CELLULARIZATION OF CLASSIFYING SPACES AND FUSION PROPERTIES OF FINITE GROUPS

RAMÓN J. FLORES AND JÉRÔME SCHERER

Abstract. One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their Bℤ/p-cellularization. In the easiest cases this is a classifying space of a finite group (and then always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears, and compute explicitly the cellularization.

Introduction

Let A be a pointed space. Although the idea of building a space as a homotopy colimit of copies of another fixed space A goes back to Adams ([Ada78]) in the framework of the classification of the acyclics of a certain generalized cohomology theory, it was in the early 90’s that Dror-Farjoun ([Far96]) and Chachólski ([Cha96]) formalized and developed this idea in the wider context of cellularity classes. Thus a space will be called A-cellular if it can be built from A by means of pointed homotopy colimits. There exists an A-cellularization functor CW_{A} that provides the best possible A-cellular approximation: the natural map CW_{A}X → X induces a weak equivalence between pointed mapping spaces Map_{∗}(A, CW_{A}X) ≃ Map_{∗}(A, X).

Our interest lies essentially in using the cellularization functor to study the p-primary part of the homotopy of the classifying space of a finite group G. This approach was already suggested by Dror-Farjoun in [Far96, Example 3.C.9], and proved to be very fruitful in the last years; see for instance work of Bousfield ([Bou97]) on cellularization of nilpotent spaces with regard to Moore spaces M(ℤ/p, n), or the relationship recently discovered ([RS01]) between the Bℤ/p-cellularization of spaces and the ℤ/p-cellularization in the category of groups.

In the present paper we focus our attention in describing the Bℤ/p-cellularization of BG, where G is a finite group. From the above description of the cellularization functor, we know this is a space which can be built from Bℤ/p by means of push-outs, wedges, and telescopes, and which encodes all the information about the pointed mapping space Map_{∗}(Bℤ/p, BG), which is weakly equivalent to Hom (ℤ/p, G). A first attempt to understand CW_{Bℤ/p}BG was

Date: June 8, 2005.

Both authors are supported by MEC grant MTM2004-06686 and the second author by the Ramón y Cajal program, MEC, Spain.

2 RAMÓN J. FLORES AND JÉRÔME SCHERER

undertaken in [Flo], where in particular the fundamental group of the cellularization was described as an extension of the group-theoretical ℤ/p-cellularization of a subgroup of G by a finite p-torsion free abelian group (see Section 1 for details). In the present note we concentrate thus on the higher homotopy groups π_{n}(CW_{Bℤ/p}BG), for n ≥ 2.

Our first result establishes that, in general, the space CW_{Bℤ/p}BG has infinitely many nontrivial homotopy groups. This contrasts with the results in [CCS]: for easier spaces to work with, such as H-spaces or classifying spaces of nilpotent groups, the assumption that the mapping space Map_{∗}(Bℤ/p, X) is discrete implies that CW_{Bℤ/p}X is aspherical.

Proposition 2.2. Let G be a finite group. Then CW_{Bℤ/p}BG is either the classifying space of a finite p-group, or it has infinitely many non-trivial homotopy groups.

An important role in the proof of this result is played by Levi’s dichotomy theorem [Lev95, Theorem 1.1.4] about the homotopy structure of the Bousfield-Kan p-completion of BG. We remark here that the homotopy groups of a Bℤ/p-cellular space are not necessarily p-groups. We wish therefore to understand the p-primary part on one hand, but also the more accessible p′-primary part. The latter is indeed closely related to the Bℤ/p-nullification of BG, due to Chachólski’s description [Cha96, 20.10] of the cellularization as a small variation of the homotopy fiber of the Bℤ/p-nullification map. The first author identified in [Flo, 3.5] the Bℤ/p-nullification P_{Bℤ/p}BG, when G is generated by elements of order p, as the product (taken over all primes q different from p) of the q-completions BG_{q}^{∧}.

Theorem 2.4. Let G be a finite group generated by elements of order p. Then either the cellularization CW_{Bℤ/p}BG has infinitely many homotopy groups containing p-torsion or it fits in a fibration

CW_{Bℤ/p}BG ⟶ BG ⟶ ∏_{q≠p} BG_{q}^{∧},

where the (finite) product is taken over all primes q dividing the order of G, and the right map is the product of the completions.

Note that in the second case the higher homotopy groups of the cellularization are those of Ω(∏_{q≠p} BG_{q}^{∧}). In particular, if G is generated by elements of order p, CW_{Bℤ/p}BG coincides with the homotopy fiber P̄_{Bℤ/p}BG of the nullification map unless there appears (a lot of) p-torsion in the higher homotopy groups. The main question which remains unanswered up to this point is thus to determine if there actually exist groups for which CW_{Bℤ/p}BG does contain p-torsion in its higher homotopy groups. This turns out to depend heavily on the p-complete classifying space BG_{p}^{∧}.

CELLULARIZATION AND FUSION 3

Proposition 3.3. Let G be a group generated by order p elements which is not a p-group. Then the universal cover of CW_{Bℤ/p}BG is p-torsion free if and only if the p-completion of BG is Bℤ/p-cellular.

We notice that the presence of p-torsion in the higher homotopy groups of the cellularization of BG depends on the fusion properties of G. If S is a Sylow p-subgroup, define Cl(S) as the smallest strongly closed subgroup of S containing all elements order p in S. This means that whenever a conjugate gxg^{−1} of an element x ∈ Cl(S) by an element g ∈ G lies in S, then in fact gxg^{−1} ∈ Cl(S). Our next result shifts the problem to the study of fusion systems.

Theorem 4.2. Let G be a group generated by elements of order p. If S is equal to Cl(S), then the universal cover of CW_{Bℤ/p}BG is p-torsion free.

There are very few groups containing a proper strongly closed subgroup. This yields many examples of Bℤ/p-cellular spaces whose higher homotopy is p-torsion free, including the cellularization of classifying spaces of the symmetric groups, see Example 4.8. It is a delicate problem to obtain an explicit description of the p-primary part of CW_{Bℤ/p}BG. For this purpose we focus on groups in which the normalizer of a Sylow p-subgroup controls the fusion.

Theorem 5.6. Let G be a finite group generated by elements of order p and S a Sylow p-subgroup. Assume that the normalizer N_{G}(S) controls the fusion in G. Then CW_{Bℤ/p}BG fits in a fibration

CW_{Bℤ/p}BG ⟶ BG ⟶ BΓ_{p}^{∧} × ∏_{q≠p} BG_{q}^{∧},

where Γ = N_{G}(S)/Cl(S).

