{E_{6}}^{2a}) = E_{E_{6}}^{2a} × PSL_{3}(ℂ). In particular Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{2a})) = E_{E_{6}}^{2a}. By abusing the notation slightly, we let \overline{g} denote the image of g ∈ SL_{3}(ℂ) in the quotient PSL_{3}(ℂ). From Lemma (1.4.2) we then see that the elements \overline{a}, \overline{y_{2}} and \overline{x_{2}x_{1}^{−1}} in C_{E_{6}(ℂ)}(E_{E_{6}}^{2a}) equal the elements \overline{β}, \overline{γ^{2}} and \overline{τ_{2}} in the PSL_{3}(ℂ) component of C_{E_{6}(ℂ)}(E_{E_{6}}^{2a}). From this we now immediately get

C_{E_{6}(ℂ)}(E_{E_{6}}^{3a}) = E_{E_{6}}^{2a} × C_{PSL_{3}(ℂ)}(\overline{β}),  C_{E_{6}(ℂ)}(E_{E_{6}}^{3d}) = E_{E_{6}}^{2a} × C_{PSL_{3}(ℂ)}(\overline{τ_{2}}),
C_{E_{6}(ℂ)}(E_{E_{6}}^{4a}) = E_{E_{6}}^{2a} × C_{PSL_{3}(ℂ)}(\overline{β}, \overline{γ^{2}}),  C_{E_{6}(ℂ)}(E_{E_{6}}^{4b}) = E_{E_{6}}^{2a} × C_{PSL_{3}(ℂ)}(\overline{β}, \overline{τ_{2}}).

Note that C_{PSL_{3}(ℂ)}(\overline{β}) = \overline{⟨𝕋_{2}, γ⟩}, from which we get C_{PSL_{3}(ℂ)}(\overline{β}, \overline{γ^{2}}) = ⟨\overline{β}, \overline{γ}⟩ and C_{PSL_{3}(ℂ)}(\overline{β}, \overline{τ_{2}}) = \overline{𝕋_{2}}. We thus have

C_{E_{6}(ℂ)}(E_{E_{6}}^{3a}) = E_{E_{6}}^{2a} × (𝕋_{2} : 3),  Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{3a})) = E_{E_{6}}^{3a},
C_{E_{6}(ℂ)}(E_{E_{6}}^{4a}) = E_{E_{6}}^{4a},  Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{4a})) = E_{E_{6}}^{4a},
C_{E_{6}(ℂ)}(E_{E_{6}}^{4b}) = E_{E_{6}}^{2a} × 𝕋_{2},  Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{4b})) = E_{E_{6}}^{2a} × 𝕋_{2}.

In particular _{3}Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{4b})) = E_{E_{6}}^{4b}. For E_{E_{6}}^{3d} we have C_{PSL_{3}(ℂ)}(\overline{τ_{2}}) ≅ GL_{2}(ℂ) so

C_{E_{6}(ℂ)}(E_{E_{6}}^{3d}) = E_{E_{6}}^{2a} × GL_{2}(ℂ),  Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{3d})) = E_{E_{6}}^{2a} × 𝕋_{1}

and _{3}Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{3d})) = E_{E_{6}}^{3d}.

We now consider the group E_{E_{6}}^{3b}. Since C_{E_{6}(ℂ)}(\overline{a}) = \overline{⟨s_{1}, SL_{3}(ℂ)^{3}/C_{3}⟩} we get

C_{E_{6}(ℂ)}(\overline{a}, \overline{x_{2}}) = \overline{⟨s_{1}, y_{1}, y_{2}, (𝕋_{2} × 𝕋_{2} × 𝕋_{2}) /C_{3}⟩},
C_{E_{6}(ℂ)}(\overline{a}, \overline{x_{2}}, \overline{y_{2}}) = \overline{⟨s_{1}, y_{1}, y_{2}, [I_{3}, β, β^{2}] , x_{2}, (⟨ωI_{3}⟩ × ⟨ωI_{3}⟩ × ⟨ωI_{3}⟩) /C_{3}, ⟩}

which is a finite group of order 3^{6}. It is not hard to check that the center is the subgroup ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩ and we conclude that Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{3b})) = E_{E_{6}}^{3b}.

Finally consider the group E_{E_{6}}^{3c}. We obtain

C_{E_{6}(ℂ)}(\overline{a}, \overline{x_{1}}) = \overline{⟨y_{1}, (SL_{3}(ℂ) × 𝕋_{2} × 𝕋_{2}) /C_{3}⟩},
C_{E_{6}(ℂ)}(\overline{a}, \overline{x_{1}}, \overline{y_{1}}) = \overline{⟨y_{1}, x_{1}, (SL_{3}(ℂ) × ⟨ωI_{3}⟩ × ⟨ωI_{3}⟩) /C_{3}⟩}.

Thus C_{E_{6}(ℂ)}(E_{E_{6}}^{3c}) = SL_{3}(ℂ) ∘_{⟨\overline{a}⟩} E_{E_{6}}^{3c} is a central product, and from this we find Z(C_{E_{6}(ℂ)}(E_{E_{6}}^{3c})) = E_{E_{6}}^{3c}.

(1.7.4) Centralizers in E_{8}(ℂ). In section 1.6 we saw that E_{E_{8}}^{3a} is conjugate to ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩

and that

C_{E_{8}(ℂ)}(\overline{a}, \overline{x_{2}}) = \overline{⟨y_{2}, \{A ⊕ B ⊕ C | det ABC = 1\}⟩}.

From this we directly get

C_{E_{8}(ℂ)}(\overline{a}, \overline{x_{2}}, \overline{y_{2}}) = \overline{⟨x_{2}, y_{2}, \{A ⊕ A ⊕ A | (det A)^{3} = 1\}⟩}
= \overline{⟨x_{2}, y_{2}, a, \{A ⊕ A ⊕ A | det A = 1\}⟩}
≅ ⟨\overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩ × PSL_{3}(ℂ).

Thus C_{E_{8}(ℂ)}(E_{E_{8}}^{3a}) = E_{E_{8}}^{3a} × PSL_{3}(ℂ) and Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{3a})) = E_{E_{8}}^{3a}.

From section 1.6 we see that the elements \overline{x_{2}}, \overline{x_{3}a^{−1}} and \overline{y_{2}} in C_{E_{6}(ℂ)}(E_{E_{8}}^{3a}) equal the elements \overline{β^{2}}, \overline{τ_{2}^{−1}} and \overline{γ^{2}} in the PSL_{3}(ℂ) component of C_{E_{8}(ℂ)}(E_{E_{8}}^{3a}). Now the computations in PSL_{3}(ℂ) done above in the case of E_{6}(ℂ) immediately show

C_{E_{8}(ℂ)}(E_{E_{8}}^{4a}) = E_{E_{8}}^{3a} × GL_{2}(ℂ),  Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{4a})) = E_{E_{8}}^{3a} × 𝕋_{1},
C_{E_{8}(ℂ)}(E_{E_{8}}^{4b}) = E_{E_{8}}^{3a} × (𝕋_{2} : 3),  Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{4b})) = E_{E_{8}}^{4b},
C_{E_{8}(ℂ)}(E_{E_{8}}^{5a}) = E_{E_{8}}^{3a} × 𝕋_{2},  Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{5a})) = E_{E_{8}}^{3a} × 𝕋_{2},
C_{E_{8}(ℂ)}(E_{E_{8}}^{5b}) = E_{E_{8}}^{5b},  Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{5b})) = E_{E_{8}}^{5b},

In particular _{3}Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{4a})) = E_{E_{8}}^{4a} and _{3}Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{5a})) = E_{E_{8}}^{5a}.

We now turn to the nontoral subgroups of 2E_{7}(ℂ) and the two remaining subgroups E_{E_{8}}^{3b} and E_{E_{8}}^{4c} in E_{8}(ℂ).

(1.7.5) Setup. Consider the description of the root systems Φ(E_{7}) ⊆ Φ(E_{8}) from [3, p. 264–270]. Let H denote a maximal torus of E_{8}(ℂ) and fix a realization u_{α} of Φ(E_{8}) in E_{8}(ℂ) as in section 1.6. The subgroup generated by the root subgroups u_{α}, α ∈ Φ(E_{7}) is then isomorphic to 2E_{7}(ℂ) having as maximal torus the subgroup generated by h_{α_{i}}(t), 1 ≤ i ≤ 7, t ∈ ℂ^{×}. Thus we have an embedding 2E_{7}(ℂ) ⊆ E_{8}(ℂ) which is consistent with the embedding Φ(E_{7}) ⊆ Φ(E_{8}).

In particular define the element g_{1} = h_{α_{3}}(ω)h_{α_{1}}(ω^{2}) ∈ 2E_{7}(ℂ). By using [13, Table VI] and [7, Table 6] we get that g_{1} belongs to the conjugacy class 3C[2E_{7}(ℂ)] ⊆ 3B[E_{8}(ℂ)] and that the centralizers C_{2E_{7}(ℂ)}(g_{1}) and C_{E_{8}(ℂ)}(g_{1}) have types A_{5}A_{2} and E_{6}A_{2} respectively. The precise structure of these centralizers may be found as follows. Since 2E_{7}(ℂ) and E_{8}(ℂ) are both simply connected, Lemma (1.1.3(2)) implies that both C_{2E_{7}(ℂ)}(g_{1}) and C_{E_{8}(ℂ)}(g_{1}) are connected. Setting

α′ = α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + α_{6},

we see that {α_{5}, α_{6}, α_{7}, α′, α_{2}} ∪ {α_{1}, α_{3}} is a basis of the root system of C_{2E_{7}(ℂ)}(g_{1}) and that {α_{5}, α_{8}, α_{6}, α_{7}, α′, α_{2}} ∪ {α_{1}, α_{3}} is a basis of the root system of C_{E_{8}(ℂ)}(g_{1}). Here the bases of the components of type A_{2}, A_{5} and E_{6} have been ordered so that the numbering is consistent with [3, p. 250–251,260–262]. From this we get explicit homomorphisms SL_{6}(ℂ) × SL_{3}(ℂ) → 2E_{7}(ℂ) and 3E_{6}(ℂ) × SL_{3}(ℂ) → E_{8}(ℂ) onto the centralizers C_{2E_{7}(ℂ)}(g_{1}) and C_{E_{8}(ℂ)}(g_{1}). The kernels of these maps are given by ⟨(ωI_{6}, ω^{2}I_{3})⟩ and ⟨(z, ω^{2}I_{3})⟩ respectively, where z ∈ 3E_{6}(ℂ) denotes the central element defined in section 1.4. Thus the centralizers are given by the central products in the following diagram

\xymatrix{SL_{6}(ℂ) ∘_{C_{3}} SL_{3}(ℂ) ≅ C_{2E_{7}(ℂ)}(g_{1}) \ar[r] \ar[d] & 2E_{7}(ℂ) \ar[d] \\ 3E_{6}(ℂ) ∘_{C_{3}} SL_{3}(ℂ) ≅ C_{E_{8}(ℂ)}(g_{1}) \ar[r] & E_{8}(ℂ),}

where the homomorphisms are defined from the identifications of root systems given above. In particular we have a homomorphism Ψ : SL_{6}(ℂ) → 3E_{6}(ℂ).

We write elements in these central products as A·B with A ∈ SL_{6}(ℂ) or A ∈ 3E_{6}(ℂ) and B ∈ SL_{3}(ℂ). Then ωA · B = A · ωB in SL_{6}(ℂ) ∘_{C_{3}} SL_{3}(ℂ) ⊆ 2E_{7}(ℂ), and zA · B = A · ωB in 3E_{6}(ℂ) ∘_{C_{3}} SL_{3}(ℂ) ⊆ E_{8}(ℂ). Moreover, under the inclusion 2E_{7}(ℂ) ⊆ E_{8}(ℂ), the element A · B in 2E_{7}(ℂ) corresponds to the element Ψ(A) · B in E_{8}(ℂ). With this notation we have g_{1} = ωI_{6} = ωI_{3} in 2E_{7}(ℂ) and g_{1} = z = ωI_{3} in E_{8}(ℂ). We also introduce the elements

g_{2} = (β ⊕ β) · β,  g_{3} = (γ ⊕ γ) · γ^{2},  g_{4} = ωI_{3} ⊕ ω^{2}I_{3}

in 2E_{7}(ℂ) ⊆ E_{8}(ℂ). Since g_{1}, g_{2}, g_{3} and g_{4} all have order 3 and are pairwise commuting, we see that the groups E^{3} = ⟨g_{1}, g_{2}, g_{3}⟩ and E^{4} = ⟨g_{1}, g_{2}, g_{3}, g_{4}⟩ are elementary abelian 3-groups of rank 3 and 4 respectively.

(1.7.6) Lemma. We have

C_{2E_{7}(ℂ)}(E^{3}) ≅ E^{3} × SL_{2}(ℂ), C_{2E_{7}(ℂ)}(E^{4}) ≅ E^{3} × 𝕋_{1}, Z(C_{2E_{7}(ℂ)}(E^{4})) = E^{4}

and moreover W_{E_{8}(ℂ)}(E^{3}) ≠ GL_{3}(𝔽_{3}).

