A UNIQUENESS RESULT FOR ORTHOGONAL

GROUPS AS 2-COMPACT GROUPS

D. Notbohm

Abstract. Two connected compact Lie groups are isomorphic if and only if their maximal torus normalizer are isomorphic. It is conjectured that this result generalizes to p-compact groups. Here, we prove the generalization for orthogonal groups O(n), the special orthogonal groups SO(2k + 1) and the spinor groups Spin(2k + 1) considered as 2-compact groups.

1. Introduction.

A p-compact group is the homotopy theoretic generalization of a compact Lie group. This concept was introduced by Dwyer and Wilkerson [7]. And it turned out that p-compact groups behave astonishingly similar as compact Lie groups; e.g. there exist maximal tori, Weyl groups and normalizer of maximal tori. Since their existence, the classification problem was one of the main questions. The classification scheme of connected p-compact groups is of the same form as for connected compact Lie groups. Every connected p-compact group has a finite covering which is a product of torus and a simply connected p-compact group [18], and every simply connected p-compact group splits into a product of finitely many simple simply connected p-compact groups [9] [22]. The classification boils down to the case of simple p-compact groups [14]. And in this situation, a case by case checking based on the Clark-Ewing list of rational pseudo reflection groups has solved the problem for odd primes ([2], [3], [4], [10], [16], [17], [19], [23]). For the prime 2, the known results in this direction concern unitary and special unitary groups [19], 2-compact groups with Weyl group isomorphic to an elementary abelian 2-group [10] and the case of the exceptional Lie group G_{2} [25].

In this work we consider products of orthogonal groups and characterize them as 2-compact groups by their associated normalizer of the maximal torus. Before we state our main results, we first have to recall some basic notation of the theory of p-compact groups from [7]. For further notation and definition as well as for the theory of p-compact groups see the survey articles [13] or [21] or the references given there.

A loop space X is a pair (X, BX) of spaces, such that such that BX is pointed and such that the loop space ΩBX is equivalent to X. The loop space inherits

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 55R35, 55P15.

Key words and phrases. compact Lie groups, p-compact groups, classification.

Date: January 6, 2000.

properties from the space X, e.g. X is called connected if the space X is so and finite or ℤ-finite if the integral cohomology H^{∗}(X; ℤ) vanishes in large degrees and is in any degree a finitely generated abelian group. The loop space X is called a p-compact group, if X is 𝔽_{p}-finite and if X and BX are p-complete. Every p-compact group X has a maximal torus T_{X} ⊂ X, a maximal torus normalizer N_{X} ⊂ X which is a finite extension of T_{X} and a Weyl group W_{X} which acts on the p-adic lattice L_{X} := π_{2}(BT_{X}). The maximal torus normalizer fits into a fibration BT_{X} ⟶ BN_{X} ⟶ BW_{X} which also determines the action of W_{X} on BT_{X} respectively on L_{X}. Moreover, the composition N_{X} ⟶ X ⟶ π_{0}(X) factors through a homomorphism W_{X} ⟶ π_{0}(X) and is always an epimorphism. The kernel of N_{X} ⟶ π_{0}(X) ≅ π_{1}(BX) respectively the fiber of the map BN_{X} ⟶ Bπ_{0}(X) is a maximal torus normalizer of the component X_{0} of the unit of X and the kernel of W_{X} ⟶ π_{0}(X) is the Weyl group of X_{0}. Examples of p-compact groups are given by the p-adic completion of compact Lie groups. If the group of components π_{0}(G) of a compact Lie group G is a finite p-group, then the pair (G_{p}^{∧}, BG_{p}^{∧}) gives rise to a p-compact group, also denoted by G. In particular, the orthogonal groups O(n) give rise to 2-compact groups.

1.1 Definition.

(i) Two loop spaces or p-compact groups X and Y are called isomorphic if the associated classifying spaces BX and BY are homotopy equivalent. (ii) Two p-compact groups X and Y are called N-isomorphic if the there exist isomorphisms N_{X} ⟶^{≅} N_{Y} and π_{0}(X) ⟶^{≅} π_{0}(Y) such that the diagram

BN_{X} ⟶ BN_{Y}
↓  ↓
Bπ_{0}(X) ⟶ Bπ_{0}(Y)

commutes up to homotopy.

