Biset functors and genetic sections for p-groups

Serge Bouc

Abstract: In this note I show that if F is a biset functor defined over finite p-groups, then for each finite p-group P, there is a direct summand of F (P) admitting a natural direct sum decomposition indexed by the irreducible rational representations of P, or equivalently, by the equivalence classes of origins in P, or also by equivalence classes of genetic sections of P. This leads to a description of the torsion part of the group of relative syzygies in the Dade group of P, and to a conjecture on the structure of the torsion part of the whole Dade group of P.

AMS Subject Classification : 20C20, 20D15, 18B99

1. Basic subgroups and origins

1.1. In Sections 1 to 3 of this paper I recall some notation and definitions which are used in the statement of the main theorem (Theorem 3.2). Section 4 is devoted to some properties of origins and genetic sections. Section 5 is the proof of Theorem 3.2, and Section 6 is an application to the Dade group. In Section 7, I use the notion of rational biset functor to give a description of the torsion part of the functor of relative syzygies in the Dade group.

1.2. Throughout this paper, the letter p denotes a fixed prime number. If P is a finite p-group, denote by R_{ℚ}(P) the Grothendieck group of finitely generated ℚP -modules. There is a natural bilinear form on R_{ℚ}(P), with values in ℤ, defined by

⟨V, W ⟩_{P} = dim_{ℚ} Hom_{ℚP} (V, W)

for ℚP -modules V and W.

Recall some definitions from [1] :

1.3. Definition : [[1] 2.3 ]Let P be a finite p-group. A subgroup Q of P is called basic if the following two conditions hold :

1. The quotient N_{P} (Q)/Q is cyclic or generalized quaternion.

2. If R is any subgroup of P such that R ∩ N_{P} (Q) ⊆ Q, then |R| ≤ |Q|.

Date : May 27, 2004

1.4. Associated simple modules. If Q is a proper basic subgroup of P, then there is a unique subgroup Q̃ ⊃ Q of P with |Q̃ : Q| = p, and the kernel of the projection map

ℚP/Q → ℚP/Q̃

is an irreducible ℚP -module, denoted by V_{Q}.

The group P itself is a basic subgroup of P, and by convention V_{P} is the trivial ℚP -module ℚ. With this notation, any irreducible ℚP -module is isomorphic to V_{Q}, for some basic subgroup Q of P.

If Q and Q′ are basic subgroups of P, then the ℚP -modules V_{Q} and V_{Q′} are isomorphic if and only if Q ≑_{P} Q′, where ≑_{P} is the relation defined in [1] 2.7 by

Q ≑_{P} Q′ ⇔ \{ \begin{matrix}|Q| = |Q′| \\ ∃x ∈ P, Q^{x} ∩ N_{P} (Q′) ⊆ Q′ \end{matrix} .

1.5. Notation and Definition : [[1] 2.4] If Q and R are subgroups of P, set

I_{P} (Q, R) = {x ∈ P | Q^{x} ∩ N_{P} (R) ⊆ R} I_{P} (Q, R) = Q\I_{P} (Q, R)/N_{P} (R) .

If V is a simple ℚP -module, and W is a finitely generated ℚP -module, denote by m(V, W) the multiplicity of V as a direct summand of W.

If Q is a basic subgroup of P, and R is any subgroup of P, then (see [1] 2.5)

m(V_{Q}, ℚP/R) = \frac{|I_{P} (R, Q)|}{|I_{P} (Q, Q)|} .

In particular, if this is non-zero, then |R| ≤ |Q|.

1.6. Definition : [[1] 7.3] If Q is a basic subgroup of P, set

T_{Q} = <I_{P} (Q, Q)> S_{Q} = ∩_{x∈T_{Q}} ^{x}Q .

The basic subgroup Q of P is called an origin (in P) if T_{Q}/S_{Q} has normal p-rank 1, i.e. if it is cyclic, generalized quaternion, semi-dihedral, or dihedral of order at least 16.

If V is a simple ℚP -module, there exists an origin Q in P such that V ≅ V_{Q} (see [1] 7.5). Then Q is called an origin of V, and V is said to have type T_{Q}/S_{Q}. One can show that the type of V is well defined up to isomorphism (see [1] 7.5). If T_{Q}/S_{Q} is cyclic or generalized quaternion, then S_{Q} = Q and T_{Q} = N_{P} (Q) (see [1] 7.6). If T_{Q}/S_{Q} is dihedral or semi-dihedral, then |Q : S_{Q}| = 2 (see [1] 7.4).

If R is a finite p-group of normal p-rank 1, then there is a unique faithful irreducible ℚR-module Φ_{R}, up to isomorphism (see [1] 3.7).

1.7. Definition : A pair (T, S) of subgroups of P with S \underline{◁} T is called a section of P. Two sections (T, S) and (T′, S′) are said to be linked (notation (T, S) (T′, S′)) if

T ∩ S′ = S ∩ T′ and T.S′ = S.T′ .

They are linked modulo P (notation (T, S) _{P} (T′, S′)) if there exists x ∈ P such that (T, S) (^{x}T′,^{x}S′).

The section (T, S) is called genetic if T/S has normal p-rank 1, and if setting

V = Ind_{T}^{P} Inf_{T/S}^{T}Φ_{T/S} ,

there is a ℚ-algebra isomorphism

End_{ℚP} (V) ≅ End_{ℚT/S}(Φ_{T/S}) .

In particular, it follows that End_{ℚP} (V) is a (skew) field, hence V is an irreducible ℚP -module, called the irreducible ℚP -module associated to (T, S). The pair (T, S) is called an genetic section of P for V, or associated to V. If Q is an origin in P, then (T_{Q}, S_{Q}) is an genetic section of P associated to V_{Q} (see [1] 7.4).

The section (P, P) of P is called the trivial genetic section of P. It is the unique genetic section (T, S) of P for which T = S : indeed Φ_{1} = ℚ, and Ind_{T}^{P} Inf_{1}^{T} Φ_{1} = ℚP/T is simple if and only if T = P, since ℚ is always a direct summand of ℚP/T.

If (T, S) and (T′, S′) are genetic sections of P, and if V and V′ are the associated simple ℚP -modules, then V ≅ V′ if and only if (T, S) _{P} (T′, S′) (see [1] 7.11). In particular, the relation _{P} is an equivalence relation on the set of genetic sections of P. Recall moreover that if (T, S) _{P} (T′, S′), then there exists a unique double coset T xT′ in P such that (T, S) ^{x}(T′, S′).

2. Biset functors

2.1. Notation and Definition : Let k be a commutative ring. Denote by C_{p,k} the following category :

• The objects of C_{p,k} are the finite p-groups.

• If P and Q are finite p-groups, then Hom_{C_{p,k}}(P, Q) = k ⊗_{ℤ} B(Q × P^{op}), where B(Q × P^{op}) is the Burnside group of finite (Q, P)-bisets.

• The composition of morphisms is k-bilinear, and if P, Q, R are finite p-groups, if U is a finite (Q, P)-biset, and V is a finite (R, Q)-biset, then the composition of (the isomorphism classes of) V and U is the (isomorphism class) of V ×_{Q} U. The identity morphism Id_{P} of the p-group P is the class of the set P, with left and right action by multiplication.

Let F_{p,k} denote the category of k-linear functors from C_{p,k} to the category k-Mod of kmodules. An object of F_{p,k} is called a biset functor (defined over p-groups, with values in k-Mod).

If F is an object of F_{p,k}, if P and Q are finite p-groups, and if φ ∈ Hom_{C_{p,k}}(P, Q), then the image of w ∈ F (P) by the map F (φ) will generally be denoted by φ(w). The composition ψ ∘ φ of morphisms φ ∈ Hom_{C_{p,k}}(P, Q) and ψ ∈ Hom_{C_{p,k}}(Q, R) will also be denoted by ψ ×_{Q} φ.

2.2. Examples. Recall that this formalism of bisets gives a single framework for the usual operations of induction, restriction, inflation, deflation, and transport by isomorphism via the following correspondences :

• If Q is a subgroup of P, then let Ind_{Q}^{P} ∈ Hom_{C_{p,k}}(Q, P) denote the set P, with left action of P and right action of Q by multiplication.

