K-theory, genotypes, and biset functors

Serge Bouc

Abstract: Let p be an odd prime number. In this paper, we show that the genome Γ(P) of a finite p-group P, defined as the direct product of the genotypes of all rational irreducible representations of P, can be recovered from the first group of K-theory K_{1}(ℚP). It follows that the assignment P ↦ Γ(P) is a p-biset functor. We give an explicit formula for the action of bisets on Γ, in terms of generalized transfers associated to left free bisets. Finally, we show that Γ is a rational p-biset functor, i.e. that Γ factors through the Roquette category of finite p-groups.

AMS Subject Classification: 19B28, 20C05, 18A99.

Keywords: K-theory, genotype, Whitehead group, biset functor, Roquette category, transfer.

1. Introduction

Let p be a prime number. This article originates in a joint work with Nadia Romero ([4]), when we started considering the possible applications of genetic bases to the computation of Whitehead groups of finite p-groups. Indeed, after the comprehensive book of B. Oliver ([7]), it became clear to N. Romero that these questions have close links to rational representations of p-groups. So the idea emerged that possibly genetic bases would be a natural tool in this context, and a first use of this is made in [8].

In particular, when trying to compute various groups related to the Whitehead group of a finite p-group P (for odd p), a specific product appears, defined in terms of the fields of endomorphisms of the irreducible ℚP -modules. After some non trivial reformulation using genetic bases, this product can be viewed as

Γ(P) = ∏_{S∈B} (N_{P} (S)/S) ,

where B is a genetic basis of P. As the groups N_{P} (S)/S are called the types or genotypes of the irreducible ℚP -modules, we call Γ(P) the genome of P. It is the main subject of this paper.

The connection of Γ(P) with Whitehead groups and K-theory is established in Theorem 4.3: the genome of P can be recovered as the p-torsion part of K_{1}(ℚP). This induces a structure of p-biset functor on the correspondence P ↦ Γ(P), which we try to make explicit in Section 5, by giving formulae to compute the action of a (Q, P)-biset on Γ(P) (Theorem 5.9). Finally, we show that Γ is a rational p-biset functor, hence it factors through the Roquette category of finite p-groups introduced in [3].

2. Review of K_{1}

2.1. Let A be a ring (with 1). Let GL(A) denote the colimit of the linear groups GL_{n}(A), for n ∈ ℕ_{>0}, where the inclusion GL_{n}(A) ↪ GL_{n+1}(A) is

M ∈ GL_{n}(A) ↦ ( \begin{matrix}M & 0 \\ 0 & 1 \end{matrix} ) ∈ GL_{n+1}(A) .

The group K_{1}(A) is defined as the abelianization of GL(A), namely

K_{1}(A) = GL(A)^{ab} = GL(A)/[GL(A), GL(A)] .

2.2. Remark : In particular there is a canonical group homomorphism from the group A^{×} = GL_{1}(A) of invertible elements of A to K_{1}(A), which factors as

\xymatrix{A^{×} \ar[r] & A^{×}/[A^{×}, A^{×}] \ar[r]^{α_{A}} & K_{1}(A)}

2.3. There is an alternative definition of K_{1}(A): let P(A) denote the category of pairs (P, a) of a finitely generated projective (left) A-module P, and an automorphism a of P. A morphism (P, a) → (Q, b) in P(A) is a morphism of A-modules f : P → Q such that b ∘ f = f ∘ a.

Let [P, a] denote the isomorphism class of (P, a) in P(A), and let K_{det}(A) denote the Grothendieck group with generators the set of these equivalence classes, and the relations of the following two forms

• [P, a ∘ a′] = [P, a] + [P, a′], for any a, a′ ∈ Aut_{A}(P),

• [Q, b] = [P, a] + [R, c] whenever there are morphisms f : [P, a] → [Q, b] and g : [Q, b] → [R, c] in P(A) such that the sequence

0 → P →^{f} Q →^{g} R → 0

is an exact sequence of A-modules (in particular, since R is projective, this sequence splits).

If n ∈ ℕ_{>0} and m ∈ GL_{n}(A), one can view m as an automorphism of the free module A^{n}. Let λ(m) = [A^{n}, m] ∈ K_{det}(A).

2.4. Theorem: The assignment m ↦ λ(m) induces a group isomorphism

K_{1}(A) ≅ K_{det}(A).

Proof : See [5] Theorem 40.6.

2.5. Let now A and B be two rings, and let L be a (B, A)-bimodule which is finitely generated and projective as a left B-module. If P is a finitely generated projective A-module, then P is a direct summand of some free Amodule A^{n}, and then L ⊗_{A} P is a direct summand of L ⊗_{A} A^{n} ≅ L^{n} as a left B-module. Hence L ⊗_{A} P is a finitely generated projective left B-module. Then the functor L⊗_{A} − : P ↦ L⊗_{A} P induces a functor T_{L} : P(A) → P(B) such that

T_{L}((P, a)) = (L ⊗_{A} P, L ⊗_{A} a) .

One checks easily that the defining relations of K_{det}(A) are preserved by this functor, hence there is a well-defined induced group homomorphism

t_{L} : K_{det}(A) → K_{det}(B)

sending the class [P, a] to the class [L ⊗_{A} P, L ⊗_{A} a]. This group homomorphism is called the (generalized) transfer associated to the bimodule L.

The properties of the tensor product of bimodules now translate to properties of this transfer homomorphism:

2.6. Proposition: Let A, B, C be rings. In the following assertions, assume that the bimodules involved are finitely generated and projective as left modules. Then:

1. if L ≅ L′ as (B, A)-bimodules, then t_{L} = t_{L′}.

2. if L is the (A, A)-bimodule A, then t_{L} = Id_{K_{det}(A)}.

3. if L ≅ L_{1} ⊕ L_{2} as (B, A)-bimodules, then t_{L} = t_{L_{1}} + t_{L_{2}}.

4. if L is a (B, A)-bimodule and M is a (C, B)-bimodule, then

t_{M} ∘ t_{L} = t_{M⊗_{B}L} .

It follows in particular from (2) and (4) that if L is a (B, A)-bimodule inducing a Morita equivalence from A to B, then t_{L} is an isomorphism (more precisely, if M is an (A, B)-bimodule such that M ⊗_{B} L ≅ A and L⊗_{A}M ≅ B as bimodules, then t_{L} and t_{M} are inverse to one another).

2.7. The group K_{1}(A) has been determined for a number of rings A. In particular:

2.8. Theorem:

1. Let D be a division ring. Then K_{1}(D) ≅ D^{×}/[D^{×}, D^{×}].

2. Let F be a field. Then the determinant homomorphism

m ∈ GL_{n}(F) → det(m) ∈ F^{×}

induces an isomorphism K_{1}(F) ≅ F^{×}.