In analogy to compact Lie groups, the classification conjecture reads as

1.2 Conjecture. Two p-compact groups X and Y are isomorphic if and only if they are N-isomorphic.

As already mentioned, for odd primes, this conjecture is known to be true. In this case, one even can proof a weaker version, namely, X and Y are isomorphic if and only if N_{X} and N_{Y} are isomorphic, and, if in addition X is connected, then it suffices to have isomorphic Weyl groups (see the above mentioned references). But, as a comparison of O(2) and SO(3) and of SO(2n + 1) and Sp(n) show, a weaker version cannot be true for the prime 2.

For p = 2, we want to prove the conjecture for finite products of orthogonal groups.

1.3 Theorem. Let G = ∏_{i} O(n_{i}) be a finite product of orthogonal groups and let X be a 2-compact group. Then, G and X are isomorphic if and only if they are N-isomorphic.

The proof of this theorem is based on homotopy uniqueness results for products of orthogonal groups [20] in terms of mod-2 cohomology and 1.4 Theorem. Let G = ∏_{i} O(n_{i}) be a finite product of orthogonal groups and let X be a 2-compact group. If G and X are N-isomorphic, then there exist an isomorphism H^{∗}(BX) ≅ H^{∗}(BG) of algebras over the Steenrod algebra.

Proof of Theorem 1.3. As shown in [20], every 2-complete space with the same mod-2 cohomology as BG is homotopy equivalent to the 2-adic completion BG_{2}^{∧}. □

As a corollary we will also see that the groups Spin(2k + 1) and SO(2k + 1) satisfy the above conjecture.

1.5 Corollary. Let G = Spin(2k + 1) or SO(2k + 1) and let X be a 2-compact group. Then X and G are isomorphic if and only if they are N-isomorphic.

Proof. Let X be a 2-compact group which is N isomorphic to SO(2k + 1). Since N_{O(2k+1)} ≅ N_{SO(2k+1)} × ℤ/2, the 2-compact group X × ℤ/2 is N-isomorphic to O(2k + 1) and therefore, by Theorem 1.3, isomorphic to O(2k + 1). The projection BX × Bℤ/2 ⟶ Bℤ/2 on the second factor is given by the classifying map of the non trivial 1-dimensional mod-2 cohomology class. The fiber of this map is BX. the same holds for BO(2k + 1)_{2}^{∧}. This shows that BX and BSO(2k + 1)_{2}^{∧} are equivalent and that X and SO(2k + 1) are isomorphic as 2-compact groups.

Let X be a 2-compact group, which is N-isomorphic to Spin(2k + 1). The center of X can be read off from the maximal torus normalizer [8]. Hence, X and Spin(2k + 1) have the same center equal to ℤ/2 and the quotients \overline{X} := X/ℤ/2 and SO(2k + 1) are N-isomorphic and therefore isomorphic. Since X is the simply connected cover of \overline{X} and Spin(2k + 1) the simply connected cover of SO(2k + 1), the 2-compact groups X and Spin(2k + 1) are isomorphic. □

Starting from Corollary 1.5, we plan in further paper to prove uniqueness results for the 2-compact group DI(4) constructed in [6] in terms of the maximal torus normalizer as well as in terms of the mod-2 cohomology of BDI(4).

The paper is organized as follows. In Section 2 we provide some facts for orthogonal groups as 2-compact groups and fix some notation. Section 3, the last section, contains the tedious proof of Theorem 1.4. We also note that cohomology is always taken with ℤ/2-coefficients; i.e. H^{∗}(−) = H^{∗}(−; 𝔽_{2}).

We will use the language and theory of p-compact groups all over the places. For references, about p-compact groups we refer the reader to the survey articles [13] and [21] or the references mentioned there.

We thank the Max-Planck-Institut für Mathematik in Bonn for the hospitality when this work was done.

2. Products of orthogonal type.

In this section we provide necessary facts about products of orthogonal groups and fix some notation.

