Relative B-groups

Serge Bouc

Abstract : This paper extends the notion of B-group to a relative context. For a finite group K and a field 𝔽 of characteristic 0, the lattice of ideals of the Green biset functor 𝔽B_{K} obtained by shifting the Burnside functor 𝔽B by K is described in terms of B_{K}-groups. It is shown that any finite group (L, φ) over K admits a largest quotient B_{K}- group β_{K}(L, φ). The simple subquotients of 𝔽B_{K} are parametrized by B_{K}-groups, and their evaluations can be precisely determined. Finally, when p is a prime, the restriction 𝔽B_{K}^{(p)} of 𝔽B_{K} to finite p-groups is considered, and the structure of the lattice of ideals of the Green functor 𝔽B_{K}^{(p)} is described in full detail. In particular, it is shown that this lattice is always finite.

AMS Subject classification : 18B99, 19A22, 20J15.

Keywords : B-group, Burnside ring, biset functor, shifted functor.

1. Introduction

In the study of the lattice of biset-subfunctors of the Burnside functor 𝔽B over a field 𝔽 of characteristic 0 (cf. Section 7.2 of [1], or Chapter 5 of [2]), a special class of finite groups, called B-groups, plays an important role: indeed, the simple subquotients of the biset functor 𝔽B are exactly the functors S_{H,𝔽}, where H is such a B-group. It was shown moreover that each finite group G has a largest quotient B-group β(G).

Let K be a fixed finite group. This paper proposes a generalization of the above methods and notions, in order to study the lattice of ideals of the shifted Burnside functor 𝔽B_{K}. We start by introducing a category grp_{⇓K} of groups over K, similar to the comma category of finite groups over K, in which morphisms are obtained by allowing diagrams to commute up to inner automorphisms of K.

To each such group (L, φ), where φ : L → K, is attached a specific ideal e_{L,φ} of 𝔽B_{K}, and it is shown that every ideal of 𝔽B_{K} is equal to the sum of the ideals e_{L,φ} it contains. A special class of groups over K is introduced, called B_{K}-groups, and it is shown that for each group (L, φ) over K, there exists a largest B_{K}-group β_{K}(L, φ) quotient of (L, φ). Moreover e_{L,φ} = e_{β_{K}(L,φ)}. It follows that the lattice of ideals of 𝔽B_{K} can be described in terms of closed families of B_{K}-groups.

Moreover, each ideal e_{L,φ} associated to a B_{K}-group (L, φ) has a unique maximal proper subideal e_{L,φ}^{0}. The quotient S_{L,φ} = e_{L,φ}/e_{L,φ}^{0} is a simple 𝔽B_{K}-module. The evaluations of this simple module can be precisely described, as well as its minimal groups, and this yields a new example of a simple module over a Green biset functor with several isomorphism classes of minimal groups.

Finally, when p is a prime number, we consider the restriction 𝔽B_{K}^{(p)} of 𝔽B_{K} to finite p-groups, and we describe completely the lattice of ideals of this Green biset functor. We show in particular that this lattice is always finite. As a byproduct, we get some examples of Green p-biset functors without non zero proper ideals.

2. Review of shifted Green biset functors

We quickly recall some definitions and basic notions on biset functors for finite groups, and refer to [2] for details. Let 𝔽 be a field of characteristic 0. The biset category 𝔽C of finite groups has all finite groups as objects. If G and H are finite groups, then Hom_{𝔽C}(G, H) = 𝔽 ⊗_{ℤ} B(H, G), where B(H, G) is the Grothendieck group of finite (H, G)-bisets. Composition in 𝔽C is induced by the product (V, U) ↦ V ×_{H} U = (V × U)/H, where V is a (K, H)-biset and U a (H, G)-biset, and H acts on (V × U) by (v, u) · h = (vh, h^{−1}u). A biset functor over 𝔽 is an 𝔽-linear functor from 𝔽C to the category of 𝔽-vector spaces.

Any biset is a disjoint union of transitive ones, and any transitive (H, G)- biset is of the form (H × G)/L, where L is a subgroup of (H × G). Denoting by p_{1} : H × G → H and p_{2} : H × G → G the first and second projections, we set k_{1}(L) = p_{1}(L ∩ Ker p_{2}) and k_{2}(L) = p_{2}(L ∩ Ker p_{1}). The biset (H × G)/L factors as the composition

(H × G)/L ≅ Ind_{p_{1}(L)}^{H} ∘ Inf_{p_{1}(L)/k_{1}(L)}^{p_{1}(L)} ∘ Iso(α) ∘ Def_{p_{2}(L)/k_{2}(L)}^{p_{2}(L)} ∘ Res_{p_{2}(L)}^{G}

of elementary bisets called induction, inflation, isomorphism, deflation, and restriction, where α : p_{2}(L)/k_{2}(L) → p_{1}(L)/k_{1}(L) is the canonical isomorphism sending bk_{2}(L) to ak_{1}(L) for (a, b) ∈ L. These elementary morphisms generate all morphisms in the category 𝔽C.

A Green biset functor A over 𝔽 (cf. Section 8.5 of [2]) is a biset functor with additional bilinear products A(G) × A(H) → A(G × H), denoted by (α, β) ↦ α × β, which are associative and bifunctorial. There is also an identity element ε_{A} ∈ A(1).

A left A-module M is then defined similarly as a biset functor with products A(G) × M(H) → M(G × H) which are associative, bifunctorial, and unital. Left A-modules form an abelian category denoted by A-Mod. A left ideal of A is an A-submodule of the left A-module A.

When A is a Green functor, each evaluation A(G) is an 𝔽-algebra for the product

α, β ∈ A(G) ↦ α · β = A(Iso(δ) ∘ Res_{Δ}^{G×G})(α × β) ,

where Δ is the diagonal subgroup of G × G, and δ : Δ → G the canonical isomorphism. The identity element of this algebra is A(Inf_{1}^{G})(ε_{A}). If M is an A-module, each evaluation M(G) is endowed with an A(G)-module structure defined similarly. By Proposition 2.16 of [7], a biset subfunctor I of A is an ideal if and only if I(G) is an ideal of the algebra A(G), for any finite group G.

A Green biset functor A is called commutative (cf. [3] for details) if the algebra A(G) is commutative, for any G.

A fundamental example of Green biset functor is the Burnside functor sending a finite group G to 𝔽B(G) = 𝔽B(G, 1), where B(G) is the Burnside group of G. The products maps 𝔽B(G) × 𝔽B(H) → 𝔽B(G × H) are induced by the cartesian product sending a G-set X and an H-set Y to the (G×H)-set X × Y. An 𝔽B-module is precisely a biset functor over 𝔽.

Let K be a finite group. A Green biset functor A over 𝔽 can be shifted by K. This gives a new Green biset functor A_{K} defined for a finite group G by

A_{K}(G) = A(G × K) .

For finite groups G and H and a finite (H, G)-biset U, the map

A_{K}(U) : A_{K}(G) → A_{K}(H)

is the map A(U × K), where U × K is viewed as a (H × K, G × K)-biset in the obvious way, letting K act on both sides on U × K by multiplication on the second component. For an arbitrary element α ∈ 𝔽B(H, G), that is an 𝔽-linear combination of (H, G)-bisets, the map A_{K}(α) : A_{K}(G) → A_{K}(H) is defined by 𝔽-linearity.

This endows A_{K} with a biset functor structure. Moreover, for finite groups G and H, the product

×_{A_{K}} : A_{K}(G) × A_{K}(H) → A_{K}(G × H)

is defined as follows: if α ∈ A_{K}(G) = A(G×K) and β ∈ A_{K}(H) = A(H×K), then α × β ∈ A(G × K × H × K). We set

α ×_{A_{K}} β = A(Iso(δ) ∘ Res_{Δ}^{G×K×H×K})(α × β) ,

where Δ = {(g, k, h, k) | g ∈ G, h ∈ H, k ∈ K}, and δ is the isomorphism Δ → G × H × K sending (g, k, h, k) to (g, h, k). The identity element ε_{A_{K}} is A(Inf_{1}^{K})(ε_{K}).

For a finite group G, the algebra structure on A_{K}(G) is simply the algebra structure on A(G × K) defined for the Green functor A.

All these notion can be extended to functors from an admissible subcategory D of the biset category (cf. Chapter 4 of [2]), which is moreover closed under taking direct products of finite groups. We have then the notions of D-biset functors and D-Green biset functors, as well as modules over them.

In this paper, we will consider the shifted Burnside functor 𝔽B_{K}, and its restriction 𝔽B_{K}^{(p)} to finite p-groups, for a prime p. A fundamental classical result is that for any finite group G, the algebra 𝔽B_{K}(G) is a split semisimple commutative algebra, with primitive idempotents e_{L}^{G×K} indexed by subgroups L of G × K, up to conjugation. The explicit formula for e_{L}^{G×K}, due to Gluck ([4]) and Yoshida ([8]) is

e_{L}^{G×K} = \frac{1}{|N_{G×K}(L)|} ∑_{X≤L} |X| μ(X, L) [(G × K)/X] ,

where X runs through all subgroups of L, where μ is the Möbius function of the poset of subgroups of G × K, and [(G × K)/X] is the isomorphism class of the transitive (G × K)-set (G × K)/X.

2.1. Notation: When N is a normal subgroup of a finite group L, let

m_{L,N} = \frac{1}{|L|} ∑_{\begin{matrix}X≤L \\ XN=L \end{matrix}} |X|μ(X, L) .

2.2. Lemma: Let G be a finite group, and L be a subgroup of G × K. If N is a normal subgroup of G, then

𝔽B_{K}(Def_{G/N}^{G})(e_{L}^{G×K}) = λ m_{L,L∩(N×1)} e(G/N)×K ; \overline{L} ,

where \overline{L} is the image of L by the projection G × K → (G/N) × K, and

λ = \frac{|N_{(G/N)×K}(\overline{L}):\overline{L}|}{|N_{G×K}(L):L|}.

Proof: Indeed

𝔽B_{K}(Def_{G/N}^{G})(e_{L}^{G×K}) = 𝔽B(Def_{(G×K)/(N×1)}^{G×K})(e_{L}^{G×K}) .

The result now follows from Assertion 4 of Theorem 5.2.4 of [2].

3. Ideals generated by idempotents

We now introduce a category grp_{⇓K}, similar to the comma category over K: its objects are the same, but morphisms are slightly different.

3.1. Definition:

• For a finite group K, let grp_{⇓K} denote the following category:

– The objects are finite groups over K, i.e. pairs (L, φ), where L is a finite group and φ : L → K is a group homomorphism.

