GLUING BOREL-SMITH FUNCTIONS AND THE GROUP OF ENDO-TRIVIAL MODULES

OLCAY COŞKUN

Boğaziçi Üniversitesi, Matematik Bölümü 80815 Bebek, İstanbul, Turkey. olcay.coskun@boun.edu.tr

Abstract. The aim of this paper is to describe the group of endo-trivial modules for a p-group P, in terms of the obstruction group for the gluing problem of Borel-Smith functions. Explicitly, we shall prove that there is a split exact sequence

0 \overline{>} ℤ \overline{>} T (P) \overline{>} Obs(C_{b}(P)) \overline{>} 0

of abelian groups where T (P) is the endo-trivial group of P, and C_{b}(P) is the group of Borel-Smith functions on P. As a consequence, we obtain a set of generators of the group T (P) that coincides with the relative syzygies found by Alperin. In order to prove the result, we solve gluing problems for the functor B^{∗} of super class functions, the functor R_{ℚ}^{∗} of rational class functions and the functor C_{b} of Borel-Smith functions.

Contents

1. Introduction 1 2. Gluing problem in general 3 3. Gluing super class functions 5 4. Gluing rational class functions for p-groups 6 5. Gluing Borel-Smith functions 10 6. The Main Result 12 References 15

1. Introduction

In [13] and [14] Dade introduced the class of endo-permutation modules for pgroups together with a certain sub-class of endo-trivial modules and classified them for abelian p-groups. He also raised the question of the classification of endopermutation modules for all p-groups. A systematic study of the endo-permutation modules is done by considering the group of the so-called capped endo-permutation modules, now called the Dade group and denoted by D(P) for the group P. The subgroup of endo-trivial modules is denoted by T (P).

The classification problem for endo-permutation modules is solved, after 25 years since it was raised, in several steps. One of the most important steps in this direction is the description of the group T (P) of the endo-trivial modules by Carlson and Thévenaz in [11] and [10]. The final classification was done by Bouc in [4] where he used the description of the group T (P) and the theory of biset functors, introduced and developed by him. See [2] for the theory of biset functors.

The author is supported by Tübitak through Kariyer Programı, (Tübitak/3501/109T661).

This paper is a contribution to the description of the group T (P). Regarding the original description, in [1], Alperin constructed a certain set of relative syzygies associated to a finite p-group P and proveed that they generate a subgroup of finite index in the group T (P) of all endo-trivial modules. Later Carlson and Thévenaz in [10] showed that this subgroup is equal to the torsion free part of the group T (P). Also in [11], they describe the torsion part of the group T (P) completing the classification of endo-trivial modules for p-groups. The above set of generators is also constructed by Carlson in [9] using cohomology. We refer to [9], [10] or [18], for further details.

The aim of this paper is to give an another construction of the above set of relative syzygies and to prove that they generate the group T (P). To construct the relative syzygies we relate the group of endo-trivial modules to the group of obstructions for the gluing problem for Borel-Smith functions and hence use the theory of biset functors for the construction. We do use the classification theorem of Carlson and Thévenaz regarding the torsion part of the group T (P) but we do not need the full classification of endo-permutation (or endo-trivial) modules. See the end of the paper for a revision of the results that we have used throughout the paper.

Returning to the new construction of the relative syzygies, the gluing problem for a biset functor F at a finite group G is the problem of finding an element f ∈ F (G) associated to a given sequence (f_{H})_{1<H≤G} where f_{H} is an element in F (N_{G}(H)/H), if it exists, see Section 2 for details. This problem is first considered by Bouc and Thévenaz [7] for the functor of torsion endo-permutation modules and later by Bouc [5] for arbitrary endo-permutation modules, both for p-groups with p odd. The first problem is solved whereas the latter is incomplete. In this paper, we consider the gluing problem for three more biset functors, namely for the functor of super class functions, for the functor of rational class functions and for the functor of Borel-Smith functions.

The relevance of the gluing problem of the functor of Borel-Smith functions to the group T (P) is given by our main result, Theorem 6.1, which proves that there is a split exact sequence of abelian groups

0  \overline{>} ℤ \overline{>} T (P) \overline{>} Obs(C_{b}(P)) \overline{>} 0  (∗)

where T (P) is the endo-trivial group of P, and C_{b}(P) is the group of Borel-Smith functions on P.

To obtain this exact sequence, we use the exact sequences

0  \overline{>} C_{b}  \overline{>} B^{∗}  \overline{>} D^{Ω}  \overline{>} 0

and

0  \overline{>} C_{b}  \overline{>} R_{ℚ}^{∗}  \overline{>} D_{t}^{Ω}  \overline{>} 0

of biset functors of Theorem 1.2 and Theorem 1.3 of [8], and the result of Puig which determines the kernel of the map D(P) → \underleftarrow{lim}_{1<Q≤P} D(N_{P} (Q)/Q) as the endo-trivial group T (P). Finally we need the solutions of the other to gluing problem mentioned above.

Now the exact sequence (∗) is the kernel-cokernel sequence of the following commutative diagram

0  \overline{>} C_{b}(P)  \overline{>} B^{∗}(P)  \overline{Ψ_{P}  >} D^{Ω}(P)  \overline{>} 0

0 \overline{>} \underleftarrow{lim}_{1<Q≤P} C_{b}(W_{P} (Q))^{\begin{aligned}r_{P}^{C_{b}} \\ & ∨ \end{aligned}}  \overline{>} \underleftarrow{lim}_{1<Q≤P} B^{∗}(W_{P} (Q))^{\begin{aligned}r_{P}^{B^{∗}} \\ & ∨ \end{aligned}}  \overline{Ψ̃_{P} ; >} \underleftarrow{lim}_{1<Q≤P} D^{Ω}(W_{P} (Q))^{\begin{aligned}r_{P}^{D} \\ & ∨ \end{aligned}}

with exact rows, where we put W_{P} (Q) := N_{P} (Q)/Q and the splitting can be constructed by considering a similar commutative diagram with first row equal to the exact sequence of Theorem 1.3 in [8]. See Section 6 for further details.