(1.7.7) Proof: Since C_{2E_{7}(ℂ)}(g_{1}) = SL_{6}(ℂ) ∘_{C_{3}} SL_{3}(ℂ) we get

C_{2E_{7}(ℂ)}(g_{1}, g_{2}) = ⟨g_{3}, C_{SL_{6}(ℂ)}(β ⊕ β) ∘_{C_{3}} 𝕋_{2}⟩ ,
C_{2E_{7}(ℂ)}(g_{1}, g_{2}, g_{3}) = ⟨g_{2}, g_{3}, C_{SL_{6}(ℂ)}(β ⊕ β, γ ⊕ γ) ∘_{C_{3}} ⟨ωI_{3}⟩⟩ .

Then C_{SL_{6}(ℂ)}(β ⊕ β, γ ⊕ γ) = Δ_{2,3}({A ∈ GL_{2}(ℂ)|(det A)^{3} = 1}) is generated by Δ_{2,3}(ω^{2}I_{2}) and Δ_{2,3}(SL_{2}(ℂ)). From this we get

C_{2E_{7}(ℂ)}(E^{3}) = ⟨g_{1}, g_{2}, g_{3}, Δ_{2,3}(SL_{2}(ℂ))⟩ ≅ E^{3} × SL_{2}(ℂ).

Under this isomorphism g_{4} corresponds to the element diag(ω, ω^{2}) ∈ SL_{2}(ℂ) and thus we get C_{2E_{7}(ℂ)}(E^{4}) ≅ E^{3} × 𝕋_{1}. As g_{4} corresponds to the element ω ∈ 𝕋_{1} we conclude _{3}Z(C_{2E_{7}(ℂ)}(E^{4})) = E^{4}.

To prove the last statement recall that C_{E_{8}(ℂ)}(g_{1}) = 3E_{6}(ℂ) ∘_{C_{3}} SL_{3}(ℂ) and that we have g_{2} = Ψ(β ⊕ β) · β and g_{3} = Ψ(γ ⊕ γ) · γ^{2} in E_{8}(ℂ). Thus we get

C_{E_{8}(ℂ)}(g_{1}, g_{2}) = ⟨g_{3}, C_{3E_{6}(ℂ)}(Ψ(β ⊕ β)) ∘_{C_{3}} 𝕋_{2}⟩ .

Note that no element in 𝕋_{2} conjugates γ^{2} into an element of the form ω^{k}γ. Thus g_{3} is not conjugate to g_{3}^{−1} in C_{E_{8}(ℂ)}(g_{1}, g_{2}). Hence diag(1, 1, −1) ∉ W_{E_{8}(ℂ)}(E^{3}) and we are done. □

(1.7.8) Proposition. The elementary abelian 3-groups E^{3} and E^{4} are nontoral in 2E_{7}(ℂ) and E_{8}(ℂ). In particular E^{3} is conjugate to E_{2E_{7}}^{3} and E^{4} is conjugate to E_{2E_{7}}^{4} in 2E_{7}(ℂ). In E_{8}(ℂ) the group E^{3} is conjugate to E_{E_{8}}^{3b} and E^{4} is conjugate to E_{E_{8}}^{4c}. Moreover we have

C_{2E_{7}(ℂ)}(E_{2E_{7}}^{3}) = E_{2E_{7}}^{3} × SL_{2}(ℂ),  Z(C_{2E_{7}(ℂ)}(E_{2E_{7}}^{3})) = E_{2E_{7}}^{3} × 2,
C_{2E_{7}(ℂ)}(E_{2E_{7}}^{4}) = E_{2E_{7}}^{3} × 𝕋_{1},  Z(C_{2E_{7}(ℂ)}(E_{2E_{7}}^{4})) = E_{2E_{7}}^{3} × 𝕋_{1},
C_{E_{8}(ℂ)}(E_{E_{8}}^{4c}) = E_{E_{8}}^{4c} ∘_{C_{3}} SL_{3}(ℂ),  Z(C_{E_{8}(ℂ)}(E_{E_{8}}^{4c})) = E_{E_{8}}^{4c}.

In particular _{3}Z(C_{2E_{7}(ℂ)}(E)) = E for E = E_{2E_{7}}^{3} and E_{2E_{7}}^{4}.

(1.7.9) Proof: Note first that if E ⊆ G is a toral subgroup, then C_{G}(E) contains a maximal torus of G. From this we see that E^{3} is nontoral in 2E_{7}(ℂ) since the rank of the centralizer C_{2E_{7}(ℂ)}(E^{3}) is less than 7 by Lemma (1.7.6). As E^{4} contains E^{3} we see that E^{4} is also nontoral in 2E_{7}(ℂ). From Theorem (1.5.3) we then get that E^{3} is conjugate to E_{2E_{7}}^{3} and E^{4} is conjugate to E_{2E_{7}}^{4} in 2E_{7}(ℂ). We also see that the Weyl groups W_{2E_{7}(ℂ)}(E^{3}) and W_{2E_{7}(ℂ)}(E^{4}) have orders divisible by 13. Since W_{2E_{7}(ℂ)}(E) ⊆ W_{E_{8}(ℂ)}(E) for any elementary abelian group E ⊆ 2E_{7}(ℂ) we conclude in particular that the Weyl groups of E^{3} and E^{4} in E_{8}(ℂ) also have orders divisible by 13. Since the Weyl group of E_{8}(ℂ) has order 2^{14} · 3^{5} · 5^{2} · 7 which is not divisible by 13, it follows from Lemma (1.1.3(4)) that E^{3} and E^{4} are nontoral in E_{8}(ℂ).

From the above and Theorem (1.5.3) we see that the class distributions of E^{3} and E^{4} are 3C^{26} and 3B^{2}3C^{78} respectively in 2E_{7}(ℂ). Now using [7, Table 6] we see that we have the inclusions

3B[2E_{7}(ℂ)] ⊆ 3B[E_{8}(ℂ)],  3C[2E_{7}(ℂ)] ⊆ 3B[E_{8}(ℂ)]

of conjugacy classes corresponding to the inclusion 2E_{7}(ℂ) ⊆ E_{8}(ℂ). Therefore the class distributions of E^{3} and E^{4} in E_{8}(ℂ) are 3B^{26} and 3B^{80} respectively. Since E^{3} and E^{4} are nontoral in E_{8}(ℂ), Theorem (1.6.4) shows that E^{3} is conjugate to E_{E_{8}}^{3b} and E^{4} is conjugate to E_{E_{8}}^{4c} in E_{8}(ℂ).

The assertions about the structure of C_{2E_{7}(ℂ)}(E), Z(C_{2E_{7}(ℂ)}(E)) and _{3}Z(C_{2E_{7}(ℂ)}(E)) for E = E_{2E_{7}}^{3} and E_{2E_{7}}^{4} follows directly from Lemma (1.7.6) using what we proved above.

Finally consider the inclusion SL_{3}(ℂ) ∘_{C_{3}} 3E_{6}(ℂ) ⊆ E_{8}(ℂ) from above. In particular we may consider the subgroups of 3E_{6}(ℂ) as subgroups of E_{8}(ℂ). Since the Weyl group of E_{3E_{6}}^{3} in 3E_{6}(ℂ) has order divisible by 13 by Theorem (1.4.5), we see as above that E_{3E_{6}}^{3} is nontoral in E_{8}(ℂ). Hence E_{3E_{6}}^{4} is also nontoral in E_{8}(ℂ). Using the informations about the inclusions of conjugacy classes given above and the ones given in (1.5.2) we see that E_{3E_{6}}^{4} has class distribution 3B^{80} in E_{8}(ℂ). Thus Theorem (1.6.4) shows that E_{3E_{6}}^{4} is conjugate to E_{E_{8}}^{4c} in E_{8}(ℂ).

Now since E_{3E_{6}}^{4} contains the element g_{1} = z we get

\begin{aligned}C_{E_{8}(ℂ)}(E_{3E_{6}}^{4}) & = C_{C_{E_{8}(ℂ)}(g_{1})}(E_{3E_{6}}^{4}) = C_{SL_{3}(ℂ)∘_{C_{3}}3E_{6}(ℂ)}(E_{3E_{6}}^{4}) \\ & = SL_{3}(ℂ) ∘_{C_{3}} C_{3E_{6}(ℂ)}(E_{3E_{6}}^{4}) = SL_{3}(ℂ) ∘_{C_{3}} E_{3E_{6}}^{4}, \end{aligned}

using the computation above of C_{3E_{6}(ℂ)}(E_{3E_{6}}^{4}). From this we get

Z(C_{E_{8}(ℂ)}(E_{3E_{6}}^{4})) = C_{3} ∘_{C_{3}} E_{3E_{6}}^{4} = E_{3E_{6}}^{4}.

Since E_{E_{8}}^{4c} is conjugate to E_{3E_{6}}^{4} in E_{8}(ℂ) the results on E_{E_{8}}^{4c} now follow immediately. □

(1.7.10) Corollary. The Weyl groups of the elementary abelian subgroups E_{3E_{6}}^{3}, E_{3E_{6}}^{4} ⊆ 3E_{6}(ℂ), E_{E_{6}}^{3b}, E_{E_{6}}^{4a} ⊆ E_{6}(ℂ), E_{2E_{7}}^{3}, E_{2E_{7}}^{4} ⊆ 2E_{7}(ℂ) and E_{E_{8}}^{3b}, E_{E_{8}}^{4c} ⊆ E_{8}(ℂ) equal the lower bounds given earlier, i.e. we have

W (E_{3E_{6}}^{3}) = W (E_{E_{6}}^{3b}) = W (E_{2E_{7}}^{3}) = W (E_{E_{8}}^{3b}) = SL_{3}(𝔽_{3}),

W (E_{3E_{6}}^{4}) = [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] , W (E_{E_{6}}^{4a}) = [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det & ∗ \\ 0 0 & 0 & det \end{matrix} ] ,

W (E_{2E_{7}}^{4}) = W (E_{E_{8}}^{4c}) = [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] .

(1.7.11) Proof: By Theorem (1.6.4) we know that W (E_{E_{8}}^{3b}) contains SL_{3}(𝔽_{3}) and by Proposition (1.7.8) and Lemma (1.7.6) we have W (E_{E_{8}}^{3b}) ≠ GL_{3}(𝔽_{3}). Thus we conclude that W (E_{E_{8}}^{3b}) = SL_{3}(𝔽_{3}). Note that by Theorem (1.6.4) W (E_{E_{8}}^{4c}) contains the group

W′ = [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] ,

whose only minimal overgroup in GL_{4}(𝔽_{3}) equals

[ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & GL_{3}(𝔽_{3}) \end{matrix} ] .

However, since E_{E_{8}}^{3b} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩ and E_{E_{8}}^{4c} = ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩, W (E_{E_{8}}^{4c}) cannot contain this group as this would imply that W (E_{E_{8}}^{3b}) = GL_{3}(𝔽_{3}). We conclude that W (E_{E_{8}}^{4c}) = W′ as claimed.

Using the inclusion 2E_{7}(ℂ) ⊆ E_{8}(ℂ) and Proposition (1.7.8) we get

W_{2E_{7}(ℂ)}(E_{2E_{7}}^{3}) ⊆ W_{E_{8}(ℂ)}(E_{E_{8}}^{3b}),  W_{2E_{7}(ℂ)}(E_{2E_{7}}^{4}) ⊆ W_{E_{8}(ℂ)}(E_{E_{8}}^{4c}).

Combining this and Theorem (1.5.3) with the results just proved for E_{E_{8}}^{3b} and E_{E_{8}}^{4c} we conclude that W (E_{2E_{7}}^{3}) = SL_{3}(𝔽_{3}) and W (E_{2E_{7}}^{4}) = W′.

In the proof of Proposition (1.7.8) we saw that E_{3E_{6}}^{4} is conjugate to E_{E_{8}}^{4c} in E_{8}(ℂ) under the inclusion 3E_{6}(ℂ) ⊆ E_{8}(ℂ) from above. Similarly we obtain that E_{3E_{6}}^{3} is conjugate to E_{E_{8}}^{3b} in E_{8}(ℂ). In particular we have W (E_{3E_{6}}^{3}) = SL_{3}(𝔽_{3}) by Theorem (1.4.5), and W (E_{3E_{6}}^{4}) ⊆ W′. Since z is central in 3E_{6}(ℂ) we also have

W (E_{3E_{6}}^{4}) ⊆ [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & GL_{3}(𝔽_{3}) \end{matrix} ] ,

and combining this with W (E_{3E_{6}}^{4}) ⊆ W′ and the lower bound from Theorem (1.4.5), it follows that

W (E_{3E_{6}}^{4}) = [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(𝔽_{3}) \end{matrix} ] .

Thus we get W (E_{E_{6}}^{3b}) = SL_{3}(𝔽_{3}) since E_{3E_{6}}^{4} = ⟨z, a, x_{2}, y_{2}⟩ and E_{E_{6}}^{3b} = ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩.

Finally consider the group E_{E_{6}}^{4a} = ⟨\overline{a}, \overline{y_{2}}, \overline{y_{1}}, \overline{x_{2}}⟩ which contains the groups E_{E_{6}}^{2a} = ⟨\overline{y_{1}}, \overline{x_{2}}⟩ and E_{E_{6}}^{3b} = ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩. From Theorem (1.4.11) we know that W (E_{E_{6}}^{4a}) contains the group

W′(E_{E_{6}}^{4a}) = [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det & ∗ \\ 0 0 & 0 & det \end{matrix} ] .