2.1 Notation and Remarks. Let n = 2k + θ with θ ∈ {0, 1}. We use Θ to denote the group ℤ/2^{θ}. The maximal torus normalizer of O(n) is isomorphic to O(2) ≀ Σ_{k} × Θ. The group O(2) contains a maximal elementary abelian 2-subgroup E_{O(2)} of rank 2. We denote by (ℤ/2)^{k} ≅ t_{O(n)} ⊂ T_{O(n)} the maximal elementary abelian 2-subgroup of the maximal torus of O(n). We have a sequence

t_{O(n)} ⊂ (ℤ/2)^{n} ≅ E_{O(n)} := E_{O(2)}^{k} × Θ ⊂ N_{O(n)} ⊂ O(n)

of inclusions.

Every elementary abelian 2-subgroup of O(n) is contained in E_{O(n)}. The subgroup E_{O(n)} is self centralizing, i.e. the inclusion E_{O(n)} ⊂ C_{O(n)}(E_{O(n))} induces an isomorphism of E_{O(n)} with the centralizer of E_{O(n)}. The Weyl group \overline{W}_{O(n)} := W_{O(n)}(E_{O(n)}) is isomorphic to Σ_{n} which acts by permutations on E_{O(n)}. Actually, the normalizer N_{O(n)}(E_{O(n)}) of E_{O(n)} is isomorphic to ℤ/2 ≀ Σ_{n} ⊂ O(n). All this follows from classical representation theory. Moreover, H^{∗}(BO(n); ℤ/2) ≅ H^{∗}(BE_{O(n)}; ℤ/2)^{Σ_{n}}.

For G = ∏_{i} O(n_{i}), taking products establishes the same type of groups which fit into the diagram of inclusions

t_{G} ⟶ T_{G}
↓  ↓
E_{G} ⟶ N_{G} ⟶ N_{G} ⟶ G

and satisfy the same properties. Among other things we collect these facts in the next proposition. And \overline{W}_{G} := W_{G}(E_{G}) denotes the Weyl group of E_{G} in G.

By SG we denote the component of the unit of G. Then SG = ∏_{i} SO(n_{i}) is a product of special orthogonal groups. We will also use the notation SX for the component of the unit of a p-compact group X.

2.2 Proposition. Let G ≅ ∏_{i} O(n_{i}) be a finite product of orthogonal groups. Then, the following holds:

(i) E_{G} is self centralizing. (ii) The mod-2 cohomology of BN_{G} is detected by elementary abelian 2-subgroups and so is the mod-2 cohomology of BG. (iii) Up to conjugation, every elementary abelian subgroup of G is contained in E_{G}. (iv) \overline{W}_{G} := ∏_{i} \overline{W}_{O(n_{i})} ≅ ∏_{i} Σ_{n_{i}}.

(v) H^{∗}(BG; ℤ/2) ≅ H^{∗}(BE_{G}; ℤ/2)^{\overline{W}^{G}}. (vi) For any elementary abelian 2-group E, the inclusion E_{G} ⟶ G induces a bijection [BE, BE_{G}]/\overline{W}_{G} ≅ [BE, BG]. (vii) For any elementary abelian 2-subgroup E ⊂ G, the centralizer C_{G}(E) is again a finite product of orthogonal groups.

Proof. : The proof is obvious and/or a straight forward application of real representation theory. It only might be necessary to make some comments for (ii). As a product of Θ and a wreath product of O(2) with a symmetric group, the mod-2 cohomology of BN_{O(n)} is detected by elementary abelian 2-subgroups [11] and so is the mod-2 cohomology of BN_{G}. □

3. Proof of Theorem 1.4.

Let G ≅ ∏_{i} O(n_{i}) be a finite product of orthogonal groups and let X be a 2compact group which is N-isomorphic to G. In this section we want to prove a version of Proposition 2.2 for X. It will include Theorem 1.4.

Since X and G are N-isomorphic, we have the following diagram of inclusions

t_{G} ⟶ T_{G}
↓  ↓
E_{G} ⟶ N_{G} ⟶ N_{G} ⟶ G

We denote by i_{X} : E_{G} ⟶ X the inclusion given by the bottom line. As already mentioned, SX denotes the component of the unit of X.

3.1 Proposition.

(i) The subgroup E_{G} ⊂ X is self centralizing; i.e. C_{X}(E_{G}) = E_{G}. (ii) The mod-2 cohomology of BX is detected by elementary abelian 2-subgroups. (iii) Every elementary abelian 2-subgroup is subconjugated to E_{G}. (iv) For every abelian 2-group E the map [BE, BE_{G}] ⟶ [BE, BX] factors through the bijection [BE, BE_{G}]/\overline{W}_{G} ⟶^{≅} [BE, BX].