Regarding the gluing problems, the solution for the gluing problem for super class functions is in Section 3. We prove, as Theorem 3.1, that there is an exact sequence

0  \overline{>} ℤ \overline{>} B^{∗}(G) \overline{>} \underleftarrow{lim}_{1<H≤G} B^{∗}(N_{G}(H)/H) \overline{>} 0

of abelian groups. In particular, the gluing problem for super class functions always has infinitely many solutions. Moreover, it is clear from the proof of this theorem, given in Section 3, that solutions to a given gluing data differ only by their evaluations at the trivial subgroup.

The solution of the problem for rational class functions for a p-group P of rank at least 2 is given by the exact sequence

0 \overline{>} R_{ℚ}^{∗}(P) \overline{\begin{matrix}r_{P} \\ > \end{matrix}} \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q) \overline{d̃_{P} ; >} H̃^{0}(A_{≥2}(P), pℤ)^{P}  \overline{>} 0

of abelian groups. See Section 4 for the details of the notation. Note that if the rank of P is 1 then R_{ℚ}^{∗} = B^{∗} and the problem is reduced to the previous case.

The gluing problem for Borel-Smith functions is solved in Section 5 where we prove that for a p-group of rank at least 2, there is an exact sequence

0 \overline{>} C_{b}(P) \overline{\begin{matrix}r_{P} \\ > \end{matrix}} \underleftarrow{lim}_{1<Q≤P} C_{b}(N_{P} (Q)/Q) \overline{d̃_{P} ; >} H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P}  \overline{>} 0

of abelian groups.

When the group P has rank 1, we obtain the exact sequence

0  \overline{>} ℤ \overline{>} C_{b}(P) \overline{\begin{matrix}r_{P} \\ > \end{matrix}} \underleftarrow{lim}_{1<Q≤P} C_{b}(N_{P} (Q)/Q) \overline{>} ℤ/m_{P} ℤ \overline{>} 0

of abelian groups where m_{P} = 2 if P is cyclic and m_{P} = 4 if P is generalized quaternion.

Finally, to simplify our notation, we introduce a general setup for the gluing problem for an arbitrary biset functor in Section 2.

2. Gluing problem in general

Let F be a biset functor and G be a finite group. A gluing data for F (G) is a sequence (f_{H})_{1<H≤G} of elements f_{H} ∈ F (N_{G}(H)/H) satisfying the following compatibility conditions:

(i) (Conjugation invariance) For any g ∈ G and H ≤ G, we have

gf_{H} = f_{^{g}H}.

(ii) (Destriction invariance) For any pair H ⊴ K of subgroups of G, we have

Des_{N_{G}(H,K)/K}^{N_{G}(H)/H}f_{H} = Res_{N_{G}(H,K)/K}^{N_{G}(K)/K}f_{K}.

Here we use the notation Des_{N_{G}(H,K)/K}^{N_{G}(H)/H} for the composition of deflation and restriction maps. Note that the same composition is written as Defres in [5]. We call the composite map Des the destriction map.

Now the gluing problem for the biset functor F at G is the problem of finding an element f ∈ F (G) such that for any non-trivial subgroup H of G, we have

Des_{N_{G}(H)/H}^{G}f = f_{H}.

If such an element exists, we call it a solution to the given gluing data. Following Bouc and Thévenaz [7], we denote by

\underleftarrow{lim}_{1<H≤G} F (N_{G}(H)/H)

the set of all gluing data for F (G).

It is clear that given a biset functor F, there is a homomorphism

r_{G} := r_{G}^{F} : F (G) → \underleftarrow{lim}_{1<H≤G} F (N_{G}(H)/H)

given by associating an element f ∈ F (G) to the sequence (Des_{N_{G}(H)/H}^{G}f)_{1<H≤G}. Now the gluing problem is asking if this map is surjective, cf. [7, Section 2]. A complete solution to the gluing problem is an exact sequence

0 \overline{>} Ker(r_{G}) \overline{>} F (G) \overline{\begin{matrix}r_{G} \\ > \end{matrix}} \underleftarrow{lim}_{1<H≤G} F (N_{G}(H)/H) \overline{>} Obs(F (G)) \overline{>} 0

where Obs(F (G)) denotes the cokernel of the map r_{G} and it is the group of obstructions for the gluing data to have a solution.

An easy (but very useful) observation about the general gluing problem is that we can lift any short exact sequence of biset functors to an exact sequence of the corresponding groups of gluing data and hence relate the obstruction groups for the solutions. This can be done as follows.

Proposition 2.1. Let

0  \overline{>} M  \overline{>} F \overline{Φ >} N  \overline{>} 0

be an exact sequence of biset functors. Then the diagram

0  \overline{>} M(G)  \overline{>} F (G)  \overline{Φ^{G}  >} N(G)  \overline{>} 0

0 \overline{>} \underleftarrow{lim}_{1<H≤G} M(W_{G}(H))^{\begin{aligned}r_{G}^{M} \\ & ∨ \end{aligned}}  \overline{>} \underleftarrow{lim}_{1<H≤H} F (W_{G}(H))^{\begin{aligned}r_{G}^{F} \\ & ∨ \end{aligned}}  \overline{Φ̃^{G} ; >} \underleftarrow{lim}_{1<H≤G} N(W_{G}(H))^{\begin{aligned}r_{G}^{N} \\ & ∨ \end{aligned}}

commutes and has exact rows. Here Φ̃^{G} is defined as the sequence (Φ^{W_{G}(H)})_{1<H≤G} of maps and we put W_{G}(H) := N_{G}(H)/H.