– A morphism f : (L, φ) → (L′, φ′) of groups over K in the category grp_{⇓K} is a group homomorphism f : L → L′ such that there exists some inner automorphism i of K with i ∘ φ = φ′ ∘ f.

– The composition of morphisms in grp_{⇓K} is the composition of group homomorphisms, and the identity morphism of (L, φ) is the identity automorphism of L.

• If (L, φ) and (L′, ϕ′) are groups over K, we say that (L′, φ′) is a quotient of (L, φ), and we note (L, φ) ↠ (L′, φ′), if there exists a morphism f ∈ Hom_{grp_{⇓K}}((L, φ), (L′, φ′)) with f : L → L′ surjective. In this case, we will say that f is a surjective morphism from (L, φ) to (L′, φ′).

3.2. Remarks:

1. Using the well known fact that the epimorphisms in the category of (finite) groups are the surjective group homomorphisms (cf. [5] I.5 Exercise 5), one can show that a morphism f ∈ Hom_{grp_{⇓K}}((L, φ), (L′, φ′)) is an epimorphism in grp_{⇓K} if and only if f : L → L′ is surjective, that is, if f is a surjective morphism. We will not use this fact here, except as a motivation to the use of the word “quotient” in Definition 3.1.

2. A morphism f : (L, φ) → (L′, φ′) in grp_{⇓K} is an isomorphism if and only if f : L → L′ is an isomorphism of groups.

3. If (L′, φ′) is a quotient of (L, φ), and if (L, φ) is a quotient of (L′, φ′), then (L, φ) and (L′, φ′) are isomorphic in grp_{⇓K}. Indeed any surjective morphism from (L, φ) to (L′, φ′) is an isomorphism, for L and L′ have the same order.

4. Clearly, the relation “being quotient of” on the class of groups over K is transitive. In particular, any group over K isomorphic in grp_{⇓K} to a quotient of (L, φ) is itself a quotient of (L, φ), and also a quotient of any group over K isomorphic to (L, φ) in grp_{⇓K}.

3.3. Notation: When (L, φ) is a group over K, we denote by L_{φ} the subgroup of L × K defined by

L_{φ} = {(l, φ(l)) | l ∈ L} .

3.4. Theorem: Let I be an ideal of the Green biset functor 𝔽B_{K}. If G is a finite group and L is a subgroup of G × K, the following conditions are equivalent:

1. The idempotent e_{L}^{G×K} belongs to I(G).

2. The idempotent e_{L_{p_{2}}}^{L×K} belongs to I(L), where p_{2} : L → K is the restriction to L of the second projection homomorphism G × K → K.

Proof: 1 ⇒ 2 Let L̂ = L_{p_{1}} ⊆ L × G, where p_{1} : L → G is the restriction to L of the first projection homomorphism G × K → G. Thus

p_{1}(L̂) = L, k_{1}(L̂) = 1 × k_{2}(L), p_{2}(L̂) = p_{1}(L), k_{2}(L̂) = 1 .

It follows that the (L, G)-biset U = (L × G)/L̂ factors as

U ≅ Inf_{L/N}^{L} ∘ Iso(θ^{−1}) ∘ Res_{p_{1}(L)}^{G} ,

where N = 1×k_{2}(L) ⊴ L and θ : L/N → p_{1}(L) is the canonical isomorphism induced by the first projection p_{1} : L → G.

Now if e_{L}^{G×K} belongs to I(G), its restriction 𝔽B_{K}(Res_{p_{1}(L)}^{G})(e_{L}^{G×K}) belongs to I(G). But

𝔽B_{K}(Res_{p_{1}(L)}^{G})(e_{L}^{G×K}) = 𝔽B(Res_{p_{1}(L)×K}^{G×K})(e_{L}^{G×K})

= ∑_{L′} e_{L′}^{p_{1}(L)×K},

where L′ runs through a set of representatives of (p_{1}(L) × K)-conjugacy classes of subgroups of p_{1}(L) × K which are conjugate to L in G × K (cf. [2], Theorem 5.2.4, Assertion 1). In particular, the group L is one of them, and

e_{L}^{p_{1}(L)×K} · 𝔽B_{K}(Res_{p_{1}(L)}^{G})(e_{L}^{G×K}) = e_{L}^{p_{1}(L)×K} ∈ I(p_{1}(L)) .

It follows that 𝔽B_{K}(Iso(θ^{−1}))(e_{L}^{p_{1}(L)×K}) ∈ I(L/N).

But 𝔽B_{K}(Iso(θ^{−1})) = 𝔽B(Iso(θ_{K}^{−1})), where θ_{K} = θ × Id_{K} is the isomorphism from (L/N) × K to p_{1}(L) × K deduced from θ. It follows that

e(L/N)×K ; \overline{L} ∈ I(L/N), where \overline{L} = θ_{K}^{−1}(L) = \{(lN, p_{2}(l)) | l ∈ L\}. Now

𝔽B_{K}(Inf_{L/N}^{L})(e(L/N)×K ; \overline{L}) = 𝔽B(Inf_{(L/N)×K}^{L×K})(e(L/N)×K ; \overline{L})

= ∑_{X} e_{X}^{L×K} ∈ I(L) ,

where X runs through a set of representatives of (L × K)-conjugacy classes of subgroups of L × K which map to a conjugate of \overline{L} through the surjection L × K → (L/N) × K (cf. [2], Theorem 5.2.4, Assertion 3).

The group L_{p_{2}} is one of these subgroups, hence

e_{L_{p_{2}}}^{L×K} · 𝔽B_{K}(Inf_{L/N}^{L})(e(L/N)×K ; \overline{L}) = e_{L_{p_{2}}}^{L×K} ∈ I(L) ,

as was to be shown.

2 ⇒ 1 We now consider the opposite (G, L)-biset U^{op} ≅ (G × L)/L̃, where L̃ = {(p_{1}(l), l) | l ∈ L}, which factors as

U^{op} ≅ Ind_{p_{1}(L)}^{G} ∘ Iso(θ) ∘ Def_{L/N}^{L} .

If e_{L_{p_{2}}}^{L×K} ∈ I(L), then u = 𝔽B_{K}(U^{op})(e_{L_{p_{2}}}^{L×K}) belongs to I(G). By Lemma 2.2

𝔽B_{K}(Def_{L/N}^{L})(e_{L_{p_{2}}}^{L×K}) = λ m_{L_{p_{2}},L_{p_{2}}∩(N×1)} e(L/N)×K ; \overline{L_{p_{2}}} ,

where \overline{L_{p_{2}}} is the image of L_{p_{2}} by the projection L × K → (L/N) × K, and λ is some non zero rational number. Now the intersection

L_{p_{2}} ∩ (N × 1) = {(a, b), b) | (a, b) ∈ L} ∩ ((1 × k_{2}(L)) × 1)

is trivial. It follows that m_{L_{p_{2}},L_{p_{2}}∩(N×1)} = 1, and

u = λ 𝔽B_{K}(Ind_{p_{1}(L)}^{G} ∘ Iso(θ))(e(G/N)×K) ; \overline{L_{p_{2}}})

= λ 𝔽B(Ind_{p_{1}(L)×K}^{G×K} ∘ Iso(θ_{K}))(e(G/N)×K) ; \overline{L_{p_{2}}}) .

Now for (a, b) ∈ L, the image by θ_{K} = θ × Id_{K} of ((a, b), b)(N × 1) ∈ \overline{L_{p_{2}}} is the element (p_{1}(a, b), b) = (a, b) of p_{1}(L) × K. Hence θ_{K}(\overline{L_{p_{2}}}) identifies with L, viewed as a subgroup of p_{1}(L) × K, and

u = λ 𝔽B(Ind_{p_{1}(L)×K}^{G×K})(e_{L}^{p_{1}(L)×K}) = λλ′e_{L}^{G×K} ,

for some non zero rational number λ′ (cf. [2], Theorem 5.2.4, Assertion 2). Since u ∈ I(G) and λλ′ ≠ 0, it follows that e_{L}^{G×K} ∈ I(G), as was to be shown.

3.5. Corollary: Let G be a finite group, and L be a subgroup of G × K. Then the ideal of 𝔽B_{K} generated by e_{L}^{G×K} is equal to the ideal of 𝔽B_{K} generated by e_{L_{p_{2}}}^{L×K}

Proof: Indeed, denoting by I the ideal generated by e_{L}^{G×K}, and by J the ideal generated by e_{L_{p_{2}}}^{L×K}, we have

e_{L}^{K×G} ∈ I(G) ⇒ e_{L_{p_{2}}}^{L×K} ∈ I(L) ⇒ J ⊆ I ,
e_{L_{p_{2}}}^{L×K} ∈ J(L) ⇒ e_{L}^{G×K} ∈ J(G) ⇒ I ⊆ J ,

so I = J.

3.6. Notation: Let (L, φ) be a group over K. We denote by e_{L,φ} the ideal of 𝔽B_{K} generated by e_{L_{φ}}^{L×K} ∈ 𝔽B_{K}(L).

3.7. Lemma: Let (L, φ) and (M, ψ) be groups over K.

1. If (M, ψ) ↠ (L, φ), then e_{M,ψ} ⊆ e_{L,φ}.

2. In particular, if (M, ψ) is isomorphic to (L, φ), then e_{M,ψ} = e_{L,φ}.

Proof: 1. Let s : M → L be a surjective group homomorphism, and i be an inner automorphism of K such that i ∘ ψ = φ ∘ s. Let U denote the set L, viewed as an (M, L)-biset for the action given by m · u · l = s(m)ul, for m ∈ M and u, l ∈ L. There is an isomorphism of (M, L)-bisets

U ≅ Inf_{M/Ker s}^{M} ∘ Iso(α^{−1}) ,

where α : \overline{M} = M/Ker s → L is the group isomorphism induced by s. Let u = 𝔽B_{K}(U)(e_{L_{φ}}^{L×K}) ∈ e_{L,φ}(M). Then

u = 𝔽B(Inf_{\overline{M}×K}^{M×K} ∘ Iso(α_{K}^{−1}))(e_{L_{φ}}^{L×K}) ,

where α_{K} = α × Id_{K} : \overline{M} × K → L × K. Then

α_{K}^{−1}(L_{φ}) = {(α^{−1}(l), φ(l)) | l ∈ L} = {(mKer s, φ ∘ s(m)) | m ∈ M} .