Using our computer program we obtain that it has the following 3 minimal overgroups in GL_{4}(𝔽_{3}):

[ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det & ∗ \\ 0 0 & 0 & ε \end{matrix} ] , [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & ε & ∗ \\ 0 0 & 0 & det \end{matrix} ] , [ \begin{matrix}GL_{2}(𝔽_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & ε & ∗ \\ 0 0 & 0 & ε \end{matrix} ] .

Thus if W (E_{E_{6}}^{4a}) ≠ W′(E_{E_{6}}^{4a}) then W (E_{E_{6}}^{4a}) would have to contain one of the diagonal matrices diag(1, 1, 1, −1), diag(1, 1, −1, 1) or diag(1, 1, −1, −1). However the first two of these fix the subgroup E_{E_{6}}^{2a} and induce the automorphisms diag(1, −1) and diag(−1, 1) on E_{E_{6}}^{2a}. Since W (E_{E_{6}}^{2a}) = SL_{2}(𝔽_{3}) we conclude that diag(1, 1, 1, −1) and diag(1, 1, −1, 1) does not belong to W (E_{E_{6}}^{4a}). The element diag(1, 1, −1, −1) fixes the subgroup E_{E_{6}}^{3b} and induces the automorphism diag(1, −1, 1) on E_{E_{6}}^{3b}. Since W (E_{E_{6}}^{3b}) = SL_{3}(𝔽_{3}) we conclude that diag(1, 1, −1, −1) ∉ W (E_{E_{6}}^{4a}) and thus W (E_{E_{6}}^{4a}) = W′(E_{E_{6}}^{4a}) as was to be proved. □

1.8 Corrections and misprints

In this section we have gathered some corrections to the papers [7], [8] and [13].

(1.8.1) Remark. There are some minor misprints in [7]. On p. 374, the term 30t^{7} should be replaced by 39t^{7} in the power series expansion of the generating function for the number of elements of order a divisor of i.

In table 4, p. 375–376, the following corrections should be made: For the class 6J the power maps should be [B, C] and for the class 6L it should be [A, C]. The representative of the class 7G should be replaced by (2, 4, 0, 0, 0, 0, 2, 0).

In table 6, p. 384–385, the following corrections should be made: The centralizers types of the classes 3A, 3E, 5E and 6Q should be A_{6}T_{1}, D_{6}T_{1}, A_{3}A_{2}A_{1}T_{1} and A_{3}A_{1}A_{1}A_{1}T_{1} respectively. The power maps for the class 6V should be [A, D].

(1.8.2) Remark. There are some minor misprints in [8]. On p. 108 the formula defining the multiplication on K (the second formula) should read

⟨x × y, z⟩ = 6 ⟨x, y, z⟩ .

On the bottom of p. 109, the formula for the action of n_{1} should read

h(α, β, γ, δ, ε, ζ)^{n_{1}} = h(α, β, γ, (γδ)^{−1}, ε, ζ).

On p. 110, line 5, the element n_{1}n_{7} should be replaced by n_{1}n_{7}h(−1, −1, 1, −1, 1, 1), since n_{1}n_{7} is not an element of F = F_{4}(ℂ). In the third displayed formula on p. 110 giving the generating function for the number of conjugacy classes of elements of finite order in F, the first term 1 should be removed. In the next formula the extra factor (1 − xt) should be inserted in denominator.

In table 2, p. 111–112, the following corrections should be made: The representative of the class 4H should be (1, 1, 1, 0, 2, 3). The g-multiplicity for the class 5D should be 20, 13, 16, 16, 13. The representative of the class 6C should be (1, 0, 1, 0, 4, 3). The centralizer type of the class 6R should be A_{2}A_{2}T_{2}. The representative of the class 7A should be (1, 5, 1, 0, 1, 6). The centralizer type of the class 7E should be A_{3}A_{1}T_{2}. The representatives of the classes 7Q and 7R should be (3, 1, 6, 1, 3, 4) and (1, 1, 2, 1, 4, 5).

Finally, on p. 149, the reference for [5] should be J. Algebra 131 (1990) 23–39.

(1.8.3) Remark. There are some errors and misprints in [13]. We consider as above the case of the ground field ℂ.

The group 2E_{7}(ℂ) for p = 3 is considered in Table II, p. 259 and in Theorem (11.16), p. 296. Here the centralizer of the rank 3 subgroup should have the shape 3^{3} × SL_{2}(ℂ). Since the Weyl group of the rank 3 group is SL_{3}(𝔽_{3}) the shape of its normalizer is wrong as well. The claims about the normalizers of both the rank 3 and the rank 4 groups mapping onto AGL_{3}(𝔽_{3}) also seem inconsistent with our Weyl group computations, although we do not know the exact structure of the normalizers. Finally, in the table it is stated that the factor 2 in the normalizers splits, but in the theorem and its proof it is claimed not to split.

The group E_{8}(ℂ) for p = 3 is considered in Table II, p. 259 and in Lemma (11.5), p. 289–290. The information for the maximal elementary abelian 3-subgroup of type 1 should be as follows. The centralizer should be 3^{3} × 𝕋_{2} (this is given wrongly in the table) and the Weyl group should have the shape 3^{3} : GL(3, 3) × Σ_{3}. Thus the stated structure of the normalizer is wrong. Furthermore for the type 1 group there is a unique subgroup of order 3 which is normal in N(E), namely E_{00} = ⟨\overline{x_{2}}⟩. However we have E_{0} ∩ 𝕋_{2} = ⟨\overline{x_{2}}, \overline{x_{3}a^{−1}}⟩, so E_{00} ≠ E_{0} ∩ 𝕋_{2} contrary to the claim on p. 290. For the group of type 2 group, the statement about the centralizer types of the rank 2 subgroups of E_{0} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{2}}, \overline{y_{2}}⟩ is wrong since not every such subgroup has a centralizer of type T_{4}D_{4}. For example the centralizer type of the group ⟨\overline{x_{1}}, \overline{x_{2}}⟩ is seen to be A_{2}^{4} (this also follows by the information given about the rank 2 subgroups in the type 1 case).

In Proposition (2.13)(vi), p. 265, the condition “H is semisimple” should be corrected to “H is simply connected”. The numbering in the proof is not correct, there is never a reference to (iv), but there are two references to (vii). Moreover the references [BoLAG] and [St2] does not occur in the bibliography.

On the bottom of p. 265 and in Table VI, p. 266, the centralizer type of an element in the class 3E[2E_{7}(ℂ)] should be D_{6}T_{1}.

In Table VI, p. 266, the eigenvalue multiplicities for the classes in 2E_{7}(𝕂) should be as follows. For 3C it should be 43, 45, 45, for 3E it should be 67, 33, 33, for 4A it should be 63, 0, 70, 0, for 4B it should be 39, 32, 30, 32 and for 4I it should be 39, 32, 30, 32.

On p. 268, line 10, 3E(𝕂) should be corrected to 3E_{6}(𝕂), and on line 12 from the bottom the word “subgroups” should be replaced by “2-subgroups”.

The statement of Theorem (11.14), p. 294, is somewhat misleading since the groups occurring in part (ii) are not maximal.

On p. 295, line 17, Σ_{3} should be replaced by the dihedral group Dih_{12}. On line 16 from the bottom, the minimum weight of the tetracode should be 3, and on line 5 from the bottom the equation should read 3C ∩ E = Z(E)v ∪ Z(E)v^{−1}.

In remark (13.2), p. 298 the Weyl group of F_{3} should be SL_{3}(𝔽_{3}) and not GL_{3}(𝔽_{3}).

Finally, in remark (A2.6), p. 303, the reference [Gr7] does not occur in the bibliography.

Bibliography

[1] Kasper K. S. Andersen, Jesper Grodal, Jesper M. Møller, and Antonio Viruel. The classification of p-compact groups for odd p. In preparation.

[2] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).

[3] N. Bourbaki. Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines. Hermann, Paris, 1968. Actualités Scientifiques et Industrielles, No. 1337.

[4] N. Bourbaki. Éléments de mathématique. Hermann, Paris, 1975. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364.

[5] Carles Broto and Antonio Viruel. Projective unitary groups are totally Ndetermined p-compact groups. Preprint.

[6] Roger W. Carter. Simple groups of Lie type. John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28.

[7] Arjeh M. Cohen and Robert L. Griess, Jr. On finite simple subgroups of the complex Lie group of type E_{8}. In The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), pages 367–405. Amer. Math. Soc., Providence, RI, 1987.

[8] Arjeh M. Cohen and David B. Wales. Finite subgroups of F_{4}(ℂ) and E_{6}(ℂ). Proc. London Math. Soc. (3), 74(1):105–150, 1997.

[9] Leonard Eugene Dickson. Linear groups: With an exposition of the Galois field theory. Dover Publications Inc., New York, 1958. with an introduction by W. Magnus.

[10] W. G. Dwyer. Lie groups and p-compact groups. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, pages 433–442 (electronic), 1998.

[11] Hans Freudenthal. Beziehungen der 𝔈_{7} und 𝔈_{8} zur Oktavenebene. VIII. Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math., 21:447–465, 1959.

[7] Daniel Gorenstein. Finite groups. Harper & Row Publishers, New York, 1968.

[13] Robert L. Griess, Jr. Elementary abelian p-subgroups of algebraic groups. Geom. Dedicata, 39(3):253–305, 1991.

[14] J. E. Humphreys. Reflection groups and Coxeter Groups, volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1992.

[15] James E. Humphreys. Linear algebraic groups. Springer-Verlag, New York, 1975. Graduate Texts in Mathematics, No. 21.

[16] Martin W. Liebeck and Gary M. Seitz. On finite subgroups of exceptional algebraic groups. J. Reine Angew. Math., 515:25–72, 1999.

[17] T. A. Springer. Linear algebraic groups. Birkhäuser Boston Inc., Boston, MA, second edition, 1998.

[18] T. A. Springer and R. Steinberg. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), pages 167–266. Springer, Berlin, 1970. Lecture Notes in Mathematics, Vol. 131.

Part II

Fusion, Solvability and Swan’s theorem Fusion, Solvability and Swan’s theorem

The purpose of this note is to discuss the relations between various notions of solvability of finite groups and control of fusion. More precisely, we prove the following result.

(2.0.1) Theorem. Let G be a finite group, p a prime number and P a Sylow p-subgroup of G.

(1) If G is p-supersolvable then N_{G}(P) controls p-fusion strongly in G.

(2) If G is p-solvable and P controls p-fusion weakly in G then G is p-supersolvable.

(3) If N_{G}(P) controls p-fusion weakly in G then N_{G}(P) controls p-fusion strongly in G.

We also give a counterexample to the original version of Swan’s theorem [15; 16] on the cohomology of p-normal groups. The results of this note represent work in progress, and we hope to return later to some of the questions that we leave open. It should also be mentioned that work of Stammbach [14] gives cohomological characterizations of solvable and supersolvable groups and it would be interesting to find relations between this and the present work.

I would like to thank Antonio Viruel for showing me another way to prove part (1). I am also happy to thank Jesper Grodal for pointing out the reference [5], which led me to prove part (3), and for many stimulating conversations. In particular, Theorem (2.1.22) (which is the key to proving part (2)) is really joint work.

Finally it should be added that the theorem is inspired by numerous calculations using the computer algebra system MAGMA [2].

2.1 Solvability and fusion

In the following all groups are finite and p denotes a prime. We start by recalling the definitions of p-nilpotence, p-solvability, p-supersolvability and control of p-fusion and p-transfer.

(2.1.1) Definition. A series of G is a series of subgroups 1 = G_{0} ⊴ G_{1} ⊴ … ⊴ G_{s} = G. The quotients G_{i}/G_{i−1}, 1 ≤ i ≤ s, are called the factors of the series. The series is called proper if G_{i−1} ⊲ G_{i} for 1 ≤ i ≤ s, i.e. if the factors all are nontrivial. A factor G_{i}/G_{i−1}, 1 ≤ i ≤ s, is called central if G_{i−1} ⊴G and G_{i}/G_{i−1} ≤ Z(G/G_{i−1}). A series without proper refinements is called a composition series. Equivalently the factors should be simple groups. The factors are called the composition factors. A series 1 = G_{0} ⊴ G_{1} ⊴ … ⊴ G_{s} = G satisfying G_{i} ⊴ G, 0 ≤ i ≤ s, is called a normal series of G. A normal series having no normal series as a proper refinement, is called a chief series. The factors are called the chief factors.

(2.1.2) Remark. The chief factors are characteristically simple groups (see [12, 7.38]). By [7, Theorem 2.1.4], [8, Satz I.9.12] or [12, Theorem 8.10] a group is characteristically simple if and only if it is a direct product H^{r} of finitely many copies of a single simple group H.

(2.1.3) Definition. A group G is called p-nilpotent if G has a normal p-complement, i.e. a normal subgroup K such that KP = G and K ∩ P = 1 for a Sylow p-subgroup P of G. Equivalently every chief factor of G of order divisible by p should be central ([8, Satz IV.4.4] or [12, Exercise 563]). A group is called p-supersolvable if its chief factors are either cyclic of order p or p′-groups. Finally a group G is called p-solvable if G has a series were all factors are either p-groups or p′-groups. Equivalently the composition factors of G are either p-groups or p′-groups. This is again equivalent to demanding that the chief factors of G are either p-groups or p′-groups.