The proof of the above proposition is trivial since Φ is a morphism of biset functors. As a corollary, applying the snake lemma, we get an exact sequence of abelian groups

0 \overline{>} K(r_{G}^{M}) \overline{>} K(r_{G}^{F}) \overline{>} K(r_{G}^{N}) \overline{>} Obs(M(G)) \overline{>} Obs(F (G)) \overline{>} Obs(N(G))

where we put K(r_{G}^{F}) := Ker(r_{G}^{F}).

Remark 1. Given a biset functor F, it is clear that the functor associating a group G to its gluing data \underleftarrow{lim}_{1<H≤G} F (W_{G}(H)) is not a biset functor. Nevertheless this functor has a structure of a destriction functor. By a destriction functor, we mean a biset functor without induction and inflation and in this sense, they are generalizations of restriction functors from the context of Mackey functors, see [12] for details. Indeed, the maps

Res_{K}^{G} : \underleftarrow{lim}_{1<H≤G} F (W_{G}(H)) → \underleftarrow{lim}_{1<H≤K} F (W_{K}(H))

for any pair of groups K ≤ G are defined in [7] and in a similar way, we can define deflation map and transport of structure by a group isomorphism. We skip the details of the definition of these maps.

We end the section by the following generalization of a result of [7], which we will use later. It shows that the gluing problem for an arbitrary biset functor F at an elementary abelian group is always solvable. We skip the proof since it is almost identical to the proof of Lemma 2.2 of [7].

Lemma 2.2 (Bouc-Thévenaz). Let E be an elementary abelian group. Then the map

r_{E}^{F} : F (E) → \underleftarrow{lim}_{1<F ≤E} F (E/F)

is surjective with a section given by

(u_{F})_{1<F ≤E} ↦ − ∑_{1<F ≤E} μ(1, F)Inf_{E/F}^{E} u_{F} .

where μ is the Mobius function of the poset of subgroups of E.

3. Gluing super class functions

Let G be a finite group. We denote by B(G) the Burnside ring of G. We recall that the Burnside ring of a finite group G is the Grothendieck ring of the category of finite G-sets and as a free abelian group, it is generated by the set of conjugacy classes of subgroups of G. By B^{∗}(G) we denote the ℤ-dual of the Burnside ring. It is well-known that the group B^{∗}(G) can be identified with the ring of super class functions S(G) → ℤ. Here S(G) is the set of all subgroups of G and a super class function is a function which is constant on the conjugacy classes of subgroups of G. The functor sending a group G to the dual Burnside ring B^{∗}(G) is a biset functor with the usual actions of bisets. We refer to [8] for further details.

The following theorem shows that the answer to the the gluing problem for super class functions is always affirmative.

Theorem 3.1. There is an exact sequence of abelian groups

0  \overline{>} ℤ \overline{>} B^{∗}(G) \overline{\begin{matrix}r_{G} \\ > \end{matrix}} \underleftarrow{lim}_{1<H≤G} B^{∗}(N_{G}(H)/H) \overline{>} 0

where the map r_{G} is as defined in the previous section.

Proof. The proof has two parts. First we prove that the kernel of r_{G} is isomorphic to ℤ and second we show that the map r_{G} is surjective.

For the first part, let f be a super class function with r_{G}(f) = 0. By the definition of the map r_{G}, we have

Des_{N_{G}(H)/H}^{G}f = 0

for any non-trivial subgroup H of G. But by the definition of the action of destriction on B^{∗}, we have

0 = (Des_{N_{G}(H)/H}^{G}f)(L/H) = f(L)

for any L/H ≤ N_{G}(H)/H. Therefore the super class function f is zero at all nontrivial subgroups of G. Also it is clear that its value at the trivial subgroup can be chosen freely. Therefore we have proved that a super class function f is in the kernel of r_{G} if and only if it is zero at any non-trivial subgroup of G, as required.

To prove that r_{G} is surjective, let (f_{H})_{1<H≤G} be a gluing data. We claim that the super class function f ∈ B^{∗}(G) defined by f(H) = f_{H}(H/H) and f(1) = 0 is a pre-image of the given gluing data. Firstly, f is a super class function since for any H ≤ G and g ∈ G, we have

f(^{x}H) = f_{^{x}H}(^{x}H/^{x}H) = (^{x}f_{H})(^{x}H/^{x}H) = ^{x}(f_{H}(H/H)) = f_{H}(H/H) = f(H).

Here we use the conjugation invariance of the gluing data. Now it remains to show that for any 1 < H ≤ G, we have

Des_{N_{G}(H)/H}^{G}f = f_{H}.

Notice that by the destriction invariance of the gluing data, for any pair H ⊴ K, we have

f_{H}(K/H) = f_{K}(K/K).

Indeed we have

f_{H}(K/H) = (Des_{N_{G}(H,K)/K}^{N_{G}(H)/H}f_{H})(K/K) = (Res_{N_{G}(H,K)/K}^{N_{G}(K)/K}f_{K})(K/K) = f_{K}(K/K).

Now it follows that

(Des_{N_{G}(H)/H}^{G}f)(K/H) = f(K) = f_{K}(K/K) = f_{H}(K/H),

for any K/H ≤ N_{G}(H)/H, as required. □

Corollary 3.2. A gluing problem for super class functions at a finite group G has infinitely many solutions.

Proof. Let (f_{H})_{1<H≤G} be a gluing data. Then the super class function f defined by putting f(H) := f_{H}(H/H) if H ≠ 1 and f(G/1) = a is a solution of the gluing data for any a ∈ ℤ. □

4. Gluing rational class functions for p-groups

Let p be a prime number and P be a finite p-group. Denote by R_{ℚ}(P) the rational character ring of P. This is the Grothendieck ring of the category of ℚP - modules. With the usual definitions of induction, restriction, inflation, deflation and transport of structure, the assignment P ↦ R_{ℚ}(P) is a biset functor. We denote the ℤ-dual of this biset functor by R_{ℚ}^{∗}. We call an element of R_{ℚ}^{∗}(P) a rational class function for P. We refer to [2] for further details.