At the prime 2, Foote actually characterized in [Foo97] the groups containing a proper strongly closed subgroup. We apply in Corollary 5.8 the previous theorem to obtain a description of the only two families of simple groups for which 2-torsion appears in the higher homotopy groups of CW_{Bℤ/2}BG. For example, in the case of the Suzuki groups Sz(2^{n}), there is a fibration

CW_{Bℤ/2}BSz(2^{n}) → BSz(2^{n}) → B((ℤ/2)^{n} ⋊ ℤ/(2^{n} − 1))_{2}^{∧} × ∏_{q≠2} BSz(2^{n})_{q}^{∧}

and a similar description holds for the groups U_{3}(2^{n}). At odd primes we offer another concrete calculation in Corollary 5.9 for certain projective special linear groups, but we do not claim that they are the unique simple groups for which p-torsion appears. In the last section we find explicit “exotic” representations (which are not induced by a group homomorphism) of these groups in compact Lie groups.

Acknowledgements. We would like to thank Wojciech Chachólski, Jesper Grodal, Ran Levi, Michel Matthey, and Guido Mislin for helpful discussions during the Arolla conference in August 2004. We appreciate the help of Jacques Thévenaz, who constructed for us the group in Example 4.6, and Albert Ruiz and Jesper Møller, with whom we computed a few fusion systems. We warmly thank Carles Broto for suggesting an explicit description of the cellularization and many other valuable comments, Bob Oliver, who hinted at the Suzuki groups for their beautiful fusion properties, and Antonio Viruel for recognizing the same features in the projective special linear groups.

1. Background

There are two main ingredients used to analyze the cellularization CW_{Bℤ/p}BG. The first one is a general recipe, the second one a specific simplification in the setting of classifying spaces of finite groups.

One of the most efficient tools to compute cellularization functors is Chachólski’s construction of CW_{A} out of the nullification functor P_{A}, see [Cha96]. His main theorem states that CW_{A}X can be constructed in two steps. First consider the evaluation map ∨_{[A,X]_{∗}}A → X (the wedge is taken over representatives of all pointed homotopy classes of maps) and let C denote its homotopy cofiber. Then CW_{A}X is the homotopy fiber of the composite map X → C → P_{ΣA}C. We will actually need a small variation of Chachólski’s description of the cellularization.

Lemma 1.1. Let A and X be pointed connected spaces and choose a pointed representative A → X for each unpointed homotopy class in [A, X]. Denote by D the homotopy cofiber of the evaluation map ev : ⋁_{[A,X]} A → X. Then CW_{A}X is weakly equivalent to the homotopy fiber of the composite X → D → P_{ΣA}D.

Proof. According to [Cha96, Theorem 20.3] one has to check that composing any map f : A → X with ev : X → D is null-homotopic. But such a map f is freely homotopic to one in [A, X]. Thus ev ∘f is freely homotopic to the constant map, and so it must also be homotopic to the constant map in the pointed category. □

In [Flo], the first author focused on the situation when A = Bℤ/p and X is the classifying space of a finite group G. A first reduction can always be done by considering the subgroup Ω_{1}(G)_{p} of G generated by the elements of order p. This notation, or the shorter notation Ω_{1}(G) when the prime p is understood, is the standard one in group theory,

CELLULARIZATION AND FUSION 5

see for example [Gor80]. In [Flo] and [RS01] the terminology “socle” and the corresponding notation S_{ℤ/p}G was used instead. We have an equivalence of pointed mapping spaces Map_{∗}(Bℤ/p, BG) ≃ Map_{∗}(Bℤ/p, BΩ_{1}(G)_{p}), which means that BΩ_{1}(G)_{p} → BG is a Bℤ/pcellular equivalence. A second simplification consists then in computing the group theoretical cellularization CW_{ℤ/p}G ≅ CW_{ℤ/p}Ω_{1}(G)_{p}, which is known to be a (finite) central extension of Ω_{1}(G)_{p} by a group of order coprime with p. Since BCW_{ℤ/p}G → BG is a Bℤ/p-cellular equivalence, one could assume that G is a finite ℤ/p-cellular group (it can be constructed out of the cyclic group ℤ/p by iterated colimits). Examples of such groups are provided at the prime 2 by dihedral groups, Coxeter groups, and in general by finite p-groups generated by order p elements.

The first author described the fundamental group of CW_{Bℤ/p}BG:

Proposition 1.2. [Flo, Theorem 4.14] Let G be a finite ℤ/p-cellular group. Then the fundamental group π = π_{1}CW_{Bℤ/p}BG is an extension H ↪ π→ →G of G by a finite p-torsion free abelian group H. □

He also showed that BG is Bℤ/p-cellular if and only if G is a finite ℤ/p-cellular p-group. This paper is an attempt to understand CW_{Bℤ/p}BG when G is not a p-group.

2. A cellular dichotomy

We establish in this section that the space CW_{Bℤ/p}BG has infinitely many non-trivial homotopy groups unless it is aspherical. We will say more about the higher homotopy groups in the next section. Recall that C is Chachólski cofiber introduced in the previous section.

Lemma 2.1. Let G be a finite group generated by elements of order p. Then P_{ΣBℤ/p}C is homotopy equivalent to (P_{ΣBℤ/p}C)_{p}^{∧} × ∏_{q} BG_{q}^{∧}, where the product is taken over all primes q ≠ p.

Proof. Since G is generated by order p elements, the evaluation map ∗ℤ/p → G taken over all homomorphisms ℤ/p → G is surjective. Hence the homotopy cofiber C of the evaluation map ∨_{[Bℤ/p,BG]}Bℤ/p → BG is a simply-connected space, and so is its nullification P_{ΣBℤ/p}C.

We want to analyze the q-completion of this last space for a prime q ≠ p. But since Bℤ/p is Hℤ/q-acyclic, we see that the composite

BG ⟶ C ⟶ P_{ΣBℤ/p}C

is an equivalence in mod q homology, which implies that (P_{ΣBℤ/p}C)_{q}^{∧} ≃ BG_{q}^{∧}. Finally, as ⋁ Bℤ/p and BG are rationally trivial, we use Sullivan’s arithmetic square to establish that P_{ΣBℤ/p}C splits as the product of its q-completions, taken over all primes q. □

Proposition 2.2. Let G be a finite group. Then CW_{Bℤ/p}BG is either the classifying space of a finite p-group, or it has infinitely many non-trivial homotopy groups.