• If Q is a subgroup of P, then let Res_{Q}^{P} ∈ Hom_{C_{p,k}}(P, Q) denote the set P, with left action of Q and right action of P by multiplication.

• If N \underline{◁} P, and Q = P/N, then let Inf_{Q}^{P} ∈ Hom_{C_{p,k}}(Q, P) denote the set Q, with left action of P by projection and multiplication, and right action of Q by multiplication.

• If N \underline{◁} P, and Q = P/N, then let Def_{Q}^{P} ∈ Hom_{C_{p,k}}(P, Q) denote the set Q, with left action of Q by multiplication, and right action of P by projection and multiplication.

• If φ : P → Q is a group isomorphism, then let Iso_{P}^{Q} = Iso_{P}^{Q}(φ) ∈ Hom_{C_{p,k}}(P, Q) denote the set Q, with left action of Q by multiplication, and right action of P by taking image by φ, and then multiplying in Q.

2.3. Sections and isomorphisms. Let (T, S) and (T′, S′) be sections of the group P, and let x ∈ P be an element such that (T, S) (^{x}T′,^{x}S′). Let U denote the set S\T xT′/S′, viewed as a (T/S, T′/S′)-biset. If t′ ∈ T′, then there exists t ∈ T ∩^{x}T′ and s′ ∈ S′ such that t′ = t^{x}s′. The correspondence u : t′S′ ↦ tS is well defined, and it is a group isomorphism from T′/S′ to T/S. Moreover, the correspondence

∀(t, t′) ∈ T × T′, Stxt′S′ ∈ S\T xT′/S′ ↦ tS.u(t′S′) ∈ T/S

is an isomorphism of (T/S, T′/S′)-bisets between U and Iso_{T′/S′}^{T/S}(u).

2.4. Opposite bisets. If P and Q are finite p-groups, and if U is a finite (Q, P)-biset, then let U^{op} denote the opposite biset : as a set, it is equal to U, and it is a (P, Q)-biset for the following action

∀h ∈ Q, ∀u ∈ U, ∀g ∈ P, g.u.h (in U^{op}) = h^{−1}ug^{−1} (in U) .

This definition can be extended by linearity, to give an isomorphism

φ ↦ φ^{op} : Hom_{C_{p,k}}(P, Q) → Hom_{C_{p,k}}(Q, P) .

It is easy to check that (φ ∘ ψ)^{op} = ψ^{op} ∘ φ^{op}, with obvious notation, and the functor

\{ \begin{matrix}P ↦ P \\ φ ↦ φ^{op} \end{matrix}

is an equivalence of categories from C_{p,k} to the dual category.

2.5. The functor P ↦ R_{ℚ}(P). If P and Q are finite p-groups, if V is a finitely generated ℚP -module, if U is a finite (Q, P)-biset, then ℚU is a (ℚQ, ℚP)-bimodule, and ℚU ⊗_{ℚP} V is a finitely generated ℚQ-module. This definition can be extended by linearity, and gives a structure of biset functor (with values in ℤ-Mod) to the correspondence P ↦ R_{ℚ}(P).

This functorial structure is well-behaved with respect to the bilinear forms ⟨ , ⟩_{P}, in the following sense : if P and Q are finite p-groups, if a ∈ R_{ℚ}(P) and b ∈ R_{ℚ}(Q), if φ ∈ Hom_{C_{p,k}}(P, Q), then

⟨b, φ(a)⟩_{Q} = ⟨φ^{op}(b), a⟩_{P} .

2.6. Some idempotents in End_{C_{p,k}}(P). Let P be a finite p-group, and let N \underline{◁} P. Then it is clear from the definitions that

Def_{P/N}^{P} ∘ Inf_{P/N}^{P} = (P/N) ×_{P} (P/N) = Id_{P/N} .

It follows that the composition

e_{N}^{P} = Inf_{P/N}^{P} ∘ Def_{P/N}^{P}

is an idempotent in End_{C_{p,k}}(P). It is the class of the set P/N, viewed as a (P, P)- biset by left and right action given by projection on P/N, and multiplication in P/N. Moreover, if M and N are normal subgroups of P, then

e_{N} ∘ e_{M} = (P/N) ×_{P} (P/M) ,

which is easily seen to be isomorphic to P/NM as (P, P)-biset. In other words

e_{N}^{P} ∘ e_{M}^{P} = e_{NM}^{P} .

Moreover e_{1}^{P} = Id_{P}. The following Lemma is well-known (and can be proved for instance using Lemme 8 of [3]) :

2.7. Lemma : If N \underline{◁} P, define f_{N}^{P} ∈ End_{C_{p,k}}(P) by

f_{N}^{P} = ∑_{{M \underline{◁} P \atop N⊆M}} μ_{\underline{◁} P} (N, M)e_{M}^{P} ,

where μ_{\underline{◁} P} denotes the Möbius function of the poset of normal subgroups of P. Then the elements f_{N}^{P}, for N \underline{◁} P, are orthogonal idempotents of End_{C_{p,k}}(P), and their sum is equal to Id_{P}.

Now if N \underline{◁} P, the poset ]1, N[^{P} of normal subgroups of P which are non-trivial and properly contained in N is contractible, unless the action of P on N is semi-simple. Since P is a p-group, this implies that if μ_{\underline{◁} P} (1, N) ≠ 0, then N is central in P. In this case moreover μ_{\underline{◁} P} (1, N) is equal to the Möbius function μ(1, N) of the poset of all proper non-trivial subgroups of N. This is zero if N is not elementary abelian. This gives finally

f_{1}^{P} = ∑_{N⊆Ω_{1}Z(P)} μ(1, N)P/N ,

where Ω_{1}Z(P) is the subgroup of the center of P consisting of elements of order at most p.

2.8. Faithful elements for biset functors. If F is an object of F_{p,k}, and if P is a finite p-group, then the idempotent f_{1}^{P} of End_{C_{p,k}}(P) acts on F (P). Its image

∂F (P) = F (f_{1}^{P})F (P)

is a direct summand of F (P) as k-module : it will be called the set of faithful elements of F (P), because any element u ∈ F (P) which is inflated from a proper quotient of P is such that F (f_{1}^{P})u = 0. In the case k = ℤ and F = R_{ℚ}, the module ∂F (P) has a basis over ℤ consisting of the faithful irreducible rational representations of P.

3. Bisets and genetic sections

3.1. Notation : Let P be a finite p-group, and (T, S) be an genetic section of P, with S ≠ T. Then (T/S) has a unique central subgroup of order p, and this subgroup is equal to Ŝ/S, for some well defined subgroup Ŝ of T. Then P/S and P/Ŝ are (P, T/S)-bisets, and I denote by a_{T,S} the element of Hom_{C_{p,k}}(T/S, P) defined by

a_{T,S} = P/S − P/Ŝ .

I denote by b_{T,S} the element of Hom_{C_{p,k}}(P, T/S) corresponding to the opposite bisets, i.e.

b_{T,S} = S\P − Ŝ\P .

If (T, S) = (P, P) is the trivial genetic section of P, I denote by a_{T,S} the image in Hom_{C_{p,k}}(1, P) of a (P, 1)-biset of cardinality 1, and by b_{T,S} the image in Hom_{C_{p,k}}(P, 1) of a (1, P)-biset of cardinality 1.

The main theorem can now be stated :

3.2. Theorem : Let F be a biset functor defined over p-groups, with values in k-Mod. Then for any finite p-group P, and any set S of representatives of genetic sections of P modulo the relation _{P} , the map

I_{S} = ⊕_{(T,S)∈S} F (a_{T,S}) : ⊕_{(T,S)∈S} ∂F (T/S) → F (P) ,

is a split injection of k-modules, with left inverse

D_{S} = ⊕_{(T,S)∈S} F (b_{T,S}) : F (P) → ⊕_{(T,S)∈S} ∂F (T/S) .