Recall that by a theorem of Ritter and Segal, the linearization map lin: B(P) → R_{ℚ}(P), associating a finite P -set X to the permutation ℚP -module ℚX with basis X, is surjective. Hence its transpose gives an injective map R_{ℚ}^{∗}(P) → B^{∗}(P), cf [3]. Note also that the linearization map is a morphism of biset functors. Therefore the biset functor R_{ℚ}^{∗} can be identified with a subfunctor of the dual B^{∗} of the Burnside functor B, via the transpose of the linearization map.

Therefore this inclusion gives rise to an inclusion of the group of gluing data for R_{ℚ}^{∗} to that of B^{∗}. Hence by Corollary 3.2, a gluing data for R_{ℚ}^{∗} can always be glued to a super class function. Thus we need to determine when the solution is a rational class function. We recall the following result from [8, Lemma 4.2]. In the following, we denote the unique group of order p by C_{p}.

Lemma 4.1 (Bouc-Yalçın [8]). Let f be a super class function for the p-group P. Then f is a rational class function if and only if for any subquotient T/S of P, isomorphic to C_{p} × C_{p}, the condition

f(P/S) − f(P/T) = ∑_{S<X<T}(f(P/X) − f(P/T)) (∗).

is satisfied.

Now let (f_{Q})_{1<Q≤P} be a gluing data for R_{ℚ}^{∗}(P) and let f be a solution of the data in B^{∗}(P). Also let T/S be a subquotient isomorphic to C_{p} × C_{p}. We claim that the equality

f(P/Y) = f_{S}((N_{P} (S)/S)/(Y/S))

is satisfied for any S ⊴ Y ≤ P. Indeed, by the definition of f, we have

f(P/Y) = f_{Y} ((N_{P} (Y)/Y)/(Y/Y))

and by the destriction invariance of the gluing data we have

f_{Y} ((N_{P} (Y)/Y)/(Y/Y)) = f_{S}((N_{P} (S)/S)/(Y/S)).

Here we apply the condition with S ⊴Y. Therefore we can write the above equality (∗) as

f_{S}(N_{P} (S)/S) − f_{S}(N_{P} (S)/S) = ∑_{S<X<T}(f_{S}(N_{P} (S)/X) − f_{S}(N_{P} (S)/T)).

Here we shorten (N_{P} (S)/S)/(Y/S) to N_{P} (S)/Y. Now if S is not the trivial subgroup, this condition is satisfied since f_{S} ∈ R_{ℚ}^{∗}(N_{P} (S)/S). Thus we have proved the following lemma.

Lemma 4.2. Let (f_{Q})_{1<Q≤P} be a gluing data for R_{ℚ}^{∗}(P) and let f be a solution of the data in B^{∗}(P). Then f is a rational class function if and only if for any elementary abelian subgroup E of P of rank 2, the condition (∗) of Lemma 4.1 is satisfied.

Our next aim is to determine the group of obstructions for a gluing data for R_{ℚ}^{∗}. First note that if P has rank 1, then the condition of the above Lemma is trivially satisfied and hence we get that R_{ℚ}^{∗} = B^{∗}. Therefore we reduced to the previous case.

Therefore we assume that P has rank at least 2, hence P is neither cyclic nor generalized quaternion. Then by the previous lemma, a solution f of the gluing data (f_{Q})_{1<Q≤P} in the dual Burnside group B^{∗}(P) is not a rational class function if there is an elementary abelian subgroup E of P of rank 2 such that the condition (∗) of Lemma 4.1 is not satisfied. Notice that the condition (∗) actually determines the value at the trivial group of the solution, since for E, the condition (∗) can equivalently be written as

f(P/1) = ∑_{1<X<T} f(P/X) − pf(P/T)

and the values f(P/X) and f(P/T) are determined by the gluing data. Hence the obstruction group should check whether the value at the trivial subgroup is well-defined, that is, if for all possible choices of the subgroup E, the value f(P/1) is constant.

To determine the obstruction group, first we introduce a notation. We denote by A_{≥2}(P) the P -poset of all elementary abelian subgroups of P of rank at least 2, and where P acts by conjugation. It is clear that any maximal elementary abelian subgroup of P of rank 2 is an isolated vertex in A_{≥2}(P). It is also easy to see that the rest of the elementary abelian subgroups lie in a connected component, which is called the big component of A_{≥2}(P), see [10, Lemma 2.1]. We refer to Lemma 2.2 in [10] and Lemma 3.1 in [7] for further properties of this poset.

Now following Bouc and Thévenaz [7], we write H̃^{0}(A_{≥2}(P), ℤ)^{P} for the group of P -invariant functions f from A_{≥2} to the additive group ℤ of integers, satisfying the condition that f(E) = f(F) if E ≤ F, modulo the constant functions. By Lemma 3.1 in [7], the group H̃^{0}(A_{≥2}(P), ℤ)^{P} is generated by the characteristic functions of the classes of maximal elementary abelian subgroups of P of rank 2. Now we define a map, denoted by d̃_{P}, cf. [7, Section 3],

d̃_{P} : \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q) → H̃^{0}(A_{≥2}(P), ℤ)^{P}

by associating the data (f_{Q})_{1<Q≤P} to the class of the function f̃ which maps a subgroup E of order p^{2} to ∑_{1<X<E} f(P/X) − pf(P/E) and a subgroup F of order more than p^{2} to f̃(E) for some subgroup E of F of order p^{2}.