Proof : See [5] Theorem 38.32.

2.9. Proposition: Let 𝔽 be a field and G be a finite group of order prime to the characteristic of 𝔽. Let Irr_{𝔽}(G) denote a set of representatives of isomorphism classes of irreducible 𝔽G-modules, and for V ∈ Irr_{𝔽}(G), let D_{V} = End_{𝔽G}(V) denote the skew field of endomorphisms of V.

Then V is an (𝔽G, D_{V}^{op})-bimodule, where the action of g ∈ G and f ∈ D_{V} on v ∈ V is given by g · v · f = gf(v) = f(gv). Let V^{∗} denote the 𝔽-dual of V, considered as a (D_{V}^{op}, 𝔽G)-bimodule.

Then the map

\xymatrix{τ : K_{1}(𝔽G) \ar[r]^{∏_{V} t_{V^{∗}}} & ∏_{V ∈Irr_{𝔽}(G)} K_{1}(D_{V}^{op})}

is a well defined isomorphism of abelian groups, with inverse

\xymatrix{τ′ : ∏_{V ∈Irr_{𝔽}(G)} K_{1}(D_{V}^{op}) \ar[r]^{∏_{V} t_{V}} & K_{1}(𝔽G) .}

Proof : As |G| is invertible in 𝔽, the group algebra 𝔽G is semisimple. Moreover for each V ∈ Irr_{𝔽}(V), the skew field D_{V}^{op} is also a semisimple 𝔽-algebra. This shows that V is projective and finitely generated as an 𝔽G-module, and that V^{∗} is projective and finitely generated as a D_{V}^{op}-module (that is V^{∗} is a finite dimensional D_{V} -vector space). Hence the generalized transfer maps t_{V} : K_{1}(𝔽G) → K_{1}(D_{V}^{op}) and t_{V^{∗}} : K_{1}(D_{V}^{op}) → K_{1}(𝔽G) are well defined.

Now for any two finitely generated 𝔽G-modules V and W, the map

α ⊗ w ↦ (v ∈ V ↦ α(v)w ∈ W )

extends to an isomorphism (see e.g. [6] (2.32))

V^{∗} ⊗_{𝔽G} W → Hom_{𝔽G}(V, W)

of ((End_{𝔽G}V)^{op}, (End_{𝔽G}W^{op}))-bimodules, where the bimodule structure on the right hand side is given by

∀h ∈ (End_{𝔽G}V)^{op}, ∀ψ ∈ Hom_{𝔽G}(V, W), ∀k ∈ (End_{𝔽G}W)^{op}, h·ψ·k = k∘ψ∘h.

In case V, W ∈ Irr_{𝔽}(G) and V ≠ W, this yields V^{∗} ⊗_{𝔽G} W = 0. And if V = W, we have an isomorphism V^{∗} ⊗_{𝔽G} V ≅ D_{V}^{op} of (D_{V}^{op}, D_{V}^{op})-bimodules. Then by Assertions (2) and (4) of Proposition 2.6

t_{V^{∗}} ∘ t_{W} = \{ \begin{matrix}0 & if V ≠ W \\ Id_{K_{1}(D_{V}^{op})} & if V = W. \end{matrix}

In other words τ ∘ τ′ is the identity map of ∏_{V ∈Irr_{𝔽}(G)} K_{1}(D_{V}^{op}). Conversely

τ′ ∘ τ = ∑_{V ∈Irr_{𝔽}(G)} t_{V} ∘ t_{V^{∗}} = t_{L} ,

where L is the (𝔽G, 𝔽G)-bimodule ⊕_{V ∈Irr_{𝔽}(G)}(V ⊗_{D_{V}^{op}}V^{∗}). For each V ∈ Irr_{𝔽}(G),

the bimodule V ⊗_{D_{V}^{op}} V^{∗} ≅ End_{D_{V}^{op}}(V) is isomorphic to the Wedderburn component of 𝔽G corresponding to the simple module V, and the semisimple algebra 𝔽G is equal to the direct sum of its Wedderburn components. Thus L ≅ 𝔽G, and t_{L} is equal to the identity map of K_{1}(𝔽G).

2.10. Corollary: Under the assumptions of Proposition 2.9, there is a group isomorphism

K_{1}(𝔽G) ≅ ∏_{V ∈Irr_{𝔽}(G)} D_{V}^{×}/[D_{V}^{×}, D_{V}^{×}] .

Proof : This follows from Proposition 2.9 and Theorem 2.8, since x ↦ x^{−1} is a group isomorphism D^{×} → (D^{op})^{×}, for any skew field D.

2.11. Recall ([2] Chapter 3) that the biset category C of finite groups has all finite groups as objects, the set of morphisms in C from a group G to a group H being the Grothendieck group of (finite) (H, G)-bisets, i.e. the Burnside group B(H, G). The composition of morphisms in C is the linear extension of the product (V, U) ↦ V ×_{H} U, for a (K, H)-biset V and an (H, G)-biset U.

A biset functor is an additive functor from C to the category Ab of abelian groups.

For a prime number p, a p-biset functor is an additive functor from the full subcategory C_{p} of C consisting of p-groups to Ab.

Let _{1}C denote the (non full) subcategory of C with the same objects, but where the set of morphisms from a group G to a group H is the Grothendieck group _{1}B(H, G) of left free (H, G)-bisets. A deflation biset functor is an additive functor from _{1}C to Ab.

2.12. Proposition:

1. Let R be a commutative ring. The assignment G ↦ K_{1}(RG) is a deflation functor.

2. The assignment G ↦ K_{1}(ℚG) is a biset functor.

Proof : For Assertion 1, if G and H are finite groups, and if U is a finite left free (H, G)-biset, then the corresponding permutation (RH, RG)-bimodule RU is free and finitely generated as a left RH-module. Hence the transfer t_{RU} : K_{1}(RG) → K_{1}(RH) is well defined. If U′ is an (H, G)-biset isomorphic to U, then RU′ ≅ RU as bimodules, hence t_{RU′} = t_{RU}. And if U is the disjoint unions of two (H, G)-bisets U_{1} and U_{2}, then RU ≅ RU_{1} ⊕ RU_{2}, thus t_{RU} = t_{RU_{1}} + t_{RU_{2}}. This shows that one can extend linearly this transfer construction U ↦ t_{RU} to a group homomorphism

u ∈ _{1}B(H, G) ↦ K_{1}(u) ∈ Hom_{Ab}(K_{1}(RG), K_{1}(RH)) .