It follows that 𝔽B(Iso(α_{K}^{−1}))(e_{L_{φ}}^{L×K}) = e_{\overline{M}_{θ}}^{\overline{M}×K}, where θ : \overline{M} → K is defined

by θ(mKer s) = φ ∘ s(m). In particular e_{\overline{M}_{θ}}^{\overline{M}×K} ∈ e_{L,φ}(\overline{M}). Now

u = 𝔽B(Inf_{\overline{M}×K}^{M×K})(e_{\overline{M}_{θ}}^{\overline{M}×K}) = ∑_{X} e_{X}^{M×K} ,

where X runs through a set of representatives of conjugacy classes of subgroups of M × K such that the projection of X in \overline{M} × K is conjugate to \overline{M}_{θ}. The subgroup M_{φ∘s} is one of these subgroups, so e_{M_{φ∘s}}^{M×K} · u is a non zero scalar multiple of e_{M_{φ∘s}}^{M×K} lying in e_{L,φ}(M). Hence e_{M_{φ∘s}}^{M×K} ∈ e_{L,φ}(M). Now φ ∘ s = i ∘ ψ, where i is an inner automorphism of K. This implies readily that the subgroups M_{i∘ψ} and M_{ψ} of M × K are conjugate. It follows that

e_{M_{ψ}}^{M×K} = e_{M_{i∘ψ}}^{M×K} = e_{M_{φ∘s}}^{M×K} ∈ e_{L,φ}(M) ,

that is e_{M,ψ} ⊆ e_{L,φ}, proving Assertion 1.

Now if f : (L, φ) → (M, ψ) is an isomorphism in grp_{⇓K}, the group homomorphism f : M → L is an isomorphism. Then (M, ψ) and (L, φ) are quotient of one another, so e_{M,ψ} = e_{L,φ}, proving Assertion 2.

3.8. Notation: We fix a set S_{K} of representatives of isomorphism classes of objects in the category grp_{⇓K}.

3.9. Proposition: Let I be an ideal of 𝔽B_{K}. Then I is equal to the sum of the ideals e_{L,φ} it contains. More precisely, if

A_{I} = {(L, φ) ∈ S_{K} | e_{L,φ} ⊆ I} ,

we have I = ∑_{(L,φ)∈A_{I}} e_{L,φ}. It follows that the ideals of 𝔽B_{K} form a set.

Proof: Let J = ∑_{\begin{matrix}(L,φ)∈S_{K} \\ e_{L,φ}⊆I \end{matrix}} e_{L,φ}. Then obviously J ⊆ I. Moreover, if (M, ψ)

is a group over K such that e_{M,ψ} ⊆ I, then e_{M,ψ} ⊆ J: indeed, there is some (L, φ) ∈ S_{K} isomorphic to (M, ψ), and e_{M,ψ} = e_{L,φ} by Lemma 3.7. Conversely, let G be a finite group, and u ∈ I(G). Then u is a linear combination

u = ∑_{L} λ_{L}e_{L}^{G×K}

with coefficients λ_{L} in 𝔽, of idempotents e_{L}^{G×K}, where L runs through a set S of representatives of conjugacy classes of subgroups of G × K. Then for any L ∈ S, we have e_{L}^{G×K} · u = λ_{L}e_{L}^{G×K} ∈ I(G), hence e_{L}^{G×K} ∈ I(G) if λ_{L} ≠ 0. So in this case, the ideal of 𝔽B_{K} generated by e_{L}^{G×K} is contained in I. This ideal is equal to e_{L,p_{2}}, by Corollary 3.5, thus e_{L,p_{2}} ⊆ J by the above observation. Hence e_{L}^{G×K} ∈ e_{L,p_{2}}(G) ⊆ J(G). It follows that

u = ∑_{\begin{matrix}L∈S \\ λ_{L}≠0 \end{matrix}} λ_{L}e_{L}^{G×K}

also belongs to J(G). Hence I(G) ⊆ J(G), so I(G) = J(G) since J ⊆ I. As G was arbitrary, it follows that I = J.

Now an ideal I of 𝔽B_{K} is determined by the subset A_{I} of S_{K}, so the class of ideals of 𝔽B_{K} is in one to one correspondence with a set of subsets of S_{K}. Hence this class is a set.

3.10. Lemma: Let A be a set of ideals of 𝔽B_{K}, and (M, ψ) be a group over K. The following are equivalent:

1. e_{M,ψ} ⊆ ∑_{I∈A} I.

2. There exists I ∈ A such that e_{M,ψ} ⊆ I.

Proof: Clearly 2 implies 1. Now 1 is equivalent to saying that

e_{M_{ψ}}^{M×K} ∈ ∑_{I∈A} I(M) .

If this holds, there exists I ∈ A and u ∈ I(M) such that e_{M_{ψ}}^{M×K} · u ≠ 0. Now e_{M_{ψ}}^{M×K} · u ∈ I(M), and moreover there is a scalar λ ∈ 𝔽 such that e_{M_{ψ}}^{M×K} · u = λe_{M_{ψ}}^{M×K} ≠ 0. Hence λ ≠ 0, and e_{M_{ψ}}^{M×K} ∈ I(M). In other words e_{M,ψ} ⊆ I, so 1 implies 2.

4. B_{K}-groups

In view of Proposition 3.9, every ideal of 𝔽B_{K} is a sum of ideals e_{L,φ}, where (L, φ) runs in some subset of S_{K}. In view of Lemma 3.10, to describe the inclusions between such sum of ideals e_{L,φ}, it suffices to describe elementary inclusions of the form e_{M,ψ} ⊆ e_{L,φ}, where (L, φ) and (M, ψ) are groups over K. Lemma 3.7 shows that it is the case if (M, ψ) ↠ (L, φ). Moreover:

4.1. Theorem: Let s : (M, ψ) → (L, φ) be a surjective morphism in grp_{⇓K}. If m_{M,Ker s} ≠ 0, then e_{M,ψ} = e_{L,φ}.

Proof: We already know from Lemma 3.7 that e_{M,ψ} ⊆ e_{L,φ}, so it suffices to prove the reverse inclusion. We first observe that since there exists an inner automorphism i of K such that i ∘ ψ = φ ∘ s, we have Ker s ≤ Ker (i ∘ ψ) = Ker ψ. So there is a group homomorphism \overline{ψ} : \overline{M} = M/Ker s → K such that ψ = \overline{ψ} ∘ π, where π : M → \overline{M} is the projection map.

Now let V be the set L, viewed as an (L, M)-biset for the action defined by l ·v ·m = lvs(m), for l, v ∈ V and m ∈ M (in other words V = U^{op}, where U is the (M, L)-biset introduced in the proof of Lemma 3.7). Then there is an isomorphism of (L, M)-bisets

V ≅ Iso(α) ∘ Def_{M/Ker s}^{M} ,

where α : \overline{M} → L is the group isomorphism induced by s, i.e. such that s = α ∘ π.

Let v = 𝔽B_{K}(V)(e_{M_{ψ}}^{M×K}) ∈ e_{M,ψ}(L). By Lemma 2.2

𝔽B_{K}(Def_{M/Ker s}^{M})(e_{M_{ψ}}^{M×K}) = λ m_{M_{ψ},M_{ψ}∩(Ker s×1)} e\overline{M}×K ; \overline{M_{ψ}} ,

where \overline{M_{ψ}} is the image of M_{ψ} by the projection M × K → \overline{M} × K, and λ is some non zero rational number. Then

v = λ 𝔽B_{K}(Iso(α))(e\overline{M}×K ; \overline{M_{ψ}}) = λ 𝔽B(Iso(α_{K}))(e\overline{M}×K ; \overline{M_{ψ}}) ,

where α_{K} = α × Id_{K} : \overline{M} × K → L × K. The image of \overline{M}_{\overline{ψ}} under α_{K} is the subgroup

α_{K}(\overline{M}_{\overline{ψ}}) = \{(α(\overline{m}), \overline{ψ}(\overline{m})) | \overline{m} ∈ \overline{M}\} = \{(l, \overline{ψ} ∘ α^{−1}(l)) | l ∈ L\} .

Moreover, we have a diagram

\xymatrix{M \ar[ddr]_{ψ} \ar[dr]^{π} \ar[drr]^{s} \\  & \overline{M} \ar[r]_{α} \ar[d]^{\overline{ψ}} & L \ar[d]_{φ} \\  & K \ar[r]^{i} & K}

where the two triangles and the outer “square” commute. It follows that

φ ∘ α ∘ π = φ ∘ s = i ∘ ψ = i ∘ \overline{ψ} ∘ π ,

hence φ ∘ α = i ∘ \overline{ψ} since π is surjective. Hence \overline{ψ} ∘ α^{−1} = i^{−1} ∘ φ, and α_{K}(\overline{M}_{\overline{ψ}}) = L_{i^{−1}∘φ}.

It follows that v = λ e_{L_{i^{−1}∘φ}}^{L×K}, and moreover e_{L_{i^{−1}∘φ}}^{L×K} = e_{L_{φ}}^{L×K} since L_{i^{−1}∘φ} and L_{φ} are conjugate in L×K. Finally v = λ e_{L_{φ}}^{L×K}, so e_{L_{φ}}^{L×K} ∈ e_{M,ψ}(L), since v ∈ e_{M,ψ}(L) and λ ≠ 0. In other words e_{L,φ} ⊆ e_{M,ψ}, and finally e_{L,φ} = e_{M,ψ}, as was to be shown.

4.2. Notation: When (M, ψ) is a group over K, and Q is a normal subgroup of M with Q ≤ Ker ψ, let ψ/Q : M/Q → K be the group homomorphism defined by ψ = (ψ/Q) ∘ π, where π is the projection M → M/Q.

Thus for any group (M, ψ) over K, if Q is a normal subgroup of M contained in Ker ψ, we get a surjective morphism π : (M, ψ) → (M/Q, ψ/Q) in grp_{⇓K}, with Ker π = Q. If moreover m_{M,Q} ≠ 0, we have e_{M,ψ} = e_{M/Q,ψ/Q}. This motivates the following:

4.3. Definition: Let (L, φ) be a group over K. We say that (L, φ) is a B_{K}-group, or a B-group relative to K, if m_{L,N} = 0 for every non-trivial normal subgroup N of L contained in Ker φ.

4.4. Examples:

1. If φ : L → K is injective, then (L, φ) is a B_{K}-group.

2. On the other hand, if K = 1, then a group over K is a pair (L, φ), where L is a finite group and φ : L → 1 is the unique morphism. Moreover the category grp_{⇓1} clearly identifies with the usual category of finite groups. With this identification, a B_{1}-group is just a B-group (cf. Section 7.2 of [1], or Chapter 5 of [2]).

4.5. Lemma: Let (L, φ) be a B_{K}-group. If (M, ψ) is a group over K, and (M, ψ) is isomorphic to (L, φ) in grp_{⇓K}, then (M, ψ) is a B_{K}-group.