Here we denote by f the solution of (f_{Q})_{1<Q≤P} in B^{∗}(P). Here after, we write [ϕ] to denote the image of ϕ ∈ H^{0}(A_{≥2}(P), ℤ)^{P} in the quotient group H̃^{0}(A_{≥2}(P), ℤ)^{P}.

Lemma 4.3. The map

d̃_{P} : \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q) → H̃^{0}(A_{≥2}(P), ℤ)^{P}

as defined above is a group homomorphism.

Proof. First we need to check that the map is well-defined, that is, we need to check that for any (f_{Q})_{1<Q≤P} ∈ \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q), the function f̃ :=

d̃_{P} ((f_{Q})_{1<Q≤P}) is in H̃^{0}(A_{≥2}(P), ℤ)^{P}. First, it is clear, by conjugation invariance of the gluing data, that the resulting function is P -invariant. To prove the condition that f(E) = f(F) if E ≤ F, note that we only need to check this condition for the big component of A_{≥2}(P), which contains a normal elementary abelian p-subgroup. By Lemma 2.1 in [10], this component is unique. So it suffices to show that the function f̃ restricted to the set of subgroups of order p^{2} in the big component is constant. Let E be a maximal element of the big component. Then by Lemma 2.2, the gluing data Res_{E}^{P}((f_{Q})_{1<Q≤P}) has a solution which implies, by Lemma 4.2, that for any pair of subgroups T, S of rank 2 of E, we have f̃(T) = f̃(S). Now the result follows since for any pair T, S of subgroups of rank 2 in the big component, T S is elementary abelian and contains both T and S, as required. The claim that d̃_{P} is a group homomorphism is trivial. □

Unfortunately d̃_{P} is not surjective but it is possible to determine the image. First of all, if A_{≥2}(P) is connected, then the group H̃^{0}(A_{≥2}(P), ℤ)^{P} vanishes and hence the map d̃_{P} becomes the zero map. Thus, without loss of generality, we can assume that the poset A_{≥2} contains at least two connected components.

Then by [7, Lemma 3.1], there is a unique central subgroup Z of P of order p contained in each maximal elementary abelian subgroup of rank 2 and moreover given such a group E, all the other subgroups of order p are P -conjugate.

On the other hand, by [16], if p is odd, and if the rank of P is at least 3, then by Proposition 2.5 in [15], there is a unique normal elementary abelian subgroup E_{0} of P of rank 2 which is non-central and characteristic, which implies that E_{0} contains the subgroup Z and all the other subgroups of order p are P -conjugate.

Similarly, if p = 2 and the rank of P is at least 3, then there is a non-maximal normal elementary abelian subgroup E_{0} of rank 2 whose two non-central elements of order 2 are P -conjugate. (Note that in this case, as in the previous one, we have |P/C_{P} (E_{0})| = 2.)

Thus if E is either an isolated vertex in A_{≥2}(P) or is equal to E_{0}, defined above, let Y be a subgroup of order p in E different from Z. Then we have

∑_{1<X<E} f(P/X) − pf(P/E) = f(P/Z) + p(f(P/Y) − f(P/E))

where we use the conjugation invariance of the gluing data to collect together all the terms f(P/X) with non-central X. Now it is clear from this equation that

[E ↦ ∑_{1<X<E} f(P/X) − pf(P/E)] = [E ↦ p(f(P/Y) − f(P/E))]

for some non-central subgroup Y of order p in E. Therefore the image of d̃_{P} is contained in the subgroup H̃^{0}(A_{≥2}(P), pℤ)^{P} of H̃^{0}(A_{≥2}(P), ℤ)^{P} consisting of functions having values in pℤ.

Next we show that this subgroup is equal to the image of d̃_{P}. Note that the subgroup H̃^{0}(A_{≥2}(P), pℤ)^{P} is generated by functions pχ_{E} as E runs over a complete set of representatives of conjugacy classes of maximal elementary abelian subgroups of rank 2 and where χ_{E} is the characteristic function of the class of E.

Let E be a maximal elementary abelian subgroup of P of rank 2 and let χ_{E} be the characteristic function of the conjugacy class of E. We define a sequence (f_{Q})_{1<Q≤P} as follows. If Q is not P -conjugate to Y, then put f_{Q} = 0. If Q is conjugate to Y, then define f_{Q} as the super class function for N_{P} (Q)/Q given by f_{Q}(N_{P} (Q)/T) = δ_{T,Q}, where δ_{?,?} is the Kronecker’s delta. We claim that the sequence is a gluing data for R_{ℚ}^{∗}(P) and its image is equal to pχ_{E}. The first claim is almost trivial since most of the f_{Q}’s are zero and the group N_{P} (Y)/Y is cyclic or generalized quaternion by [7, Lemma 3.1]. So we need to check that d̃_{P} ((f_{Q})_{1<Q≤P}) = pχ_{E} but this follows easily from the definition of the map d̃_{P}.

The above arguments prove that there is a surjective group homomorphism

d̃_{P} : \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q) → H̃^{0}(A_{≥2}(P), pℤ)^{P} .

Next we show that the kernel of this map is exactly the image of the map r_{P}. For this aim, let f be a rational class function and (f_{Q})_{1<Q≤P} be its image under r_{P}.

Then

d̃_{P} ((f_{Q})_{1<Q≤P})(E) = p(f(P/Y) − f(P/E))

where E and Y are as above. But by Lemma 4.1, we have p(f(P/Y) − f(P/E)) = f(P/1) − f(P/Z) where Z is as before. Since Z is contained in any maximal elementary abelian subgroup of rank 2, the right hand side of the above equality is independent of the choice of E, that is, d̃_{P} (r_{P} (f)) is constant. Hence d̃_{P} (r_{P} (f)) = 0 as required. Similarly, if (f_{Q})_{1<Q≤Q} is a sequence in the kernel of d̃_{P}, then a solution of (f_{Q})_{1<Q≤P} in B^{∗}(P) can be chosen to be a rational class function, by choosing the value at the trivial subgroup to be the constant that is determined by the gluing data under the map d̃_{P} (so that the condition of Lemma 4.2 is satisfied).