Moreover, if K is a third group, and V is a finite left free (K, H)-biset, then t_{RV} ∘ t_{RU} = t_{RV ⊗_{RH}RU} = t_{R(V ×_{H}U)} since the bimodules RV ⊗_{RH} RU and R(V ×_{H} U) are isomorphic. Finally, if U is the identity biset at G, namely the set G acted on by left and right multiplication, then RU ≅ RG as (RG, RG)-bimodule, thus t_{RU} = Id_{K_{1}(RG)}. This completes the proof of Assertion (1).

The proof of Assertion (2) is the same, except that the transfer t_{ℚU} : K_{1}(ℚG) → K_{1}(ℚH) is well defined for an arbitrary finite (H, G)-biset U: indeed ℚU is always finitely generated and projective as a ℚH-module.

3. Review of genetic subgroups

3.1. Let p be a prime number. A finite p-group is called a Roquette p-group if it has normal rank 1, i.e. if all its normal abelian subgroups are cyclic. The Roquette p-groups (see [9]) are the cyclic groups C_{p^{n}}, for n ∈ ℕ, if p is odd. The Roquette 2-groups are the cyclic groups C_{2^{n}}, for n ∈ ℕ, the generalized quaternion groups Q_{2^{n}}, for n ≥ 3, the dihedral groups D_{2^{n}}, for n ≥ 4, and the semidihedral groups SD_{2^{n}}, for n ≥ 4.

If P is a Roquette p-group, then P admits a unique faithful irreducible rational representation Φ_{P} ([2] Proposition 9.3.5). 3.2. If S is a subgroup of a finite p-group P, denote by Z_{P} (S) the subgroup of N_{P} (S) defined by Z_{P} (S)/S = Z(N_{P} (S)/S). The subgroup S is called genetic if it fulfills the following two conditions:

1. if x ∈ P, then S^{x} ∩ Z_{P} (S) ≤ S if and only if S^{x} = S.

2. the group N_{P} (S)/S is a Roquette p-group.

When S is a genetic subgroup of P, let V (S) = Indinf_{N_{P} (S)/S}^{P}Φ_{N_{P} (S)/S} denote the ℚP -module obtained by inflation of Φ_{N_{P} (S)/S} to N_{P} (S) followed by induction to P.

Two genetic subgroups S and T of P are said to be linked modulo P (notation S _{P} T) if there exists an element x ∈ P such that S^{x}∩Z_{P} (T) ≤ T and ^{x}T ∩ Z_{P} (S) ≤ S (where as usual S^{x} = x^{−1}Sx and ^{x}T = xT x^{−1}).

3.3. Theorem: Let p be a prime number and P be a finite p-group.

1. If V is a simple ℚP -module, then there exists a genetic subgroup S of P such that V ≅ V (S).

2. If S is a genetic subgroup of P, then there is an isomorphism of ℚ-

algebras

End_{ℚP} V (S) ≅ End_{ℚN_{P} (S)/S}Φ_{N_{P} (S)/S}

induced by the induction-inflation functor from ℚN_{P} (S)/S-modules to

ℚP -modules.

3. If S and T are genetic subgroups of P, then V (S) ≅ V (T) if and only if S _{P} T. In this case, the groups N_{P} (S)/S and N_{P} (T)/T are isomorphic.

Proof : See Theorem 9.4.1, Lemma 9.4.3, Definition 9.4.4, Corollary 9.4.5, Theorem 9.5.6 and Theorem 9.6.1 of [2].

It follows in particular that the relation _{P} is an equivalence relation on the set of genetic subgroups of P. A genetic basis of P is by definition a set of representatives of genetic subgroups of P for this equivalence.

It also follows that if V is a simple ℚP -module, and if S is a genetic subgroup of P such that V ≅ V (S), then the group N_{P} (S)/S does not depend on the choice of such a genetic subgroup S. This factor group is called the type of V ([2] Definition 9.6.8). Laurence Barker ([1]) has introduced the word genotype instead of type, and we will follow this terminology.

3.4. Definition: Let p be a prime number and P be a finite p-group. The genome Γ(P) of P is the product group

Γ(P) = ∏_{S∈B} (N_{P} (S)/S) ,

where B is a genetic basis of P. It is well defined up to isomorphism.

More precisely, suppose that B and B′ are genetic bases of a p-group P. Then for S ∈ B, there exists a unique S′ ∈ B′ such that there exists some x ∈ P with

(3.5)  S^{x} ∩ Z_{P} (S′) ≤ S′ and ^{x}S′ ∩ Z_{P} (S) ≤ S ,

and the correspondence S ↦ S′ is a bijection from B to B′. Moreover, for each S ∈ B corresponding to S′ ∈ B′, the set D of elements x satisfying (3.5) is a single (N_{P} (S), N_{P} (S′))-double coset in P ([2], Proposition 9.6.9).

Let x ∈ D. Then for each n ∈ N_{P} (S)/S, there is a unique element n′ ∈ N_{P} (S′)/S′ such that nSx = xS′n′, and the map n ↦ n′ is a group isomorphism N_{P} (S)/S → N_{P} (S′)/S′, which only depends on x up to interior automorphism of N_{P} (S)/S. In particular, when p is odd, the group N_{P} (S)/S is cyclic, so this group isomorphism does not depend on x.

Thus for odd p, this yields a canonical group isomorphism

(3.6)  \xymatrix{∏_{S∈B} (N_{P} (S)/S) \ar[r]^{γ_{B′,B}} & ∏_{S′∈B′} (N_{P} (S′)/S′) .}

3.7. Remark : Let p be a prime number, and P be a finite p-group. Since the Roquette p-groups are all indecomposable (that is, they cannot be written as a direct product of two non-trivial of their subgroups), the genotypes of the simple ℚP -modules are determined by the group Γ(P): by the Krull-Remak-Schmidt theorem, the group Γ(P) can be written as a direct product of indecomposable groups Γ_{1}, …, Γ_{r}, and such a decomposition is unique (up to permutation and isomorphism of the factors). Then Γ_{1}, …, Γ_{r} are the genotypes of the simple ℚP -modules.

In terms of the Roquette category R_{p} (see Section 7, or [3]), this means that two finite p-groups P and Q become isomorphic in R_{p} if and only if their genomes Γ(P) and Γ(Q) are isomorphic (as groups) (see [3] Proposition 5.14).

4. K-theory and genome

4.1. Lemma: Let p be a prime, and C be a cyclic p-group. Recall that Φ_{C} is the unique faithful irreducible rational representation of C, up to isomorphism.

1. If C = 1, then Φ_{C} = ℚ.

2. If C ≠ 1, let Z be the unique subgroup of order p of C. Then there is an exact sequence

(4.2)  0 → Φ_{C} → ℚC → ℚ(C/Z) → 0 ,

of (ℚC, ℚC)-bimodules, where ℚC → ℚ(C/Z) is the canonical surjection.