4. Some combinatorial properties of origins and genetic sections

4.1. Lemma : Let P be a finite p-group, and Q be an origin in P, with associated genetic section (T_{Q}, S_{Q}), and corresponding simple ℚP -module V_{Q}. Then if R is any subgroup of P, the multiplicity m(V_{Q}, ℚP/R) of V_{Q} in the permutation module ℚP/R is equal to

m(V_{Q}, ℚP/R) = |Q : S_{Q}| a + b ,

where

a = card{x ∈ T_{Q}\P/R | T_{Q} ∩^{x}R ⊆ S_{Q}}
b = card{x ∈ T_{Q}\P/R | |(T_{Q} ∩^{x}R)S_{Q}/S_{Q}| = p, (T_{Q} ∩^{x}R)S_{Q}/S_{Q} ⊈ Z(T_{Q}/S_{Q})} .

Proof: First, the result holds if Q = P, for in this case V_{Q} = ℚ and T_{Q} = S_{Q} = P, thus m(V_{Q}, ℚP/R) = 1, and moreover a = 1 and b = 0.

Suppose now that Q is a proper subgroup of P. Set (T, S) = (T_{Q}, S_{Q}), and D = T/S. Then :

⟨V_{Q}, ℚP/R⟩_{P} = ⟨Ind_{T}^{P} Inf_{D}^{T}Φ_{D}, ℚP/R⟩_{P}
= ⟨Φ_{D}, Def_{D}^{T}Res_{T}^{P} ℚP/R⟩_{D} .

Now Def_{D}^{T}Res_{T}^{P} ℚP/R is isomorphic to ℚS\P/R as ℚD-module. In other words

Def_{D}^{T}Res_{T}^{P} ℚP/R ≅ ⊕_{x∈T \P/R} ℚS\T xR/R ,

and the ℚD-module ℚS\T xR/R is isomorphic to ℚD/E_{x}, where E_{x} = (T ∩^{x}R)S/S, for each x ∈ P : indeed S\T xR/R is a transitive D-set, and the stabilizer of SxR in D is equal to E_{x}. It follows that

⟨V_{Q}, ℚP/R⟩_{P} = ∑_{x∈T \P/R}⟨Φ_{D}, ℚD/E_{x}⟩_{D} .

Now there are two cases : if E_{x} contains the unique central subgroup Ŝ/S of order p of D, then the permutation module ℚD/E_{x} is inflated from a proper quotient of D. Hence it is a direct sum of non-faithful irreducible ℚD-modules. Hence

⟨Φ_{D}, ℚD/E_{x}⟩_{D} = 0

in this case. And if E_{x} does not contain Ŝ/S, then it is either trivial, or non-central of order p (this can only happen of course if P is dihedral or semi-dihedral, hence if p = 2).

If E_{x} is trivial, i.e. if T ∩^{x}R ⊆ S, then

⟨Φ_{D}, ℚD/E_{x}⟩_{D} = dim_{ℚ} Φ_{D} ,

And if E_{x} is non-central of order p in D (hence if p = 2 and D is dihedral or semidihedral), then

⟨Φ_{D}, ℚD/E_{x}⟩_{D} = ⟨ℚD/E_{x}, ℚD/E_{x}⟩_{D} = ⟨Φ_{D}, Φ_{D}⟩_{D}m(Φ_{D}, ℚD/E_{x}) .

But in this case E = E_{x} is a basic subgroup of D, and Φ_{D} is the corresponding simple ℚD-module V_{E} (see [1] 3.7). By assertion 2 of Proposition 2.5 of [1]

m(V_{E}, ℚD/E) = \frac{|I_{D}(E, E)|}{|I_{D}(E, E)|} = 1 ,

thus

⟨Φ_{D}, ℚD/E_{x}⟩_{D} = ⟨Φ_{D}, Φ_{D}⟩_{D} .

This gives finally

⟨V_{Q}, ℚP/R⟩_{P} = a dim_{ℚ} Φ_{D} + b⟨Φ_{D}, Φ_{D}⟩_{D}

where

a = card{x ∈ T \P/R | E_{x} = 1}
b = card{x ∈ T \P/R | |E_{x}| = 2, E_{x} ⊈ Z(D)}

On the other hand

⟨V_{Q}, ℚP/R⟩_{P} = ⟨V_{Q}, V_{Q}⟩_{P} m(V_{Q}, ℚP/R) = ⟨Φ_{D}, Φ_{D}⟩_{D}m(V_{Q}, ℚP/R) ,

hence

m(V_{Q}, ℚP/R) = \frac{dim_{ℚ} Φ_{D}}{⟨Φ_{D}, Φ_{D}⟩_{D}} a + b .

Now there are two cases : if D is cyclic or generalized quaternion, then

dim_{ℚ} Φ_{D} = (p − 1)|D|/p = ⟨Φ_{D}, Φ_{D}⟩_{D}

(see [1] 3.7 and 5.5), and Q = S_{Q} in this case. Lemma 4.1 holds in this case. And if D is dihedral or semi-dihedral, then dim_{ℚ} Φ_{D} = |D|/4 and ⟨Φ_{D}, Φ_{D}⟩_{D} = |D|/8, thus

\frac{dim_{ℚ} Φ_{D}}{⟨Φ_{D}, Φ_{D}⟩_{D}} = 2. But |Q : S_{Q}| = 2 in this case, and this completes the proof.

4.2. Lemma : Let Q be an origin in P, with associated section (T_{Q}, S_{Q}) and corresponding simple ℚP -module V_{Q}. Then the multiplicity m(V_{Q}, ℚP/S_{Q}) is equal to |Q : S_{Q}|, and the kernel of the projection map ℚP/S_{Q} → ℚP/Ŝ_{Q} is isomorphic to a direct sum of |Q : S_{Q}| copies of V_{Q}. Proof: Let D = T_{Q}/S_{Q}. If D is cyclic or generalized quaternion, then Q = S_{Q},

Q̃ = Ŝ_{Q}, and m(V_{Q}, ℚP/Q) = \frac{|I_{P} (Q, Q)|}{|I_{P} (Q, Q)|} = 1, and the result holds in this case, since

V_{Q} is the kernel of the projection map ℚP/Q → ℚP/Q̃.

Suppose that D is dihedral or semi-dihedral (hence p = 2), and let Z denote the unique central subgroup of order 2 of D. There is a unique faithful irreducible ℚDmodule Φ_{D}, such that Def_{D/Z}^{D}Φ_{D} = 0 (see [1] 3.12). Any other irreducible ℚD-module W is inflated from D/Z, i.e. such that W = Inf_{D/Z}^{D}Def_{D/Z}^{D}W. Now ℚD/1 splits as

ℚD/1 = mΦ_{D} ⊕ D′ ,

for some m ∈ ℕ, where D′ is a sum of simple modules which are inflated from D/Z. The above considerations show that D′ = Inf_{D/Z}^{D}Def_{D/Z}^{D}ℚD = ℚD/Z. Taking dimensions in the previous equality gives

|D| = m dim_{ℚ} Φ_{D} + |D|/2 = m|D|/4 + |D|/2 ,

and it follows that m = 2, thus

ℚD/1 = 2Φ_{D} ⊕ ℚD/Z .

Taking inflation from D to T_{Q}, and then induction to P gives

ℚP/S_{Q} = 2V_{Q} ⊕ ℚP/Ŝ_{Q} ,

thus

m(V_{Q}, ℚP/S_{Q}) = 2 + m(V_{Q}, ℚP/Ŝ_{Q}) .

Since Q/S_{Q} is a non-central subgroup of order 2 of T_{Q}/S_{Q}, it follows that Q ⊈ Ŝ_{Q}, and |Q.Ŝ_{Q}| > |Q| (actually |Q.Ŝ_{Q}| = 2|Q| since Q ∩ Ŝ_{Q} = S_{Q}). Since Q is basic, it follows that I_{P} ((Q.Ŝ_{Q}), Q) = ∅, hence that m(V_{Q}, ℚP/(Q.Ŝ_{Q})) = 0.