We remark that a similar argument shows that the map r_{P} is injective. Indeed a rational class function f is in the kernel of r_{P} if and only if it is zero at any nontrivial subgroup of P. But by Lemma 4.2, the value at the trivial group is uniquely determined by the values at nontrivial subgroups. Hence f should be the zero function, as required. With this remark, we have finished the proof of the following theorem.

Theorem 4.4. Let p be a prime number and P be a p-group of rank at least 2. Then there is an exact sequence of abelian groups

0 \overline{>} R_{ℚ}^{∗}(P) \overline{\begin{matrix}r_{P} \\ > \end{matrix}} \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q) \overline{d̃_{P} ; >} H̃^{0}(A_{≥2}(P), pℤ)^{P}  \overline{>} 0

where the maps r_{P} and d̃_{P} are as defined above.

Note that when the poset A_{≥2}(P) is connected, the group H̃^{0}(A_{≥2}(P), pℤ)^{P} becomes the trivial group and we get that the gluing problem always has a unique solution, which proves the following corollary.

Corollary 4.5. Let P be a p-group of rank at least 2 such that the poset A_{≥2}(P) is connected. Then the map

r_{P}^{R_{ℚ}^{∗}} : R_{ℚ}^{∗}(P) → \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q)

is an isomorphism.

Finally, the following corollary is an easy consequence of the above proof, completing the solution of the gluing problem for rational class functions for p-groups.

Corollary 4.6. Let P be a p-group of rank 1. Then there is an exact sequence

0  \overline{>} ℤ \overline{>} R_{ℚ}^{∗}(P) \overline{\begin{matrix}r_{P} \\ > \end{matrix}} \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q) \overline{>} 0

of abelian groups.

5. Gluing Borel-Smith functions

Let p be a prime number and P be a p-group. As in Section 3, let B^{∗}(P) denotes the group of super class functions for P. Following tom Dieck [19], we call a super class function f ∈ B^{∗}(P) a Borel-Smith function if the following conditions are satisfied:

(1) If p is odd, then for any subquotient T/S of P, of order p, the number f(T) − f(S) is even.

(2) If p = 2, then for any sequence of subgroups H ⊴ K ⊴ L ≤ N_{P} (H), with |K : H| = 2, the value f(K) − f(H) is even if L/K is cyclic of order 4 and is divisible by 4 if L/K is quaternion of order 8.

(3) For any elementary abelian subquotient T/S of P, of rank 2, the equality

f(S) − f(T) = ∑_{S<X<T}(f(X) − f(T))

holds. It is proved in [8] that the subgroups C_{b}(Q) of Borel-Smith functions as Q runs over all p-groups together with the induced actions of bisets is a subfunctor of the biset functor B^{∗}.

To solve the gluing problem for Borel-Smith functions, we introduce the following group. Define the group

H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P}

as the subgroup of H̃^{0}(A_{≥2}(P), pℤ)^{P} generated by the elements pn_{E}χ_{E} where E runs over a complete set of representatives of conjugacy classes of maximal elementary abelian subgroups of P of rank 2 and for such a subgroup E, the integer n_{E} is given by

n_{E} = \{ \begin{matrix}1, & if |C_{P} (Y) : Y | = 2; \\ 2, & if C_{P} (Y)/Y is cyclic of order ≥ 3; \\ 4, & if C_{P} (Y)/Y is generalized quaternion \end{matrix}

where Y is a non-central proper subgroup of E and recall that by Lemma 2.2 in [10], the group C_{P} (Y)/Y is either cyclic or generalized quaternion.

Now we define a map

d̃_{P} : \underleftarrow{lim}_{1<Q≤P} C_{b}(N_{P} (Q)/Q) → H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P}

by associating a given sequence (b_{Q})_{1<Q≤P} to the class of the function E ↦ p(b_{S}(W_{P} (S)/(S/S)) − b_{S}(W_{P} (S)/(E/S))) where S is a non-central proper subgroup of E, and E is chosen as in the case of the rational class functions. With this notation, we can prove the following theorem.

Theorem 5.1. Let P be a p-group of rank at least 2. There is an exact sequence of abelian groups

0 \overline{>} C_{b}(P) \overline{\begin{matrix}r_{P} \\ > \end{matrix}} \underleftarrow{lim}_{1<Q≤P} C_{b}(N_{P} (Q)/Q) \overline{d̃_{P} ; >} H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P}  \overline{>} 0

where the maps r_{P} and d̃_{P} are as defined above.

Proof. The proof of the injectivity of r_{P} is similar to the proof of the injectivity of r_{P}^{R_{ℚ}^{∗}}. To prove that d̃_{P} is surjective, note that we can embed H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P} into H̃^{0}(A_{≥2}(P), pℤ)^{P}. Now if n_{E}χ_{E} is the characteristic function of the class of the maximal elementary abelian subgroup E of P of rank 2, then we regard pn_{E}χ_{E} as a rational class function and using the surjectivity of d̃_{P}^{R_{ℚ}^{∗}}, we construct the preimage (f_{Q})_{1<Q≤P} of n_{E}χ_{E} in \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(N_{P} (Q)/Q). Now this class is a gluing data for Borel-Smith functions since f_{Q} is non-zero only if Q is P -conjugate to a non-central proper subgroup of E in which case it is either 1 or 2 or 4 according to whether C_{P} (Q)/Q is cyclic of order 2 or cyclic of order at least 3 or generalized quaternion and in this case it is clear that the function f_{Q} is a Borel-Smith function, proving the surjectivity of the map d̃_{P}.