(2.1.4) Remark. A group is nilpotent if and only if it is p-nilpotent for all primes p. Similarly G is solvable if and only if G is p-solvable for all primes p, and G is supersolvable if and only if G is p-supersolvable for all primes p. We also have the implications

G p-nilpotent ⟹ G p-supersolvable ⟹ G p-solvable.

These implications can not be reversed. The group Σ_{3} is supersolvable but not 3-nilpotent, and the group A_{4} is solvable but not 2-supersolvable.

We next turn to the definitions of control of p-fusion in a group.

(2.1.5) Definition (Fusion). A subgroup H ≤ G is said to control p-fusion weakly in G if H contains a Sylow p-subgroup of G and whenever Q, Q^{g} ≤ H for some p-subgroup Q and some g ∈ G we may write g = nh with n ∈ N_{G}(Q) and h ∈ H. The subgroup H ≤ G is said to control p-fusion strongly in G if H contains a Sylow p-subgroup of G and whenever Q, Q^{g} ≤ H for some p-subgroup Q and some g ∈ G we may write g = ch with n ∈ C_{G}(Q) and h ∈ H.

(2.1.6) Remark. Obviously, if H controls p-fusion strongly in G, then H controls p-fusion weakly in G. Furthermore if H ≤ K ≤ G and H controls p-fusion weakly or strongly respectively in G, then K also controls p-fusion weakly or strongly respectively in G.

Our interest in control of fusion stems from the following result which relates it to group cohomology. Here, for an abelian group M, we denote by M_{(p)} its p-primary submodule, i.e. the submodule consisting of the elements of p-power order.

(2.1.7) Theorem. Let G be a finite group, H a subgroup of G and p a prime number. Then the following conditions are equivalent.

(1) H controls p-fusion strongly in G.

(2) The restriction map H^{∗}(G, A)_{(p)} ⟶ H^{∗}(H, A)_{(p)} is an isomorphism for any trivial ℤ[G]-module A.

(3) The restriction map H^{∗}(G, 𝔽_{p}) ⟶ H^{∗}(H, 𝔽_{p}) is an isomorphism.

(2.1.8) Proof: The implication (1) ⟹ (2) is well known ([10, Proposition 2.1], [9, Theorem 12.8]) and goes back to [15]. The implication (2) ⟹ (3) is obvious and (3) ⟹ (1) is due to Mislin [11]. □

Denote by O^{p}(G) the smallest normal subgroup K of G, such that G/K is a p-group. The following result is due to Tate.

(2.1.9) Theorem (Transfer). Let H be a subgroup of G containing a Sylow p-subgroup P of G. Then the following conditions are equivalent:

(1) G/O^{p}(G) ≅ H/O^{p}(H)

(2) |G/O^{p}(G)| = |H/O^{p}(H)|

(3) H ∩ O^{p}(G) = O^{p}(H)

(4) G/(O^{p}(G)G′) ≅ H/(O^{p}(H)H′)

(5) |G/(O^{p}(G)G′)| = |H/(O^{p}(H)H′)|

(6) H ∩ O^{p}(G)G′ = O^{p}(H)H′

(7) P ∩ G′ = P ∩ H′

(8) P G′/G′ ≅ P H′/H′ (2.1.10) Proof: See [9, Lemma X.6.15 and Theorem X.6.18]. □

(2.1.11) Remark. In case the conditions are satisfied we say that H controls p-transfer in G. By condition (7) we see that if H ≤ K ≤ G and H controls p-transfer in G then K also controls p-transfer in G. Note also that by the Focal Subgroup Theorem ([9, X.6.2] and [12, 10.34]) the subgroup P ∩ G′ occurring in (7) is the focal subgroup of P in G.

(2.1.12) Proposition. Let H be a subgroup of G containing a Sylow p-subgroup P of G. If H controls p-fusion strongly in G then H controls p-transfer in G.

(2.1.13) Proof: See [9, Lemma X.9.2]. A cohomological proof runs as follows. Since H controls p-fusion strongly in G we have H^{2}(G, ℤ)_{(p)} ≅ H^{2}(H, ℤ)_{(p)}. However for any group K we have H^{2}(K, ℤ)_{(p)} = H_{1}(K, ℤ)_{(p)} ≅ K/(O^{p}(K)K′) and we conclude that condition (4) is satisfied. □

(2.1.14) Definition. Let G be a finite group, p a prime number and P a Sylow p-subgroup of G. Then a p-subgroup Q of P is said to be weakly closed in P with respect to G if Q^{g} ≤ P implies Q^{g} = Q, i.e. if Q is the only conjugate of Q contained in P. The group G is called p-normal if Z(P) is weakly closed in P with respect to G.

(2.1.15) Theorem. Let G be a finite group, p a prime number and P a Sylow p-subgroup of G. Then N_{G}(P) controls p-fusion strongly in G if and only if P has a central series consisting of subgroups which are weakly closed in P with respect to G.

(2.1.16) Proof: This is a special case of the main result in [5] translated to our notation (the definition of strong control of fusion in [5] differs slightly from ours). □

(2.1.17) Remark. This theorem generalizes work of Thevénaz [19], whose main theorem states that if P has a strongly characteristic central series 1 = P_{0}⊴P_{1}⊴…⊴P_{s} = P, then N_{G}(P) controls p-fusion strongly in G. Actually the original inductive proof in [19] fails, but if one in addition assumes that P_{i}/P_{j} is strongly characteristic in P/P_{j} for 1 ≤ j ≤ i ≤ s, then the inductive argument does go through [18]. However, as already mentioned, due to Theorem (2.1.15) one does not need this stronger hypothesis.

(2.1.18) Lemma. Let G be a finite group, p a prime number and P a Sylow p-subgroup of G. If G is p-nilpotent then P controls p-fusion strongly in G.

(2.1.19) Proof: We first give a group theoretical proof, similar to [8, proof of Satz IV.4.9]. Let K ⊴ G be a normal p-complement of P, and suppose Q, Q^{g} ≤ P. We can then write g = kp where k ∈ K and p ∈ P. For x ∈ Q we have x^{k} = (x^{g})^{p^{−1}} ≤ (Q^{g})^{p^{−1}} ≤ P^{p^{−1}} = P so x^{−1}x^{k} ∈ P. On the other hand we also have x^{−1}x^{k} = (x^{−1}k^{−1}x)k ∈ K since K ⊴ G. So x^{−1}x^{k} ∈ P ∩ K = 1 for any x ∈ Q, which proves k ∈ C_{G}(Q). Since g = kp we get that P controls p-fusion strongly in G. A cohomological proof runs as follows. Let ι : P → G be the inclusion and let π : G → G/K ≅ P be the projection. Consider the diagram

\xymatrix{H^{∗}(P, 𝔽_{p}) \ar[r]^{π^{∗}} & H^{∗}(G, 𝔽_{p}) \ar[r]^{ι^{∗}} & H^{∗}(P, 𝔽_{p}) \ar[r]^{^{corG P}} & H^{∗}(G, 𝔽_{p})}

where cor_{P}^{G} is the corestriction. Since the composition of the first two is the identity and the composition of the last two is multiplication by [G : P] which is prime to p, we see that ι^{∗} is an isomorphism. Thus P controls p-fusion strongly in G by Theorem (2.1.7). □

(2.1.20) Lemma. Let G be a finite group, p a prime number and P a Sylow p-subgroup of G. If G has a normal p-nilpotent subgroup of index prime to p, then N_{G}(P) controls p-fusion strongly in G.

(2.1.21) Proof: Let K ⊴ G be a p-nilpotent subgroup of index prime to p. Then P ≤ K and by the Frattini argument [8, Satz I.7.8] we have G = KN_{G}(P). Assume now that Q, Q^{g} ≤ P, and write g = kn with k ∈ K and n ∈ N_{G}(P). Then Q^{k} = Q^{gn^{−1}} ≤ P^{n^{−1}} = P. By Lemma (2.1.18) P controls p-fusion strongly in K, so we may write k = cp with c ∈ C_{K}(Q) and p ∈ P. Thus g = cpn with c ∈ C_{K}(Q) and pn ∈ N_{G}(P) proving that N_{G}(P) controls p-fusion strongly in G. A cohomological proof runs as follows. As before we have P ≤ K and G = KN_{G}(P) by the Frattini argument. From the Lyndon-Hochschild-Serre spectral sequence [4] we get that the restriction H^{∗}(G, 𝔽_{p}) ⟶ H^{∗}(K, 𝔽_{p})^{N_{G}(P)} is an isomorphism since K has index prime to p. By Lemma (2.1.18) and Theorem (2.1.7) the restriction H^{∗}(K, 𝔽_{p}) ⟶ H^{∗}(P, 𝔽_{p}) is also an isomorphism. Finally H^{∗}(N_{G}(P), 𝔽_{p}) ⟶ H^{∗}(P, 𝔽_{p})^{N_{G}(P)} is an isomorphism, so the commutative diagram

\xymatrix{H^{∗}(G, 𝔽_{p}) \ar[r]^{≅} \ar[d] & H^{∗}(K, 𝔽_{p})^{N_{G}(P)} \ar[d]^{≅} \\ H^{∗}(N_{G}(P), 𝔽_{p}) \ar[r]^{≅} & H^{∗}(P, 𝔽_{p})^{N_{G}(P)}}

shows that H^{∗}(G, 𝔽_{p}) ⟶ H^{∗}(N_{G}(P), 𝔽_{p}) is an isomorphism, and we conclude that N_{G}(P) controls p-fusion strongly in G by Theorem (2.1.7). □

(2.1.22) Theorem. Let G be a finite group, p a prime number and K⊴G a normal subgroup. Assume that H ≤ G controls p-fusion weakly or strongly respectively in G. If K is p-solvable, then HK/K controls p-fusion weakly or strongly respectively in G/K.

(2.1.23) Proof: We start by considering the case where K is either a p-group or a p′-group. Note that we may assume K ≤ H. Let P be a Sylow p-subgroup of G contained in H. Then P K/K is a Sylow p-subgroup of G/K contained in H/K. Let R, R\overline{g} ≤ P K/K be conjugate in G/K. We may then find Q_{1}, Q_{2} ≤ P such that R = Q_{1}K/K and R\overline{g} = Q_{2}K/K, i.e. we have Q_{1}^{g}K = Q_{2}K.

If K is a p′-group, then both Q_{1}^{g} and Q_{2} are Sylow p-subgroups of Q_{1}^{g}K = Q_{2}K. We can thus find k ∈ K, such that (Q_{1}^{g})^{k} = Q_{2}. In the case where K is a p-group we have K ≤ P since K is normal. We thus conclude that Q_{1}^{g} ≤ Q_{1}^{g}K = Q_{2}K ≤ P.

Thus setting k = 1 ∈ K in the last case, we see that we have Q_{1}, Q_{1}^{gk} ≤ P in both cases. We may then write gk = xh where h ∈ H and x ∈ N_{G}(Q_{1}) if H controls p-fusion weakly in G or x ∈ C_{G}(Q_{1}) if H controls p-fusion strongly in G. As R = Q_{1}K/K we see that \overline{x} ∈ N_{G/K}(R) or \overline{x} ∈ C_{G/K}(R) respectively. Since \overline{g} = \overline{x}\overline{h} we conclude that H/K controls p-fusion weakly or strongly respectively in G/K.

The special case where K is a p′-group and we want to prove that H/K controls p-fusion strongly in G/K may also be handled directly by cohomological means. Consider the diagram

\xymatrix{H^{∗}(G, 𝔽_{p}) \ar[r]^{≅} & H^{∗}(H, 𝔽_{p}) \\ H^{∗}(G/K, 𝔽_{p}) \ar[r] \ar[u]^{≅} & H^{∗}(H/K, 𝔽_{p}) \ar[u]_{≅}}

where the vertical maps are isomorphisms by the Lyndon-Hochschild-Serre spectral sequence since K is a p′-group. Thus the restriction H^{∗}(G/K, 𝔽_{p}) ⟶ H^{∗}(H/K, 𝔽_{p}) is an isomorphism and by Theorem (2.1.7) we conclude that H/K controls p-fusion strongly in G/K.

We now consider the general case, i.e. the case where K is p-solvable. Then the upper p-series of K,

1 = K_{0} ⊴ K_{1} = O_{p}(K) ⊴ K_{2} = O_{p,p′}(K) ⊴ … ⊴ K_{n} = K

terminates by [7, Theorem 6.3.1]. By construction the subgroups K_{i} are characteristic in K, and since K is normal in G we have K_{i} ⊴ G. Since the factors K_{i}/K_{i−1} are either p-groups or p′-groups, the general case follows by induction from the special case considered above. □

(2.1.24) Remark. We do not know if the assumption that K is p-solvable can be removed. Assume that G is a minimal counterexample, i.e. that the theorem does not hold for the group G and that the order of G is chosen minimal with respect to this property. Choose K ⊴ G and H ≤ G such that H controls p-fusion weakly or strongly respectively in G, but HK/K does not control p-fusion weakly or strongly respectively in G/K. Now the Fitting subgroup F (K) is a nilpotent characteristic subgroup of K [8, §III.4]. In particular F (K) is p-solvable, so since G is a minimal counterexample we conclude that F (K) = 1 by the theorem.