Proof: Since (M, ψ) is isomorphic to (L, φ) in grp_{⇓K}, there exists a group isomorphism f : L → M and an inner automorphism i of K such that ψ ∘ f = i ∘ φ. If P is a normal subgroup of M contained in Ker ψ, then f^{−1}(P) is a normal subgroup of L contained in Ker φ, because

i ∘ φ(f^{−1}(P)) = ψ ∘ f(f^{−1}(P)) = ψ(P) = 1

and i is an automorphism. Moreover m_{L,f^{−1}(P)} = m_{M,P}. If P is non trivial, then f^{−1}(P) is non trivial, so m_{L,f^{−1}(P)} = m_{M,P} = 0, as was to be shown.

4.6. Theorem: Let (L, φ) be a group over K.

1. If Q is a normal subgroup of L, contained in Ker φ, and maximal such that m_{L,Q} ≠ 0, then (L/Q, φ/Q) is a B_{K}-group, quotient of (L, φ).

2. If (P, ψ) is a B_{K}-group which is quotient of (L, φ), and if N is a normal subgroup of L contained in Ker φ and such that m_{L,N} ≠ 0, then (P, ψ) is a quotient of (L/N, φ/N).

3. In particular, if P and Q are normal subgroups of L, contained in Ker φ, and maximal such that m_{L,P} ≠ 0 ≠ m_{L,Q}, then (L/P, φ/P) and (L/Q, φ/Q) are isomorphic in grp_{⇓K}.

Proof: 1. Let P/Q be a normal subgroup of L/Q contained in Ker (φ/Q) = Ker φ/Q. Then P is a normal subgroup of L, and Q ≤ P ≤ Ker φ. If P/Q ≠ 1, i.e. if Q < P, then by maximality of Q and Proposition 5.3.1 of [2]

m_{L,P} = 0 = m_{L,Q}m_{L/Q,P/Q} .

Since m_{L,Q} ≠ 0, it follows that m_{L/Q,P/Q} = 0, so (L/Q, φ/Q) is a B_{K}-group, quotient of (L, φ).

2. Since (P, ψ) is a quotient of (L, φ), there exists a surjective group homomorphism s : L → P and an inner automorphism i of K such that ψ ∘ s = i ∘ φ. It follows that M = Ker s is a normal subgroup of L contained in Ker (i ∘ φ) = Ker φ.

We have a diagram

\xymatrix{L \ar[ddr]_{φ} \ar[dr]^{π_{M}} \ar[drr]^{s} \\  & L/M \ar[r]_{\overline{s}} \ar[d]^{φ/M} & P \ar[d]_{ψ} \\  & K \ar[r]^{i} & K}

where the two triangles and the outer “square” commute, and \overline{s} is an isomorphism, the map π_{M} : L → L/M being the projection. As in the proof of Theorem 4.1, we have

ψ ∘ \overline{s} ∘ π_{M} = ψ ∘ s = i ∘ φ = i ∘ (φ/M) ∘ π_{M} ,

so ψ ∘ \overline{s} = i ∘ (φ/M) since π_{M} is surjective. It follows that \overline{s} is an isomorphism from (L/M, φ/M) to (P, ψ) in grp_{⇓K}, so (L/M, φ/M) is a B_{K}-group by Lemma 4.5.

Now by Proposition 5.3.3 of [2]

m_{L,N} = \frac{1}{|L|} ∑_{Y N=Y M=L} |Y |μ(Y, L)m_{L/M,(Y ∩N)M/M} .

In particular, if m_{L,N} ≠ 0, there exists Y ≤ L such that Y N = Y M = L and m_{L/M,(Y ∩N)M/M} ≠ 0. But since N ⊆ Ker φ, the group (Y ∩ N)M/M is a normal subgroup of L/M contained in Ker (φ/M) = Ker φ/M. Then since m_{L/M,(Y ∩N)M/M} ≠ 0 and since (L/M, φ/M) is a B_{K}-group, we have (Y ∩ N)M/M = 1, i.e. Y ∩ N ⊆ Y ∩ M.

Consider now the following diagram:

\xymatrix{ &  & K \\ L \ar[urr]^{φ} \ar[r]^{π_{N}} & L/N \ar@{-->}[ur]_{φ/N} \ar@{-->}[rr]^{θ} &  & L/M \ar@{-->}[ul]^{φ/M} & L \ar[l]_{π_{M}} \ar[ull]_{φ} \\  & Y/(Y ∩ N) \ar[u]_{v} \ar[rr]^{σ} &  & Y/(Y ∩ M) \ar[u]^{u} \\  &  & Y \ar[uull]^{j} \ar[ul]^{ρ_{N}} \ar[ur]_{ρ_{M}} \ar[uurr]_{j}}

where

• j : Y → L is the inclusion map,

• ρ_{N} : Y → Y/(Y ∩ N) and ρ_{M} : Y → Y/(Y ∩ M) are the projection maps,

• u : Y/(Y ∩ M) → L/M and v : Y/(Y ∩ N) → L/N are the canonical isomorphisms Y/(Y ∩ M) ≅ Y M/M = L/M and Y/(Y ∩ N) ≅ Y N/N = L/N, respectively,

• σ : Y/(Y ∩N) → Y/(Y ∩M) is the projection map (as Y ∩N ⊆ Y ∩M),

• θ : L/N → L/M is defined as θ = u ∘ σ ∘ v^{−1}. In particular θ is surjective.

We have π_{N} ∘ j = v ∘ ρ_{N}, since for any y ∈ Y

π_{N} ∘ j(y) = π_{N}(y) = yN = v(y(Y ∩ N)) = v ∘ ρ_{N}(y) .

Similarly π_{M} ∘ j = u ∘ ρ_{M}. We also have σ ∘ ρ_{N} = ρ_{M}. Then

θ ∘ π_{N} ∘ j = θ ∘ v ∘ ρ_{N} = u ∘ σ ∘ ρ_{N} = u ∘ ρ_{M} = π_{M} ∘ j .

Hence

(φ/M) ∘ θ ∘ π_{N} ∘ j = (φ/M) ∘ π_{M} ∘ j = φ ∘ j = (φ/N) ∘ π_{N} ∘ j .

Since π_{N} ∘ j = v ∘ ρ_{N} : Y → L/N is surjective, it follows that

(φ/M) ∘ θ = (φ/N) .

Hence θ is a surjective morphism from (L/N, φ/N) to (L/M, φ/M) in grp_{⇓K}. As the latter is isomorphic to (P, ψ) in grp_{⇓K}, it follows that (P, ψ) is a quotient of (L/N, φ/N), as was to be shown.

3. If P and Q are normal subgroups of L, contained in Ker φ, and maximal such that m_{L,P} ≠ 0 ≠ m_{L,Q}, then (L/P, φ/P) and (L/Q, φ/Q) are both B_{K}- groups by Assertion 1, and they are quotient of one another by Assertion 2. Hence they are isomorphic in grp_{⇓K}.

4.7. Notation: Let (L, φ) be a group over K. If Q is a normal subgroup of L, contained in Ker φ, and maximal such that m_{L,Q} ≠ 0, we denote by β_{K}(L, φ) the quotient (L/Q, φ/Q) of (L, φ).

4.8. Remark: As observed in Example 4.4, when K is trivial, a B_{K}-group is simply a B-group. Moreover, for any finite group L, if u : L → 1 is the unique group homomorphism, then β_{1}(L, u) = β(L).

The following corollary shows that β_{K}(L, φ) is the largest B_{K}-group quotient of (L, φ):

4.9. Corollary: Let (L, φ) be a group over K.

1. β_{K}(L, φ) is well defined up to isomorphism in grp_{⇓K}.

2. β_{K}(L, φ) is a B_{K}-group, quotient of (L, φ).

3. If (P, ψ) is a B_{K}-group, quotient of (L, φ), then (P, ψ) is a quotient of

β_{K}(L, φ).

4. e_{L,φ} = e_{β_{K}(L,φ)}.

Proof: 1. This follows from Assertion 3 of Theorem 4.6.

2. This follows from Assertion 1 of Theorem 4.6.

3. This follows from Assertion 2 of Theorem 4.6.

4. This follows from Theorem 4.1, by definition of β_{K}(L, φ)

4.10. Corollary: Let s : (M, ψ) ↠ (L, φ) be a surjective morphism in grp_{⇓K}. Then β_{K}(M, ψ) ≅ β_{K}(L, φ) if and only if m_{M,Ker s} ≠ 0.

Proof: Indeed β_{K}(L, φ) is a quotient of (M, ψ), as it is a quotient of (L, φ) and s is surjective. Hence β_{K}(L, φ) is a quotient of β_{K}(M, ψ). Set N = Ker s, so that (L, φ) ≅ (M/N, ψ/N).

If m_{M,N} ≠ 0, then since β_{K}(M, ψ) is a B_{K}-group quotient of (M, ψ), Assertion 2 of Theorem 4.6 implies that β_{K}(M, ψ) is a quotient of (M/N, ψ/N) ≅ (L, φ), hence of β_{K}(L, φ). It follows that β_{K}(M, ψ) ≅ β_{K}(L, φ), as they are quotient of one another.

Conversely, suppose that β_{K}(M, ψ) ≅ β_{K}(L, φ), and let P/N be a normal subgroup of M/N contained in Ker (ψ/N) = Ker ψ/N and maximal such that m_{M/N,P/N} ≠ 0. Then the quotient ((M/N)/(P/N), (ψ/N)/(P/N)) ≅ (M/P, ψ/P) is isomorphic to β_{K}(M/N, P/N) ≅ β_{K}(L, φ), hence to β_{K}(M, ψ). Now if Q is a normal subgroup of M contained in Ker ψ and maximal such that m_{M,Q} ≠ 0, then the quotient (M/Q, ψ/Q) is isomorphic to β_{K}(M, ψ) ≅ (M/P, ψ/P). In particular M/Q ≅ M/P, and then m_{M,P} = m_{M,Q} by Proposition 5.3.4 of [2], so m_{M,P} ≠ 0. But m_{M,P} = m_{M,N}m_{M/N,P/N}, so m_{M,N} ≠ 0, as was to be shown.

5. The ideals of 𝔽B_{K}

5.1. Notation and Definition:

1. We let B_{K}-gr denote the subset of S_{K} consisting of B_{K}-groups.

2. A subset P of B_{K}-gr is said to be closed if

∀(L, φ) ∈ P, ∀(M, ψ) ∈ B_{K}-gr, (M, ψ) ↠ (L, φ) ⟹ (M, ψ) ∈ P .

5.2. Proposition: Let I be an ideal of 𝔽B_{K}, and

P_{I} = {(L, φ) ∈ B_{K}-gr | e_{L,φ} ⊆ I} .