(v) H^{∗}(BX) ≅ H^{∗}(BE_{G})^{\overline{W}^{G}}.

The proof of this proposition splits into several parts and is worked out in the rest of this section. In particular, the parts (iii) and (iv) are proved via inductions, which themselves split into several steps.

3.2 Remark. According to the splitting of G = ∏_{i} O(n_{i}), the maximal torus normalizer N_{G} = ∏ N_{O(n_{i})} splits into a product. By [22], this implies a splitting X ≅ ∏_{i} X_{i} such that N_{O(n_{i})} ⊂ X_{i} is a maximal torus normalizer and such that the diagram

∏_{i} N_{O(n_{i})} ⟶^{≅} N_{G}
↓  ↓
∏_{i} X_{i}  ⟶^{≅} X

commutes. In particular, X_{i} and O(n_{i}) are N-isomorphic. Moreover, all statements of Proposition 3.1 also split in the same manner. Hence, on the way of proving the proposition, we always can assume that G = O(n) is an orthogonal group. This even is possible for the inductions necessary for the proofs of (iii) and (iv).

3.3 Proof of (i) and (ii): Part (ii) follows from Proposition 2.3, since N_{G} is the maximal torus normalizer of X and since H^{∗}(BX) ⟶ H^{∗}(BN_{G}) is a monomorphism [7].

Since E_{G} is self centralizing in G, we have C_{N_{G}}(E_{G}) = E_{G}, too. By reasons of size, every monomorphism of E_{G} ⟶ N_{G} takes image in E_{G} ⊂ N_{G}. By [15], there exist a lift E_{G} ⟶ N_{G} of i_{X} : E_{G} ⟶ X such that C_{N}(E_{G}) ⊂ C_{X}(E_{G}) is the maximal torus normalizer. Hence, the maximal torus normalizer of C_{X}(E_{G}) is E_{G}.

This implies that C_{X}(E_{G}) is discrete, therefore its own maximal torus normalizer and congruent to E_{G}. This proves the first part.

The second part is a consequence of Proposition 2.2 (ii) and the fact that for every p-compact group X the map BN_{X} ⟶ BX induces an monomorphism in modp cohomology [7], in particular H^{∗}(BX) ⟶ H^{∗}(BN_{G}) is a monomorphism. □

Before we continue with the proof we digress for the following lemma:

3.4 Lemma. Let E ⊂ T_{G} be an elementary abelian 2-subgroup. Then C_{G}(E) and C_{X}(E) are N-isomorphic.

Proof. By [8, 7.6], both C_{G} := C_{G}(E) and C_{X} := C_{X}(E) have the same Weyl groups and therefore the same maximal torus normalizer, namely C_{N} := C_{N_{G}}(E). We only have to show that the components of the unit are N-isomorphic.

Since E is a toral subgroup and therefore E ⊂ SX and since SC_{X}(E) is contained in SX, we have SC_{X}(E) = SC_{SX}(E). But the Weyl group of SC_{SX}(E) is completely determined by the Weyl group action of W_{SX} on the maximal torus T_{G} = T_{X} [8, 7.6]. Since SG and SX are N-isomorphic, and since the same holds for G, both centralizers, C_{G} and C_{X}, have N-isomorphic components of the unit. □

3.5 Claim. Proposition 3.1 (iii) is true for E ≅ ℤ/2.

Proof. We prove this via an induction over the order of the Weyl group of G. Because of Remark 3.2 we can assume that G = O(n). If G ≅ O(2), then X = N_{G} = G. And for O(2) all of Proposition 3.1 is true. This is the beginning of the induction.