Now consider the generalized Fitting subgroup F^{∗}(K), cf. [9, §X.13]. Then F^{∗}(K) is quasinilpotent by [9, Theorem 13.10] and thus Z_{∞}(K) = F (K) = 1 by [9, Corollary 13.7]. By [9, Theorem 13.6] we then conclude that F^{∗}(K) is semisimple, i.e. the direct product of nonabelian simple groups. Writing F^{∗}(K) = S_{1}^{e_{1}} × … × S_{n}^{e_{n}} where the groups S_{i} are isomorphic nonabelian simple groups, we see that the subgroup S_{1}^{e_{1}} is characteristic in F^{∗}(K). Since F^{∗}(K) is characteristic in K and K ⊴ G we conclude that S_{1}^{e_{1}} ⊴ G. Thus to prove the theorem without the assumption of psolvability, it suffices to consider the case where K is a direct product of isomorphic nonabelian simple groups. Our main result Theorem (2.0.1) is contained in the following.

(2.1.25) Theorem. Let G be a finite group, p a prime and P a Sylow p-subgroup of G. Consider the following properties:

(1) G is p-nilpotent.

(2) P controls p-fusion strongly in G.

(3) P controls p-fusion weakly in G.

(4) N_{G}(P) controls p-fusion strongly in G.

(5) N_{G}(P) controls p-fusion weakly in G.

(6) G is p-normal.

(7) N_{G}(Z(P)) controls p-fusion strongly in G.

(8) N_{G}(Z(P)) controls p-fusion weakly in G.

(9) G is p-supersolvable.

(10) G is p-solvable.

(11) P controls p-transfer in G.

(12) N_{G}(P) controls p-transfer in G.

(13) N_{G}(Z(P)) controls p-transfer in G.

Then we have the following implications:

\xymatrix{(1) ⇔ (2) ⇔ (11) \ar@{=>}[r] \ar@{=>}[d] & (3) \ar@{=>}[d] \\ (9) \ar@{=>}[r] \ar@{=>}[d] & (4) ⇔ (5) \ar@{=>}[r] \ar@{=>}[d] & (12) \ar@{=>}[d] \\ (10) & (6) ⇔ (7) ⇔ (8) \ar@{=>}[r] & (13) \\ (3) + (10) \ar@{=>}[r] & (9)}

(2.1.26) Proof: Most of the implications are trivial. By Remark (2.1.6) we get (2) ⟹ (3), (4) ⟹ (5) and (7) ⟹ (8). Since Z(P) is a characteristic subgroup of P, we get P ≤ N_{G}(P) ≤ N_{G}(Z(P)). Thus by Remark (2.1.6) we also have (3) ⟹ (5) ⟹ (8) and similarly (12) ⟹ (13) by Remark (2.1.11).

Proposition (2.1.12) shows (2) ⟹ (11), (4) ⟹ (12) and (7) ⟹ (13). The implications (1) ⟹ (9) ⟹ (10) are obvious from the definitions, cf. Remark (2.1.4).

The implication (6) ⟹ (7) is proved in [9, proof of corollary X.12.9] and the implication (8) ⟹ (6) follows directly from the definitions. Namely, assume that (8) is satisfied, and let g ∈ G with Z(P)^{g} ≤ P. We may then write g = nh with n, h ∈ N_{G}(Z(P)) so we get g ∈ N_{G}(Z(P)) and thus Z(P)^{g} = Z(P) as desired.

Assume that P controls p-transfer in G. By Theorem (2.1.9) we have |G/O^{p}(G)| = |P |. In particular P has a normal p-complement O^{p}(G), so G is p-nilpotent. This argument, which is due to Tate [17], proves (11) ⟹ (1). The implication (1) ⟹ (2) follows from Lemma (2.1.18).

We now prove have (9) ⟹ (4). Assume that G is p-supersolvable and let G_{i}/G_{i−1} be a chief factor of G of order divisible by p. By definition, G_{i}/G_{i−1} is then cyclic of order p, and Aut(G_{i}/G_{i−1}) is cyclic of order p − 1. Since it is abelian of order prime to p, both G′ and P acts trivially on G_{i}/G_{i−1}. Since this holds for all chief factors, we see that P G′ is p-nilpotent by [8, Satz VI.5.4(b)]. Thus P G′ is a normal p-nilpotent subgroup of G of index prime to p, and N_{G}(P) controls p-fusion strongly in G by Lemma (2.1.20).

To prove (5) ⟹ (4), we note that by Theorem (2.1.15) P has a central series consisting of subgroups which are weakly closed in P with respect to N_{G}(P), since N_{G}(P) obviously controls p-fusion strongly in N_{G}(P). Now assume that Q ≤ P is any subgroup which is weakly closed in P with respect to N_{G}(P), and that Q^{g} ≤ P for some g ∈ G. By assumption we may write g = nh, with n ∈ N_{G}(Q) and h ∈ N_{G}(P). Thus Q^{h} = Q^{g} ≤ P. Since Q is weakly closed in P with respect to N_{G}(P) we then get Q^{h} = Q. Thus Q^{g} = Q^{h} = Q and we conclude that Q is weakly closed in P with respect to G. Thus the central series of P from above actually consists of subgroups which are weakly closed in P with respect to G, so N_{G}(P) controls p-fusion strongly in G by Theorem (2.1.15).

Finally, we prove that (3) + (10) ⟹ (9). Let H/K be a chief factor of G of order divisible by p. Then H/K is by assumption a p-group and we have to prove that |H/K| = p. Assume to the contrary that |H/K| > p. Note that H/K ⊴P K/K, and since P K/K is a p-group we have H/K ∩ Z(P K/K) ≠ 1 by [8, Satz I.6.9]. Thus we may find R such that R ⊴ P K/K and 1 < R < H/K. Since P controls p-fusion weakly in G it follows from Theorem (2.1.22) that P K/K controls p-fusion weakly in G/K. Now let \overline{g} ∈ G/K. Then we have R\overline{g} < (H/K)\overline{g} = H/K ≤ P K/K. Thus we may write \overline{g} = n̄p̄ with \overline{n} ∈ N_{G/K}(R) and \overline{p} ∈ P K/K. Thus since R ⊴ P K/K we get R\overline{g} = R^{n̄p̄} = R\overline{p} = R. Thus R ⊴ G/K, but since 1 < R < H/K this contradicts the fact that H/K is a chief factor of G. We conclude that |H/K| = p and we are done. □

(2.1.27) Remark. Let P be a p-group and let G be a finite group such that P is a Sylow p-subgroup of G. If the restriction H^{∗}(G, 𝔽_{p}) ⟶ H^{∗}(P, 𝔽_{p})^{N_{G}(P)} is an isomorphism for any such group G, then P is called a Swan group. By Theorem (2.1.7) this condition is equivalent to N_{G}(P) controlling p-fusion strongly in G for any such group G. It is well known that if P is abelian then P is a Swan group [15, Lemma 1]. This follows also from Theorem (2.1.25) since in this case G is obviously p-normal. Since Z(P) = P we then obtain condition (4).

If P is cyclic we obtain directly from the definition that P controls p-fusion weakly in G, since P then contains at most one subgroup of a given order.

Finally, if P is a T.I. set (trivial intersection set), i.e. if P^{g} = P or P^{g} ∩ P = 1 for all g ∈ G, then we also get directly that N_{G}(P) controls p-fusion strongly in G. In fact, if P is a T.I. set, then we have an even stronger cohomological result, namely that the restriction map H^{∗}(G, A)_{(p)} ⟶ H^{∗}(H, A)_{(p)} is an isomorphism for any ℤ[G]-module A [1, Corollary 3.6.19].

The following table gives all possible combinations of truth values allowed by Theorem (2.1.25)). For each case we also list a group with given properties. In the examples Σ̃_{5} denotes either of the two nonisomorphic double covers of Σ_{5} ([13]) and G_{576} denotes the solvable group of order 576 with the presentation:

G_{576} = ⟨x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}|x_{1}^{2} = x_{4}^{2} = x_{5}^{2} = x_{6}^{2} = x_{7}^{2} = x_{8}, x_{2}^{3} = x_{3}^{3} = 1, x_{8}^{2} = 1, x_{3}^{x_{1}} = x_{3}^{2}, x_{4}^{x_{1}} = x_{6}x_{8}, x_{7}^{x_{2}} = x_{7}^{x_{3}} = x_{6}, x_{4}^{x_{2}} = x_{5}^{x_{3}} = x_{4}x_{5}x_{8}, x_{4}^{x_{3}} = x_{5}, x_{5}^{x_{2}} = x_{4}, x_{5}^{x_{4}} = x_{7}^{x_{1}} = x_{5}x_{8}, x_{5}^{x_{1}} = x_{7}^{x_{6}} = x_{7}x_{8}, x_{6}^{x_{1}} = x_{4}x_{8}, x_{6}^{x_{2}} = x_{6}^{x_{3}} = x_{6}x_{7},

[x_{1}, x_{2}] = [x_{2}, x_{3}] = [x_{4}, x_{6}] = [x_{5}, x_{6}] = [x_{4}, x_{7}] = [x_{5}, x_{7}] = [x_{1}, x_{8}] =

[x_{2}, x_{8}] = [x_{3}, x_{8}] = [x_{4}, x_{8}] = [x_{5}, x_{8}] = [x_{6}, x_{8}] = [x_{7}, x_{8}] = 1⟩ .

This group is an extension of C_{3} ≀ C_{2} by the extraspecial 2-group 2_{+}^{1+4} generated by x_{4}, x_{5}, x_{6}, x_{7} and x_{8}. The center of G_{576} is the cyclic group of order 2 generated by x_{8}, and we let G_{288} = G_{576}/Z(G_{576}).

The group ASL denotes the affine special linear group, which also occur is [6] where it is called the quadratic group Qd.

(1), (2), (11) (3) (4), (5) (6), (7), (8) (9) (10) (12) (13) Groups
+  +  +  +  + +  +  + G abelian
–  +  +  +  + +  +  + Σ_{3}, p = 3
–  +  +  +  –  –  +  + A_{5}, p = 3, 5
–  –  +  +  + +  +  + C_{3} ≀ C_{2}, p = 3
–  –  +  +  –  +  +  + A_{4}, p = 2
–  –  +  +  –  –  +  + A_{5}, p = 2
–  –  –  +  –  +  +  + G_{576}, p = 2
–  –  –  +  –  +  –  + GL_{2}(𝔽_{3}), p = 2
–  –  –  +  –  –  +  + A_{5} × G_{576}, p = 2
–  –  –  +  –  –  –  + Σ̃_{5}, p = 2
–  –  –  –  –  +  +  + G_{288}, p = 2
ASL_{2}(𝔽_{3}), p = 3
–  –  –  –  –  +  –  + GL_{2}(𝔽_{3}) × G_{288}, p = 2
–  –  –  –  –  +  –  – Σ_{4}, p = 2
–  –  –  –  –  –  +  + A_{5} × ASL_{2}(𝔽_{3}), p = 3
–  –  –  –  –  –  –  + Σ̃_{5} × G_{288}, p = 2
–  –  –  –  –  –  –  – Σ_{5}, p = 2

(2.1.28) Remark. A finite group G is called a monomial group (or simply an M-group) if every irreducible complex character of G is induced from a linear character of a subgroup of G. Then every supersolvable group is an M-group and every M-group is solvable [8, Satz V.18.5(a) and Satz V.18.6(b)]. However, there seems to be no good notion of a p-monomial group. The obvious definition would require that every irreducible complex character of G of degree divisible by p is induced from a linear character of a subgroup of G. With this definition G is an M-group if and only if G is p-monomial for all p, but it is not true that any p-monomial group is p-solvable. To see this consider the group A_{5}, whose irreducible characters have degrees 1, 3, 3, 4 and 5 [3, §14D]. Since the character of degree 5 is induced from any of the two nontrivial linear characters of the subgroup A_{4}, we see that A_{5} is 5-monomial. However it is not 5-solvable since A_{5} is simple.

We end this section by mentioning a question inspired by Theorem (2.1.25) and the table above.

(2.1.29) Question. Let G be a finite group, p a prime number, P a Sylow p-subgroup and H a subgroup containing N_{G}(P). Is it then true that H controls p-fusion strongly in G if H controls p-fusion weakly in G ? By Theorem (2.1.25) this holds for N_{G}(P) and N_{G}(Z(P)). We have checked that this is true for all group of order less than 384.

2.2 Swan’s theorem

In this section we give a counterexample to the original version of Swan’s theorem on the cohomology of p-normal group [15]. The theorem was later corrected [16] due to a counterexample of Evens, however the example was never published. The corrected version of the theorem is as follows:

(2.2.1) Theorem. Let G be p-normal and let P be a Sylow p-subgroup of G. If A is a trivial ℤ[G]-module, then the restriction map

H^{∗}(G, A)_{(p)} → H^{∗}(N_{G}(Z(P)), A)_{(p)}

is an isomorphism.