Then P_{I} is a closed subset of B_{K}-gr, and I = ∑_{(L,φ)∈P_{I}} e_{L,φ}

Proof: The subset P_{I} of B_{K}-gr is closed by Lemma 3.7. The second assertion follows from Proposition 3.9 and Assertion 4 of Corollary 4.9.

5.3. Theorem: Let (L, φ) be a B_{K}-group. Then for any finite group G

e_{L,φ}(G) = ∑_{X} 𝔽e_{X}^{G×K} ,

where X runs through all subgroups of G × K such that (X, p_{2}) ↠ (L, φ).

Proof: If X ≤ G×K and (X, p_{2}) ↠ (L, φ), then e_{X,p_{2}} ⊆ e_{L,φ} by Lemma 3.7. Equivalently e_{X_{p_{2}}}^{X×K} ∈ e_{L,φ}(X), which is equivalent to e_{X}^{G×K} ∈ e_{L,φ}(G), by Theorem 3.4. This proves that for each finite group G, the sum E(G) = ∑_{X} 𝔽e_{X}^{G×K}, where X ≤ G × K and (X, p_{2}) ↠ (L, φ), is a subset of e_{L,φ}(G).

Moreover the map (l, φ(l)) ∈ L_{φ} ↦ l ∈ L is clearly an isomorphism (L_{φ}, p_{2}) → (L, φ) in grp_{⇓K}. In particular (L_{φ}, p_{2}) ↠ (L, φ), and then by definition e_{L_{φ}}^{L×K} ∈ E(L). If we can prove that G ↦ E(G) defines an ideal E of 𝔽B_{K}, then we are done, because E ⊆ e_{L,φ} since E(G) ⊆ e_{L,φ}(G) for any G, and e_{L,φ} ⊆ E because the generator e_{L_{φ}}^{L×K} of e_{L,φ} belongs to E(L).

Since E(G) is obviously an ideal of the algebra 𝔽B_{K}(G), for any G, all we have to do is to show that E is a biset subfunctor of 𝔽B_{K}, in other words that it is preserved by the elementary biset operations of induction, restriction, inflation, deflation, and transport by group isomorphism. For this, in what follows, we refer to Theorem 5.2.4 of [2].

Let X ≤ G × K be such that (X, p_{2}) ↠ (L, φ), and suppose first that G is a subgroup of a group H. Then

𝔽B_{K}(Ind_{G}^{H})(e_{X}^{G×K}) = 𝔽B(Ind_{G×K}^{H×K})(e_{X}^{G×K}) = λ e_{X′}^{H×K}

for some scalar λ, where X′ is the group X, viewed as a subgroup of H × K. Clearly (X′, p_{2}) = (X, p_{2}), so (X′, p_{2}) ↠ (L, φ) and e_{X′}^{H×K} ∈ E(H). Hence E is preserved by induction.

Assume now that H is a subgroup of G. Then

𝔽B_{K}(Res_{H}^{G})(e_{X}^{G×K}) = 𝔽B(Res_{H×K}^{G×K})(e_{X}^{G×K}) = ∑_{Y} e_{Y}^{H×K} ,

where Y runs through a set of representatives of (H × K)-conjugacy classes of subgroups of H × K which are conjugate to X in G × K. If Y is such a subgroup, there exists (g, k) ∈ G × K such that Y = X^{(g,k)}. Then we have a commutative diagram

\xymatrix{Y \ar[d]_{p_{2}} \ar[r]^{α} & X \ar[d]^{p_{2}} \\ K \ar[r]^{β} & K}

where α is (left-)conjugation by (g, k) and β is (left-)conjugation by k. Since β is an inner automorphism of K, and since α is a group isomorphism, it follows that α : (Y, p_{2}) → (X, p_{2}) is an isomorphism in grp_{⇓K}. Hence (Y, p_{2}) ↠ (L, φ), and e_{Y}^{H×K} ∈ E(H). It follows that E is preserved by restriction.

Assume next that G is a quotient of a group H by a normal subgroup N. Then

𝔽B_{K}(Inf_{G}^{H})(e_{X}^{G×K}) = 𝔽B(Inf_{G×K}^{H×K})(e_{X}^{G×K}) = ∑_{Y} e_{Y}^{H×K} ,

where Y runs through a set of (H×K) conjugacy classes of subgroup of H×K which map to a conjugate of X under the projection π×Id_{K} : H×K → G×K, where π : H → G is the projection. Replacing Y by a conjugate, which does not change e_{Y}^{H×K}, we can assume that Y is mapped to X by π × Id_{K}. This gives a commutative diagram

\xymatrix{Y \ar@{->>}[rr]^{π×Id_{K}} \ar[dr]_{p_{2}} &  & X \ar[dl]^{p_{2}} \\  & K}

showing that (Y, p_{2}) ↠ (X, p_{2}). Hence (Y, p_{2}) ↠ (L, φ), so e_{Y}^{H×K} ∈ E(H), and E is preserved by inflation.

As for deflation, we assume now that H = G/N, where N ⊴ G. Let π : G → H be the projection map. Then by Lemma 2.2

𝔽B_{K}(Def_{H}^{G})(e_{X}^{G×K}) = λ m_{X,X∩(N×1)} eH×K ; \overline{X} ,

where \overline{X} is the image of X under the projection π × Id_{K} : G × K → H × K, and λ is some non zero scalar. As above, we get a commutative diagram

\xymatrix{X \ar@{->>}[rr]^{s} \ar[dr]_{p_{2}} &  & \overline{X} \ar[dl]^{p_{2}} \\  & K}

where s is the restriction of π × Id_{K} to X. Then s : (X, p_{2}) → (\overline{X}, p_{2}) is a surjective morphism in grp_{⇓K}. Setting P = Ker s = X ∩ (N × 1), we get an isomorphism (\overline{X}, p_{2}) ≅ (X/P, p_{2}/P) in grp_{⇓K}. Moreover (L, φ) is a B_{K}-group quotient of (X, p_{2}) by assumption. Then there are two cases: either m_{X,P} = 0, and then 𝔽B_{K}(Def_{H}^{G})(e_{X}^{G×K}) = 0 ∈ E(H). Or m_{X,P} ≠ 0, and then (L, φ) is a quotient of (X/P, p_{2}/P) ≅ (\overline{X}, p_{2}), by Assertion 2 of Theorem 4.6. It follows that eH×K ; \overline{X} ∈ E(H), so 𝔽B_{K}(Def_{H}^{G})(e_{X}^{G×K}) ∈ E(H) as well. This shows that E is preserved by deflation.

Finally, it is clear that E is preserved by group isomorphisms. This completes the proof of Theorem 5.3.

5.4. Remark: Theorem 5.3 implies that the set of idempotents e_{X}^{G×K}, where X runs through a set of representatives of conjugacy classes of subgroups of G × K such that (X, p_{2}) ↠ (L, φ), is an 𝔽-basis of e_{L,φ}(G).

5.5. Corollary: Let (L, φ) be a B_{K}-group, and (M, ψ) be a group over K. Then e_{M,ψ} ⊆ e_{L,φ} if and only if (M, ψ) ↠ (L, φ).

Proof: Indeed e_{M,ψ} ⊆ e_{L,φ} if and only if e_{M_{ψ}}^{M×K} ∈ e_{L,φ}(M), i.e. if and only if (M_{ψ}, p_{2}) ↠ (L, φ). But we have already noticed at the beginning of the proof of Theorem 5.3 that the map (m, ψ(m)) ∈ M_{ψ} ↦ m ∈ M is an isomorphism from (M_{ψ}, p_{2}) to (M, ψ) in grp_{⇓K}.

5.6. Remark: It was shown in Section 5.2.2 of [3] that the category 𝔽B_{K}-Mod splits as a product

𝔽B_{K}-Mod ≅ ∏_{H} = e_{H}^{K}𝔽B_{K}-Mod ,

of categories of modules over smaller Green biset functors e_{H}^{K}𝔽B_{K}, where H runs through a set of representatives of conjugacy classes of subgroups of K. The functor e_{H}^{K}𝔽B_{K} is the direct summand of 𝔽B_{K} obtained from the idempotent e_{H}^{K} of 𝔽B_{K}(1) ≅ 𝔽B(K). Its value at a group G is the set of 𝔽-linear combinations of idempotents e_{L}^{G×K} associated to subgroups L for which p_{2}(L) is conjugate to H in K. This condition is equivalent to the existence of a surjective morphism (L, p_{2}) ↠ (H, j_{H}), where j_{H} : H ↪ K is the inclusion morphism. Since (H, j_{H}) is a B_{K}-group by Example 4.4, it follows that e_{H}^{K}𝔽B_{K} = e_{H,j_{H}}.

5.7. Theorem: Let I_{𝔽B_{K}} be the lattice of ideals of 𝔽B_{K}, ordered by inclusion of ideals, and Cl_{B_{K}-gr} be the lattice of closed subsets of B_{K}-gr, ordered by inclusion of subsets. Then the map

I ∈ I_{𝔽B_{K}} ↦ P_{I} = {(L, φ) ∈ B_{K}-gr | e_{L,φ} ⊆ I}

is an isomorphism of lattices from I_{𝔽B_{K}} to Cl_{B_{K}-gr}. The inverse isomorphism is the map

P ∈ Cl_{B_{K}-gr} ↦ I_{P} = ∑_{(L,φ)∈P} e_{L,φ} .

In particular I_{𝔽B_{K}} is completely distributive.

Proof: By Proposition 5.2, if I is an ideal of 𝔽B_{K}, then P_{I} is a closed subset of B_{K}-gr, so the map α : I ↦ P_{I} from I_{𝔽B_{K}} to Cl_{B_{K}-gr} is well defined. It is moreover clearly order preserving. The map β : P ↦ P_{I} from Cl_{B_{K}-gr} is also well defined and order preserving. By Proposition 5.2 again, the composition β ∘ α is the identity map of I_{𝔽B_{K}}. Conversely, if P ∈ Cl_{B_{K}-gr}, then

α ∘ β = {(M, ψ) ∈ B_{K}-gr | e_{M,ψ} ⊆ ∑_{(L,φ)∈P} e_{L,φ}} .

Then clearly P ⊆ α ∘ β(P). Conversely, if e_{M,ψ} ⊆ ∑_{(L,φ)∈P} e_{L,φ}, then by

Lemma 3.10 there exists (L, φ) ∈ P such that e_{M,ψ} ⊆ e_{L,φ}. Then (L, φ) is a B_{K}-group, and by Corollary 5.5, this implies (M, ψ) ↠ (L, φ). Hence (M, ψ) ∈ P, since P is closed. Thus α ∘ β(P) ⊆ P, proving that α ∘ β is the identity map of Cl_{B_{K}-gr}. The last assertion follows from the fact that Cl_{B_{K}-gr} is clearly completely distributive, since its join and meet operation are union and intersection of closed subsets, respectively, and since arbitrary unions (resp. intersections) distribute over arbitrary intersections (resp. unions).