3. If C has order p^{n}, then the algebra End_{ℚC}(Φ_{C}) is isomorphic to the cyclotomic field ℚ(ζ_{p^{n}}), and if p > 2, the map sending c ∈ C to the endomorphism φ ↦ φc of Φ_{C} is a group isomorphism from C to the p-torsion part _{p}ℚ(ζ_{p^{n}})^{×} of the multiplicative group ℚ(ζ_{p^{n}})^{×}.

Proof : Assertion 1 is trivial. Assertion 2 follows e.g. from [2], Proposition 9.3.5. A different proof consists in observing that if C has order p^{n}, then the algebra ℚC is isomorphic to ℚ[X]/(X^{p^{n}} − 1), and the projection map ℚC → ℚ(C/Z) becomes the canonical map

ℚ[X]/(X^{p^{n}} − 1) → ℚ[X]/(X^{p^{n−1}} − 1) .

The kernel of this map is now clearly isomorphic to ℚ[X]/(γ_{p^{n}}), where γ_{p^{n}} is the p^{n}-th cyclotomic polynomial, that is, the p^{n}-th cyclotomic field, which is clearly a simple faithful module for the cyclic group generated by X in the algebra ℚ[X]/(X^{p^{n}} − 1). Observe moreover that the exact sequence 4.2 is indeed a sequence of (ℚC, ℚC)-bimodules.

The first part of Assertion 3 follows easily. For the last part, let ζ_{p^{n}} be a primitive p^{n}-th root of unity. Observe that a p-torsion element in ℚ(ζ_{p^{n}})^{×} is a p^{n}-th root of unity. Hence the p-torsion part of ℚ(ζ_{p^{n}})^{×} is cyclic of order p^{n}, generated by ζ_{p^{n}}.

4.3. Theorem: Let p be an odd prime, and P be a finite p-group, and B be a genetic basis of P. If S is a genetic subgroup of P, and a ∈ N_{P} (S)/S, view a as an automorphism of Φ_{N_{P} (S)/S}, and let ã denote the corresponding automorphism of V (S) = Indinf_{N_{P} (S)}^{P}Φ_{N_{P} (S)/S}.

1. The group homomorphism

\xymatrix{Γ(P) = ∏_{S∈B} (N_{P} (S)/S) \ar[r]^{ν_{B}} & K_{1}(ℚP)}

sending a ∈ N_{P} (S)/S, for S ∈ B, to the class [V (S), ã] in K_{1}(ℚP) is an isomorphism of the genome Γ(P) onto the p-torsion part _{p}K_{1}(ℚP)

of K_{1}(ℚP).

2. If B′ is another genetic basis of P, and γ_{B′,B} is the canonical isomorphism defined in 3.6, then

ν_{B′} ∘ γ_{B′,B} = ν_{B} .

Proof : Since p is odd, the Roquette p-groups are the cyclic p-groups. Assertion 1 now follows from Proposition 2.9, Theorem 3.3, and Lemma 4.1.

For Assertion 2, let S ∈ B and let S′ be the unique element of B′ such that S′ _{P} S. Let φ : N_{P} (S)/S → N_{P} (S′)/S′ be the restriction of γ_{B′,B} to N_{P} (S)/S. If a ∈ N_{P} (S)/S, let a′ = φ(a). Then φ induces an isomorphism of ℚP -modules φ̃ : V (S) → V (S′) such that the diagram

\xymatrix{V (S) \ar[d]_{φ̃} \ar[r]^{ã} & V (S) \ar[d]^{φ̃} \\ V (S′) \ar[r]^{ã′} & V (S′)}

is commutative. Hence (V (S), ã) ≅ (V (S′), ã′) in P(ℚP), thus [V (S), ã] = [V (S′), ã′] in K_{1}(ℚP), as was to be shown.

4.4. Remark : The elements of odd order of ℚ(ζ_{p^{n}})^{×} are the p^{n}-th roots of unity. So Γ(P) is also the odd-torsion part of K_{1}(ℚP).

4.5. Corollary: Let p be an odd prime. Then the correspondence sending a finite p-group P to its genome Γ(P) is a p-biset functor.

Proof : Indeed by Proposition 2.12, the assignment P ↦ K_{1}(ℚP) is a p-biset functor. So its p-torsion part is also a p-biset functor.

5. Explicit transfer maps

We begin with a slight generalization of the transfer homomorphism, associated to a left-free biset:

5.1. Lemma and Definition: Let G and H be finite groups, and let Ω be a left free (H, G)-biset. Let [H\Ω] be a set of representatives of H-orbits on Ω. For g ∈ G, and x ∈ Ω, let h_{g,x} ∈ H and σ_{g}(x) ∈ [H\Ω] be the elements defined by xg = h_{g,x}σ_{g}(x).

1. The map g ∈ G ↦ ∏_{x∈[H\Ω]} h_{g,x} (in any order) induces a well defined

group homomorphism

Ver_{Ω} : G/[G, G] → H/[H, H]

called the (generalized) transfer associated to Ω.

2. If Ω′ ≅ Ω as (H, G)-bisets, then Ver_{Ω′} = Ver_{Ω}.

3. If Ω = Ω_{1} ⊔ Ω_{2} as (H, G)-bisets, then Ver_{Ω} = Ver_{Ω_{1}} + Ver_{Ω_{2}}.

4. If K is another finite group, and Ω′ is a finite left free (K, H)-biset, then Ω′ ×_{H} Ω is a finite left free K-set, and

Ver_{Ω′} ∘ Ver_{Ω} = Ver_{Ω′×_{H}Ω} .

The notation and terminology comes from the classical transfer from G/[G, G] to H/[H, H], when H is a subgroup of G: the corresponding biset Ω is the set G itself, in this case.

Proof : Changing the set of representatives [H\Ω] amounts to replacing each x ∈ [H\Ω] by η_{x}x, for some η_{x} ∈ H. This changes the element h_{g,x} to h′_{g,x} = η_{x}h_{g,x}η_{σ_{g}(x)}^{−1}, so the product over x ∈ [H\Ω] of the elements h′_{g,x} is equal to the product of the elements h_{g,x} in the abelianization H/[H, H]. Hence Ver_{Ω} does not depend on the choice of a set of representatives.

It follows moreover from the definition that for g, g′ ∈ G and x ∈ [H\Ω], we have h_{gg′,x} = h_{g,x}h_{g′,σ_{g}(x)}. Hence

∏_{x∈[H\Ω]} h_{gg′,x} = ∏_{x∈[H\Ω]} h_{g,x} ∏_{x∈[H\Ω]} h_{g′,σ_{g}(x)} = ∏_{x∈[H\Ω]} h_{g,x} ∏_{x∈[H\Ω]} h_{g′,x}

in H/[H, H], so Ver_{Ω} is a group homomorphism. This proves Assertion 1.