Up to conjugation E ⊂ N_{G}. Since N_{G} = O(2)≀Σ_{k}×Θ, the generator of E is given by an element x = ((A_{1}, ..., A_{k}; σ), a) ∈ N_{G} with A_{i} ∈ O(2), σ ∈ Σ_{k} and a ∈ Θ. Since E ≅ ℤ/2, the permutation σ is of order two. Without loss of generality we can assume that σ = ∏_{j=1}^{r} τ_{2j−1,2j} is a product of commuting transpositions. Therefore x ∈ (O(2) ≀ ℤ/2)^{r} × O(2)^{k−2r} × Θ. If r = 0, then E obviously is subconjugated to E_{G}. If r ≥ 1 then there exists a ℤ/2 ≅ E′ ⊂ (T_{G})^{E} which is neither central in N_{G} nor in G or in X. By Lemma 3.4, C_{G}(E′) and C_{X}(E′) are N-isomorphic. Moreover, they have smaller Weyl groups than G. Since E_{G} and E are contained in C_{N_{G}}(E′), E is subconjugated to E_{G} by induction hypothesis. □

3.6 Claim. Proposition 3.1 (iii) is true for any general elementary abelian 2-group E.

Proof. This time we make an induction over the order of E. The starting point of the induction is given by Claim 3.5. Again, we can assume that G = O(n). If the dimension of E is greater than 1, the composition E ⊂ X ⟶ π_{0}(X) has a non trivial kernel and we can split E ≅ E_{0} ⊕ E_{1} such that ℤ/2 ≅ E_{0} ⊂ SX. Since E_{0} is cyclic, it is subconjugated to T_{G}, in fact to t_{G}. The centralizers C_{X}(E_{0}) and C_{G}(E_{0}) are N-isomorphic (Lemma 3.4), and E_{G} ⊂ C_{X}(E_{0}). By induction hypothesis, E_{1} is subconjugated to E_{G} inside of C_{X}(E_{0}), which shows that E is subconjugated to E_{G} inside X. □

We make the map BE_{G} ⟶ BX into a fibration and denote the fiber by X/E_{G}. The inclusion i_{X} : E_{G} ⟶ X establishes a proxy action of E_{G} on X/E_{G} (see [7]) and the homotopy fixed point set (X/E_{G})^{E_{G}} fits into a fibration

(X/E_{G})^{E_{G}} ⟶ map(BE_{G}, BE_{G})_{lifts} ⟶ map(BE_{G}, BX)_{Bi_{X}} .

The total space is given by all self maps of BE_{G} which are lifts of Bi_{X}. Since E_{G} ⊂ X is self centralizing (Proposition 3.1 (i)) the fiber is homotopically discrete and the set of components \overline{W}_{X} := π_{0}((X/E_{G})^{E_{G}}) is a group, the Weyl group W_{X}(E_{G}) of E_{G} ⊂ X. Since π_{0}(map(BE_{G}, BE_{G}) = Hom(E_{G}, E_{G}), every element of \overline{W}_{X} is represented by an automorphism of E_{G}. This establishes an action of \overline{W}_{X} on E_{G} as well as on BE_{G}.

3.7 Claim.

(i) The Weyl group \overline{W}_{X} is finite and acts faithfully on E_{G}. (ii) H^{∗}(BX) ≅ H^{∗}(BE_{G})^{\overline{W}^{X}}.

Proof. The Weyl group \overline{W}_{X} is finite since (X/E_{G})^{E_{G}} is 𝔽_{2}-finite [7]. The action is obviously faithful.

By Proposition 3.1 (ii) and (iii), H^{∗}(BX) ⟶ H^{∗}(BE_{G}) is the Adams-Wilkerson embedding [1]. As a consequence of [26, Theorem II], [5, 3.6] and properties of Lannes’ T -functor [12] we get H^{∗}(BX) ≅ H^{∗}(BC_{X}(E_{G}))^{\overline{W}^{X}}, which is Part (ii). □

3.8 Claim. The Weyl group \overline{W}_{X} contains \overline{W}_{G}.

Proof. Again we can assume that G = O(n). That is that \overline{W}_{G} = Σ_{n} For n ≥ 3 the two subgroups Σ_{2} × Σ_{n−2} and Σ_{n−2} × Σ_{2} generate Σ_{n}. The inclusions of both subgroups are induced by the Weyl group inclusion \overline{W}_{C_{G}(E)} ⊂ \overline{W}_{G} for an elementary abelian 2-subgroup ℤ/2 ≅ E ⊂ t_{G}. By Lemma 3.4, C_{G}(E) and C_{X}(E) are N-isomorphic. Now, the claim follows by an induction over the order of the Weyl group. □

The next step is a proof of Proposition 3.1 (iv). We need some further lemmas.