Proof. If CW_{Bℤ/p}BG is the classifying space of a group, then it must be a finite p-group as shown by the first author in [Flo, Theorem 4.14]. Assume therefore that it is not so, i.e. there exists a prime q different from p dividing the order of G. By considering the subgroup Ω_{1}(G) we might also assume that G is generated by elements of order p.

By Levi’s result [Lev95, Theorem 1.1.4] for q-completions of classifying spaces of finite groups one infers that BG_{q}^{∧} must have infinitely many non-trivial homotopy groups (because G is q-perfect as its abelianization must be p-torsion). Consider now the long exact sequence in homotopy of the fibration from [Cha96]

CW_{Bℤ/p}BG ⟶ BG ⟶ P_{ΣBℤ/p}C

and conclude by using Lemma 2.1. □

In view of the proof of the preceding theorem, the main question is to determine whether or not the higher homotopy groups of the cellularization of classifying spaces contain p-torsion. We make next an observation about ΣBℤ/p-local spaces which play such an important role for the Bℤ/p-cellularization because of Lemma 1.1. The proof is very much in the spirit of the original theorem of Serre [Ser53, Théorème 10], the Dwyer-Wilkerson result [DW90, Theorem 1.3], Levi’s one [Lev95, Theorem 1.1.4]; compare also with Grodal’s work on Postnikov pieces [Gro98].

Proposition 2.3. Let X be a simply-connected torsion ΣBℤ/p-local space. Then it is either p-torsion free or has infinitely many homotopy groups containing p-torsion.

Proof. Let n be an integer ≥ 2 and consider the Postnikov fibration

X⟨n⟩ ⟶ X⟨n − 1⟩ ⟶ K(π_{n}X, n).

Since the base point component of the iterated loop space Ω^{n−1}X and Ω^{n−1}(X⟨n − 1⟩) are weakly equivalent there is a fibration

Ω_{0}^{n−1}X ⟶ K(π_{n}X, 1) ⟶ Ω^{n−2}(X⟨n⟩)

in which the fiber is Bℤ/p-local. Thus the total space is Bℤ/p-local if and only if the base is so, i.e. the homotopy group π_{n}X contains p-torsion if and only if the n-connected cover X⟨n⟩ has some p-torsion. We use here that a simply-connected p′-torsion space Y has trivial p-completion, and therefore the pointed mapping space Map_{∗}(Bℤ/p, Y) is contractible. □

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We turn now to a more detailed study of the case when G is not a p-group. We have seen that CW_{Bℤ/p}BG has infinitely many non-trivial homotopy groups, but they can arise in two different ways, because the space P_{ΣBℤ/p}C may contain p-torsion or not.

Theorem 2.4. Let G be a finite group generated by order p elements. Then either the cellularization CW_{Bℤ/p}BG has infinitely many homotopy groups containing p-torsion or it fits in a fibration

CW_{Bℤ/p}BG ⟶ BG ⟶ ∏_{q≠p} BG_{q}^{∧},

where the (finite) product is taken over all primes q dividing the order of G, and the right map is the product of the completions.

Proof. Because the group G is finite, its integral homology groups are finite. The space P_{ΣBℤ/p}C is constructed out of BG by taking iterated homotopy cofibers of maps out of (finite) wedges of suspensions of Bℤ/p, whose homology groups are all torsion, and thus P_{ΣBℤ/p}C satisfies the conditions of the above proposition.

Assume thus that P_{ΣBℤ/p}C is p-torsion free. We see then by Lemma 2.1 that P_{ΣBℤ/p}C is weakly equivalent to the product ∏_{q≠p} BG_{q}^{∧}. □

Remark that the computation in [Flo, Section 3] allows to identify the cellularization of BG with P̄_{Bℤ/p}BG, the homotopy fiber of the nullification map, when the space P_{ΣBℤ/p}C is p-torsion free.

Example 2.5. Consider the C_{2}-cellular group Σ_{3}, the symmetric group on three letters. The choice of any transposition yields a map f : Bℤ/2 → BΣ_{3} which induces an isomorphism in mod 2 homology. Therefore the homotopy cofiber C_{f} of f is 2-torsion free and so (P_{ΣBℤ/2}C_{f})_{2}^{∧} is contractible. By Lemma 2.1, CW_{Bℤ/2}BΣ_{3} is a space whose fundamental group is Σ_{3} and its universal cover is Ω(BΣ_{3})_{3}^{∧}, a space weakly equivalent to S^{3}{3}, the homotopy fiber of the degree 3 map on the sphere S^{3}, compare with [BK72, VII, 4.1] and [RS01, Theorem 7.5].

3. The p-completed classifying space

The example of the symmetric group Σ_{3} might lead us to think that there will be very few cases when the universal cover of CW_{Bℤ/p}BG is p-torsion free. In this section we will see that there are surprisingly many groups for which this occurs and that the properties of the cellularization we are looking for are reflected in the classifying space completed at p.

We start with a general result that tells us how to detect the presence of p-torsion in the cellularization of BG.

Proposition 3.1. Let G be a finite group generated by elements of order p. For CW_{Bℤ/p}BG to have infinitely many homotopy groups with p-torsion, there must exist a p-complete space Z which is ΣBℤ/p-local and a map f : BG → Z which is not null-homotopic such that the restriction Bℤ/p → BG → Z to any cyclic subgroup of order p is null-homotopic.

Proof. Consider again the cofibration ∨Bℤ/p → BG → C. The cofiber C is a simply connected torsion space, hence equivalent to the finite product ∏ C_{q}^{∧}, where q runs over the primes dividing the order of G. We need to understand the ΣBℤ/p-nullification of C. Since nullification commutes with finite products, we only look at C_{p}^{∧} and infer that CW_{Bℤ/p}BG has infinitely many homotopy groups with p-torsion if and only if P_{ΣBℤ/p}(C_{p}^{∧}) does so, which happens if and only if P_{ΣBℤ/p}(C_{p}^{∧}) is not contractible by Theorem 2.4.