6. Some simple 𝔽B_{K}-modules

6.1. Theorem:

1. Let (L, φ) be a B_{K}-group. Then e_{L,φ} admits a unique maximal proper subideal e_{L,φ}^{0}, defined by

e_{L,φ}^{0} = ∑_{\begin{matrix}(M,ψ)∈B_{K}-gr \\ (M,ψ)↠(L,φ) \\ (M,ψ)≇(L,φ) \end{matrix}} e_{M,ψ} .

2. The quotient S_{L,φ} = e_{L,φ}/e_{L,φ}^{0} is a simple 𝔽B_{K}-module.

3. For any finite group G, let A_{G} be a set of representatives of conjugacy classes of subgroups X of G × K such that β_{K}(X, p_{2}) ≅ (L, φ). Then the set {e_{X}^{G×K} | X ∈ A_{G}} maps to an 𝔽-basis of S_{L,φ}(G) under the projection map e_{L,φ}(G) → S_{L,φ}(G).

4. If I′ ⊂ I are ideals of 𝔽B_{K} such that I/I′ is a simple 𝔽B_{K}-module, then there exists a B_{K}-group (L, φ) such that I/I′ ≅ S_{L,φ}.

Proof: 1. Without loss of generality, we can assume that (L, φ) ∈ B_{K}-gr. Using Theorem 5.7, saying that e_{L,φ} admits a unique maximal proper subideal is equivalent to saying that the closed subset P_{e_{L,φ}} contains a unique maximal proper closed subset. But

P_{e_{L,φ}} = {(M, ψ) ∈ B_{K}-gr | (M, ψ) ↠ (L, φ)} ,

so P^{0} = P_{e_{L,φ}} −{(L, φ)} is the unique maximal proper closed subset of P_{e_{L,φ}}. It follows that I_{P^{0}} = e_{L,φ}^{0} is the unique maximal proper subideal of e_{L,φ}.

2. This is clear, from 1.

3. We know from Remark 5.4 that e_{L,φ}(G) has a basis consisting of the idempotents e_{X}^{G×K}, for X in a set of representatives of conjugacy classes of subgroups of G×K such that (X, p_{2}) ↠ (L, φ), or equivalently, by Corollary 4.9, such that β_{K}(X, p_{2}) ↠ (L, φ). Now saying that e_{X}^{G×K} ∈ e_{L,φ}^{0}(G) amounts to saying that e_{X_{p_{2}}}^{X×K} ∈ e_{L,φ}^{0}(X), by Theorem 3.4, i.e. that e_{X,p_{2}} ⊆ e_{M,ψ} for some (M, ψ) ∈ B_{K}-gr such that (M, ψ) ↠ (L, φ), but (M, ψ) ≇ (L, φ). This in turn is equivalent to saying that β_{K}(X, p_{2}) ↠ (L, φ), but β_{K}(X, p_{2}) ≇ (L, φ). Hence S_{L,φ}(G) has a basis consisting of the idempotents e_{X}^{G×K}, for X in a set of representatives of conjugacy classes of subgroups of G × K such that β_{K}(X, p_{2}) ≅ (L, φ). Assertion 2 follows.

4. Let I′ ⊂ I be ideals of 𝔽B_{K} such that S = I/I′ is a simple 𝔽B_{K}-module, or equivalently, such that I′ is a maximal subideal of I. Then there exists (L, φ) ∈ B_{K}-gr such that e_{L,φ} ⊆ I but e_{L,φ} ⊈ I′. Hence e_{L,φ} + I′ = I, and S = I/I′ ≅ e_{L,φ}/(e_{L,φ} ∩ I′). Then e_{L,φ} ∩ I′ is a proper subideal of e_{L,φ}, so e_{L,φ} ∩ I′ ⊆ e_{L,φ}^{0}, and then S maps surjectively onto e_{L,φ}/e_{L,φ}^{0} = S_{L,φ}. Since S and S_{L,φ} both are simple 𝔽B_{K}-modules, the surjection S → S_{L,φ} is an isomorphism.

6.2. Remark: By Corollary 4.10, the condition β_{K}(X, p_{2}) ≅ (L, φ) in Assertion 3 is equivalent to the existence of a surjective morphism s from (X, p_{2}) to (L, φ) such that m_{X,Ker s} ≠ 0. By Theorem 5.4.11 of [2], or by Corollary 4.10 applied to the case K = 1, this is equivalent to the condition β(X) ≅ β(L).

6.3. Corollary: Let (L, φ) and (M, ψ) be B_{K}-groups. Then the simple 𝔽B_{K}-modules S_{L,φ} and S_{M,ψ} are isomorphic if and only if (L, φ) and (M, ψ) are isomorphic in grp_{⇓K}.

Proof: Clearly if (L, φ) ≅ (M, ψ) in grp_{⇓K}, then S_{L,φ} ≅ S_{M,ψ}. Conversely, if θ : S_{L,φ} → S_{M,ψ} is an isomorphism of 𝔽B_{K}-modules, then for any finite group G, we get an isomorphism θ_{G} : S_{L,φ}(G) → S_{M,ψ}(G) of 𝔽B_{K}(G)- modules. Choose G such that S_{L,φ}(G) ≠ 0 (e.g. G = L), and a subgroup X of G × K such that β_{K}(X, p_{2}) ≅ (L, φ). Then the image u of a = e_{X}^{G×K} ∈ 𝔽B_{K}(G) in S_{L,φ}(G) is non zero, and moreover a · u = u. It follows that θ_{G}(a · u) = a · θ_{G}(u) = θ_{G}(u) is also non zero in S_{M,ψ}(G). So there is a subgroup Y ≤ G×K with β_{K}(Y, p_{2}) ≅ (M, ψ), such that the image v of e_{Y}^{G×K} in S_{M,ψ}(G) satisfies a · v ≠ 0. This forces X and Y to be conjugate in G × K, so (L, φ) ≅ β_{K}(X, p_{2}) ≅ β_{K}(Y, p_{2}) ≅ (M, ψ) in grp_{⇓K}, as was to be shown.

Recall that a minimal group for a (non zero) biset functor F is a finite group G of minimal order such that F (G) ≠ {0}.

6.4. Lemma: Let (L, φ) be a group over K.

1. If N ⊴ L, and N ∩ Ker φ = 1, then e_{L,φ}(L/N) ≠ {0}.

2. If moreover (L, φ) is a B_{K}-group, then S_{L,φ}(L/N) ≠ {0}.

Proof: Indeed the map

θ : l ∈ L ↦ (lN, φ(l)) ∈ (L/N) × K

is injective. Let \overline{L} ≤ (L/N) × K denote the image of θ. Then we have a commutative diagram

\xymatrix{ & \overline{L} \ar[dl]_{p_{1}} \ar[d]^{p_{2}} \ar@{->>}[r]^{t} & L \ar[d]^{φ} \\ L/N & K \ar[r]^{i} & K,}

where t : \overline{L} → L is the inverse of the isomorphism L → \overline{L} induced by θ, and i is the identity map of K. Hence (\overline{L}, p_{2}) ≅ (L, φ) in grp_{⇓K}, and e_{L,φ} = e_{\overline{L},p_{2}}. In particular e_{\overline{L}_{p_{2}}}^{(L/N)×K} ∈ e_{L,φ}(L/N) by Theorem 3.4, hence e_{L,φ}(L/N) ≠ \{0\}. This proves 1.

If moreover (L, φ) is a B_{K}-group, then β_{K}(\overline{L}, p_{2}) ≅ (L, φ). It follows from Theorem 6.1 that e(L/N)×K ; \overline{L} ∈ e_{L,φ}(L/N) maps to an element of a basis of S_{L,φ}(L/N), so S_{L,φ}(L/N) ≠ {0}, proving 2.

6.5. Theorem: Let (L, φ) be a B_{K}-group, and G be a finite group. The following are equivalent:

1. The group G is a minimal group for S_{L,φ}.

2. The group G is isomorphic to L/N, where N is a normal subgroup of L

of maximal order such that N ∩ Ker φ = 1.

Moreover in this case, the images in S_{L,φ}(G) of the idempotents e_{X}^{G×K}, where X runs through a set of representatives of conjugacy classes of subgroups of G × K such that (X, p_{2}) ≅ (L, φ), form an 𝔽-basis of S_{L,φ}(G).

Proof: By Theorem 6.1, saying that S_{L,φ}(G) ≠ {0} for a finite group G amounts to saying that there exists a subgroup X of G × K such that β_{K}(X, p_{2}) ≅ (L, φ) in grp_{⇓K}. Equivalently, there is a commutative diagram

(6.6)  \xymatrix{ & X \ar[dl]_{p_{1}} \ar[d]^{p_{2}} \ar@{->>}[r]^{s} & L \ar[d]^{φ} \\ G & K \ar[r]^{i} & K,}

where

• s is surjective and m_{X,Ker s≠0},

• i is an inner automorphism of K,

• the map (p_{1}, p_{2}) : X → (G × K) is injective.

Now we proceed with the proof of Theorem 6.5. 1 ⇒ 2 If G is minimal for S_{L,φ}, then S_{L,φ}(G) ≠ {0}, so we have a diagram (6.6). Let H = p_{1}(G). Replacing G by H in this diagram gives a diagram for the group H with the same properties, so S_{L,φ}(H) ≠ 0. Hence H = G by minimality of G. In other words p_{1} is surjective, so G ≅ X/Ker p_{1}. Let N = s(Ker p_{1}). If u ∈ N ∩ Ker φ, then u = s(x) for some x ∈ X, and then φ ∘ s(x) = i ∘ p_{2}(x) = 1, so p_{2}(x) = 1. Thus x = 1 since Ker p_{1} ∩ Ker p_{2} = 1. Moreover N is normal in L, since s is surjective. Lemma 6.4 shows that S_{L,φ}(L/N) ≠ {0}, and by minimality of G, the surjection \overline{s} : G ≅ X/Ker p_{1} ↠ L/N induced by s must be an isomorphism. Lemma 6.4 also implies that N is a normal subgroup of maximal order of L such that N ∩ Ker φ. Hence 2 holds.

Observe that it also follows that Ker s ≤ Ker p_{1}, so Ker s = 1 since Ker s ≤ Ker p_{2} as φ ∘ s = i ∘ p_{2}, and Ker p_{1} ∩ Ker p_{2} = 1. So s is an isomorphism X → L. This proves the last assertion of the theorem.