Now if m(V_{Q}, ℚP/Ŝ_{Q}) ≠ 0, it follows that V_{Q} appears as a direct summand of the kernel K_{Q} of the projection map

ℚP/Ŝ_{Q} → ℚP/(Q.Ŝ_{Q}) .

The dimension of K_{Q} is equal to

dim_{ℚ} K_{Q} = \frac{|P : S_{Q}|}{2} − \frac{|P : S_{Q}|}{4} = \frac{|P : S_{Q}|}{4} = dim_{ℚ} V_{Q} .

Hence K_{Q} ≅ V_{Q} is an irreducible ℚP -module. By Proposition 4 of [4], it follows that Ŝ_{Q} is a basic subgroup of P, and in particular the group W = N_{P} (Ŝ_{Q})/Ŝ_{Q} is cyclic or generalized quaternion. But W contains the group T_{Q}/Ŝ_{Q} ≅ D/Z, which is dihedral. This cannot happen, and it follows that m(V_{Q}, ℚP/Ŝ_{Q}) = 0, thus m(V_{Q}, ℚP/S_{Q}) = 2 in this case, as was to be shown.

Now the kernel L_{Q} of the projection map ℚP/S_{Q} → ℚP/Ŝ_{Q} has a direct summand isomorphic to a direct sum of 2 copies of V_{Q}. But

dim_{ℚ} L_{Q} = |P : S_{Q}| − \frac{1}{2}|P : S_{Q}| = 2 dim_{ℚ} V_{Q} ,

hence L_{Q} ≅ 2V_{Q}, completing the proof.

4.3. Corollary : Let (T, S) be a non-trivial genetic section of P, and let x ∈ P.

1. If Ŝ ∩^{x}S ⊆ S, then x ∈ T. In particular T = N_{P} (S).

2. The group (T ∩^{x}S)S/S cannot be a non-central subgroup of order p of T/S.

Proof: There exists an origin Q in P such that (T, S) = (T_{Q}, S_{Q}). With the notation of the two previous lemmas, one has that

|Q : S|a + b = |Q : S| ,

where

a = card{x ∈ T \P/S | T ∩^{x}S ⊆ S}
b = card{x ∈ T \P/S | |(T ∩^{x}S)S/S| = p, (T ∩^{x}S)S/S ⊈ Z(T/S)}

Thus a ≥ 1, and b ≥ 0. Since |Q : S|(a − 1) + b = 0, it follows that b = 0 and a = 1. Assertion 2 follows from b = 0. For Assertion 1, observe that the equality a = 1 means that

∀x ∈ P, T ∩^{x}S ⊆ S ⟹ x ∈ T .

Set L = (T ∩^{x}S)S. Then L/S is a subgroup of T/S. If Ŝ∩^{x}S ⊆ S, then L/S ∩Ŝ/S = 1. Since L/S cannot be non-central of order 2 in T/S, it follows that L/S = 1. Thus T ∩^{x}S ⊆ S, and x ∈ T.

Corollary 4.3 leads to the following combinatorial characterization of genetic sections :

4.4. Proposition : Let P be a finite p-group, and let (T, S) be a section of P. Let Z_{P} (S) denote the subgroup of P defined by

Z_{P} (S)/S = Z(N_{P} (S)/S) .

Then the following conditions are equivalent :

1. The section (T, S) is an genetic section of P.

2. The group N_{P} (S)/S has normal p-rank 1, the group T is equal to N_{P} (S), and if x ∈ P is such that S^{x} ∩ Z_{P} (S) ⊆ S, then x ∈ N_{P} (S).

Proof: Suppose first that (T, S) is an genetic section of P. Then either T = S = P, and Assertion 2 holds trivially, or (T, S) is a non-trivial genetic section. In this case T = N_{P} (S) by Corollary 4.3, thus N_{P} (S)/S = T/S has normal p-rank 1. Let x ∈ P. Since Ŝ/S = Ω_{1}(Z_{P} (S)/S), the hypotheses S^{x} ∩ Ŝ ⊆ S and S^{x} ∩ Z_{P} (S) ⊆ S are equivalent, and Assertion 2 holds by Corollary 4.3.

Conversely, suppose that Assertion 2 holds. Then either S = T = P, and (T, S) is an genetic section of P, or S is a proper subgroup of P, thus also a proper subgroup of T = N_{P} (S). Let Ŝ be the subgroup of P defined by Ŝ/S = Ω_{1}(Z(T/S)).

Let Q/S be a basic subgroup of T/S, such that Q/S ∩Z(T/S) = 1. Such a subgroup exists because T/S has a faithful irreducible rational representation. Then in particular Ŝ ⊆ N_{P} (Q) and Q ∩ Ŝ = S. Now if x ∈ I_{P} (Q, Q), then

S^{x} ∩ Ŝ ⊆ Q^{x} ∩ N_{P} (Q) ⊆ Q ∩ Ŝ = S ,

hence S^{x} = S. Thus I_{P} (Q, Q) ⊆ N_{P} (S) = T, and in particular N_{P} (Q) = N_{T} (Q). Let \overline{I} denote the set of elements x ∈ T such that xS ∈ I_{T/S}(Q/S, Q/S). If x ∈ \overline{I}, then

Q^{x} ∩ N_{P} (Q) = Q^{x} ∩ N_{T} (Q) ⊆ Q ,

thus x ∈ I_{P} (Q, Q). Moreover \overline{I} generates T, since I_{T/S}(Q/S, Q/S) generates T/S by Lemma 7.2 of [1]. It follows that the group T_{Q} generated by I_{P} (Q, Q) contains T, and is contained in N_{P} (S). Hence T_{Q} = T = N_{P} (S).

Now S_{Q} is the largest normal subgroup of T contained in Q, hence it is equal to S, since Q/S intersects the center of T/S trivially. Hence (T_{Q}, S_{Q}) = (T, S), and T_{Q}/S_{Q} has normal p-rank 1. Thus Q is an origin in P, and (T, S) = (T_{Q}, S_{Q}) is an genetic section of P by Proposition 7.4 of [1].

4.5. Lemma : Let (T, S) and (T′, S′) be non-trivial genetic sections of P. Suppose that there exists x and y in P such that

Ŝ ∩^{x}S′ ⊆ S  and  Ŝ′ ∩^{y}S ⊆ S′ .

Then (T, S) _{P} (T′, S′). Proof: Choose origins R and R′ in P such that (T, S) = (T_{R}, S_{R}) and (T′, S′) = (T_{R′}, S_{R′}). Consider the subgroup L = (T ∩^{x}S′).S/S of T/S. Since

L ∩ (Ŝ/S) = ((T ∩^{x}S′).S ∩ Ŝ)/S = (T ∩^{x}S′ ∩ Ŝ).S/S = (Ŝ ∩^{x}S′).S/S = 1 ,

the group L is trivial or non-central of order 2. By Lemma 4.1, the multiplicity m(V_{R}, ℚP/S′) is non-zero. Since R is basic, it follows that |S′| ≤ |R|. By symmetry m(V_{R′}, ℚP/S) ≠ 0 and |S| ≤ |R′|.

Suppose that |R′| < |R|. Then |R′| ≤ |R|/p ≤ |S| ≤ |R′|. Thus |R′| = |R|/p = |S| in this case, and this can only happen if p = 2 and T/S is dihedral or semi-dihedral.

Since |R| ≠ |R′|, the modules V_{R} and V_{R′} are not isomorphic. Hence there is a decomposition

ℚP/S = ℚ ⊕ 2V_{R} ⊕ V_{R′} ⊕ W ,

for some ℚP -module W, since m(V_{R}, ℚP/S) = 2 by Lemma 4.2. It follows that

|P : S| > 2\frac{|P : S|}{4} + \frac{|P : R′|}{2} = \frac{|P : S|}{2} + \frac{|P : R′|}{2} ,

hence |R′| > |S| = |R|/2. It follows that |R′| ≥ |R|, contradicting the assumption |R′| < |R|.