For Assertion 2, let f : Ω → Ω′ be an isomorphism of (H, G)-bisets. Then the set f([H\Ω]) is a set of representatives of the H-orbits on Ω′. Moreover for x ∈ [H\Ω] and g ∈ G,

f(x)g = f(xg) = f(h_{g,x}σ_{g}(x)) = h_{g,x}f(σ_{g}(x)) ,

so Ver_{Ω′}(g) = ∏_{x∈Ω} h_{g,x} = Ver_{Ω}(g), which proves Assertion 2.

Assertion 3 is clear, since [H\Ω] = [H\Ω_{1}] ⊔ [H\Ω_{2}].

For Assertion 4, it is straightforward to check that Ω′ ×_{H} Ω is left free. Moreover, the set of pairs (x′, x) ∈ Ω′ ×_{H} Ω, for x′ ∈ [K\Ω′] and x ∈ [H\Ω], is a set of representatives of K-orbits on Ω′ ×_{H} Ω. Then for x′ ∈ [K\Ω′] and x ∈ [H\Ω], and g ∈ G

(x′, x)g = (x′, xg) = (x′, h_{g,x}σ_{g}(x))
= (x′h_{g,x}, σ_{g}(x)) = (k_{h_{g,x},x′}τ_{h_{g,x}}(x′), σ_{g}(x))
= k_{h_{g,x},x′}(τ_{h_{g,x}}(x′), σ_{g}(x)) ,

where k_{h,x′} ∈ K and τ_{h}(x′) ∈ [K\Ω′] are defined by x′h = k_{h,x′}τ_{h}(x′), for h ∈ H and x′ ∈ [K\Ω′].

It follows that

Ver_{Ω′×_{H}Ω}(g) = ∏_{{x∈[H\Ω] \atop x′∈[K\Ω′]}} k_{h_{g,x},x′} = Ver_{Ω′}( ∏_{x∈[H\Ω]} h_{g,x}) = Ver_{Ω′} ∘ Ver_{Ω}(g) ,

which completes the proof.

5.2. Corollary 4.5 shows that there exists a p-biset functor structure on the assignment P ↦ Γ(P) for p-groups, when p is odd. This raises the following question: suppose that P and Q are finite p-groups, that B_{P} is a genetic basis of P, and B_{Q} is a genetic basis of Q. When U is a finite (Q, P)-biset, how can we compute the map

Γ(U) : Γ(P) = ∏_{S∈B_{P}} (N_{P} (S)/S) → Γ(Q) = ∏_{T ∈B_{Q}} (N_{Q}(T)/T )

giving the action of the biset U?

This amounts to finding the map

Γ(U)_{T,S} : \overline{N}_{P} (S) = N_{P} (S)/S → \overline{N}_{Q}(T) = N_{Q}(T)/T

for each pair (T, S) of a genetic subgroup T of Q and a genetic subgroup S of P, defined as follows: if a ∈ \overline{N}_{P} (S), then a can be viewed as an automorphism of the ℚ\overline{N}_{P} (S)-module Φ_{\overline{N}_{P} (S)}, viewed as an ideal of ℚ\overline{N}_{P} (S) as in 4.2. Then ã = Indinf_{\overline{N}_{P} (S)}^{P}a is an automorphism of V (S) = Indinf_{\overline{N}_{P} (S)}^{P}Φ_{\overline{N}_{P} (S)}, hence an element â = [V (S), ã] of K_{1}(ℚP). This element is mapped by t_{ℚU} to the element

t_{ℚU}(â) = [ℚU ⊗_{ℚP} V (S), ℚU ⊗_{ℚP} ã]

of K_{1}(ℚQ). This in turn is mapped to the element t_{V (T)^{∗}} ∘ t_{ℚU}(â) of the direct summand K_{1}(F_{T}) of K_{1}(ℚQ) corresponding to the simple ℚQ-module V (T) as in Proposition 2.9, where F_{T} is the field D_{V (T)} = End_{ℚQ}V (T).

Thus to find Γ(U)_{T,S}(a), we have to compute the element

[V (T)^{∗} ⊗_{ℚQ} ℚU ⊗_{ℚP} V (S), V (T)^{∗} ⊗_{ℚQ} ℚU ⊗_{ℚP} ã]

of K_{1}(F_{T}) ≅ F_{T}^{×}, and identify it as an element of \overline{N}_{Q}(T).

We set L(U)_{T,S} = V (T)^{∗}⊗_{ℚQ}ℚU⊗_{ℚP} V (S) for simplicity. First we observe that the induction-inflation functor Indinf_{\overline{N}_{P} (S)}^{P} is isomorphic to the functor ℚ(P/S) ⊗_{\overline{N}_{P} (S)} (−), where ℚ(P/S) is endowed with its natural structure of

(ℚP, ℚ(\overline{N}_{P} (S)))-bimodule. Hence

ℚU ⊗_{ℚP} V (S) = ℚU ⊗_{ℚP} Indinf_{\overline{N}_{P} (S)}^{P}Φ_{\overline{N}_{P} (S)}
≅ ℚU ⊗_{ℚP} ℚ(P/S) ⊗_{ℚ\overline{N}_{P} (S)} Φ_{\overline{N}_{P} (S)}
≅ ℚ(U/S) ⊗_{ℚ\overline{N}_{P} (S)} Φ_{\overline{N}_{P} (S)} ,

where ℚ(U/S) is given its natural structure of (ℚQ, ℚ(\overline{N}_{P} (S)))-bimodule.

Tensoring on the left with V (T)^{∗}, and using a similar argument, we get that

L(U)_{T,S} ≅ Φ_{\overline{N}_{Q}(T)}^{∗} ⊗_{ℚ\overline{N}_{Q}(T)} ℚ(T \U/S) ⊗_{ℚ\overline{N}_{P} (S)} Φ_{\overline{N}_{P} (S)} ,

where ℚ(T \U/S) is the permutation (ℚ\overline{N}_{Q}(T), ℚ\overline{N}_{P} (S))-bimodule associated to the (\overline{N}_{Q}(T), \overline{N}_{P} (S))-biset T \U/S. Moreover Φ_{\overline{N}_{Q}(T)} is self dual, since it is the unique faithful rational irreducible representation of \overline{N}_{Q}(T), so we can replace Φ_{\overline{N}_{Q}(T)}^{∗} by Φ_{\overline{N}_{Q}(T)} in the right hand side of the previous isomorphism.