(2.2.2) Proof: Since G is p-normal, we know by Theorem (2.1.25) that N_{G}(Z(P)) controls p-fusion strongly in G. Thus the theorem follows by Theorem (2.1.7). □

(2.2.3) Example. We now give an example showing that the original version of Swan’s theorem which left out the condition of trivial coefficients, does not hold. Let p = 2 and

G = D_{12} = ⟨σ, τ|σ^{6} = τ^{2} = 1, σ^{τ} = σ^{−1}⟩

be the dihedral group of order 12. Then P = ⟨σ^{3}, τ⟩ is a Sylow 2-subgroup, and since P is abelian, G is 2-normal. Moreover N_{G}(Z(P)) = P. Let C_{2} =< σ^{3} >, C_{3} =< σ^{2} > and C_{6} =< σ > be the cyclic subgroups of the subgroup of rotations in G. Let M be an arbitrary ℤ[G]-module. By [20, 7.3, p. 158] we then have

H^{∗}(G, M)_{(2)} = H^{∗}(P, M)_{(2)} + \frac{1}{2} [H^{∗}(C_{6}, M)_{(2)} − H^{∗}(C_{2}, M)_{(2)}] .

The first term is the one predicted by Swan’s theorem. Therefore the original version of Swan’s theorem does not hold if we can construct M with H^{∗}(C_{2}, M)_{(2)} ≅ H^{∗}(C_{6}, M)_{(2)}.

Now C_{2} is normal in G and G/C_{2} ≅ Σ_{3} ≅ GL(2, 𝔽_{2}) acts naturally on M = 𝔽_{2}^{2}. Using this, M gets the structure of a ℤ[G]-module. The action of σ and τ is up to conjugation given by

σ ↦ ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ) , τ ↦ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

Using the Lyndon-Hochschild-Serre spectral sequence we get the isomorphism

H^{∗}(C_{6}, M) ≅ H^{∗}(C_{6}/C_{3}, M^{C_{3}}).

A direct computation shows that M^{C_{3}} = 0, so H^{∗}(C_{6}, M) = 0. However the action of C_{2} on M is trivial, so H^{n}(C_{2}, M) = 𝔽_{2}^{2} for all n.

(2.2.4) Remark. As remarked above, by [1, Corollary 3.6.19] the restriction map

H^{∗}(G, A)_{(p)} → H^{∗}(N_{G}(P), A)_{(p)}

is an isomorphism for any ℤ[G]-module A if the Sylow p-subgroup P is a T.I. set. In particular this happens if |P | = p, and thus we see that the order of a counterexample has to be at least 2^{2} · 3 = 12.

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Part III

On the Poincaré series of cohomology rings of finite groups On the Poincaré series of cohomology rings of finite groups

Let G be a finite group and let p be a prime. By the Venkov-Evens theorem [13; 29], the cohomology ring H^{∗}(G, 𝔽_{p}) is a finitely generated 𝔽_{p}-algebra. If the degrees of the generators are n_{1}, …, n_{s} then by the Hilbert-Serre theorem ([2, Theorem 2.1.1] or [26, Theorem 4.1.1]), the Poincaré series P_{G}(t) = ∑_{n=0}^{∞} dim_{𝔽_{p}} H^{n}(G, 𝔽_{p})t^{n} is a rational function of the form

P_{G}(t) = \frac{f(t)}{∏_{i=1}^{s}(1 − t^{n_{i}})}  (i)

where f(t) is a polynomial in t with integer coefficients. The purpose of this paper is to discuss some of the properties of P_{G}(t) and their relations to properties of the cohomology ring H^{∗}(G, 𝔽_{p}). Along the way we correct a formula for the Poincaré series of wreath products given by Webb [31].

3.1 Zeroes and poles of the Poincaré series

The series P_{G}(t) reflects some of the properties of the cohomology ring H^{∗}(G, 𝔽_{p}). For example the order of the pole of P_{G}(t) at t = 1 equals the Krull dimension of H^{∗}(G, 𝔽_{p}) by [2, Theorem 2.2.7] or [26, Theorem 5.3.3]. By Quillen’s theorem [23] this also equals the p-rank of G (i.e. the rank of the largest elementary abelian p-subgroup of G).

Concerning the other poles of P_{G}(t) we have less information. Of course by (i) they must be roots of unity as well. In [24, p. 367] it is claimed that if G is a 2-group and p = 2 then all the poles are |G|’th roots of unity, but no proof is given. For odd primes p, this result is not true. Consider the semidirect product of cyclic groups of order p^{α} and p^{β}:

G = ⟨x, y|x^{p^{α}} = y^{p^{β}} = 1, x^{−1}yx = y^{1+u}⟩ ,  (ii)

where 1 + u satisfies (1 + u)^{p^{α}} − 1 = m · p^{β} for some m ∈ ℤ. Then if α, β ≥ 1 and p ∤ m we have

P_{G}(t) = \frac{1 + t}{(1 − t)(1 − t^{2p})}  (iii)

by [17]. Note that the condition is satisfied if α = 1, β = 2 and u = p, in which case G = p_{−}^{1+2}, the extraspecial group of order p^{3} and exponent p^{2}. Unfortunately no details of the computation are given in [17], however their result may be verified by using [30] ^{1} which gives the additive structure of the integral cohomology ring for any split metacyclic group. We will return to this example again later.

Lacking evidence to the contrary, we conjecture that for p = 2 the poles of P_{G}(t) are all |G|’th roots of unity and that for p odd they are 2|G|’th roots of unity. By the Hilbert-Serre theorem such a result pertains to the degrees of the generators of the cohomology ring H^{∗}(G, 𝔽_{p}) and thus potentially interesting for concrete computations.

The location of the zeros of P_{G}(t) is more mysterious. By the Hilbert-Serre theorem they are algebraic numbers. In [24, p. 367] Rusin gives an example showing that they need not be roots of unity, and asks whether they are always algebraic integers. In Example (3.1.8) below we show that this is not the case. In general we have the following result linking ring theoretic properties of H^{∗}(G, 𝔽_{p}) to properties of P_{G}(t). The commutative algebra we will need may be found in [7] and [19].

(3.1.1) Theorem. Let G be a finite group, p a prime number, P_{G}(t) the Poincaré series of H^{∗}(G, 𝔽_{p}) and d the Krull dimension of H^{∗}(G, 𝔽_{p}). Consider the following conditions:

(1) H^{∗}(G, 𝔽_{p}) is a complete intersection.

(2) H^{∗}(G, 𝔽_{p}) is Gorenstein.

(2′) H^{∗}(G, 𝔽_{p}) is Cohen-Macaulay.

(3) The roots of P_{G}(t) are roots of unity.

(4) P_{G}(t) satisfies P_{G}(1/t) = (−t)^{d}P_{G}(t).

(5) P_{G}(t) satisfies P_{G}(1/t) = (−1)^{d}t^{ρ}P_{G}(t) for some ρ ∈ ℤ.

1The summations in the formulas for f_{2n}d_{1} and k_{m} [30, p. 254] should start at j = 0 instead of j = 1.

(6) The roots of P_{G}(t) are algebraic integers.

We then have the implications:

\xymatrix{(1) \ar@{=>}[r] \ar@{=>}[dr] & (2) ⟺ (2′) \ar@{=>}[r] & (4) \ar@{=>}[d] \\  & (3) \ar@{=>}[r] & (5) \ar@{=>}[r] & (6)}

(3.1.2) Remark. In [15, p. 2079] the properties (4) and (5) are called strongly quasi-Gorenstein and quasi-Gorenstein respectively.

(3.1.3) Proof: The implications (1) ⟹ (2) ⟹ (2′) are well known and are proved in [7, Proposition 3.1.20] for strictly commutative rings, but the proof carries over verbatim to graded commutative rings. The implication (4) ⟹ (5) is obvious, and (2′) ⟹ (4) and (2′) ⟹ (2) are [3, Theorem 1.1]. From [27, Corollary 3.3] we see that if (1) is satisfied we may write P_{G}(t) in the form

P_{G}(t) = \frac{∏_{j=1}^{r}(1 − t^{f_{j}})}{∏_{i=1}^{s}(1 − t^{e_{i}})},  (i)

for some positive integers f_{1}, …, f_{r}, e_{1}, … e_{s} (note that even though [27] is written for strictly commutative rings, the proof immediately carries over to graded commutative rings as well). This immediately implies (3). If the roots of P_{G}(t) are roots of unity we see by comparison with (i) that all roots of f(t) are roots of unity as well. Note that f(0) = 1 since H^{0}(G, 𝔽_{p}) = 𝔽_{p}. Since f(t) has integer coefficients it then follows that f(t) is a product of cyclotomic polynomials. Since these may be written in the form (i) it then follows that P_{G}(t) may also be written in the form (i). Then by direct computation we get that

\frac{P_{G}(1/t)}{P_{G}(t)} = (−1)^{r−s}t^{ρ},

where ρ = ∑_{i=1}^{s} e_{i} − ∑_{j=1}^{r} f_{j}. Since the Krull dimension is the order of the pole of P_{G}(t) at t = 1 we get d = r − s. This proves (3) ⟹ (5). To show that (5) ⟹ (6) we write P_{G}(t) in the form (i). We then have

\frac{P_{G}(1/t)}{P_{G}(t)} = \frac{f(1/t)}{f(t)}(−1)^{s}t^{∑_{i=1}^{s} n_{i}},

Thus if (5) is satisfied, we see that \frac{f(1/t)}{f(t)} must have the form ±t^{r} for some r ∈ ℤ. Now write f(t) = a_{0} + a_{1}t + … + a_{m}t^{m}, where a_{m} ≠ 0. Note that a_{0} = 1 since f(0) = 1. We then get

±t^{r} = \frac{f(1/t)}{f(t)} = \frac{a_{m} + a_{m−1}t + … + a_{0}t^{m}}{a_{0} + a_{1}t + … + a_{m}t^{m}} t^{−m}.

Since a_{0} = 1 and a_{m} ≠ 0 we conclude that a_{m} = ±a_{0} = ±1. In particular all roots of f(t) must be algebraic integers, which proves (6). □

(3.1.4) Remark. The Cohen-Macaulay property is satisfied in many interesting cases. Let P be a Sylow p-subgroup of G. Then if H^{∗}(P, 𝔽_{p}) is Cohen-Macaulay then so is H^{∗}(G, 𝔽_{p}) [3, Proposition 6.8]. This condition is satisfied if P is abelian or more generally if all elements of order p in P are central [12]. If P is an extraspecial 2-group then H^{∗}(P, 𝔽_{2}) is Cohen-Macaulay by [22]. However if p is odd and P is an extraspecial p-group, then H^{∗}(P, 𝔽_{p}) is not Cohen-Macaulay unless p = 3 and P = 3_{+}^{1+2} is the extraspecial 3-group of order 27 and exponent 3 [20].

We now proceed to give examples showing that the theorem is the best possible in the sense that no further implications hold. For this we need the following result on the cohomology of wreath products.

(3.1.5) Theorem. Let G be a finite group and p a prime number. The cohomology ring H^{∗}(G ≀ C_{p}, 𝔽_{p}) of the wreath product G ≀ C_{p} has Poincaré series

P_{G≀C_{p}}(t) = \frac{1}{p} (P_{G}(t)^{p} − P_{G}(t^{p})) + \frac{P_{G}(t^{p})}{1 − t}.

The Krull dimension and depth is given by

dim H^{∗}(G ≀ C_{p}, 𝔽_{p}) = max{1, p · dim H^{∗}(G, 𝔽_{p})},
depth H^{∗}(G ≀ C_{p}, 𝔽_{p}) = depth H^{∗}(G, 𝔽_{p}) + 1.

(3.1.6) Proof: The result on the Poincaré series follows from [5] (an translation may be found in [6]). We will return to Webb’s generalization [31] in section 3.2. The result on the depth is [9, Theorem 2.1]. Finally, the Krull dimension may be computed as follows. Let x ∈ G ≀ C_{p} be an element which permutes the factors of G^{p} ⊆ G ≀ C_{p} cyclically. Let d denote the p-rank of G, and let E be an elementary abelian psubgroup of G ≀ C_{p}. If E ⊆ G^{p} then E ⊆ π_{1}(E) × … × π_{p}(E) and thus rank E ≤ pd, where π_{i} : G^{p} → G is the projection on the i’th factor. If E ⊈ G^{p} then E contains an element of the form y = (g_{1}, …, g_{p})x^{i}, 1 ≤ i ≤ p − 1. We may assume that i = 1 after replacing y by a suitable power of itself. Since y has order p we get g_{1}·. . .·g_{p} = 1. Conjugating E by the element h = (h_{1}, …, h_{p}) where h_{i} = g_{i} ·. . .·g_{p} we get that E^{h} contains x. Thus E^{h} ⊆ C_{G≀C_{p}}(x) = ΔG × ⟨x⟩, where ΔG = {(g, …, g)|g ∈ G} ≅ G is the diagonal subgroup. From this we get rank E = rank E^{h} ≤ d + 1. Conversely, let E be an elementary abelian p-subgroup of G of rank d. Then the subgroups E^{p} and ΔE × ⟨x⟩ are elementary abelian p-subgroups of G ≀ C_{p} of rank pd and d + 1 respectively. Thus the p-rank of G ≀ C_{p} is exactly max{pd, d + 1} = max{1, pd}. By Quillen’s theorem the p-rank equals the Krull dimension of the cohomology ring and we are done. Alternatively the formula for the Krull dimension may be proved by combining the formula for the Poincaré series of the wreath product with the fact that the Krull dimension is the order of the pole of the Poincaré series at t = 1. □

We can now give the promised examples, which will mainly be wreath products of 2-groups. For our main examples we will not need to know very much about the cohomology of these groups except for Theorem (3.1.5). For completeness we will also give some other examples which have lower order, however their verification requires a more detailed knowledge of the cohomology rings involved. For 2-groups of order at most 32 these results may be found in [24]. The cohomology of the 2-groups of order at most 64 (except for a few) may be found on Carlson’s web page [8]. These groups were first classified by Hall and Senior ([14]) and we will use their names for the groups. The following table (which gives all possible combination of truth values allowed by Theorem (3.1.1)), summarizes our examples. Incidentally we do not know if any of the implications (1) ⟹ (2) ⟹ (4) or (3) ⟹ (5) can be reversed in odd characteristic. However we do not believe that this is the case, and that the lack of examples is probably only due to the fact that very few computations have been carried out in odd characteristic compared to characteristic 2.