In other words we wish to know when the map C_{p}^{∧} → ∗ is not a ΣBℤ/p-equivalence. This means by definition that there exists some ΣBℤ/p-local space Z for which the pointed mapping space Map_{∗}(C_{p}^{∧}, Z) is not contractible. Because C_{p}^{∧} is a p-torsion space (being a homotopy cofiber of such spaces) which is simply connected, we can assume, using Sullivan’s arithmetic square, that Z is p-complete.

The cofibration sequence ∨Bℤ/p → BG → C yields a fibration

Map_{∗}(C, Z) → Map_{∗}(BG, Z) → ∏ Map_{∗}(Bℤ/p, Z)

in which the loop space of the base is trivial since Z is ΣBℤ/p-local. The existence of the map f tells us that there is a component of the total space (different from the component of the constant map) which lies over the component of the base point in the base. Therefore Map_{∗}(C, Z) has at least two components as well. As Z is p-complete, Map_{∗}(C, Z) is weakly equivalent to Map_{∗}(C_{p}^{∧}, Z).

On the other hand, if no such map f exists, Map_{∗}(C, Z) is weakly equivalent to the connected component of the constant Map_{∗}(BG, Z)_{c}, which is contractible by Dwyer’s result [Dwy96, Theorem 1.2]. □

We compare now the cellularization of the classifying space BG and of its p-completion. Recall that the fundamental group of BG_{p}^{∧} is always isomorphic to the group theoretical p-completion, i.e. the quotient of G by O^{p}(G), the maximal p-perfect subgroup of G.

Proposition 3.2. Let G be a finite group generated by elements of order p. Then the space (CW_{Bℤ/p}BG)_{p}^{∧} is homotopy equivalent to CW_{Bℤ/p}(BG_{p}^{∧}).

Proof. We know for example from [DZ87] that the completion map BG ⟶ BG_{p}^{∧} induces a chain of bijections identifying the set of unpointed homotopy classes

[Bℤ/p, BG] ≃ [Bℤ/p, BG_{p}^{∧}] ≃ Rep(ℤ/p, G).

Choose a set of representatives f : Bℤ/p → BG for all conjugacy classes of elements of order p in G and write f_{p}^{∧} for the corresponding map into the p-completion of BG. We

CELLULARIZATION AND FUSION 9

have seen in Lemma 1.1 that CW_{Bℤ/p}BG can be constructed as the homotopy fiber of the composite BG → D → P_{ΣBℤ/p}D, where D is the homotopy cofiber of the evaluation map ev : ⋁ Bℤ/p → BG. Likewise CW_{Bℤ/p}(BG_{p}^{∧}) is constructed using the completed version, call the corresponding cofiber D′. In short we have a diagram of cofibrations

[figure: ⋁ Bℤ/p; ∨f; BG; D; ⋁ Bℤ/p; ∨f_{p}^{∧}; BG_{p}^{∧}; D′]

Now since G (and thus its quotient G/O^{p}(G) as well) is generated by elements of order p, the cofibers D and D′ are simply connected. As ⋁ Bℤ/p and BG are rationally trivial, we use Sullivan’s arithmetic square to establish that D = ∏_{q prime} D_{q}^{∧}. Looking next at the bottom cofibration we see that BG_{p}^{∧} is p-complete, and so is D′.

Comparing the Mayer-Vietoris sequences in mod p homology of the two cofibrations, it is easy to see that D_{p}^{∧} ≃ D′. Therefore we can compute

P_{ΣBℤ/p}D ≃ ∏_{q} P_{ΣBℤ/p}D_{q}^{∧} ≃ P_{ΣBℤ/p}D′ × ∏_{q≠p} D_{q}^{∧}.

and the first factor, namely P_{ΣBℤ/p}D′, is p-complete. Notice that, since the base spaces P_{ΣBℤ/p}D and P_{ΣBℤ/p}D′ in Chachólski’s fibrations are simply connected, the p-completion of these fibrations are again fibrations by the nilpotent fiber lemma [BK72, II.5.1]. Therefore CW_{Bℤ/p}(BG_{p}^{∧}) is p-complete and coincides with (CW_{Bℤ/p}BG)_{p}^{∧}. □

As a consequence, the existence of p-torsion in the upper homotopy of CW_{Bℤ/p}BG is strongly related with the Bℤ/p-cellularity of BG_{p}^{∧}.

Corollary 3.3. Let G be a finite group generated by order p elements. Then the universal cover of CW_{Bℤ/p}BG is p-torsion free if and only if the p-completion of BG is Bℤ/p-cellular.

Proof. By the previous proposition the p-completion of BG is Bℤ/p-cellular if and only if the p-completion of CW_{Bℤ/p}BG is so. From the fibration

CW_{Bℤ/p}(BG_{p}^{∧}) → BG_{p}^{∧} → P_{ΣBℤ/p}D′

we see that this happens precisely when the space D′ is ΣBℤ/p-acyclic, i.e. when the space P_{ΣBℤ/p}D is p-torsion free. □

4. Strongly closed subgroups

We recall now (see for example [Foo97]) the notion of strongly closed subgroup. The main result of the section is that the existence of a proper strongly closed subgroup in a Sylow p-subgroup is an unavoidable condition to find p-torsion in the Bℤ/p-cellularization of BG.

10 RAMÓN J. FLORES AND JÉRÔME SCHERER

Definition 4.1. Let G be a finite group and H a subgroup of some Sylow p-subgroup S of G. Then H is strongly closed in G if whenever h ∈ H and g ∈ G are such that ghg^{−1} ∈ S, then ghg^{−1} ∈ H.

We consider next the smallest strongly closed subgroup Cl(S) of S containing Ω_{1}(S). It can be built inductively in a finite number of steps starting from Ω_{1}(S) = Cl_{0}(S), and constructing Cl_{i+1}(S) from Cl_{i}(S) by adding all conjugates of elements of Cl_{i}(S) by elements of G which belong to S, i.e. Cl_{i+1}(S) is generated by all elements gxg^{−1} ∈ S with g ∈ G and x ∈ Cl_{i}(S). Then Cl(S) = ⋃_{i} Cl_{i}(S).

We start with an observation which leads then naturally to a closer analysis of the fusion of the groups we look at.

Theorem 4.2. Let G be a group generated by elements of order p. If S is equal to Cl(S), then the universal cover of CW_{Bℤ/p}BG is p-torsion free.