By symmetry, it follows that |R| = |R′|. Suppose that V_{R} ≅ V̸_{R′}. If T/S and T′/S′ are both cyclic or generalized quaternion, then R = S and R′ = S′. Since ℚ ⊕ V_{R} ⊕ V_{R′} is a direct summand of ℚP/S = ℚP/R, it follows that

|P : R| > (1 − \frac{1}{p})|P : R| + (1 − \frac{1}{p})|P : R′| = (2 − \frac{2}{p})|P : R| ≥ |P : R|

since p ≥ 2. This is impossible.

Hence p = 2, and at least one of T/S or T′/S′, say T/S, is dihedral or semi-dihedral. This implies in particular that |R : S| = 2.

If m(V_{R′}, ℚP/R) ≠ 0, then ℚ ⊕ V_{R} ⊕ V_{R′} is a direct summand of ℚP/R, and

|P : R| > \frac{1}{2}|P : R| + \frac{1}{2}|P : R′| = |P : R| ,

which is impossible. Thus m(V_{R′}, ℚP/R) = 0.

Now there are decompositions

ℚP/S = ℚ ⊕ 2V_{R} ⊕ V_{R′} ⊕ W
ℚP/R = ℚ ⊕ V_{R} ⊕ W′ ,

for some ℚP -modules W and W′, with m(V_{R}, W′) = m(V_{R′}, W′) = 0. Hence the kernel K of the projection map

ℚP/S → ℚP/R

has a direct summand isomorphic to V_{R} ⊕ V_{R′}. But

dim_{ℚ} K = 2|P : R| − |P : R| = |P : R| ,

whereas

dim_{ℚ}(V_{R} ⊕ V_{R′}) = \frac{1}{2}|P : R| + \frac{1}{2}|P : R′| = |P : R| .

Hence K ≅ V_{R} ⊕ V_{R′}, and

ℚP/S ≅ V_{R} ⊕ V_{R′} ⊕ ℚP/R ≅ 2V_{R} ⊕ ℚP/Ŝ

by Lemma 4.2. This shows that V_{R′} appears as a direct summand of ℚP/Ŝ. On the other hand V_{R′} cannot appear as a direct summand of ℚP/(R.Ŝ), since R′ is basic, and |R.Ŝ| > |R′|. Hence V_{R′} is a direct summand of the kernel L of the projection map ℚP/Ŝ → ℚP/(R.Ŝ). Since

dim_{ℚ} L = \frac{1}{2}|P : S| − \frac{1}{4}|P : S| = \frac{1}{4}|P : S| = \frac{1}{2}|P : R| = \frac{1}{2}|P : R′| = dim_{ℚ} V_{R′} ,

it follows that L ≅ V_{R′}.

Since L is a simple ℚP -module, it follows that Ŝ is a basic subgroup of P, hence that N_{P} (Ŝ)/Ŝ is cyclic or generalized quaternion. But this group contains T/Ŝ, which is dihedral. This cannot happen, and this contradiction shows that V_{R} ≅ V_{R′}, or equivalently (T, S) _{P} (T′, S′), completing the proof.

4.6. Theorem : Let P be a finite p-group. If σ = (T, S) and σ′ = (T′, S′) are genetic sections of P, set

π_{σ′}^{σ} = b_{T,S} ∘ a_{T′,S′} ∈ Hom_{C_{p,k}}(T′/S′, T/S) .

If π_{σ′}^{σ} ≠ 0, then σ _{P} σ′.

(Recall that in this case, the simple ℚP -modules corresponding to σ and σ′ are isomorphic).

Proof: Suppose first that σ is the trivial genetic section. If σ′ is also trivial, there is nothing to prove since then σ′ = σ, hence σ′ _{P} σ. And if σ′ is non-trivial, then

b_{T,S} ∘ a_{T′,S′} = P \P ×_{P} (P/S′ − P/Ŝ′) = P \P/S − P \P/Ŝ′ = 0 .

A similar argument, or the fact that (π_{σ′}^{σ})^{op} = π_{σ}^{σ′} shows that π_{σ′}^{σ} = 0 if σ′ is trivial and σ is not.

Hence I can suppose that σ and σ′ are both non-trivial. Then

π_{σ′}^{σ} = (S\P − Ŝ\P) ×_{P} (P/S′ − P/Ŝ′) .

Claim : Let (B, A) be a section of P such that A\P ∘(P/S−P/Ŝ) is a non-zero element of Hom_{C_{p,k}}(T/S, B/A). Then there exists an element x ∈ P such that Ŝ ∩ A^{x} ⊆ S.

Indeed the hypothesis means that A\P/S ≠ A\P/Ŝ in Hom_{C_{p,k}}(T/S, B/A). Hence there exists x ∈ P such that AxS ≠ AxŜ, or equivalently xŜ ⊈ AxS, i.e. Ŝ ⊈ A^{x}.S. Since |Ŝ : S| = p, this means that Ŝ ∩ A^{x} ⊆ S, proving the claim.

Thus if π_{σ′}^{σ} ≠ 0 in Hom_{C_{p,k}}(T′/S′, T/S), then at least one of the elements

u = S\P ∘ (P/S′ − P/Ŝ′)  or  v = Ŝ\P ∘ (P/S′ − P/Ŝ′)

is non-zero. If u ≠ 0, then there exists y ∈ P such that Ŝ′ ∩^{y}S ⊆ S′. And if v ≠ 0, then there exists y ∈ P such that Ŝ′ ∩^{y}Ŝ ⊆ S′, which implies Ŝ′ ∩^{y}S ⊆ S′. Similarly, since (π_{σ′}^{σ})^{op} = π_{σ}^{σ′} ≠ 0, it follows that there exists x ∈ P such that Ŝ ∩^{x}S′ ⊆ S. By Lemma 4.5, it follows that (T, S) _{P} (T′, S′).

4.7. Lemma : Let (T, S) and (T′, S′) be genetic sections of P, and suppose that (T, S) _{P} (T′, S′). Then :

1. There exists a unique double coset SxT′ in P such that S^{x} ∩ T′ ⊆ S′. In this case (T, S) (^{x}T′,^{x}S′).

2. If x ∈ P, the group (S^{x} ∩ T′)S′/S′ is not a non-central subgroup of order p of T′/S′.

Proof: If (T, S) is trivial, then so is (T′, S′), and there is nothing to prove. Hence I can suppose that (T, S) and (T′, S′) are both non-trivial, and choose origins R and R′ in P such that (T_{R}, S_{R}) = (T, S) and (T_{R′}, S_{R′}) = (T′, S′).

There exists a unique double coset T xT′ such that (T, S) ^{x}(T′, S′). Since the simple ℚP -modules V_{R} and V_{R′} associated to (T, S) and (T′, S′) are isomorphic, it follows that |R| = |R′|. Moreover R/S ≅ R′/S′, and T/S ≅ T′/S′.

By Lemmas 4.1 and 4.2, one has that

m(V_{R}, ℚP/S′) = m(V_{R′}, ℚP/S′) = |R′ : S′| = |R : S| = |R : S|a + b ,

where

a = card{x ∈ T′\P/S | T′ ∩^{x}S ⊆ S′}
b = card{x ∈ T′\P/S | |(T′ ∩^{x}S)S′/S′| = p, (T′ ∩^{x}S)S′/S′ ⊈ Z(T′/S′)} .

Since a ≥ 1 if (T, S) _{P} (T′, S′), it follows that a = 1 and b = 0. Assertion 2 follows from b = 0. Since a = 1, there is a unique double coset SxT′ in P such that S^{x}∩T′ ⊆ S′. This is the case in particular if (T, S) ^{x}(T′, S′). Thus SxT′ = T xT′ is also the unique double coset T xT in P such that (T, S) ^{x}(T′, S′).

4.8. Lemma : Let (T, S) be an genetic section of P. Then

a_{T,S} = a_{T,S}f_{1}^{T/S} b_{T,S} = f_{1}^{T/S}b_{T,S} .