3.9 Lemma. Let G = O(n) and Δ ⊂ E_{G} denote the diagonal. Then, Δ is a central subgroup of X.

Proof. If n is even, Δ ⊂ SX is subconjugated to T_{G} and the statement follows from Lemma 3.4. If n = 2k + 1 is odd, N_{O(n)} ≅ N_{SO(n)} × ℤ/2. By [22], X splits into the product SX × ℤ/2 and SX and SO(n) are N-isomorphic. Since the center of a connected p-compact group can be read off from the maximal torus normalizer [8], the center of SX is trivial and the center of X equals ℤ/2.

Because Lannes’ T -functor commutes with taking invariants [24] and because H^{∗}(BX) ≅ H^{∗}(BC_{X}(E_{G}))^{\overline{W}^{X}}, a central elementary abelian 2-subgroup E ⊂ E_{G} ⊂ X induces a map H^{∗}(BE_{G}) ⟶ H^{∗}(BE) which is fixed under the action of \overline{W}_{X}. Since Σ_{n} ⊂ \overline{W}_{X}, only the diagonal Δ ⊂ E_{G} can have this property. Thus, Δ ⊂ E_{G} ⊂ X has to be central. □

Let X be a p-compact group with maximal torus normalizer N_{X} ⊂ X. A elementary abelian p-subgroup E ⊂ N_{X} is called special if C_{N_{X}} (E) ⊂ C_{X}(E) is a maximal torus normalizer. As shown in [15], for every elementary abelian E ⊂ X, there exists a special monomorphism E ⟶ N_{X}, such that the composition E ⟶ N_{X} ⊂ X and E ⊂ X are conjugate. In [15], this monomorphism is called a preferred lift. For G = O(n), we denote by ℤ/2 ≅ E(l) ⊂ E_{G} ≅ (ℤ/2)^{n} the subgroup given by the diagonal of the first l-components and by E′(l) the diagonal of the last l components. If x ∈ E(l) is a generator, then −x is a generator of E′(n − l) (we write E_{G} respectively ℤ/2 multiplicatively). Moreover, since Δ ⊂ E_{G} is central in X, both E(l) and E′(n − l) have the same centralizer.

3.10 Lemma. Let G = O(n) and let n = 2k + θ.

(i) If θ = 0 (n is even) and if a subgroup ℤ/2 ≅ E ⊂ N_{G} is special for X, then there exists 1 ≤ l ≤ n such that, as subgroups of N_{G}, E and E(l) are conjugate. (ii) Every subgroup E(l) ⊂ E_{G} is special as a subgroup of X. (iii) C_{G}(E(l)) and C_{X}(E(l)) are N-isomorphic.

Proof. First we proof (i). As shown in the proof of Claim 3.5, up to conjugation, E ⊂ (O(2) ≀ ℤ/2)^{r} × O(2)^{k−2r}. That is that the generator x ∈ E has coordinates A_{j} ∈ O(2) and B_{i} ∈ O(2) ≀ ℤ/2, where B_{i} = (C_{i}, C_{i}^{−1}; σ) with C_{i} ∈ O(2) and 1 ≠ σ ∈ ℤ/2 (we write ℤ/2 multiplicatively). Conjugation by (C_{i}, I; 0) shows that B_{i} is conjugate to (I, I; σ) where I ∈ O(2) denotes the unit. We can assume that C_{i} = I for all i. We also can transform the A_{j}’s on a normal form, namely A_{j} = I,

A_{j} = −I or A_{j} is conjugate to the matrix S = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ). A straight forward

calculation shows that the centralizer C_{N}(E) is isomorphic to

(O(2) × ℤ/2) ≀ Σ_{r} × O(2) ≀ Σ_{q} × O(2) ≀ Σ_{t} × (ℤ/2 × ℤ/2) ≀ Σ_{s} ,

where q is the number of A_{j}’s equal to I, t the number of A_{j}’s equal −I and s the number of A_{j}’s conjugate to S. Moreover, 2r + q + t + s = k.