2 ⇒ 1 Suppose that 2 holds. Then S_{L,φ}(G) ≠ 0, by the above claim. By the first part of the proof, if H is a minimal group for S_{L,φ}, then H ≅ L/M, where M is a normal subgroup of maximal order such that M ∩ Ker φ = 1. Then |M| = |N|, so |G| = |H|, and S_{L,φ}(G′) = {0} for any group G′ of order smaller than |G| = |H|. Hence G is minimal for S_{L,φ}, and 1 holds.

6.7. Corollary: Let (L, φ) be a group over K. The following conditions are equivalent:

1. φ : L → K is injective.

2. (L, φ) is a B_{K}-group and S_{L,φ}(1) ≠ {0}.

Proof: 1 ⇒ 2 If φ is injective, then (L, φ) is a B_{K}-group (cf. Example 4.4). Moreover L ∩ Ker φ = 1, so S_{L,φ}(L/L) = S_{L,φ}(1) ≠ {0}.

2 ⇒ 1 If (L, φ) is a B_{K}-group and S_{L,φ}(1) ≠ {0}, then 1 is a minimal group for S_{L,φ}. So there is a normal subgroup N of L of maximal order such that N ∩ Ker φ = 1, such that moreover L/N ≅ 1. Hence N = L, and Ker φ = N ∩ Ker φ = 1.

6.8. Example: Let L = C_{2} × (C_{3} ⋊ C_{4}) be a direct product of a group of order 2, generated by the element a, and a semidirect product of a group of order 3, generated by b, and a cyclic group of order 4, generated by c (so cbc^{−1} = b^{−1}). Let P be the subgroup of L generated by a and b. Then P is cyclic of order 6, and the factor group K = L/P is cyclic of order 4, generated by the class cP. Let φ : L → K be the projection map. One can check that (L, φ) is a B_{K}-group, i.e. that m_{L,Q} = 0 when Q is any of the non trivial subgroups of P (these subgroups are all normal in L, as P is cyclic).

Then the subgroups M = ⟨ac^{2}⟩ and N = ⟨c^{2}⟩ both are normal (central, in fact) subgroups of L of maximal order (equal to 2) intersecting trivially P = Ker φ. So the groups G = L/M and H = L/N are both minimal groups^{1} for the simple 𝔽B_{K}-module S_{L,φ}, but they are not isomorphic, as G ≅ C_{3}⋊C_{4} but H ≅ C_{2} × S_{3}, where S_{3} is the symmetric group of degree 3. This gives yet another counterexample to a conjecture I made in 2010, saying that the minimal groups for a Green biset functor should form a single isomorphism class of groups. The first counterexample to this conjecture was found by Nadia Romero in 2013 (cf. [6]). Another counterexample was found recently by Ibrahima Tounkara (cf. [7]).

7. Restriction to p-groups

In this section, we fix a prime number p, and restrict the functor 𝔽B_{K} to finite p-groups. We obtain a Green p-biset functor 𝔽B_{K}^{(p)}. We do not assume that K is itself a p-group.

1One can show moreover that S_{L,φ}(G) and S_{L,φ}(H) are both one dimensional.

In order to study the ideals of 𝔽B_{K}^{(p)}, it is natural to try to determine those groups (L, φ) over K for which the restriction of e_{L,φ} to p-groups does not vanish. This motivates the following definition:

7.1. Definition: Let K be a finite group. Then a group (L, φ) over K is called p-persistent if there is a finite p-group P such that e_{L,φ}(P) ≠ {0}.

We denote by grp_{⇓K}^{(p)} the full subcategory of grp_{⇓K} consisting of ppersistent groups over K.

7.2. Remarks:

1. If X is a subgroup of P × K, where P is a p-group, then (X, p_{2}) is p-persistent: indeed e_{X}^{P ×K} ∈ e_{X,p_{2}}(P) by Corollary 3.5.

2. Any quotient of a p-persistent group over K is p-persistent: indeed is s : (M, ψ) ↠ (L, φ) is a surjective morphism in grp_{⇓K}, then e_{M,ψ} ⊆ e_{L,φ} by Lemma 3.7. It follows that e_{L,φ}(P) ≠ {0} if P is a p-group such that e_{M,ψ}(P) ≠ {0}. In particular, if (L, φ) is p-persistent, then β_{K}(L, φ) is a p-persistent B_{K}-group.

7.3. Notation: When L is a finite group, we denote by O^{p}(L) the subgroup of L generated by p′-elements, and by L^{[p]} the quotient L/O^{p}(L).

Recall that O^{p}(L) is the smallest normal subgroup N of L such that L/N is a p-group. Also recall that if s : M → L is a surjective group homomorphism, then s(O^{p}(M)) = O^{p}(L). Indeed N = s(O^{p}(M)) ⊴ L, and s induces a surjection M^{[p]} → L/N. So L/N is a p-group, thus N ≥ O^{p}(L). But N is generated by p′-elements, as O^{p}(M) is, so N ≤ O^{p}(L).

7.4. Proposition: Let (L, φ) be a group over K. The following are equivalent:

1. (L, φ) is p-persistent.

2. e_{L,φ}(L^{[p]}) ≠ {0}.

3. m_{L,O^{p}(L)∩Ker φ} ≠ 0.

Proof: Indeed if 3 holds, then setting N = O^{p}(L) ∩ Ker φ, we have e_{L,φ} = e_{L/N,φ/N} by Theorem 4.1. Moreover O^{p}(L/N) = O^{p}(L)/N, and Ker (φ/N) = Ker φ/N. Thus O^{p}(L/N) ∩ Ker(φ/N) = 1, so e_{L/N,φ/N}((L/N)/O^{p}(L/N)) is non zero by Lemma 6.4. But

e_{L/N,φ/N}((L/N)/O^{p}(L/N)) ≅ e_{L/N,φ/N}(L/O^{p}(L)) = e_{L,φ}(L^{[p]}) ,

so 2 holds. Clearly 2 implies 1, as L^{[p]} is a p-group. Now if 1 holds, let P be a p-group such that e_{L,φ}(P) ≠ {0}. Let N be a normal subgroup of L contained in Ker φ, and maximal such that m_{L,N} ≠ 0. Then setting \overline{L} = L/N and \overline{φ} = φ/N, we have β_{K}(L, φ) ≅ (\overline{L}, \overline{φ}), and e_{L,φ} = e_{\overline{L},\overline{φ}} by Theorem 4.1. Moreover as (\overline{L}, \overline{φ}) is a B_{K}-group, by Theorem 5.3, there exists a subgroup X of P × K, and a commutative diagram

\xymatrix{ & X \ar[dl]_{p_{1}} \ar[d]^{p_{2}} \ar@{->>}[r]^{s} & \overline{L} \ar[d]^{\overline{φ}} \\ P & K \ar[r]^{i} & K,}

where s is surjective and i is an inner automorphism of K. Then N = s(Ker p_{1}) is a normal subgroup of \overline{L}, as s is surjective. Moreover if l ∈ N ∩ Ker \overline{φ}, then l = s(x) for some x ∈ Ker p_{1}, so p_{1}(x) = 1 and i ∘ p_{2}(x) = φ ∘ s(x) = 1, so p_{2}(x) = 1. Hence x = 1, and l = 1, so N ∩ Ker \overline{φ} = 1. Now s induces a surjection X/Ker p_{1} ≅ p_{1}(X) ↠ L/N, so L/N is a p-group, thus N ≥ O^{p}(L). It follows that O^{p}(\overline{L}) ∩ Ker \overline{φ} = 1. Now if π : L → \overline{L} = L/N is the projection map, we have \overline{φ} ∘ π = φ, so

π(O^{p}(L) ∩ Ker φ) ≤ O^{p}(\overline{L}) ∩ Ker \overline{φ} = 1 ,

that is O^{p}(L) ∩ Ker φ ≤ N = Ker π. Then if M = O^{p}(L) ∩ Ker φ, we have m_{L,N} = m_{L,M}m_{L/M,N/M} ≠ 0, hence m_{L,M} ≠ 0, so 3 holds.

7.5. Corollary: Let (L, φ) be a p-persistent B_{K}-group. Then

O^{p}(L) ∩ Ker φ = 1 .

Proof: Indeed m_{L,O^{p}(L)∩Ker φ} ≠ 0, and (L, φ) is a B_{K}-group.

7.6. Notation: When (L, φ) is a p-persistent group over K, we denote by L_{φ}^{(p)} the subgroup of L^{[p]} × K defined by

L_{φ}^{(p)} = {(lO^{p}(L), φ(l)) | l ∈ L} .

The following theorem is analogous to Theorem 3.4:

7.7. Theorem: Let I be an ideal of the Green biset functor 𝔽B_{K}^{(p)}. If G is a finite p-group and L is a subgroup of G × K, the following conditions are equivalent:

1. The idempotent e_{L}^{G×K} belongs to I(G).

2. The idempotent e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K} belongs to I(L^{[p]}).

Proof: The proof is similar to the proof of Theorem 3.4, so we only sketch it. If L ≤ G × K, denote by L̂ the image of L in the group L^{[p]} × G by the map l ↦ (lO^{p}(L), p_{1}(l)). Recall that Ker p_{1} ≥ O^{p}(L), since G is a pgroup. Furthermore p_{1}(L̂) = L^{[p]}, k_{1}(L̂) = Ker p_{1}/O^{p}(L), p_{2}(L̂) = p_{1}(L), and k_{2}(L̂) = p_{1}(O^{p}(L)) = 1. The (L^{[p]}, G)-biset U = (L^{[p]} × G)/L̂ factors as

U ≅ Inf_{L^{[p]}/k_{1}(L̂)}^{L^{[p]}} ∘ Iso(θ^{−1}) ∘ Res_{p_{1}(L)}^{G} ,

where θ : L^{[p]}/k_{1}(L̂) → p_{1}(G) is the isomorphism induced by the map lO^{p}(L) ↦ p_{1}(l) from L^{[p]} to G.

If e_{L}^{G×K} belongs to I(G), then 𝔽B_{K}^{(p)}(U)(e_{L}^{G×K}) belongs to I(L^{[p]}). As in the proof of Theorem 3.4, one can check that the product e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K}·𝔽B_{K}^{(p)}(e_{L}^{G×K})

is non zero. As it is a scalar multiple of e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K}, we get that e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K} ∈ I(L^{[p]}),

thus 1 implies 2.

Conversely, assume that e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K} ∈ I(L^{[p]}). Then, as in the proof of The-

orem 3.4 again, the opposite biset U^{op} factors as

U^{op} ≅ Ind_{p_{1}(L)}^{G} ∘ Iso(θ) ∘ Def_{L^{[p]}/k_{1}(L̂)}^{L^{[p]}} ,

and the element 𝔽B_{K}^{(p)}(U^{op})(e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K}) belongs to I(G). One can can check

moreover that there is a non zero scalar λ such that

𝔽B_{K}^{(p)}(U^{op})(e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K}) = λ m_{L_{p_{2}}^{(p)},L_{p_{2}}^{(p)}∩(N×1)} e_{L}^{G×K} ,

where N = k_{1}(L̂) = Ker p_{1}/O^{p}(L) ≤ L^{[p]}.