Proof. By Proposition 3.1, the existence of p-torsion is detected by a non-trivial map f : BG_{p}^{∧} → Z into some ΣBℤ/p-local space Z, which is trivial when restricted to any cyclic subgroup ℤ/p in G. The composite BG → Z → K(π_{1}Z, 1) is then null-homotopic because it corresponds to a homomorphism G → π_{1}Z restricting trivially to all generators of G. Therefore the map f lifts to the universal cover of Z and so we might assume that Z is 1-connected. We can also p-complete it, if necessary, so that Z is actually Hℤ/p-local.

By Dwyer’s theorem [Dwy96, Theorem 1.4], the map f is null-homotopic if and only if the restriction f̄ to some Sylow p-subgroup S of G is so. As we assume that S = Cl(S) we will show by induction that the restriction of f to Cl_{i}(S) is null-homotopic. Clearly the restriction to Cl_{0}(S) = Ω_{1}(S) is so, because this subgroup is generated by elements of order p. Assume now that the restriction to Cl_{i}(S) is null-homotopic and consider the extension Cl_{i}(S) ↪ Cl_{i+1}(S) ↠ Cl_{i+1}(S)/Cl_{i}(S). By induction hypothesis the composite

BCl_{i}(S) → BS ⟶^{f̄} Z

is null-homotopic. Hence the map f̄ restricted to BCl_{i+1}(S) factors through a map h : B(Cl_{i+1}(S)/Cl_{i}(S)) → Z by Zabrodsky’s Lemma [Dwy96, Proposition 3.5] and it is nullhomotopic if and only if h is so. Since generators of the quotient group are classes of conjugates of elements in Cl_{i}(S), on which f̄ restricts trivially, we see that h is null-homotopic when restricted to a set of generators. Therefore h is null-homotopic (cf. [CCS, Proposition 2.4] for a more detailed account), and so must be f̄. □

In many cases the strongly closed subgroup Cl(S) is equal to S for obvious reasons, for example when S = Ω_{1}(S).

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Corollary 4.3. Let G be a finite group generated by elements of order p whose Sylow psubgroup is also generated by elements of order p. The universal cover of CW_{Bℤ/p}BG is then p-torsion free. □

Example 4.4. Consider the C_{2}-cellular group Σ_{2^{n}}, the symmetric group on 2^{n} letters with n ≥ 2. The Sylow 2-subgroup is an iterated wreath product of copies of ℤ/2 which is always generated by elements of order 2. Therefore the above corollary applies and we obtain that P_{ΣBℤ/2}C ≃ ∏_{q≠2}(BΣ_{2^{n}})_{q}^{∧}. Thus CW_{Bℤ/2}BΣ_{2^{n}} fits into a fibration

∏_{q≠2} Ω((BΣ_{2^{n}})_{q}^{∧}) → CW_{Bℤ/2}BΣ_{2^{n}} → BΣ_{2^{n}}.

Our second corollary deals with the case when the Sylow p-subgroup is possibly not generated by elements of order p, but is nevertheless equal to Cl_{1}(S).

Corollary 4.5. Let G be a finite group generated by elements of order p whose Sylow psubgroup is generated by the elements in Ω_{1}(S) and all their conjugates (by elements in G) which belong to S. The universal cover of CW_{Bℤ/p}BG is then p-torsion free. □

Example 4.6. The symmetric group Σ_{3} acts by permutation on (ℤ/4)^{3}. The diagonal is invariant, and so is the “orthogonal” subgroup ℤ/4 × ℤ/4. We define G to be the semi-direct product of ℤ/4×ℤ/4 by Σ_{3}. It is easy to check that G is generated by elements of order 2, but the Sylow 2-subgroup S = (ℤ/4 × ℤ/4) ⋊ ℤ/2 is not. The subgroup Ω_{1}(S) has index 2, and a representative of the generator of the quotient can be taken inside S to have order four (as an element in G it is a product of three elements of order 2). This element has a conjugate which lies inside Ω_{1}(S), so we may conclude by the above corollary that CW_{Bℤ/2}BG is 2-torsion free, i.e. its universal cover is Ω(BG_{3}^{∧}) by Theorem 2.4.

Example 4.7. The group G = P SL_{3}(3) presents the same features as the above example. It is generated by elements of order 2 (it is simple), but its Sylow 2-subgroup S is semi-dihedral of order 16. The subgroup Ω_{1}(S) is dihedral of order 8. Here as well the universal cover of CW_{Bℤ/2}BP SL_{3}(3) is Ω(BP SL_{3}(3)_{3}^{∧}).

Let us finish this section by showing with some examples the applicability of the last result.

Example 4.8. Consider the 2-completion of the classifying space of the symmetric group Σ_{2^{n}}. As we recalled in Example 4.4, the Sylow 2-subgroup of Σ_{2^{n}} is generated by order 2 elements, and moreover π_{k}(CW_{Bℤ/2}BΣ_{2^{n}}) is 2-torsion free if k ≥ 2. Thus by Proposition 3.3 (BΣ_{2^{n}})_{2}^{∧} is Bℤ/2-cellular.

Unlike what happens with the cellularization of BG, our study does not give in general information about CW_{Bℤ/p}(BG_{p}^{∧}) when G is not generated by order p elements. In this case, however, it is sometimes possible to reduce the problem to the case of groups for which the hypothesis of the theorem hold.

Example 4.9. Let us compute the Bℤ/2-cellularization of (BA_{4})_{2}^{∧}. It is known that the natural inclusion A_{4} < A_{5} induces an equivalence in mod 2 homology, and therefore (BA_{4})_{2}^{∧} ≃ (BA_{5})_{2}^{∧}. Now, as A_{5} is simple with 2-torsion, it is generated by order 2 elements. Recall also that the Sylow 2-subgroup of A_{5} is the Klein group ℤ/2×ℤ/2. Hence, (BA_{5})_{2}^{∧} is Bℤ/2-cellular by Proposition 3.3, and then (BA_{4})_{2}^{∧} is so as well.

The computations we were able to perform up to now depend on the absence of p-torsion in the cellularization. In the following section we start a more systematic study and include explicit examples where p-torsion appears.

5. Control of fusion

By Theorem 4.2, to find an example where CW_{Bℤ/p}BG has p-torsion, one should look for groups generated by elements of order p with a Sylow p-subgroup which is not generated by elements of order p. Moreover the smallest strongly closed subgroup containing Ω_{1}(S) should be a proper subgroup of S. A possible source of examples is given by groups where the subgroup Ω_{1}(S) has large index in S, and is preserved by fusion.