Now the biset T \U/S splits as a disjoint union

T \U/S = ⨆_{ω∈N_{Q}(T)\U/N_{P} (S)} T \ω/S

of transitive (\overline{N}_{Q}(T), \overline{N}_{P} (S))-bisets, where N_{Q}(T)\U/N_{P} (S) is the set of (N_{Q}(T), N_{P} (S))-orbits on U. This yields a decomposition

(5.3)  ℚ(T \U/S) ≅ ⨁_{ω∈N_{Q}(T)\U/N_{P} (S)} ℚ(T \ω/S)

as (ℚ\overline{N}_{Q}(T), ℚ\overline{N}_{P} (S))-bimodules.

5.4. Lemma: Let C and D be cyclic p-groups, and let Ω be a transitive (D, C)-biset. Then Φ_{D} ⊗_{ℚD} ℚΩ = 0 unless Ω is left free, and ℚΩ⊗_{ℚC} Φ_{C} = 0 unless Ω is right free.

Proof : Suppose that the action of C is not free. This means that C is non-trivial, and that the unique subgroup Z of order p of C acts trivially on Ω: indeed since Ω is a transitive biset, the stabilizers in C of the points of Ω are conjugate in C, hence equal since C is abelian. So these stabilizers all contain Z if one of them is non trivial. Then Ω is inflated from a (C/Z)- set \overline{Ω}, and then ℚΩ ≅ ℚ\overline{Ω} ⊗_{ℚ(C/Z)} ℚ(C/Z). But ℚ(C/Z) ⊗_{QC} Φ_{C} is the module of Z-coinvariants on Φ_{C}, hence it is zero, since Φ_{C} is faithful. Hence ℚΩ ⊗_{ℚC} Φ_{C} = 0 in this case. Similarly, if the action of D is not free, then Φ_{D} ⊗_{ℚD} ℚΩ = 0.

5.5. It follows from Lemma 5.4 that to compute

Φ_{\overline{N}_{Q}(T)}^{∗} ⊗_{ℚ\overline{N}_{Q}(T)} ℚ(T \U/S) ⊗_{ℚ\overline{N}_{P} (S)} Φ_{\overline{N}_{P} (S)}

using decomposition 5.3, we can restrict to orbits ω = N_{Q}(T)uN_{P} (S), where u ∈ U, for which the (\overline{N}_{Q}(T), \overline{N}_{P} (S))-biset T \ω/S is left and right free. The left stabilizer of the element T uS of this biset is equal to

\{xT ∈ \overline{N}_{Q}(T) | ∃s ∈ S, xu = us\} ,

hence T \ω/S is left free if and only if

(5.6)  ^{u}S ∩ N_{Q}(T) ≤ T ,

where ^{u}S = {x ∈ Q | ∃s ∈ S, xu = us} ([2] Notation 2.3.16). Similarly T \ω/S is right free if and only if

T^{u} ∩ N_{P} (S) ≤ S ,

where T^{u} = {x ∈ P | ∃t ∈ T, tu = ux}.

Finally, the bimodule L(U)_{T,S} is isomorphic to

⨁_{u∈[N_{Q}(T))\U/N_{P} (S)] ; {^{u}S∩N_{Q}(T)≤T \atop T^{u}∩N_{P} (S)≤S}} Φ_{\overline{N}_{Q}(T)} ⊗_{ℚ\overline{N}_{Q}(T)} ℚ(T \N_{Q}(T)uN_{P} (S)/S) ⊗_{ℚ\overline{N}_{P} (S)} Φ_{\overline{N}_{P} (S)},

where [N_{Q}(T))\U/N_{P} (S)] is a set of representatives of (N_{Q}(T), N_{P} (S))- orbits on U.

5.7. Lemma: Let p be an odd prime, and let C and D be cyclic p-groups. Let moreover Ω be a left and right free finite (D, C)-biset. Let a ∈ C, viewed as an automorphism of the ℚC-module Φ_{C}. Then the image of [Φ_{C}, a] in K_{1}(ℚD) by the transfer associated to the bimodule L = Φ_{D} ⊗_{ℚD} ℚΩ is equal to the image of Ver_{Ω}(a) ∈ D = D/[D, D] by the map α_{ℚD} of Remark 2.2.

Proof : By Lemma 5.1 and Proposition 2.6, we can assume that Ω is a transitive biset, of the form (D ×C)/B for some subgroup B of D ×C. Then Ω is left and right free if and only if there exists a subgroup E of C and an injective group homomorphism φ : E → D such that B = {(φ(e), e) |∈ E}. There are two cases:

• either E = 1: in this case Ω = D × C, so ℚΩ ≅ ℚD ⊗_{ℚ} ℚC, and Φ_{D}⊗_{ℚD}ℚΩ⊗_{ℚC} Φ_{C} ≅ Φ_{D}⊗_{ℚ}Φ_{C}. As a vector space over the cyclotomic field F of endomorphisms of Φ_{D}, it is isomorphic to F ⊗_{ℚ}Φ_{C}. The action of a ∈ C on this vector space is given by the matrix of a acting on Φ_{C}.

Suppose that a is a generator of C, of order p^{n}. Then this action is the action by multiplication of a primitive p^{n}-th root of unity ζ on the field ℚ(ζ). As an element of K_{1}(F), it is equal to the determinant of the matrix representing this multiplication, i.e. to the norm N_{ℚ(ζ)/ℚ}(ζ), which is equal to 1, as the p^{n}-th cyclotomic polynomial has even degree p^{n−1}(p − 1) and value 1 at 0. It follows that [Φ_{C}, a] is mapped to the identity element of K_{1}(ℚD) in this case. Since this holds for a generator a of C, the same is true for any element a of C.

In this case also, a set of representatives of [D\Ω] is the set 1 × C, which is invariant by right multiplication by C. It follows that the elements d_{a,x} ∈ D defined for a ∈ C and x ∈ [D\Ω] by xa = d_{a,x}x′, for x′ ∈ [D\Ω], are all equal to 1. So the transfer Ver_{Ω} is also the trivial homomorphism in this case.

• or E ≠ 1: let Z denote the unique subgroup of order p of C. Tensoring over ℚC the exact sequence of (ℚC, ℚC)-bimodules

0 → Φ_{C} → ℚC → ℚ(C/Z) → 0

with Φ_{D} ⊗_{ℚD} ℚΩ gives the exact sequence

0 → Φ_{D} ⊗_{ℚD} ℚΩ ⊗_{ℚC} Φ_{C} → Φ_{D} ⊗_{ℚD} ℚΩ → Φ_{D} ⊗_{ℚD} ℚ(Ω/Z) → 0 .

But Ω/Z is not free as a left D-set, since the unique subgroup φ(Z) of order p of D stabilizes BZ ∈ Ω/Z, as for z ∈ Z and e ∈ E

φ(z)(φ(e), e) = (φ(ze), e) = (φ(ze), ze)z .