(1) (2), (2′) (3) (4) (5) (6) Groups
+  +  + + + + G abelian
–  +  + + + + Example (3.1.10)
–  +  – + + + Example (3.1.12)
–  –  + + + + Example (3.1.7)
–  –  + – + + Example (3.1.9)
–  –  – + + + Example (3.1.7)
–  –  – – + + Example (3.1.13)
–  –  – – – + Example (3.1.11)
–  –  – – – – Example (3.1.8)

(3.1.7) Example. Consider the group G = C_{2}^{n} ≀ C_{2}, n ≥ 1 and let p = 2. The cohomology ring H^{∗}(C_{2}^{n}, 𝔽_{2}) is a polynomial algebra on n generators in degree 1 so it is Cohen-Macaulay of Krull dimension n and its Poincaré series is 1/(1 − t)^{n}. From Theorem (3.1.5) we then see that G satisfies (2′) if and only if n = 1. From Theorem (3.1.5) we also get

P_{G}(t) = \frac{1}{2} (\frac{1}{(1 − t)^{2n}} − \frac{1}{(1 − t^{2})^{n}} ) + \frac{1}{(1 − t)(1 − t^{2})^{n}}

= \frac{1}{(1 − t)^{2n}(1 + t)^{n−1}} · \frac{1}{2} [(1 + t)^{n−1} + (1 − t)^{n−1}]

= \frac{1}{(1 − t)^{2n}(1 + t)^{n−1}} · ∑_{j=0}^{[(n−1)/2]} ( \begin{matrix}n − 1 \\ 2j \end{matrix} ) t^{2j}.

The leading term in the sum is t^{n−1} for n odd and (n − 1)t^{n−2} for n even. From this we see that (6) is satisfied if and only if n is odd or n = 2. Since the Krull dimension of H^{∗}(G, 𝔽_{2}) is 2n we see from this by direct computation that (4) is satisfied exactly when n is odd and (5) holds exactly when n = 2 or n is odd. We also see that (3) holds if and only if all roots of the polynomial (1 + t)^{n−1} + (1 − t)^{n−1} are roots of unity. However it is easily seen that this is the case only for n = 1, 2 and 3.

We conclude that for n odd, n ≥ 5, the group C_{2}^{n} ≀ C_{2} satisfies (4), (5) and (6), but not (1), (2), (2′) or (3). For n = 3 the group satisfies (3), (4), (5) and (6), but not (1), (2) or (2′). Another example with the same properties is the group 32Γ_{7}a_{2}. In particular the Poincaré series of these groups satisfy the functional equation (4), but their mod-2 cohomology rings are not Cohen-Macaulay. The example 32Γ_{7}a_{2} is also given in [4], which generalizes the implication (2′) ⟹ (4) of Theorem (3.1.1).

(3.1.8) Example. From the above example we also conclude that for n even, n ≥ 4, the group C_{2}^{n} ≀ C_{2} satisfies none of the properties in Theorem (3.1.1). In particular the roots of the Poincaré series are not algebraic integers. As mentioned above, this answers a question of Rusin [24, p. 267]. From [8] we also find some examples of order 64, namely the groups 64Γ_{15}c_{3}, 64Γ_{15}c_{4}, 64Γ_{15}e_{3}, 64Γ_{15}f_{1}, 64Γ_{16}c_{1} and 64Γ_{17}b_{2}.

We can also construct some examples in odd characteristic. Consider the group G = C_{p} ≀ C_{p} which is the Sylow p-subgroup of the symmetric group Σ_{p^{2}} (see for example [1, p. 184–185]). From Theorem (3.1.5) we get that for odd p the Poincaré series of H^{∗}(G, 𝔽_{p}) is given by:

\begin{aligned}P_{G}(t) & = \frac{1}{p} (\frac{1}{(1 − t)^{p}} − \frac{1}{1 − t^{p}} ) + \frac{1}{1 − t} · \frac{1}{1 − t^{p}} \\ & = \frac{1}{(1 − t)^{p}(1 − t^{p})} [1 + ∑_{i=1}^{p−2} \frac{p − i − 1}{p} ( \begin{matrix}p \\ i \end{matrix} ) (−t)^{i}] . \end{aligned}

The leading term is \frac{p−1}{2}(−t)^{p−2}, so for p ≥ 5 we see that the roots of P_{G}(t) are not algebraic integers. Thus none of the properties in Theorem (3.1.1) are satisfied.

(3.1.9) Example. From Example (3.1.7) we see that the group C_{2}^{2} ≀ C_{2} = 32Γ_{4}a_{1} satisfies (3), (5) and (6), but not (1), (2), (2′) or (4). From [24] we also get that this holds for the groups 16Γ_{2}c_{1}, 16Γ_{2}d and 16Γ_{3}a_{2} of order 16. The group 16Γ_{3}a_{2} is the semidihedral group of order 16 and occurs as the Sylow 2-subgroup of the Mathieu group M_{11} and of GL_{2}(𝔽_{q}) for q ≡ 3 (mod 8). It also occurs as the Sylow 2-subgroup of the simple groups PSL_{3}(q) for q ≡ 3 (mod 8) and PSU_{3}(q) for q ≡ 5 (mod 8). Both 16Γ_{2}d and 16Γ_{3}a_{2} are split metacyclic groups, so the Poincaré series of their cohomology rings may also be found using [30].

In odd characteristic we have the example (ii) above. From its Poincaré series (we assume that α, β ≥ 1 and p ∤ m) we immediately see that this group satisfies (3) and hence (5) and (6), but not (4) and hence neither (1), (2) nor (2′). In particular this applies to the extraspecial groups p_{−}^{1+2} of order p^{3} and exponent p^{2}.

Another example with the same properties is C_{3} ≀ C_{3} for p = 3, which by Example (3.1.8) has the Poincaré series

P_{C_{3}≀C_{3}}(t) = \frac{1}{(1 − t)^{2}(1 − t^{3})}.

(3.1.10) Example. Consider the group G = C_{4} ≀ C_{2} = 32Γ_{3}e and let p = 2. This group occurs as the Sylow 2-subgroup of the groups GL_{2}(𝔽_{q}) for q ≡ 5 (mod 8), PSL_{3}(𝔽_{q}) for q ≡ 5 (mod 8) and PSU_{3}(𝔽_{q}) for q ≡ 3 (mod 8). Since the Poincaré of C_{4} is \frac{1}{1−t} we see from Theorem (3.1.5) that the cohomology ring H^{∗}(G, 𝔽_{2}) is Cohen-Macaulay of dimension 2 and has the Poincaré series \frac{1}{(1−t)^{2}} . Thus the conditions (2), (2′), (3), (4), (5) and (6) of Theorem (3.1.1) are all satisfied. However the cohomology ring is not a complete intersection, which may be seen from Rusin’s calculation [24]. Of course the cohomology ring may also be directly computed from Nakaoka’s theorem [21, Theorem 3.3]. From [24] we find that the groups 32Γ_{2}h, 32Γ_{3}c_{2} and 32Γ_{4}c_{3} have the same properties.

(3.1.11) Example. Consider the group G = C_{2} ≀ (C_{2} × C_{2}) = 64Γ_{25}a_{1}, where C_{2} × C_{2} acts regularly on a set of size 4 and let p = 2. This group is the Sylow 2-subgroup of the simple groups A_{8} ≅ GL_{4}(𝔽_{2}) ≅ O_{6}^{+}(𝔽_{2}), P SU_{4}(𝔽_{2^{2}}) ≅ O_{5}(𝔽_{3}) ≅ O_{6}^{−}(𝔽_{2}) and P Sp_{4}(𝔽_{5}) ≅ O_{5}(𝔽_{5}) and it is also the Sylow 2-subgroup of the affine general linear group AGL_{3}(𝔽_{2}). By [31, Example 3.4] the Poincaré series of the cohomology ring is

P_{G}(t) = \frac{1 + t^{2} − t^{3}}{(1 − t)^{3}(1 − t^{4})}.

Thus G satisfies (6), but none of the properties (1), (2), (2′), (3), (4) or (5). Other groups with the same properties are 32Γ_{4}c_{1}^{2}, 32Γ_{4}d, 32Γ_{6}a_{2}, 32Γ_{7}a_{3}, 16Γ_{2}d ≀ C_{2} and 16Γ_{3}a_{2} ≀ C_{2} (the groups 16Γ_{2}d and 16Γ_{3}a_{2} also occurred in Example (3.1.9) above).

For odd primes we have more examples. Consider the extraspecial group G = p_{+}^{1+2} of order p^{3} and exponent p. The cohomology ring of this group is given in [18] and its Poincaré series is found to be

P_{G}(t) = \frac{1 + t^{2} − t^{4} − t^{2p}}{(1 − t)^{2}(1 − t^{2p})}

(cf. [25, Theorem 7]). From this we see that if p ≥ 5, only condition (6) in Theorem (3.1.1) is satisfied.

We also have the following example. Consider again the group

P = ⟨x, y|x^{p^{α}} = y^{p^{β}} = 1, x^{−1}yx = y^{1+u}⟩

from (ii) above, where 1 + u satisfies (1 + u)^{p^{α}} − 1 = m · p^{β} for some m ∈ ℤ. Assume furthermore that p is an odd prime, that α, β ≥ 1 and that p ∤ m. Then by [10, Proposition 3.2(1)] we get Out(P) = O_{p} ⋊ C_{p−1}, where O_{p} is a Sylow p-subgroup of Out(P). Choose any divisor d of p − 1 and consider the semidirect product G = P ⋊ C_{d}, where C_{d} ≤ C_{p−1} ≤ Out(P). Then by [11, Theorem 1.1(2)] we have

H^{∗}(G, 𝔽_{p}) ≅ 𝔽_{p}[v, z] ⊗ E[b, α_{2i−1}, i = 1, …, p]/R

2The Poincaré series for this group equals \frac{1+t+t^{3}}{(1−t)^{2}(1−t^{4})}, but is given in [24, p. 367] as \frac{1+t^{2}+t^{3}}{(1−t)^{2}(1−t^{4})}.

where the relations R are given by

α_{2i−1}α_{2j−1} = 0 for 1 ≤ i, j ≤ p
α_{2i−1}v = 0 for 1 ≤ i ≤ p − 1,

with deg b = 1, deg v = 2, deg z = 2p and deg α_{2i−1} = 2i − 1 + 2pd(i), where 0 ≤ d(i) < d is the residue of −i mod d. From this we can easily compute the Poincaré series:

\begin{aligned}P_{G}(t) & = \frac{1 + t}{1 − t^{2p}} [\frac{1}{1 − t^{2}} + \frac{t^{deg α_{2p−1}}}{1 − t^{2}} + ∑_{i=1}^{p−1} t^{deg α_{2i−1}}] \\ & = \frac{1 + t^{2d−1} − t^{2d} − t^{2d+1} + t^{2pd+1} − t^{2pd+2d−1}}{(1 − t)(1 − t^{2d})(1 − t^{2p})} . \end{aligned}

If d > 1 we see from this that condition (5) is not satisfied and that condition (6) is satisfied. Thus the group satisfies none of the conditions in Theorem (3.1.1) except for (6). Note that for d = 1 we have G = P and we recover the Poincaré series (iii) given earlier. For d > 1 we also see that the a-invariant ([7, Definition 4.3.6]) of H^{∗}(G, 𝔽_{p}), which by definition is the degree of the Poincaré series as a rational function, equals 2pd − 2p − 2. Thus it can be arbitrarily large compared with the Krull dimension which is 2. Hence the analogue of [16, Conjecture 22] does not hold for cohomology rings of finite groups.

(3.1.12) Example. Let p = 2 and consider the group Q_{8} ≀ C_{2}, where Q_{8} is the quaternion group of order 8. By [1, Theorem IV.2.9] we have P_{Q_{8}}(t) = \frac{1+2t+2t^{2}+t^{3}}{1−t^{4}} . By Theorem (3.1.5) we then see that H^{∗}(Q_{8} ≀ C_{2}, 𝔽_{2}) is Cohen-Macaulay of dimension 2 and has the Poincaré series

P_{Q_{8}≀C_{2}}(t) = \frac{(1 + t + t^{2})(1 + 2t^{2} − t^{3} + 2t^{4} + t^{6})}{(1 − t)^{2}(1 + t^{2})^{2}(1 + t^{4})}.

Thus we see that Q_{8} ≀ C_{2} satisfies (2), (2′), (4), (5) and (6) but not (1) and (3). Other groups with this behavior are 64Γ_{4}k_{2}, 64Γ_{9}b_{4}, 64Γ_{9}d_{2} and 64Γ_{9}e which is the Sylow 2-subgroup of the simple group Sz(8).