Definition 5.1. Let G be a finite group and S a Sylow p-subgroup. The normalizer N_{G}(S) of the Sylow p-subgroup controls fusion if, whenever P < G is a p-subgroup and gP g^{−1} < N_{G}(S), we have that g = hc, with h ∈ N_{G}(S) and c ∈ C_{G}(P). Moreover, a p-group P is called a Swan group if the normalizer N_{G}(P) controls fusion for each group G containing P as a Sylow p-subgroup.

So far several families of Swan groups have been found, as for example abelian p-groups [Swa60], generalized extraspecial p-groups (which include extraspecial groups) [Sta02] or metacyclic p-groups for p odd [Die93]. In fact, in a certain sense, almost all p-groups are Swan groups [MP97].

It is also well known that the inclusion N_{G}(S) ↪ G induces an isomorphism in mod p cohomology if N_{G}(S) controls fusion in G, see for example [MP97, Proposition 2.1]. In other words there is an equivalence BN_{G}(S)_{p}^{∧} ≃ BG_{p}^{∧} of p-completed classifying spaces.

Next we describe some examples of these kind of groups, which will be central in our study of the Bℤ/p-cellularization.

Example 5.2. Consider the Suzuki group Sz(2^{n}), with n an odd integer ≥ 3. The section [Gor80, 16.4] is extremely useful to understand its basic subgroup and fusion properties. In particular the Sylow 2-subgroup of Sz(2^{n}) can be written as an extension (ℤ/2)^{n} ↪ S ↠

CELLULARIZATION AND FUSION 13

(ℤ/2)^{n}, where the kernel is the center of the group and contains all its order 2 elements. Observe in particular that Ω_{1}(S) is not equal to S and its index is 2^{n}. The normalizer of the Sylow 2-subgroup of Sz(2^{n}) is a semi-direct product S ⋊ ℤ/(2^{n} − 1) which is maximal in Sz(2^{n}).

There are various ways to see why this normalizer controls fusion in Sz(2^{n}). One can for example use Dwyer’s normalizer decomposition [DH01, Proposition 7.17] for the collection of 2-centric and 2-radical subgroups (which is very much in the spirit of [BLO03b]).

The group Sz(2^{n}) has the property that all of its Sylow 2-subgroups are disjoint (they have trivial intersection), hence the unique 2-centric 2-radical subgroup (up to conjugation) is S itself. The outer automorphism group of S in the fusion system of the Suzuki group, i.e. the quotient of the normalizer N_{Sz(2^{n})}(S) by S, is cyclic of order 2^{n} − 1, generated by an element ϕ which acts fixed-point free and permutes transitively the non-trivial elements of the center of S (see [Gor80, 16.4]).

Now it is clear that the inclusion S ⋊ℤ/(2^{n} −1) ↪ Sz(2^{n}) induces a homotopy equivalence f : B(S ⋊ ℤ/(2^{n} − 1))_{2}^{∧} ≃ BSz(2^{n})_{2}^{∧}.

An alternative way is to use the Cartan-Eilenberg double coset formula [AM94, Theorem II.6.6]. From the fact that the Sylow 2-subgroups are disjoint it is easy to deduce that both inclusions S ↪ Sz(2^{n}) and S ↪ N_{Sz(2^{n})}(S) induce monomorphisms in mod 2 cohomology with isomorphic images (of stable elements).

Example 5.3. Consider the simple groups U_{3}(2^{n}), with n ≥ 2. We refer again the reader to [Gor80, 16.4] for the following facts. Any Sylow 2-subgroup S of U_{3}(2^{n}) is a 2-group of order 2^{3n}. Here as well Z(S) = Ω_{1}(S) = Φ(S) is elementary abelian of order 2^{n}, and it can be seen, as in the previous case, that the unique 2-central 2-radical subgroup is the 2-Sylow. The normalizer of S is a semi-direct product S ⋊ ℤ/(2^{2n} − 1)d^{−1}, where d = 1 if n is even, and d = 3 if n is odd. The normalizer N_{U_{3}(2^{n})}(S) is a maximal subgroup in U_{3}(2^{n}).

At odd primes, the special linear groups give other examples, as was pointed out to us by Antonio Viruel.

Example 5.4. Let p be an odd prime, n an integer ≥ 2, and q = mp^{n} + 1. Consider the linear group P SL_{2}(q). According to ([Gor80, 15.1.1]), the p-Sylow subgroup of P SL_{2}(q) is cyclic of order p^{n}, and moreover its normalizer N_{P SL_{2}(q)}(S) is isomorphic to the semidirect product ℤ/p^{n} ⋊ ℤ/2, where the action is given by the change of sign. As the Sylow subgroup is abelian, it is a Swan group, and hence the inclusion N_{P SL_{2}(q)}(S) ↪ P SL_{2}(q) induces a homotopy equivalence between the p-completions of the classifying spaces.

Let G be a finite group generated by elements of order p, S a Sylow p-subgroup, and Cl(S) the smallest strongly closed subgroup containing Ω_{1}(S). Assume that N_{G}(S) controls

14 RAMÓN J. FLORES AND JÉRÔME SCHERER

fusion in G. Since every automorphism of S induced by conjugation in G restricts to an automorphism of Cl(S), we see that Cl(S) is normal in N_{G}(S). In the sequel we will denote by Γ the quotient N_{G}(S)/Cl(S). Notice that Γ is p-perfect since an epimorphism Γ ↠ ℤ/p would yield a non-trivial map BN_{G}(S)_{p}^{∧} → Bℤ/p which is null-homotopic when restricted to BCl(S). But since N_{G}(S) controls fusion in G, this corresponds to an epimorphism G ↠ ℤ/p which is trivial on all elements of order p. Such a morphism cannot exist since G is generated by its order p elements.

Proposition 5.5. Let G be a finite group generated by elements of order p and assume that the normalizer N_{G}(S) of a Sylow p-subgroup controls fusion in G. Let D′ be as above the cofiber of Chachólski’s cofibration. Then P_{ΣBℤ/p}D′ has the homotopy type of BΓ_{p}^{∧}.