By Lemma 5.4, it follows that Φ_{D} ⊗_{ℚD} ℚ(Ω/Z) = 0, hence

Φ_{D} ⊗_{ℚD} ℚΩ ⊗_{ℚC} Φ_{C} ≅ Φ_{D} ⊗_{ℚD} ℚΩ .

As a vector space over the cyclotomic field F of endomorphisms of Φ_{D}, this is isomorphic to F ⊗_{ℚ} ℚ[D\Ω]. The action of a ∈ C on this vector space is given for x ∈ [D\Ω] and λ ∈ F by

(λ ⊗ x)a = λ ⊗ xa = λ ⊗ d_{a,x}σ_{a}(x) = λd_{a,x} ⊗ σ_{a}(x) ,

where d_{a,x} ∈ D and σ_{a}(x) ∈ [D\Ω] are defined by xa = d_{a,x}σ_{a}(x). In other words, the matrix of the action of a is the product of the permutation matrix of σ_{a} with a diagonal matrix of coefficients d_{a,x}, for x ∈ [D\Ω]. In K_{1}(F), this matrix is equal to its determinant, that is the signature of σ_{a}, which is equal to 1 as σ_{a} is a product of cycles of odd length (equal to some power of p), multiplied by the product of the elements d_{a,x}, that is the image in K_{1}(ℚD) of Ver_{Ω}(a), as was to be shown.

5.8. Remark : Recall that if Q is a central subgroup of finite index n in a group G, then the transfer G/[G, G] → Q is induced by the map g ↦ g^{n} from G to Q (see [10] Theorem 7.47). It follows easily that in the situation of Lemma 5.7, if Ω = (D × C)/B, where B = {(φ(e), e) | e ∈ E} for a subgroup E of C and an injective homomorphism φ : E → D, the transfer Ver_{Ω} : C → D is given by a ↦ φ(a^{|C:E|}). Moreover |C : E| = |D\Ω|.

5.9. Theorem: Let p be an odd prime. Let P and Q be finite p-groups, and let U be a finite (Q, P)-biset.

1. Let S be a genetic subgroup of P and T be a genetic subgroup of Q. Let D(U)_{T,S} be the set of orbits N_{Q}(T)uN_{P} (S) of those u ∈ U for which T^{u} ∩ N_{P} (S) ≤ S and ^{u}S ∩ N_{Q}(T) ≤ T (see 5.6 for notation). Then for ω ∈ D(U)_{T,S}, the set T \ω/S is a left and right free (N_{Q}(T)/T, N_{P} (S)/S)-biset, and the map

Γ(U)_{T,S} : N_{P} (S)/S → N_{Q}(T)/T

sending a ∈ N_{P} (S)/S to

∏_{ω∈D(U)_{T,S}} Ver_{T \ω/S}(a)

is a well defined group homomorphism.

2. Let B_{P} and B_{Q} be genetic bases of P and Q, respectively. Then the map Γ(U) : Γ(P) → Γ(Q) giving the biset functor structure of Γ is the

map ; Γ(P) = ∏_{S∈B_{P}} (N_{P} (S)/S) → ∏_{T ∈B_{Q}} (N_{Q}(T)/T ) = Γ(Q)

with component (T, S) equal to Γ(U)_{T,S}.

Proof : This results from Paragraph 5.2, Lemma 5.4, Paragraph 5.5, and Lemma 5.7.

6. Examples

6.1. Proposition: Let P be a finite p-group, for p odd, and let B be a genetic basis of P. Let N ⊴ P, and \overline{P} = P/N. Let B_{N} be the subset of B defined by

B_{N} = {S ∈ B | S ≥ N} .

Then:

1. The set \overline{B} = \{\overline{S} = S/N | S ∈ B_{N}\} is a genetic basis of \overline{P}.

2. Up to the identification of N_{\overline{P}} (\overline{S}) with N_{P} (S)/S, for S ∈ B_{N}, the inflation morphism

Inf_{P/N}^{P} : Γ(P/N) = ∏_{\overline{S}∈\overline{B}} (N_{\overline{P}} (\overline{S})/\overline{S}) → Γ(P) = ∏_{S∈B} (N_{P} (S)/S)

is the embedding in the product of the factors of Γ(P) corresponding to genetic subgroups S containing N.

3. Similarly, the deflation morphism

Def_{P/N}^{P} : Γ(P) = ∏_{S∈B} (N_{P} (S)/S) → Γ(P/N) = ∏_{\overline{S}∈\overline{B}} (N_{\overline{P}} (\overline{S})/\overline{S})

is the projection onto the product of the factors of Γ(P) corresponding to genetic subgroups S containing N.

Proof : Assertion 1 is clear from the definitions: if N ≤ S ≤ P, then S is genetic in P if and only if S/N is genetic in P/N. Moreover the relation P/N gives the relation _{P} by inflation.

Now the inflation morphism Inf_{P/N}^{P} is defined by the (P, \overline{P})-biset U = P/N, with natural actions of P and \overline{P}. Let u = xN be an element of U, for some x ∈ P. Let T ∈ B and \overline{S} ∈ \overline{B}.

T^{u} ∩ N_{\overline{P}} (\overline{S}) = \{\overline{y} = yN ∈ N_{\overline{P}} (\overline{S}) | ∃t ∈ T, tu = u\overline{y}\}
= \{\overline{y} = yN ∈ N_{\overline{P}} (\overline{S}) | ∃t ∈ T, txN = xyN\}
= \{\overline{y} = yN ∈ N_{\overline{P}} (\overline{S}) | yN ∈ T^{x} N\}
= (T^{x} N ∩ N_{P} (S))/N = (T^{x} ∩ N_{P} (S))N/N ,

where the last two equalities hold because S ≥ N. Hence T^{u} ∩ N_{\overline{P}} (\overline{S}) ≤ \overline{S} if and only if T^{x} ∩ N_{P} (S) ≤ S.

On the other hand

u\overline{S} ∩ N_{P} (T) = \{y ∈ N_{P} (T) | ∃\overline{s} = sN ∈ \overline{S}, yu = u\overline{s}\}
= {y ∈ N_{P} (T) | ∃s ∈ S, yxN = xNs}
= {y ∈ N_{P} (T) | y ∈ ^{x}S}
= ^{x}S ∩ N_{P} (T) .

Hence ^{u}\overline{S} ∩ N_{P} (T) ⊆ T if and only if ^{x}S ∩ N_{P} (T) ≤ T.