On the other hand, as a subgroup of X, E is subconjugated to E_{G} (Proposition 3.1 (iii)), and, since Σ_{2k} ⊂ \overline{W}_{X} (Claim 3.8), E is conjugate to a subgroup E(l) ⊂ E_{G}. Let l = 2m + ν with ν ∈ {0, 1}. Then, C_{N}(E(l)) ≅ O(2) ≀ Σ_{m} × (ℤ/2 × ℤ/2)^{ν} × O(2) ≀ Σ_{k−m−ν} × Θ. By inspection, the maximal torus of C_{N}(E(l)) has dimension k − ν and the maximal torus of C_{N}(E) dimension r + q + t = k − r − s The parity of s shows whether E ⊂ SX and the parity of ν does this for E(l). Hence, ν ≡ s mod 2. Since E ⊂ N_{G} was special, k − r − s ≥ k − ν which implies that s = ν and r = 0. This shows, that, up to conjugation by elements of Σ_{k}, E = E(l′) and proves Part (i).

Now, we prove Part (ii). If n = 2k + 1, then either l or n − l is even and either E(l) or E′(n − l) is contained in SX and therefore subconjugate to T_{G}. Hence, in this case, the claim follows from Lemma 3.4 and the above remark. If n and l are even, the same argument applies.

Finally, we have to consider the case n = 2k and l = 2m+1. By (i), there exists a special subgroup E(p) ⊂ E_{G} conjugate to E(l). Since p is odd if and only if l is odd, we can assume that p = 2q + 1. The centralizers C_{N_{G}}(E(l) = O(2) ≀ Σ_{m} × (ℤ/2)^{2} × O(2) ≀ Σ_{k−l} and C_{N_{G}}(E(p) = O(2) ≀ Σ_{q} × (ℤ/2)^{2} × O(2) ≀ Σ_{k−q} have isomorphic components of the unit, i.e. are of the same rank. Therefore, π_{0}(C_{N_{G}}(E(l)) = ℤ/2 ≀ Σ_{m} × (ℤ/2)^{2} × ℤ/2 ≀ Σ_{k−l} is subconjugated to W_{X} ≅ π_{0}(C_{N_{G}}(E(q)). This implies that m = q or m = k − q, that C_{N_{G}}(E(l)) ≅ C_{N_{G}}(E(p) and, in particular, that E(l) ⊂ E_{G} ⊂ N_{G} is special, which finishes the proof of (ii).

By the above calculations and by Part (ii), C_{G}(E(l)) and C_{X}(E(l)) have isomorphic maximal torus normalizer. We have to show that both are N-isomorphic or, which is sufficient, have isomorphic groups of components.

If a quotient of ℤ/2 ≀ Σ_{r} is a 2-group then it is an elementary abelian 2-group. And the same holds for the Weyl group of C_{X}(E(l)). This implies that

\begin{aligned}π_{0}(C_{X}(E(l)) & ≅ H^{1}(BC_{X}(E(l))) \\ & ≅ H^{1}(BE_{G})^{W′} ⊂ H^{1}(BE_{G})^{Σ_{l}×Σ_{n−l}} ≅ (ℤ/2)^{2} , \end{aligned}

where W′ := Iso_{\overline{W}_{X}} (E(l)) is the isotropy group of E(l) in \overline{W}_{X}.

If l is even, N-isomorphism follows from Lemma 3.4. If l and n are odd, we can replace E(l) by E′(n − l) and, again, N-isomorphism follows from Lemma 3.4. If l is odd and n is even, the maximal torus normalizer contains a direct factor (ℤ/2)^{2} which is contained in π_{0}(C_{X}(E(l))). Hence π_{0}(C_{X}(E(l))) ≅ (ℤ/2)^{2} ≅ π_{0}(C_{G}(E(l))). This shows that C_{X}(E(l)) and C_{G}(E(l)) are N-isomorphic and finishes the proof. □

3.11 Claim. For every elementary abelian 2-group E, the inclusion E_{G} ⟶ X induces an isomorphism [BE, BE_{G}]/\overline{W}_{X} ⟶^{≅} [BE, BX].