Proof: Since b_{T,S} = (a_{T,S})^{op} and f_{1}^{T/S} = (f_{1}^{T/S})^{op}, the two assertions are equivalent. Moreover if (T, S) is the trivial genetic section, the result is trivial. And if (T, S) is non-trivial, then Ω_{1}Z(T/S) = Ŝ/S, thus

f_{1}^{T/S} = (T/S)/(S/S) − (T/S)/(Ŝ/S) = T/S − T/Ŝ ,

It follows that

(P/S) ×_{T/S} f_{1}^{T/S} = P/S − P/Ŝ = a_{T,S} ,

hence a_{T,S} = a_{T,S}f_{1}^{T/S}, since f_{1}^{T/S} is an idempotent.

4.9. Theorem : Let σ = (T, S) and σ′ = (T′, S′) be genetic sections of P, and suppose that (T, S) _{P} (T′, S′). Let T xT′ denote the unique double coset in P such that (T, S) ^{x}(T′, S′). Denote by φ_{σ′}^{σ} the (T/S, T′/S′)-biset S\T xT′/S′, viewed as an element of Hom_{C_{p,k}}(T′/S′, T/S). Then

π_{σ′}^{σ} = b_{T,S} ∘ a_{T′,S′} = f_{1}^{T/S}φ_{σ′}^{σ} = φ_{σ′}^{σ}f_{1}^{T′/S′}

in Hom_{C_{p,k}}(T′/S′, T/S). Proof: If σ is trivial, so is σ′, and the result is trivial in this case. Suppose then that σ and σ′ are non-trivial, and denote by R and R′ origins of P such that (T, S) = (T_{R}, S_{R}) and (T′, S′) = (T_{R′}, S_{R′}). Here again |R| = |R′|, the groups R/S and R′/S′ are isomorphic, as well as the groups T/S and T′/S′. In particular |S| = |S′|.

There are two steps in the proof :

Step 1 : Claim :The composition Ŝ\P ∘(P/S′ −P/Ŝ′) is zero in Hom_{C_{p,k}}(T′/S′, T/S) : indeed otherwise, by the Claim of the proof of Theorem 4.6, there exists an element x in P such that Ŝ^{x} ∩ Ŝ′ ⊆ S′. Then the subgroup L = (Ŝ^{x} ∩ T′).S′/S′ of T′/S′ is such that L ∩ Ŝ′/S = 1. Thus L is either trivial, or non-central of order p in T′/S′. By Lemma 4.1, the multiplicity m(V_{R′}, ℚP/Ŝ) is non-zero. Since R′ is basic, this implies

|R′| ≤ |Ŝ| = |Ŝ′| ≤ |R| ,

hence |R′| = |Ŝ′| = |R|. Hence |R′ : S′| = p, and this can occur only if p = 2 and T/S and T′/S′ are dihedral or semi-dihedral. Moreover |R.Ŝ| = 2|R| > |R′|, thus m(V_{R′}, ℚP/(R.Ŝ)) = 0. Hence V_{R′} appears as a direct summand of the kernel K of the projection map ℚP/Ŝ → ℚP/(R.Ŝ). This kernel has dimension

dim_{ℚ} K = \frac{1}{2}|P : S| − \frac{1}{4}|P : S| = \frac{1}{4}|P : S| = dim_{ℚ} V_{R′} ,

hence K ≅ V_{R′} is irreducible. Thus Ŝ is basic, hence N_{P} (Ŝ)/Ŝ is cyclic or generalized quaternion. But this group contains T/Ŝ, which is dihedral. This contradiction proves the claim.

Step 2 : It follows from Step 1 that

π_{σ′}^{σ} = S\P ×_{P} (P/S′ − P/Ŝ′)
= ∑_{x∈T \P/T′}(S\T xT′/S′ − S\T xT′/Ŝ′) in Hom_{C_{p,k}}(T′/S′, T/S) .

Moreover for x ∈ P

SxS′ ≠ SxŜ′ ⇔ Ŝ′ ≠ (S^{x} ∩ Ŝ′).S′ ⇔ S^{x} ∩ Ŝ′ ⊆ S′ .

In that case the subgroup L = (S^{x} ∩ T′).S′/S′ of T′/S′ is trivial or non-central of order 2, hence trivial by Lemma 4.7. There is a single double coset T x_{0}T′ of such elements x in P, and

π_{σ′}^{σ} = ( ∑_{x∈T \(P −T x_{0}T′)/T′}(S\T xT′/S′ − S\T xT′/Ŝ′))+(S\T x_{0}T′/S′−S\T x_{0}T′/Ŝ′) .

If x ∈ P − T x_{0}T′, then SyS′ = SyŜ′ for all y ∈ T xT′. It follows that

S\T xT′/S′ = S\T xT′/Ŝ′

as (T/S, T′/S′)-bisets, and finally

π_{σ′}^{σ} = S\T x_{0}T′/S′ − S\T x_{0}T′/Ŝ′ .

Now recall that f_{1}^{T′/S′} = T′/S′ − T′/Ŝ′. It clearly follows that

π_{σ′}^{σ} = (S\T x_{0}T′/S′) ×_{T′/S′} f_{1}^{T′/S′} = φ_{σ′}^{σ}f_{1}^{T′/S′} .

Exchanging σ and σ′ now gives

π_{σ}^{σ′} = φ_{σ}^{σ′}f_{1}^{T/S} ,

and taking opposite bisets gives

π_{σ′}^{σ} = f_{1}^{T/S}φ_{σ′}^{σ} ,

as was to be shown.

5. Proof of Theorem 3.2

Let F be a biset functor defined over p-groups, with values in k-Mod. Let P be a finite p-group, and let S be a set of representatives of genetic sections of P modulo the relation _{P} . Denote by I_{S} the map

I_{S} = ⊕_{(T,S)∈S} F (a_{T,S}) : ⊕_{(T,S)∈S} ∂F (T/S) → F (P) .

It follows from Lemma 4.8 that the image of the map F (b_{T,S}) is contained in ∂F (T/S), and one can define a map D_{S} by

D_{S} = ⊕_{(T,S)∈S} F (b_{T,S}) : F (P) → ⊕_{(T,S)∈S} ∂F (T/S) .

Choose σ = (T, S) ∈ S. Then for any u ∈ ∂F (T/S), and for any (T′, S′) in S, different from (T, S), one has that

F (b_{T′,S′})F (a_{T,S})(u) = 0

by Theorem 4.6. Moreover, by Theorem 4.9

F (b_{T,S})F (a_{T,S})(u) = F (φ_{σ}^{σ})F (f_{1}^{T/S})(u) = u ,

since F (f_{1}^{T/S})(u) = u for u ∈ ∂F (T/S), and since moreover

φ_{σ}^{σ} = S\T/S = T/S = Id_{T/S} .

It follows that D_{S} ∘ I_{S} is the identity map, and Theorem 3.2 follows.

6. An application to the Dade group

The definitions and notation concerning the Dade group refer to [6] (in particular, see [6] Corollary 3.10 for the definition of the map D(U) between Dade groups associated to a biset U).

6.1. Theorem : Let P be a finite p-group. Denote by D^{t}(P) the torsion subgroup of the Dade group of P, and by T^{t}(P) the subgroup of D^{t}(P) formed by the images of the torsion endo-trivial modules.

1. If Q is a non-trivial p-group of normal p-rank one, then T^{t}(Q) is equal to the kernel of the map Def_{Q/Z}^{Q} : D^{t}(Q) → D^{t}(Q/Z), where Z is the unique central

subgroup of order p in Q. It follows that T^{t}(Q) = D(f_{1}^{Q})D^{t}(Q).

2. Let S be a set of representatives of genetic sections of P, modulo the relation _{P} . Then the map

I_{S} = ⊕_{(R,S)∈S} Teninf_{R/S}^{P} : ⊕_{(R,S)∈S} T^{t}(R/S) → D^{t}(P) ,

is a split injection.

Proof: Assertion 1 follows from Dade’s Theorem on the structure of the Dade group of cyclic groups (see [9], [10]), and of Lemma 10.2 of [8] for the other cases.

Assertion 2 is not a direct consequence of Theorem 3.2, for the Dade group is not a biset functor over p-groups, because of the extra phenomenon of “Galois torsion”, described in Section 3 of [6]. However, the proof of Theorem 3.2 can be adapted to take this Galois torsion into account, in the following way :

First observe that since the Dade group of the trivial group is trivial, one can always replace S by S − {1} in the statement of the theorem, i.e. suppose that the genetic sections considered here are non-trivial ones.