Finally, the proof of the exactness at the middle term of the sequence is similar to the proof of the exactness at the middle term of the sequence of Theorem 4.4. □

As in the previous case for rational class functions, if the poset A_{≥2}(P) is connected, the group H_{b}^{0}(A_{≥2}(P), pℤ)^{P} vanishes and we obtain the following corollary.

Corollary 5.2. Let P be a p-group such that the poset A_{≥2}(P) is connected. Then the map

r_{P}^{C_{b}} : C_{b}(P) → \underleftarrow{lim}_{1<Q≤P} C_{b}(N_{P} (Q)/Q)

is an isomorphism.

Finally, the following corollary completes the solution of the gluing problem for Borel-Smith functions for p-groups. We leave the straightforward proof as an exercise.

Corollary 5.3. Let P be a p-group of rank 1. Then there is an exact sequence

0  \overline{>} ℤ \overline{>} C_{b}(P) \overline{\begin{matrix}r_{P} \\ > \end{matrix}} \underleftarrow{lim}_{1<Q≤P} C_{b}(N_{P} (Q)/Q) \overline{>} ℤ/m_{P} ℤ \overline{>} 0

of abelian groups where m_{P} = 2 if P is cyclic and m_{P} = 4 if P is generalized quaternion.

6. The Main Result

The main result of the paper is the following theorem which relates the endotrivial group to the group of obstructions of the gluing problem for the functor of Borel-Smith functions. As an immediate corollary, we will obtain a set of generators for the group T (P) of endo-trivial modules.

Theorem 6.1. Let p be a prime number and P be a p-group, which is not cyclic, quaternion or semi-dihedral. There is a split exact sequence of abelian groups

0  \overline{>} ℤ \overline{>} T (P) \overline{>} Obs(C_{b}(P)) \overline{>} 0

where T (P) is the group of endo-trivial modules and Obs(C_{b}(P)) is the group of obstructions for the gluing problem at P for the biset functor of Borel-Smith functions.

Proof. The proof consists of two parts. The first part is the proof of the existence of the exact sequence and the second part is the construction of an explicit splitting.

For the first part of the proof, recall that, by Theorem 1.2 of [8], there is an exact sequence of p-biset functors

0  \overline{>} C_{b}  \overline{>} B^{∗} \overline{Ψ>} D^{Ω}  \overline{>} 0.

Here the map Ψ is defined as the family (Ψ_{P}) where Ψ_{P} maps an element ω_{P/Q} of the basis {ω_{P/Q}|Q ≤_{P} P } of B^{∗}(P), introduced in Lemma 2.2 of [4], to the corresponding relative syzygy Ω_{P/Q}.

Now by Proposition 2.1, there is a commutative diagram

0  \overline{>} C_{b}(P)  \overline{>} B^{∗}(P)  \overline{Ψ_{P}  >} D^{Ω}(P)  \overline{>} 0

0 \overline{>} \underleftarrow{lim}_{1<Q≤P} C_{b}(W_{P} (Q))^{\begin{aligned}r_{P}^{C_{b}} \\ & ∨ \end{aligned}}  \overline{>} \underleftarrow{lim}_{1<Q≤P} B^{∗}(W_{P} (Q))^{\begin{aligned}r_{P}^{B^{∗}} \\ & ∨ \end{aligned}}  \overline{Ψ̃_{P} ; >} \underleftarrow{lim}_{1<Q≤P} D^{Ω}(W_{P} (Q))^{\begin{aligned}r_{P}^{D} \\ & ∨ \end{aligned}}

where the maps Ψ̃_{P} and r_{P} ’s are defined as in Section 2. Now by Snake Lemma together with Theorem 3.1, we obtain an exact sequence

0 \overline{>} Ker(r_{P}^{C_{b}}) \overline{>} ℤ \overline{>} T^{Ω}(P) \overline{>} Obs(C_{b}(P)) \overline{>} 0

where T^{Ω}(P) denotes the kernel of r_{P}^{D^{Ω}}. Now we claim that the equality

T^{Ω}(P) = T (P)

holds. Indeed, by [17, Section 2.1.1], the kernel of r_{P}^{D} is equal to the group T (P) and hence we have

T^{Ω}(P) = T (P) ∩ D^{Ω}(P).

But by Theorem 7.7 in [4], any class in the Dade group which is not equal to a class of relative syzygy has finite order, and by Theorem 1.1 in [11], the group T (P) is torsion free if P is not cyclic, generalized quaternion or semi-dihedral. Thence, we have T^{Ω}(P) = T (P).

On the other hand, by the proof of Theorem 3.1, the group ℤ, in the above sequence, can be identified with the subgroup of B^{∗}(P) generated by ω_{P}. Now by the definition of the map Ψ, the image of ω_{P} is equal to Ω_{P}. Moreover it is well-known that the subgroup of T (P) generated by Ω_{P} is infinite cyclic since P is not cyclic or quaternion, see [14], cf. [18]. Therefore we get that Ker(r_{P}^{C_{b}}(P)) = 0, which completes the first part of the proof of the theorem.

For the second part of the proof, first note that the surjective map T (P) → Obs(C_{b}(P)) is the connecting homomorphism of the following diagram.