Proof. Consider the composition BG_{p}^{∧} ⟶^{≃} BN_{G}(S)_{p}^{∧} → BΓ_{p}^{∧} induced by the projection N_{G}(S) ↠ Γ. As Ω_{1}(G) ≤ Cl(S), this map factors through the cofibre D′, and hence through P_{ΣBℤ/p}D′, because BΓ_{p}^{∧} is ΣBℤ/p-null. So we have a diagram

\xymatrix{BG_{p}^{∧} \ar[r]^{h} \ar[d]_{Bπ_{p}^{∧}} & D′ \ar[dl]_{g′} \ar[r]^{l} & P_{ΣBℤ/p}D′ \ar[dll]^{g} \\ BΓ_{p}^{∧}.}

On the other hand, using the same argument as in the proof of 4.2, we see that the composite

BH → BG_{p}^{∧} → D′ → P_{ΣBℤ/p}D′

is null-homotopic because so is its restriction to any generator of H. Apply now Dwyer’s version of Zabrodsky Lemma [Dwy96, Proposition 3.4] to the fibration BH → BN_{G}(S) → BΓ, taking account of the fact that P_{ΣBℤ/p}D′ is p-complete (which we proved in Proposition 3.2). There exists a map f : BΓ_{p}^{∧} → P_{ΣBℤ/p}D′ making the following diagram commutative (up to unpointed homotopy):

\xymatrix{BG_{p}^{∧} \ar[r]^{l∘h} \ar[d]_{Bπ_{p}^{∧}} & P_{ΣBℤ/p}D′ \\ BΓ_{p}^{∧}. \ar@{-->}[ur]_{f}}

Let us check that both composites f ∘ g and g ∘ f are homotopic to the identity. For the first one we only need to check that f ∘ g′ is homotopic to the coaugmentation map l by the universal property of the nullification.

First, observe that since P_{ΣBℤ/p}D′ is simply connected, it is enough to prove that l and f ∘g′ are homotopic in the unpointed category. Now, it is a consequence of the Puppe sequence of the cofibration BG_{p}^{∧} ⟶^{h} D′ → ∨ΣBℤ/p that two maps from D′ are homotopic if they are

CELLULARIZATION AND FUSION 15

so when precomposing with h. In our case f ∘ g′ ∘ h ≃ l ∘ h by commutativity of the above diagrams.

For the second one, we might as well prove that g ∘ f ∘ Bπ_{p}^{∧} is homotopic to Bπ_{p}^{∧}, because of the universal property of the quotient. The same argument as above applies since BΓ_{p}^{∧} is simply connected and we conclude by the commutativity of the diagrams that g ∘ f ∘ Bπ_{p}^{∧} ≃ g ∘ p ∘ h ≃ Bπ_{p}^{∧}. □

Theorem 5.6. Let G be a finite group generated by elements of order p and S a Sylow psubgroup. Assume that the normalizer N_{G}(S) controls fusion in G. Then CW_{Bℤ/p}BG fits in a fibration

CW_{Bℤ/p}BG ⟶ BG ⟶ BΓ_{p}^{∧} × ∏_{q≠p} BG_{q}^{∧},

where Γ = N_{G}(S)/Cl(S).

Proof. It is an immediate consequence of the previous proposition, taking into account the description of P_{ΣBℤ/p}D given in Proposition 3.2. □

In the following corollaries we give finally some explicit calculations of the homotopy type of the Bℤ/p-cellularization, in the case where p-torsion appears in the higher homotopy groups.

Corollary 5.7. Let G be a finite simple group. Then CW_{Bℤ/2}BG has 2-torsion in an infinite number of homotopy groups if and only if G is either a Suzuki group Sz(2^{n}) or a unitary group U_{3}(2^{n}).

Proof. From Foote’s classification [Foo97, Corollary 1] we see that the Suzuki and unitary groups U_{3}(2^{n}) are the only simple groups having a proper strongly closed subgroup. Theorem 4.2 implies thus that they are the only candidates for CW_{Bℤ/2}BG to have infinitely many homotopy groups with 2-torsion. Theorem 5.6 implies that 2-torsion appears indeed. □

In the only two cases where 2-torsion appears, Theorem 5.6 identifies the cellularization.

Corollary 5.8. We have two fibrations

CW_{Bℤ/2}BSz(2^{n}) → BSz(2^{n}) → B((ℤ/2)^{n} ⋊ ℤ/(2^{n} − 1))_{2}^{∧} × ∏_{q≠2} BSz(2^{n})_{q}^{∧}

CW_{Bℤ/2}BU_{3}(2^{n}) → BU_{3}(2^{n}) → B((ℤ/2)^{2n} ⋊ ℤ/(2^{2n} − 1)d^{−1})_{2}^{∧} × ∏_{q≠2} BU_{3}(2^{n})_{q}^{∧}

where d = 1 if n is even, and d = 3 if n is odd. □

For odd primes we are not aware of a classification of the simple groups having a proper strongly closed subgroup. We offer as an illustration the Bℤ/p-cellularization of the unitary groups described in Example 5.4.

16 RAMÓN J. FLORES AND JÉRÔME SCHERER

Corollary 5.9. Let p be an odd prime and q be any integer of the form mp^{k} + 1 with k ≥ 2. Then the Bℤ/p-cellularization of BP SL(q) has p-torsion in an infinite number of homotopy groups and is given by the following fibration

CW_{Bℤ/p}BP SL(q) → BP SL(q) → B(ℤ/p^{n−1} ⋊ ℤ/2)_{p}^{∧} × ∏_{r≠p} BP SL(q)_{r}^{∧} □

6. Representations

In this last section we construct explicit representations of the Suzuki groups and the special linear groups into certain unitary groups. We think that they might be of independent interest, and they also give an alternative proof of the presence of p-torsion in the Bℤ/pcellularization of the classifying spaces. The map we are looking for is best understood as a fusion preserving representation of the Sylow p-subgroup of G in some compact Lie group (this relationship is well explained by Jackson in [Jac04]). This observation definitively shifts the problem to the study of fusion systems, a notion which has recently led Broto, Levi, and Oliver to the concept of p-local finite groups in a topological context (see [BLO03a] and [BLO03b]).

For the Suzuki groups we construct below a map BSz(2^{n}) ⟶ BU(2^{n} − 1)_{2}^{∧} such that the composition Bℤ/2 ⟶ BSz(2^{n}) ⟶ BU(2^{n} −1)_{2}^{∧} is null-homotopic for every cyclic subgroup ℤ/2 in Sz(2^{n}). The loop space ΩBU(2^{n} − 1)_{2}^{∧} is Bℤ/2-local by [Mil84, 9.9]. The existence of infinitely many homotopy groups π_{n}CW_{Bℤ/2}BSz(2^{n}) containing 2-torsion is then a direct consequence of Proposition 3.1.