It follows from Assertion 1 that if Q is a non-trivial p-group of normal p-rank one, and if u ∈ T^{t}(Q), then Def_{Q/Z}^{Q}u = 0.

This means that the map I_{S} can also be defined by

I_{S} = ⊕_{(R,S)∈S} D(a_{R,S}) ,

since D(P/Ŝ) = Teninf_{R/Ŝ}^{P} ∘ Def_{R/Ŝ}^{R/S}.

Moreover, if v ∈ D^{t}(P), then the element u = D(b_{R,S})(v) is an element of D^{t}(R/S) in the kernel of the deflation map Def_{R/Ŝ}^{R/S}, hence u ∈ T^{t}(R/S). This allows to define a map

D_{S} : D^{t}(P) → ⊕_{(R,S)∈S} T^{t}(R/S) ,

and this map will be a left inverse to I_{S}.

To see this, let (R′, S′) be an element of S, and let (R, S) be any section of P. If u is an element of D^{t}(R′/S′), one has that

D(S\P)D(P/S′)(u) = Defres_{R/S}^{P}Teninf_{R′/S′}^{P}u
= Def_{R/S}^{R}Res_{R}^{P}Ten_{R′}^{P}Inf_{R′/S′}^{R′}u
= ∑_{x∈R\P/R′} γ_{|S:S∩^{x}R′|}D(S\RxR′/S′)(u) .

Similarly

D(S\P)D(P/Ŝ′)(u) = ∑_{x∈R\P/R′} γ_{|S:S∩^{x}R′|}D(S\RxR′/Ŝ′)(u)

by Proposition 3.10 of [6]. Hence

D(S\P)D(a_{R′,S′})(u) = ∑_{x∈R\P/R′} γ_{|S:S∩^{x}R′|}(D(S\RxR′/S′)(u) − D(S\RxR′/Ŝ′)(u)) .

If this is non-zero, then there exists y ∈ P such that SyŜ′ ≠ SyS′, or equivalently Ŝ′ ∩ S^{y} ⊆ S′.

Suppose now that (R, S) ∈ S. The same argument shows that if D(Ŝ\P)D(a_{R,S})(u) is non-zero, then there exists y ∈ P such that Ŝ′∩Ŝ^{y} ⊆ S′, and this implies Ŝ′∩S^{y} ⊆ S′.

Since

D(b_{R,S}) = D(S\P) − D(Ŝ\P) ,

it follows that if D(b_{R,S})D(a_{R′,S′})(u) is non-zero, then there exists y ∈ P such that Ŝ′ ∩ S^{y} ⊆ S′.

Recall that D(P/Ŝ′)(u) = 0 for u ∈ D^{t}(R′/S′). Hence

D(b_{R,S})D(a_{R,S})(u) = D(b_{R,S})D(P/S)(u) .

By the above computation, this is equal to

∑_{x∈R\P/R′} (γ_{|S:S∩^{x}R′|}D(S\RxR′/S′)(u) − γ_{|Ŝ:Ŝ∩^{x}R′|}D(Ŝ\RxR′/S′)(u)) .

If this is non-zero, then at least one of the following holds :

• There exists x ∈ P such that |S : S ∩^{x}R′| ≠ |Ŝ : Ŝ ∩^{x}R′|, or equivalently

|Ŝ : S| = p ≠ |Ŝ ∩^{x}R′ : S ∩^{x}R′| ,

hence Ŝ ∩^{x}R′ ⊆ S. This implies Ŝ ∩^{x}S′ ⊆ S.

• There exists x ∈ P such that ŜxS′ ≠ SxS′, or equivalently Ŝ ∩^{x}S′ ⊆ S.

In both cases, there exists x ∈ P with Ŝ ∩^{x}S′ ⊆ S.

Now Lemma 4.5 shows that if D(b_{R,S})D(a_{R′,S′})(u) ≠ 0, then (R, S) _{P} (R′, S′), hence (R, S) = (R′, S′) since both pairs are in S.

In this case by Corollary 4.3, there is a unique double coset RxR in P such that Ŝ ∩^{x}S ⊆ S, namely the coset R. It follows that

D(b_{R,S})D(a_{R,S})(u) = γ_{|S:S∩R|}D(S\R/S)(u) − γ_{|Ŝ:Ŝ∩R|}D(Ŝ\R/S)(u)
= D(R/S)(u) − γ_{|Ŝ:Ŝ∩R|}D(R/Ŝ)(u)
= u

since R/S = Id_{R/S} and D(R/Ŝ)(u) = 0 for u ∈ D^{t}(R/S). It follows that D_{S} ∘ I_{S} is the identity map, and this completes the proof of the theorem. 6.2. Conjecture : The map I_{S} of Theorem 6.1 is an isomorphism, for any finite p-group P, and any set S of representatives of genetic sections of P modulo the relation _{P} .

By [8], and by Theorem 6.1, this conjecture is equivalent to the following : 6.3. Conjecture : The torsion part D^{t}(P) of the Dade group of P is isomorphic to (ℤ/2ℤ)^{n_{P}} ⊕ (ℤ/4ℤ)^{m_{P}}, where m_{P} is equal to the number of rational irreducible representations of P of generalized quaternion type, and n_{P} is equal to the number of rational irreducible representations of P whose type is

• cyclic of order at least 3, or semi-dihedral, or generalized quaternion, if the ground field contains cubic roots of unity.

• cyclic of order at least 3, or semi-dihedral, or generalized quaternion of order at least 16, if the ground field does not contain cubic roots of unity.

6.4. Remark. Conjectures 6.2 and 6.3 are known to be true in the following cases :

• For p ≠ 2, by a result of J. Carlson and J. Thévenaz ([7] Theorem 13.3).

• For p ≠ 2 and metacyclic p-groups, by a result of N. Mazza, independent of Carlson and Thévenaz’s result ([11]).

• For cyclic, generalized quaternion, dihedral or semi-dihedral 2-groups, by [8].

• For some other 2-groups, such as D_{8} ∗ C_{4}, D_{8} ∗ D_{8}, or D_{8} ∗ Q_{8}, by results of J. Thévenaz, and more generally for all (almost) extraspecial p-groups, by a result of N. Mazza and the author ([5]).

7. Rational biset functors and the torsion part of D^{Ω}

The results of previous sections lead to the following definition :

7.1. Definition : Let F be a biset functor defined over p-groups, with values in k-Mod. The functor F is called rational if for any finite p-group P, there exists a set S of representatives of genetic sections of P modulo the relation _{P} , such that the map

I_{S} = ⊕_{(T,S)∈S} F (a_{T,S}) : ⊕_{(T,S)∈S} ∂F (T/S) → F (P) ,

is a k-module isomorphism.

7.2. Example : The word “rational” in the previous definition comes from the example k = ℤ, and F = R_{ℚ}. Then for any p-group Q with normal p-rank 1, there is a unique faithful irreducible rational representation Φ_{Q}, and ∂F (Q) ≅ ℤ, with basis Φ_{Q}. Moreover, any rational irreducible representation V of P is isomorphic to Ind_{T}^{P} Inf_{T/S}^{T}Φ_{T/S} for a suitable genetic section (T, S) of P, which can be chosen in a given set S of representatives of such sections. Thus V = R_{ℚ}(a_{T,S})(Φ_{T/S}), and the map I_{S} is surjective for any p-group, and any set of representatives S. Hence it is an isomorphism, and R_{ℚ} is rational.

The following lemma shows that if F is rational, then the map I_{S} is an isomorphism for any set S of representatives of equivalence classes of genetic sections of P.