If moreover T^{u}∩N_{\overline{P}} (\overline{S}) ≤ \overline{S}, i.e. T^{x}∩N_{P} (S) ≤ S, it follows that T _{P} S, hence T = S since T and S belong to the same genetic basis B. Moreover x ∈ N_{P} (S), and the induced group homomorphism N_{\overline{P}} (\overline{S})/\overline{S} → N_{P} (T)/T is the canonical isomorphism N_{\overline{P}} (\overline{S})/\overline{S} → N_{P} (S)/S. This completes the proof of Assertion 2.

For Assertion 3, we consider the deflation map Def_{P/N}^{P} : Γ(P) → Γ(P/N). It corresponds to the biset V = P/N, with left action of \overline{P} and right action of P. For v = yN ∈ V, for T ∈ B and \overline{S} ∈ \overline{B}, and with the same notation as above, the computation is similar: we have T^{v} ∩ N_{\overline{P}} (\overline{S}) ≤ \overline{S} if and only if T^{y} ∩ N_{P} (S) ≤ S, and ^{v}\overline{S} ∩ N_{P} (T) ≤ T if and only if ^{y}S ∩ N_{P} (T) ≤ T. These two conditions are fulfilled if and only if S = T and y ∈ N_{P} (S). This completes the proof.

Recall that the faithful part ∂F (P) of the evaluation of a biset functor F at a group P is the set of faithful elements of F (P), introduced in [2] Definition 6.3.1: it is the set of elements u ∈ F (P) such that Def_{P/N}^{P}u = 0 for any non trivial normal subgroup N of P. Equivalently Def_{P/Z}^{P}u = 0 for any non trivial central subgroup Z of P. The following is now clear:

6.2. Corollary: Let p be an odd prime and P be a finite p-group. Let B be a genetic basis of P. Then the faithful part ∂Γ(P) of Γ(P) is equal to

∂Γ(P) = ∏_{{S∈B \atop S∩Z(P)=1}} (N_{P} (S)/S) .

7. Genome and Roquette category

7.1. Let F be a p-biset functor. It is shown in [2] Theorem 10.1.1 that if P is a finite p-group and B is a genetic basis of P, then the map

I_{B} = ⨁_{S∈B} Indinf_{N_{P} (S)/S}^{P} : ⨁_{S∈B} ∂F (N_{P} (S)/S) → F (P)

is always split injective. When I_{B} is an isomorphism for one particular genetic basis B of P, then I_{B′} is an isomorphism for any other genetic basis B′ of P.

The functors for which I_{B} is an isomorphism for any finite p-group P and any genetic basis B of P are called rational p-biset functors. It has been shown further ([3]) that these rational p-biset functors are exactly those p-biset functors which factorize through the Roquette category R_{p} of p-groups: more precisely ([3], Definition 3.3), the category R_{p} is defined as the idempotent additive completion of a specific quotient R_{p}^{♯} of the category C_{p}, so there is a canonical additive functor π_{p} : C_{p} → R_{p}, equal to the composition of the projection functor C_{p} → R_{p}^{♯} and the inclusion functor R_{p}^{♯} → R_{p}. The rational p-biset functors are the additive functors F : C_{p} → Ab for which there exists an additive functor \overline{F} : R_{p} → Ab such that F = \overline{F} ∘ π_{p}. In this case, the functor \overline{F} is unique.

7.2. Proposition: Let p be an odd prime. Then the genome p-biset functor Γ is rational.

Proof : Let P be a p-group, and B be a genetic basis of P. If S ∈ B, then Q = N_{P} (S)/S is cyclic, so the trivial subgroup S/S of Q is the only one intersecting trivially the center of Q, and it is a genetic subgroup of Q. By Corollary 6.2, we have that

∂Γ(N_{P} (S)/S) = N_{P} (S)/S .

Now the induction-inflation map Indinf_{N_{P} (S)/S}^{P} is given by the (P, N_{P} (S)/S)- biset U = P/S. Let T ∈ B, and let u = xS ∈ U such that

T^{u} ∩ N_{Q}(S/S) ≤ S/S and ^{u}(S/S) ∩ N_{P} (T) ≤ T .

The first inclusion means that

{yS ∈ N_{P} (S)/S | ∃t ∈ T, txS = xSy} = S/S .

In other words T^{x} ∩ N_{P} (S) ≤ S. The second inclusion means similarly that

{y ∈ N_{P} (T) | ∃s ∈ S/S, txS = xSs} ≤ T ,

that is N_{P} (T) ∩^{x}S ≤ T. Hence T _{P} S, thus T = S since T and S both belong to a genetic basis of P. Moreover x ∈ N_{P} (S), and the morphism we get from N_{P} (S)/S to N_{P} (T)/T is the identity map.

In other words, the map

Indinf_{N_{P} (S)/S}^{P} : ∂Γ(N_{P} (S)/S) = N_{P} (S)/S → Γ(P)

is the canonical embedding of N_{P} (S)/S in Γ(P). It clearly follows that the map I_{B} is an isomorphism, hence Γ is rational.

7.3. Corollary: Let p be an odd prime. Then there exists a unique additive functor \overline{Γ} from the Roquette category R_{p} to Ab such that Γ = \overline{Γ}∘π_{p}. Moreover \overline{Γ}(∂P) = ∂Γ(P) for any finite p-group P, where ∂P is the edge of P in R_{p}. In particular \overline{Γ}(∂C) = C for any cyclic p-group C.

Proof : This follows from the definition and properties of the category R_{p} (for the definition of the edge ∂P of a p-group P in the Roquette category, see [3] Definition 3.7).

References

[1] L. Barker. Genotypes of irreducible representations of finite p-groups. Journal of Algebra, 306:655–681, 2007.

[2] S. Bouc. Biset functors for finite groups, volume 1990 of Lecture Notes in Mathematics. Springer, 2010.

[3] S. Bouc. The Roquette category of finite p-groups. Journal of the European Mathematical Society, 17:2843–2886, 2015.

[4] S. Bouc and N. Romero. The Whitehead group of (almost) extra-special p-groups with p-odd. J. Pure Appl. Algebra, 223 (2019) 86-107.

[5] C. Curtis and I. Reiner. Methods of representation theory with applications to finite groups and orders, volume II of Wiley classics library. Wiley, 1990.

[6] C. Curtis and I. Reiner. Methods of representation theory with applications to finite groups and orders, volume I of Wiley classics library. Wiley, 1990.

[7] R. Oliver. Whitehead groups of finite groups, volume 132 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1988.

[8] N. Romero. Computing Whitehead groups using genetic bases. Journal of Algebra, 450:646–666, 2016.

[9] P. Roquette. Realisierung von Darstellungen endlicher nilpotenter Gruppen. Arch. Math., 9:224–250, 1958.

Serge Bouc LAMFA-CNRS UMR7352 Université de Picardie Jules Verne 33 rue St Leu, Amiens Cedex 01 France

email:  serge.bouc@u-picardie.fr