Lemma 6.1. Consider the semi-direct product (ℤ/2)^{n} ⋊ ℤ/(2^{n} − 1) where a generator ϕ of ℤ/(2^{n} −1) acts on the elementary abelian group of rank n by permuting transitively the 2^{n} −1 non-trivial elements. There exists then a faithful representation σ : (ℤ/2)^{n} ⋊ ℤ/(2^{n} − 1) ↪ U(2^{n} − 1).

Proof. The representation can be induced from the trivial one to the subgroup (ℤ/2)^{n}. When n = 3 we can be very explicit. Send the first generator of the elementary abelian group to the diagonal matrix with entries (−1, 1, −1, −1, −1, 1, 1), and the other elements to the cyclic permutations of it. The standard cyclic permutation matrix of order 7 in U(7) is the image of ϕ and so σ is well-defined. □

Proposition 6.2. There exists a non-trivial map BSz(2^{n}) ⟶ BU(2^{n} − 1)_{2}^{∧} such that the composition Bℤ/2 ⟶ BSz(2^{n}) ⟶ BU(2^{n} −1)_{2}^{∧} is null-homotopic for every cyclic subgroup ℤ/2 in Sz(2^{n}).

Proof. We construct actually a map from the 2-completion of BSz(2^{n}) and the desired map is then obtained by pre-composing with the completion map BSz(2^{n}) → BSz(2^{n})_{2}^{∧}. Since

CELLULARIZATION AND FUSION 17

the normalizer of a Sylow 2-subgroup S controls fusion in the Suzuki groups, we only need to construct a map out of B((S ⋊ ℤ/(2^{n} − 1))_{2}^{∧} where S denotes the 2-Sylow subgroup of Sz(2^{n}).

Because BU(2^{n} − 1) is simply connected the sets of homotopy classes of pointed and unpointed maps into BU(2^{n}−1) agree. The fusion system of the Suzuki group (i.e. that of the semi-direct product) is reduced to the Sylow subgroup, so that the set of [BSz(2^{n}), BU(2^{n}−1)] is isomorphic to the set of fusion preserving representations of S inside U(2^{n} − 1). This means that, to find a non-trivial map f : BSz(2^{n}) → BU(2^{n} − 1)_{2}^{∧}, it is enough to find a representation ρ : S ⋊ ℤ/(2^{n} − 1) → U(2^{n} − 1) which is non-trivial when restricted to S (compare with Dwyer’s result [Dwy96, Theorem 1.4] which we have already used before).

On the other hand we want the map f to be null-homotopic when restricted to any cyclic subgroup ℤ/2. In other words the composite Ω_{1}(S) → S ⋊ ℤ/(2^{n} − 1) → U(2^{n} − 1) should be trivial. The subgroup Ω_{1}(S) is the center of S, an elementary abelian subgroup of rank n. It is normal in S ⋊ ℤ/(2^{n} − 1), and the quotient is isomorphic to (ℤ/2)^{n} ⋊ ℤ/(2^{n} − 1), where the action of the generator ϕ of order 2^{n} − 1 permutes transitively the non-trivial elements (one has a bijection between these non-trivial classes in the quotient and the squares of their representatives in the center of S). Hence we can define ρ as the composite

S ⋊ ℤ/(2^{n} − 1) ⟶ (ℤ/2)^{n} ⋊ ℤ/(2^{n} − 1) ⟶^{σ} U(2^{n} − 1)

where σ is the representation constructed in the preceding lemma. It is convenient to remark here that the existence of such a representation does not contradict the theorems in this article since S ⋊ ℤ/(2^{n} − 1) is not generated by order 2 elements.

Observe that the morphism B(S ⋊ ℤ/(2^{n} − 1))_{2}^{∧}⟶ BU(2^{n} − 1)_{2}^{∧} induced by the representation we have just constructed is clearly trivial when pre-composing with any map Bℤ/2 ⟶ B(S ⋊ ℤ/(2^{n} − 1))_{2}^{∧}, because the (unpointed) homotopy classes of these last maps can be identified with the conjugacy classes of ℤ/2 inside S ⋊ ℤ/(2^{n} − 1). □

Remark 6.3. We point out that the representation ρ constructed in the previous proposition cannot be induced by a homomorphism Sz(2^{n}) ⟶ U(2^{n} − 1), not even composed with an Adams operation, because the group Sz(2^{n}) is simple, hence generated by order 2 elements. If it were so, the homomorphism would be zero over the generators, and thus trivial. Thus our example could be compared with the map BM_{12} → BG_{2} constructed by Benson and Wilkerson in [BW95]. It is also related to work of Mislin-Thomas [MT89], and more recently of Broto-Møller [BM].

The methods above can also be used to obtain interesting representations of the groups U_{3}(2^{n}). We indicate now similar results at odd primes for the groups P SL_{2}(q) introduced in Example 5.4.

Proposition 6.4. Let p be an odd prime, n ≥ 2, and q = p^{n}. There exists a non-trivial map BP SL_{2}(q) ⟶ BU(2)_{p}^{∧} such that, for every cyclic subgroup ℤ/p in P SL_{2}(q), the composition Bℤ/p ⟶ BP SL_{2}(q) ⟶ BU(2)_{q}^{∧} is null-homotopic.

Proof. We first construct an inclusion j : ℤ/p ⋊ ℤ/2 ↪ U(2) as follows. If x = e^{2πi/p} the image of the generator of order p is the diagonal matrix with entries (x, x̄), and the image of the generator of order 2 is the standard permutation matrix. One can then take for example the composition

f : ℤ/p^{n} ⋊ ℤ/2 ⟶ ℤ/p ⋊ ℤ/2 ⟶^{j} U(2),

where the first map is the natural projection. The induced map at the level of p-completed classifying spaces

BP SL_{2}(q)_{p}^{∧} ≃ B(ℤ/p^{n} ⋊ ℤ/2)_{p}^{∧} ⟶ BU(2)_{p}^{∧}

is then essential, but homotopically trivial when precomposing with any map from Bℤ/p. □

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Ramón J. Flores, Jérôme Scherer Departament de Matemàtiques, Universitat Autónoma de Barcelona, E–08193 Bellaterra

e-mail: ramonj@mat.uab.es, jscherer@mat.uab.es