7.3. Lemma : Let F be a biset functor and P be a p-group. Let S and S′ be sets of representatives of genetic sections of P for the relation _{P} . If the map I_{S} is an isomorphism, then the map I_{S′} is an isomorphism. Proof: Showing that the map I_{S′} is an isomorphism is equivalent to showing that the map D_{S′} is injective, since this map is split surjective by Theorem 3.2. Equivalently, since I_{S} is an isomorphism by hypothesis, this amounts to showing that the map f = D_{S′} ∘I_{S} is injective. Let u = (u_{T,S})_{(T,S)∈S} an element of Ker(f), with u_{T,S} ∈ ∂F (T/S). If (T′, S′) ∈ S′, then the component of f(u) in ∂F (T′/S′) is equal to

b_{T′,S′}( ∑_{(T,S)∈S} a_{T,S}(u_{T,S})) .

Now b_{T′,S′} ∘ a_{T,S} is equal to zero by Theorem 4.6, unless (T′, S′) _{P} (T, S). There is a unique such (T, S) ∈ S, and for this one

b_{T′,S′} ∘ a_{T,S}(u_{T,S}) = φ_{T,S}^{T′,S′}(u_{T,S}) ,

by Theorem 4.9, since f_{1}^{T/S}(u_{T,S}) = u_{T,S}. It follows that this is equal to the component of f(u) in ∂F (T′/S′). Hence this is zero, and u_{T,S} = 0 since φ_{T,S}^{T′,S′} is an isomorphism. Since for any (T, S) ∈ S, there is a unique (T′, S′) ∈ S′ such that (T′, S′) _{P} (T, S), thus u_{T,S} = 0 for any (T, S) ∈ S, and u = 0.

7.4. Proposition : Let F be a biset functor over p-groups, with values in k-Mod, let F′ be a subfunctor of F, and let M be any k-module.

1. The functor F is rational if and only if F′ and F/F′ are rational.

2. If F is rational, then Hom(F, M) is rational.

The first assertion means that the class of rational biset functors is a Serre subclass of all biset functors. In the second assertion, recall that Hom(F, M) is defined by Hom(F, M)(P) = Hom_{k}(F (P), M) for any p-group P, and

Hom(F, M)(φ)(α) = α ∘ F (φ^{op})

for φ ∈ Hom_{C_{p,k}}(P, Q) and α ∈ Hom(F, M)(P).

Proof: Let P be a finite p-group, and S be a set of representatives of equivalence classes of genetic sections of P. Let F′′ = F/F′. The diagram

0  0
↓  ↓
⊕_{(T,S)∈S} ∂F′(T/S) →^{i′} F′(P)
↓  ↓
⊕_{(T,S)∈S} ∂F (T/S) →^{i} F (P)
↓  ↓
⊕_{(T,S)∈S} ∂F′′(T/S) →^{i′′} F′′(P)
↓  ↓
0  0

is commutative, where the horizontal maps are the maps I_{S} for the corresponding functors. By the snake’s lemma, and since i′, i′, and i′′ are (split) injective, there is an exact sequence

0 → Coker(i′) → Coker(i) → Coker(i′′) → 0 .

Now saying that F is rational is equivalent to saying that Coker(i) = 0 for any P and S. Thus Coker(i′) = Coker(i′′) = 0, and F′ and F′′ are rational. Conversely, if F′ and F′′ are rational, then Coker(i′) = Coker(i′′) = 0, and Coker(f) = 0, thus F is rational.

For the second assertion, denote by F̂ the functor Hom(F, M), and observe that for any genetic section (T, S) of P, one has that b_{T,S} = a_{T,S}^{op}, and that (f_{1}^{T/S})^{op} = f_{1}^{T/S}. Applying the functor Hom_{k}(−, M) to the isomorphism

I_{S} = ⊕_{(T,S)∈S} F (a_{T,S}) : ⊕_{(T,S)∈S} ∂F (T/S) → F (P) ,

gives the isomorphism

D_{S} = ⊕_{(T,S)∈S} F̂(b_{T,S}) : F̂(P) → ⊕_{(T,S)∈S} ∂F̂(T/S) ,

for the functor F̂. Thus F̂ is rational.

7.5. Corollary : The functor D_{tors}^{Ω} is rational. Proof: This follows from the fact that, by [2], the image of the subfunctor R_{ℚ}^{∗} = Hom(R_{ℚ}, ℤ) of the dual Burnside functor B^{∗} is equal to the torsion subfunctor D_{tors}^{Ω} of D^{Ω}. Hence D_{tors}^{Ω} is a quotient of Hom(R_{ℚ}, ℤ). Since R_{ℚ} is a rational ℤ-valued biset functor, so is its dual, and any quotient of it.

7.6. Corollary : Let P be a finite p-group. Then

D_{tors}^{Ω}(P) ≅ (ℤ/4ℤ)^{a_{P}} ⊕ (ℤ/2ℤ)^{b_{P}} ,

where a_{P} is equal to the number of isomorphism classes of rational irreducible representations of P whose type is generalized quaternion, and b_{P} is the number of isomorphism classes of rational irreducible representations of P whose type is cyclic of order at least 3 or semi-dihedral. Proof: This follows from the structure of the Dade group of groups of normal p-rank 1 (see [8] Theorems 5.4, 6.3, 7.1).

Note that Corollary 7.5 is actually more precise, since it gives a generating set for D_{tors}^{Ω}(P) ≅ (ℤ/4ℤ)^{a_{P}} ⊕ (ℤ/2ℤ)^{b_{P}} : if Q has normal p-rank 1, then ∂D_{tors}^{Ω}(Q) is a subgroup of the endo-trivial subgroup of D(Q). In particular D^{Ω}(a_{T,S})(u) = Teninf_{T/S}^{P}(u) for any genetic section (T, S) of P, and any u ∈ ∂D_{tors}^{Ω}(T/S).

Now if Q = T/S is cyclic or generalized quaternion, then ∂D_{tors}^{Ω}(Q) is generated by u_{T,S} = Ω_{Q/1}, and if Q is semi-dihedral, then ∂D_{tors}^{Ω} is generated by u_{T,S} = Ω_{Q/1} + Ω_{Q/R}, where R is a non-central subgroup of Q of order 2. The elements v_{T,S} = Teninf_{T/S}^{P}(u_{T,S}), for (T, S) in a set S of representatives of equivalence classes of genetic sections of P for which T/S is cyclic of order at least 3, or semi-dihedral, or generalized quaternion, are a set of generators of D_{tors}^{Ω}(P), and these elements are as linearly independent as they can be : the linear combination ∑_{(T,S)∈S} λ_{T,S}v_{T,S} is zero if

and only if for any (T, S) ∈ S, the integer λ_{T,S} is a multiple of the order of v_{T,S}, which is equal to 4 if T/S is generalized quaternion, and 2 otherwise.

References

[1] S. Bouc. The functor of rational representations for p-groups. Preprint (2002), to appear in Adv. in Maths.

[2] S. Bouc. A remark on the Dade group and the Burnside group. Preprint (2002), to appear in J. of Algebra.

[3] S. Bouc. Construction de foncteurs entre catégories de G-ensembles. J. of Algebra, 183(0239):737–825, 1996.

[4] S. Bouc. A remark on a theorem of Ritter and Segal. J. of Group Theory, 4:11–18, 2001.

[5] S. Bouc and N. Mazza. The Dade group of (almost) extraspecial p-groups. Preprint (2004), to appear in J. Pure and Applied Algebra.

[6] S. Bouc and J. Thévenaz. The group of endo-permutation modules. Invent. Math., 139:275–349, 2000.

[7] J. Carlson and J. Thévenaz. The classification of torsion endo-trivial modules. Preprint (2003), to appear in Ann. of Maths.

[8] J. Carlson and J. Thévenaz. Torsion endo-trivial modules. Algebras and representation Theory, 3:303–335, 2000.

[9] E. Dade. Endo-permutation modules over p-groups I. Ann. of Math., 107:459–494, 1978.

[10] E. Dade. Endo-permutation modules over p-groups II. Ann. of Math., 108:317–346, 1978.

[11] N. Mazza. Modules d’endo-permutation. PhD thesis, Université de Lausanne, 2003.

Serge Bouc, UFR de Mathématiques, Université Paris 7-Denis Diderot, 75251, Paris Cedex 05 , France

email:  bouc@math.jussieu.fr