Proof. This is a consequence of the isomorphisms H^{∗}(BX) ≅ H^{∗}(BE_{G})^{\overline{W}^{X}} and the facts that, for finite groups, Lannes’ T -functor commutes with taking invariants (e.g. see [24]) and that the [BE, BX] = Hom_{A_{2}}((H^{∗}(BX), H^{∗}(BE)), where Hom_{A_{2}}(−, −) denotes the set of maps of algebras over the Steenrod algebra A_{2} [12]. □

3.12 Claim. Part (iv) is true for E ≅ ℤ/2.

Proof. Again we can assume that G = O(n). Let α, β : E ⟶ E_{G} be two homomorphism such that i_{X}α, i_{X}β : E_{G} ⟶ X are conjugate. If α is the constant homomorphism then so is β. Hence, we can assume that both are monomorphisms. Up to the action of elements of Σ_{n} ≅ \overline{W}_{G} ⊂ \overline{W}_{X}, the image of α is E(l) and the image of β is E(p). If w ∈ \overline{W}_{X} conjugates E(l) into E(p), then, since the diagonal Δ ⊂ E_{G} ⊂ X is central, w also conjugates E′(n − l) into E′(n − p).

Let l = 2m + ν and p = 2q + ν′. The number l is even if and only if E(l) ⊂ SX if and only if E(p) ⊂ SX if and only if p is even. And if this is the case, then E(l) and E(p) are conjugate by an element of W_{X} = W_{G} ⊂ Σ_{n} ⊂ \overline{W}_{X} and l = p. This proves the claim in this case. Moreover, the argument shows that, in general, ν = ν′ and that, for any w ∈ W_{X} and any 2k, the subgroup w(E(2k)) ⊂ E_{G} is conjugate to E(2k) by an element of Σ_{n}.

By Claim 3.11, there exists w ∈ W_{X} such that w(E(l)) = E(p). Let σ ∈ Σ_{n} such that σw(E(2)) = E(2). Then, σw ∈ \overline{W}_{C_{X}(E(2))} and E(l) and σ(E(p)) are conjugate in C_{X} := C_{X}(E(2)) which is N-isomorphic to O(2) × O(n − 2) (Lemma 3.4). By induction, we can assume that \overline{W}_{C_{X}} = Σ_{2} × Σ_{n−2} ⊂ Σ_{n}. Hence, E(l) and σ(E(p)) are conjugate by an element of Σ_{n} (Claim 3.11) as well as E(l) and E(p). This shows that l = p and proves the claim.

For G = O(2), which is the beginning of the induction, X ≅ G, and the claim is obvious. □

3.13 Claim. Part (iv) is true for any elementary abelian 2-group.

Proof. Again we prove this via an induction over the order of the Weyl group W_{G} of G. We can assume that G = O(n). Let E be an elementary abelian 2-group and α, β : E ⟶ E_{G} such that the compositions i_{X}α, i_{X}β : E ⟶ X are conjugate. We can assume that both are monomorphism.

By Claim 3.11, there exists w ∈ W_{X} such that wα = β. We split E ≅ E_{0} ⊕ E_{1} such that E_{0} ≅ ℤ/2 As in the proof of Claim 3.12, we can assume that α(E_{0}) = E(l). Claim 3.12 also shows that, changing w by an element of Σ_{n}, β(E_{0}) = E(l). That is that w fixes E(l), that w ∈ \overline{W}_{C_{X}(E(l))} and that α and β are conjugate in C_{X} := C_{X}(E(l)). Since C_{X} and C_{G} := C_{G}(E(l)) ≅ O(l) × O(n − l) are Nisomorphic (Lemma 3.10) and have smaller Weyl group than G, there exists σ ∈ Σ_{l} × Σ_{n−l} ≅ \overline{W}_{C_{X}} ⊂ Σ_{n}, such that σα = β. This finishes the induction step. The starting point is given by the case G = O(2) where nothing is to show. □

By Claim 3.11 the next claim, which is the last one, proves Part (v) and finishes the proof of Proposition 3.1.

3.14 Claim. \overline{W}_{X} = \overline{W}_{G}.

Proof. By Claim 3.8, \overline{W}_{G} ⊂ \overline{W}_{X}. Since \overline{W}_{X} acts faithfully on E_{G} (Claim 3.7), it is the subgroup of Gl(E_{G}) of all elements w such that i_{X}w and i_{X} are conjugate. By Proposition 3.1 (iv), this subgroup is \overline{W}_{G}. □

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Max-Planck-Institut für Mathematik, P.O.Box: 7280, D-53072 Bonn, Germany. e-mail : notbohm at mpim-bonn.mpg.de