0  \overline{>} ℤ  \overline{>} T (P)

0  \overline{>} C_{b}(P)^{∨}  \overline{>} B^{∗}(P)^{∨}  \overline{Ψ_{P}  >} D^{Ω}(P)^{∨}  \overline{>} 0

0 \overline{>} \underleftarrow{lim}_{1<Q≤P} C_{b}(W_{P} (Q))^{\begin{aligned}r_{P} \\ & ∨ \end{aligned}}  \overline{>} \underleftarrow{lim}_{1<Q≤P} B^{∗}(W_{P} (Q))^{\begin{aligned}r_{P} \\ & ∨ \end{aligned}}  \overline{Ψ̃_{P} ; >} \underleftarrow{lim}_{1<Q≤P} D^{Ω}(W_{P} (Q))^{\begin{aligned}r_{P} \\ & ∨ \end{aligned}}

Obs(C_{b}(P))^{∨}  \overline{>} 0^{∨}

To be able to construct a splitting for this connecting homomorphism, notice that by Theorem 5.1, we also have the following commutative diagram

0  0  0

0  \overline{>} C_{b}(P)^{∨}  \overline{>} R_{ℚ}^{∗}(P)^{∨}  \overline{Ψ_{P}  >} D_{t}^{Ω}(P)^{∨}  \overline{>} 0

0 \overline{>} \underleftarrow{lim}_{1<Q≤P} C_{b}(W_{P} (Q))^{∨}  \overline{>} \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(W_{P} (Q))^{∨}  \overline{>} \underleftarrow{lim}_{1<Q≤P} D_{t}^{Ω}(W_{P} (Q))^{∨}

0  \overline{>} H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P}^{∨}  \overline{>} H̃^{0}(A_{≥2}(P), pℤ)^{P}^{∨}  \overline{>} Obs(D_{t}^{Ω})^{∨}

0^{∨}  0^{∨}  0^{∨}

and the identification

Obs(C_{b}(P)) = H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P} .

Here, as usual, ↪ denotes an injective map and ↠ denotes a surjective map.

Now the splitting is defined in the following way. First consider the third diagram in this proof. Let pn_{E}χ_{E} ∈ H̃_{b}^{0}(A_{≥2}(P), pℤ)^{P} be a generator of this group where E is a maximal elementary abelian subgroup E of P of rank 2. We regard it as an element in H̃^{0}(A_{≥2}(P), pℤ)^{P}. By the proof of Theorem 4.4, the pre-image of pn_{E}χ_{E} in \underleftarrow{lim}_{1<Q≤P} R_{ℚ}^{∗}(W_{P} (Q)) is the sequence (f_{Q})_{1<Q≤P} given by f_{Q} = 0 unless Q is conjugate to a non-central cyclic subgroup, say A, of E in which case f_{Q} is given by f_{Q}(T/Q) = n_{E}δ_{T,Q}. Since it is not zero, the gluing data (f_{Q})_{1<Q≤P} is not coming from a rational class function.

But we pass to the second diagram of this proof by embedding this data to the group of the gluing data of super class functions. Then by Theorem 3.1, this data can be glued to a super class function, say f, which is given by, f(P/Q) = 0 unless Q is conjugate to A in which case f(P/Q) = n_{E}. Using the basis {ω_{P/Q}|Q ≤_{P} P } of B^{∗}(P), we can write

f = n_{E}(ω_{P/A} − ω_{P/1}).

Now by the definition of the map Ψ^{P}, we have

Ψ^{P} (f) = [(Ω_{P/A} ⊗ Ω_{P/1}^{−1})^{⊗^{n_{E}}}]

where for an endo-permutation module M, we denote its class in the Dade group by [M]. Hence the above construction gives a homomorphism

σ : H_{b}^{0}(A_{≥2}(P), pℤ)^{P} → D(P).

of abelian groups. Now we claim that the class of (Ω_{P/A} ⊗ Ω_{P/1}^{−1})^{⊗^{n_{E}}} in the Dade group D(P) is in the image of the endo-trivial group. Remark that this class coincides with the class found by Alperin in [1] where he proved this claim. However we can give a direct proof of the claim.

To prove the claim, we need to prove that the class (Ω_{P/A} ⊗ Ω_{P/1}^{−1})^{⊗^{n_{E}}} is in the kernel of the map r_{P}^{D} but this is trivial since the gluing data (f_{Q})_{1<Q≤P} is a data for Borel-Smith functions, by its construction and hence the claim follows from the commutativity of the second diagram of this proof.

Hence we get that the class [(Ω_{P/A} ⊗ Ω_{P/1}^{−1})^{⊗^{n_{E}}}] is in the image of T (P) in D(P) which gives a homomorphism

σ : H_{b}^{0}(A_{≥2}(P), pℤ)^{P} → T (P).

of abelian groups. To end the proof, we need to check that this map is a splitting for the connecting homomorphism but this is straightforward from the construction.

□

The following theorem follows from the above proof. Note that although we end up with the same set of generators for the endo-trivial group found by Alperin in [1], our proof does not depend on his result, hence gives another proof of the classification theorem of Carlson and Thévenaz in [10].

Theorem 6.2 (Carlson-Thévenaz[10]). Let p > 2 be a prime number and P be a non-abelian p-group which is not semi-dihedral. Then the group T (P) of endotrivial modules for P is free abelian on the basis Ω_{P} , Γ((Ω_{P/A}⊗Ω_{P/1}^{−1})^{⊗^{n_{E}}}) as E runs over a complete set of representatives of conjugacy classes of maximal elementary abelian subgroups of P of rank 2. Here, for a kP -module M, the module Γ(M) is the sum of all indecomposable summands of M with vertex P.

We end the paper by noting that our proof does not use the classification theorem of Carlson and Thévenaz, (which means that our arguments are not circular). Indeed, we have used three results from general theory of endo-permutation modules. Namely, the theorems about two exact sequences from [8], Theorem 7.1 from [4] which states that the classes of endo-permutation modules which are not relative syzygies are torsion and the injectivity of the map r_{P}^{D_{t}} from [7]. Finally we have used the description of the torsion part of the group T (P) from [11]. A careful reading would show that all these results together with any result that they refer are independent of this classification theorem.

Acknowledgement. I would like to thank to Nadia Mazza for answering a question about the poset A_{≥2}(P) and Ergün Yalçın for his useful comments.

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