(3.1.13) Example. As a final example note that by combining Example (3.1.9) and Example (3.1.12) we obtain that the group (Q_{8} ≀ C_{2}) × ((C_{2})^{2} ≀ C_{2}) satisfies only conditions (5) and (6) of Theorem (3.1.1). Other examples are given by the groups 32Γ_{2}d ≀ C_{2}, 32Γ_{2}f ≀ C_{2}, 32Γ_{3}a_{2} ≀ C_{2}, 32Γ_{4}b_{2} ≀ C_{2}, 32Γ_{4}c_{2} ≀ C_{2} and 32Γ_{6}a_{1} ≀ C_{2}^{3}. Here 32Γ_{2}d = C_{2} × 16Γ_{2}d and 32Γ_{3}a_{2} = C_{2} × 16Γ_{3}a_{2}, where the groups 16Γ_{2}d and 16Γ_{3}a_{2} occurs in Example (3.1.9) above. Moreover the group 32Γ_{3}a_{2} is the Sylow 2-subgroup of GL_{3}(𝔽_{q}) for p ≡ q (mod 8). Note also that 32Γ_{6}a_{1} = C_{8} ⋊ Aut(C_{8}).

3The Poincaré series of the groups 32Γ_{2}d, 32Γ_{2}f, 32Γ_{3}a_{2}, 32Γ_{4}b_{2}, 32Γ_{4}c_{2} and 32Γ_{6}a_{1} is \frac{1}{(1−t)^{3}(1+t^{2})} and thus dim H^{n}(G, 𝔽_{p}) = 1, 3, 7, 10, 14, … for n = 0, . . .. These dimensions are stated wrongly in [24, p. 368].

3.2 Poincaré series of wreath products

In this section we obtain a formula for the Poincaré series of the cohomology ring of a wreath product of finite groups. Such a formula was first obtained by Webb [31], but unfortunately his formula is not correct in general. Here we show how the correct formula may be obtained by modifying Webb’s formula. To save space and time we will assume complete familiarity with the definitions and results in [31].

Now let p be a prime number, F a finite group, G a subgroup of Σ_{m} and consider the wreath product F ≀ G = F^{m} ⋊ G. We are then interested in computing the Poincaré series of H^{∗}(F ≀ G, 𝔽_{p}). As in [31, p. 450] we start by using Nakaoka’s theorem [21, Theorem 3.3], which gives an isomorphism H^{∗}(F ≀ G, 𝔽_{p}) ≅ H^{∗}(G, H^{∗}(F, 𝔽_{p})^{⊗m}) as graded rings. Here G acts on the factors of the m-fold tensor product H^{∗}(F, 𝔽_{p})^{⊗m} not simply by permuting the factors (as claimed in [31]), but also by including the appropriate sign. For example if σ = (1, 2) ∈ Σ_{2} then

σ · (ξ_{1} ⊗ ξ_{2}) = (−1)^{deg ξ_{1} deg ξ_{2}}(ξ_{2} ⊗ ξ_{1}).

Since there are no problems with the sign for p = 2 we will from now on assume that p is odd unless stated otherwise. Now let Ξ be a graded basis of H^{∗}(F, 𝔽_{p}), i.e. a graded set such that Ξ(n) is a basis of H^{n}(F, 𝔽_{p}), and let B be the set of elements ξ_{1} ⊗ … ⊗ ξ_{m}, where ξ_{i} ∈ Ξ. Then B forms a basis of H^{∗}(F, 𝔽_{p})^{⊗m}, but unfortunately G does not act on B since we also have to include the appropriate sign. To overcome this problem we introduce the ’double’

B̃ = B ⨿ −B = {±b|b ∈ B} ⊆ H^{∗}(F, 𝔽_{p})^{⊗m},

which is a G-set, and we let M̃ = 𝔽_{p} \left[{ \atop B}\right] be the associated permutation module.

Then B_{+} = {b + (−b)|b ∈ B} ⊆ M̃ is a G-set and the G-action is just given by permuting the coordinates (without the sign). Let M_{+} = 𝔽_{p} [B_{+}] be the associated permutation module. Define also B_{−} = {b − (−b)|b ∈ B} ⊆ M̃ and let M_{−} be the submodule of M̃ spanned by B_{−}. Since p is odd we find that M̃ = M_{+} ⊕ M_{−} and M_{−} ≅ H^{∗}(F, 𝔽_{p})^{⊗m} as 𝔽_{p} [G]-modules. For a subgroup J ≤ G we now define the power series f̃_{J} and f_{J} by

f̃_{J}(t) = ∑_{i=0}^{∞} |B̃(i)^{J} | t^{i}, f_{J}(t) = ∑_{i=0}^{∞} |B_{+}(i)^{J} | t^{i}.

Now using [31, Proposition 1.2] and replacing G-sets by permutation modules we obtain

PM̃(t) = ∑_{K≤J} \frac{𝔽_{p} [G/K] μ(K, J)f̃_{J}(t)}{[G : K]}, P_{M_{+}}(t) = ∑_{K≤J} \frac{𝔽_{p} [G/K] μ(K, J)f_{J}(t)}{[G : K]},

where the sums are over all pairs of subgroups K ≤ J of G and μ is the Möbius function of the poset of subgroups of G [28, 3.7]. As in [31] both sides of these equations are to be interpreted as elements in the ring of power series over the Green ring A(G). Combining this with the above we see that the series of the graded 𝔽_{p}G-module H^{∗}(F, 𝔽_{p})^{⊗m} is

∑_{K≤J} \frac{𝔽_{p} [G/K] μ(K, J) (f̃_{J}(t) − f_{J}(t))}{[G : K]}.

Thus using [31, Lemma 3.2] we obtain that the Poincaré series of the cohomology ring H^{∗}(F ≀ G, 𝔽_{p}) equals

∑_{K≤J} \frac{g_{K}(t)μ(K, J) (f̃_{J}(t) − f_{J}(t))}{[G : K]},

where g_{K}(t) is the Poincaré series of H^{∗}(K, 𝔽_{p}).

To compute the series f̃_{J} and f_{J} we may proceed as follows. Let {1, …, m} = Ω_{1} ∪ … ∪ Ω_{n} be the decomposition of {1, …, m} into J-orbits, f(t) be the Poincaré series of H^{∗}(F, 𝔽_{p}) and let b = ξ_{1} ⊗ … ⊗ ξ_{m} ∈ B, ξ_{i} ∈ Ξ. Since G acts on B_{+} simply by permuting the coordinates we see that b + (−b) ∈ B_{+} is invariant under J if and only if ξ_{i} = ξ_{j} whenever i, j ∈ {1, …, m} are in the same J-orbit. Thus we get

f_{J}(t) = f(t^{|Ω_{1}|}) · . . . · f(t^{|Ω_{n}|})

as in [31, p. 450 and Proposition 1.4(1)]. Computing the series f̃_{J} is more difficult since we have to take care of the signs. If b ∈ B̃ is invariant under the action of J then ξ_{i} = ξ_{j} whenever i, j ∈ {1, …, m} are in the same J-orbit; however this is enough to guarantee that b is J-invariant. Now assume that b satisfies this condition. From the decomposition {1, …, m} = Ω_{1} ∪ … ∪ Ω_{n} we get the homomorphism

φ : J ⟶ Σ_{Ω_{1}} × … × Σ_{Ω_{n}} ⟶^{σ_{1}×...×σ_{n}} 𝔽_{2} × … × 𝔽_{2} = V,

where σ_{i} : Σ_{Ω_{i}} → 𝔽_{2} is the sign homomorphism. Let i_{1} ∈ Ω_{1}, …, i_{n} ∈ Ω_{n} be orbit representatives and let • be the inner product on V, determined by the formula u • v = u_{1}v_{1} + … + u_{n}v_{n}. With a slight abuse of notation we then have

g · b = (−1)^{φ(g)•(deg ξ_{i_{1}},... ,deg ξ_{in})} · b

for g ∈ J. Thus b is J-invariant if and only if (deg ξ_{i_{1}}, …, deg ξ_{i_{n}}) • φ(g) = 0 for all g ∈ J, i.e. when (deg ξ_{i_{1}}, …, deg ξ_{i_{n}}) ∈ φ(J)^{⊥}. Now set

f_{0}(t) = ∑_{i=0, i even}^{∞} dim H^{i}(F, 𝔽_{p}) t^{i}= \frac{1}{2} (f(t) + f(−t)) ,

f_{1}(t) = ∑_{i=0, i odd}^{∞} dim H^{i}(F, 𝔽_{p}) t^{i} = \frac{1}{2} (f(t) − f(−t))

and let

f_{v}(t) = f_{v_{1}}(t^{|Ω_{1}|}) · . . . · f_{v_{n}}(t^{|Ω_{n}|})

for v = (v_{1}, …, v_{n}) ∈ V. Thus we get the formula

f̃_{J}(t) = 2 · ∑_{v∈φ(J)^{⊥}} f_{v}(t),

where the factor 2 comes from the fact that −b is J-invariant if b is. We summarize the results of the above discussion in the following theorem which corrects [31, Theorem 3.1] (as noted above the original formula is correct for p = 2).

(3.2.1) Theorem. Let p be a prime number, F a finite group and G a subgroup of Σ_{m}. For a subgroup J ≤ G define the series f̃_{J}(t),f_{J}(t) and g_{J}(t) as above. Then the Poincaré series of the cohomology ring H^{∗}(F ≀ G, 𝔽_{p}) is given by

P_{F ≀G}(t) = ∑_{K≤J} \frac{g_{K}(t)μ(K, J)f_{J}(t)}{[G : K]},

for p = 2 and by

P_{F ≀G}(t) = ∑_{K≤J} \frac{g_{K}(t)μ(K, J) (f̃_{J}(t) − f_{J}(t))}{[G : K]},

for p odd. □

(3.2.2) Example. Let p be any prime number and let G = C_{p} act regularly on a set of size p. Let F be any finite group with Poincaré series f(t). We then have

g_{1}(t) = 1,  f_{1}(t) = f(t)^{p}
g_{G}(t) = \frac{1}{1 − t}, f_{G}(t) = f(t^{p}).

Now assume that p is odd. Noting that (1, …, p) ∈ G is an even permutation we get

f̃_{1}(t) = 2f(t)^{p},  f̃_{G}(t) = 2f(t^{p}).

Thus the Poincaré series of H^{∗}(F ≀ C_{p}, 𝔽_{p}) is given as

\begin{aligned}P_{F ≀C_{p}}(t) & = \frac{g_{G}(t) · 1 · (f̃_{G}(t) − f_{G}(t))}{[G : G]} + \frac{g_{1}(t) · (−1) · (f̃_{G}(t) − f_{G}(t))}{[G : 1]} \\ & + \frac{g_{1}(t) · 1 · ({ \atop f_{1}(t)} − f_{1}(t))}{[G : 1]} = \frac{1}{p} (f(t)^{p} − f(t^{p})) + \frac{f(t^{p})}{1 − t}. \end{aligned}

For p = 2 it is easily seen that the same formula holds, since f̃_{J}(t) − f_{J}(t) = f_{J}(t). Thus we recover the formula of Bogačenko ([5]) from Theorem (3.1.5) above. Incidentally this is also the result obtained from Webb’s original formula.

(3.2.3) Example. Let p be an odd prime and let G = C_{2} act regularly on a set of size 2. Let F be any finite group with Poincaré series f(t). We then have

g_{1}(t) = 1,  f_{1}(t) = f(t)^{2}
g_{G}(t) = 1, f_{G}(t) = f(t^{2}).

Since σ = (1, 2) ∈ G is an odd permutation we get

f̃_{1}(t) = 2f(t)^{2},  f̃_{G}(t) = 2f_{0}(t^{2})

and hence the Poincaré series of H^{∗}(F ≀ C_{2}, 𝔽_{p}) equals

\begin{aligned}P_{F ≀C_{2}}(t) & = \frac{g_{G}(t) · 1 · (f̃_{G}(t) − f_{G}(t))}{[G : G]} + \frac{g_{1}(t) · (−1) · (f̃_{G}(t) − f_{G}(t))}{[G : 1]} \\ & + \frac{g_{1}(t) · 1 · (f̃_{1}(t) − f_{1}(t))}{[G : 1]} = \frac{1}{2} (f(−t^{2}) + f(t)^{2}) . \end{aligned}

In this case the result does not agree with Webb’s original formula which gives \frac{1}{2} (f(t^{2}) + f(t)^{2}). In the case F = C_{p} we have f(t) = \frac{1}{1−t} and we get

P_{C_{p}≀C_{2}}(t) = \frac{1 + t^{3}}{(1 − t)(1 − t^{4})}.

This may be checked easily since we have the isomorphism

C_{p} ≀ C_{2} = (C_{p} × C_{p}) ⋊ C_{2} = C_{p} × (C_{p} ⋊ C_{2}) = C_{p} × D_{2p}

because the matrices [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] and [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] are conjugate in GL_{2}(𝔽_{p}). Since

H^{∗}(D_{2p}, 𝔽_{p}) has Poincaré series \frac{1+t^{3}}{1−t^{4}} by [30] this verifies our calculation.

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