But if (lO^{p}(L), p_{2}(l)) ∈ L_{p_{2}}^{(p)} ∩ (N × 1), then l ∈ Ker p_{2} ∩ Ker p_{1} = 1. It follows that m_{L_{p_{2}}^{(p)},L_{p_{2}}^{(p)}∩(N×1)} = m_{L_{p_{2}}^{(p)},1} = 1, and e_{L}^{G×K} ∈ I(G), as λ ≠ 0. Hence 2 implies 1.

7.8. Corollary: Let G be a finite p-group, and L be a subgroup of G × K. Then the ideal of 𝔽B_{K}^{(p)} generated by e_{L}^{G×K} is equal to the ideal of 𝔽B_{K}^{(p)} generated by e_{L_{p_{2}}^{(p)}}^{L^{[p]}×K}

Proof: The proof is the same as the proof of Corollary 3.5.

7.9. Notation: Let (L, φ) be a p-persistent group over K. We denote by e_{L,φ}^{(p)} the ideal of 𝔽B_{K}^{(p)} generated by e_{L_{φ}^{(p)}}^{L^{[p]}×K} ∈ 𝔽B_{K}^{(p)}(L^{[p]}).

7.10. Theorem: Let s : (M, ψ) ↠ (L, φ) be a surjective morphism in grp_{⇓K}, and assume that (M, ψ) is p-persistent. Then:

1. (L, φ) is p-persistent, and e_{M,ψ}^{(p)} ⊆ e_{L,φ}^{(p)}.

2. If m_{M,Ker s} ≠ 0, then e_{M,ψ}^{(p)} = e_{L,φ}^{(p)}.

Proof: 1. We already observed in Remarks 7.2 that any quotient of a ppersistent group over K is itself p-persistent, hence (L, φ) is p-persistent. Let i be an inner automorphism of K such that i ∘ ψ = φ ∘ s. The surjection s : M → L induces a surjection s^{[p]} : M^{[p]} → L^{[p]}, hence a surjection

s^{[p]} × Id_{K} : M^{[p]} × K → L^{[p]} × K .

Let u = (mO^{p}(M), ψ(m)) be the image of m ∈ M in M_{ψ}^{(p)}. Then

(s^{[p]} × Id_{K})(u) = (s(m)O^{p}(L), ψ(m)) = (s(m)O^{p}(L), i^{−1} ∘ φ(s(m))) ,

which shows that s^{[p]} × Id_{K} maps M_{ψ}^{(p)} to a conjugate of L_{φ}^{(p)} in L^{[p]} × K. Then the idempotent e_{M_{ψ}^{(p)}}^{M^{[p]}×K} appears in the decomposition of

𝔽B_{K}^{(p)}(Inf_{M^{[p]}/Ker s^{[p]}}^{M^{[p]}} ∘ Iso(α^{−1}))(e_{L_{φ}^{(p)}}^{L^{[p]}×K}) ,

where α : M^{[p]}/Ker s^{[p]} → L^{[p]} is the canonical isomorphism. It follows that e_{M_{ψ}^{(p)}}^{M^{[p]}×K} ∈ e_{L,φ}^{(p)}(M^{[p]}), hence e_{M,ψ}^{(p)} ⊆ e_{L,φ}^{(p)}.

2. Consider now v = 𝔽B_{K}^{(p)}(Iso(α)∘Def_{M^{[p]}/Ker s^{[p]}}^{M^{[p]}})(e_{M_{ψ}^{(p)}}^{M^{[p]}×K}) ∈ e_{M,ψ}^{(p)}(L^{[p]}). By

Lemma 2.2, there is a non zero scalar λ such that

(7.11)  v = λ m_{M_{ψ}^{(p)},M_{ψ}^{(p)}∩(Ker s^{[p]}×1)}e_{L_{φ}^{(p)}}^{L^{[p]}×K} .

Now the projection m ∈ M ↦ (mO^{p}(M), ψ(m)) ∈ M_{ψ}^{(p)} induces an isomorphism M_{ψ}^{(p)} ≅ M/(O^{p}(M) ∩ Ker ψ). As Ker s^{[p]} = Ker s O^{p}(L)/O^{p}(L), the subgroup M_{ψ}^{(p)}∩(Ker s^{[p]}×1) maps to (Ker s O^{p}(M)∩Ker ψ)/(O^{p}(M)∩Ker ψ) under this isomorphism.

Moreover Ker s O^{p}(M)∩Ker ψ = Ker s(O^{p}(M)∩Ker ψ) as Ker s ≤ Ker ψ. It follows that

m_{M_{ψ}^{(p)},M_{ψ}^{(p)}∩(Ker s^{[p]}×1)} = m_{M/(O^{p}(M)∩Ker ψ),Ker s(O^{p}(M)∩Ker ψ)/(O^{p}(M)∩Ker ψ)} .

Multiplying by m_{M,O^{p}(M)∩Ker ψ}, which is non zero by Proposition 7.4, since (M, ψ) is p-persistent, this gives

m_{M,O^{p}(M)∩Ker ψ} m_{M_{ψ}^{(p)},M_{ψ}^{(p)}∩(Ker s^{[p]}×1)} = m_{M,Ker s(O^{p}(M)∩Ker ψ)}
= m_{M,Ker s} m_{M/Ker s,Ker s(O^{p}(M)∩Ker ψ)/Ker s}
= m_{M,Ker s} m_{L,O^{p}(L)∩Kerφ} ,

as the canonical isomorphism M/Ker s → L maps Ker s(O^{p}(M)∩Ker ψ)/Ker s to O^{p}(L) ∩ Kerφ. Since m_{L,O^{p}(L)∩Kerφ} ≠ 0 as (L, φ) is p-persistent, and since m_{M,Ker s} ≠ 0 by assumption, it follows that m_{M_{ψ}^{(p)},M_{ψ}^{(p)}∩(Ker s^{[p]}×1)} ≠ 0, hence

e_{L_{φ}^{(p)}}^{L^{[p]}×K} is a non zero scalar multiple of v, by 7.11. It follows that e_{L_{φ}^{(p)}}^{L^{[p]}×K}

belongs to e_{M,ψ}^{(p)}(L^{[p]}), so e_{L,φ}^{(p)} ⊆ e_{M,ψ}^{(p)}, and e_{L,φ}^{(p)} = e_{M,ψ}^{(p)}, as was to be shown.

7.12. Corollary: Let (L, φ) be a p-persistent group over K. Then the restriction of e_{L,φ} to finite p-groups is equal to e_{L,φ}^{(p)}.

Proof: Since e_{L,φ} = e_{β_{K}(L,φ)} by Corollary 4.9, and since e_{L,φ}^{(p)} = e_{β_{K}(L,φ)}^{(p)} by Theorem 7.10, we may assume that (L, φ) is a B_{K}-group. By Corollary 7.5, we have O^{p}(L)∩Ker φ = 1. Thus the projection L → L_{φ}^{(p)} is an isomorphism, and it induces an isomorphism (L, φ) ≅ (L_{φ}^{(p)}, p_{2}). Hence e_{L_{φ}^{(p)}}^{L^{(p]}×K} ∈ e_{L,φ}(L^{[p]}),

and e_{L,φ}^{(p)} is contained in the restriction of e_{L,φ} to p-groups.

Conversely, if G is a p-group and e_{X}^{G×K} ∈ e_{L,φ}(G), then (X, p_{2}) ↠ (L, φ) by Theorem 5.3. Then e_{X,p_{2}}^{(p)} ⊆ e_{L,φ}^{(p)}, hence e_{X}^{G×K} ∈ e_{L,φ}^{(p)} by Corollary 7.8. Hence the restriction of e_{L,φ} is contained in e_{L,φ}^{(p)}, which completes the proof.

7.13. Corollary: Let (L, φ) be a p-persistent B_{K}-group, and (M, ψ) be a p-persistent group over K. Then e_{M,ψ}^{(p)} ⊆ e_{L,φ}^{(p)} if and only if (M, ψ) ↠ (L, φ).

Proof: Indeed if (M, ψ) ↠ (L, φ), then e_{M,ψ}^{(p)} ⊆ e_{L,φ}^{(p)} by Theorem 7.10. Conversely, if e_{M,ψ}^{(p)} ⊆ e_{L,φ}^{(p)}, showing that (M, ψ) ↠ (L, φ) amounts to showing that β_{K}(M, ψ) ↠ (L, φ), because (L, φ) is a B_{K}-group. Now e_{M,ψ} = e_{β_{K}(M,ψ)}, hence e_{M,ψ}^{(p)} = e_{β_{K}(M,ψ)}^{(p)} by Corollary 7.12, and we can assume that (M, ψ) is also a B_{K}-group.

If e_{M,ψ}^{(p)} ⊆ e_{L,φ}^{(p)}, then e_{M_{ψ}^{(p)}}^{M^{[p]}×K} ∈ e_{L,φ}^{(p)}(M^{[p]}), and e_{L,φ}^{(p)}(M^{[p]}) = e_{L,φ}(M^{[p]})

by Corollary 7.12. Hence (M_{ψ}^{(p)}, p_{2}) ↠ (L, φ) by Theorem 5.3. But the projection M → M_{ψ}^{(p)} is a group isomorphism, since (M, ψ) is a B_{K}-group. It is in fact an isomorphism from (M, ψ) to (M_{ψ}^{(p)}, p_{2}) in grp_{⇓K}. It follows that (M, ψ) ↠ (L, φ).

The following is analogous to Lemma 3.10, and the proof is the same:

7.14. Lemma: Let A be a set of ideals of 𝔽B_{K}^{(p)}, and (M, ψ) be a ppersistent group over K. The following are equivalent:

1. e_{M,ψ}^{(p)} ⊆ ∑_{I∈A} I.

2. There exists I ∈ A such that e_{M,ψ}^{(p)} ⊆ I.

Proof: Clearly 2 implies 1. Now 1 is equivalent to saying that

e_{M_{ψ}^{(p)}}^{M^{[p]}×K} ∈ ∑_{I∈A} I(M^{[p]}) .

If this holds, there exists I ∈ A and u ∈ I(M^{[p]}) such that e_{M_{ψ}^{(p)}}^{M^{[p]}×K} · u ≠ 0.

Now e_{M_{ψ}^{(p)}}^{M^{[p]}×K} · u ∈ I(M^{[p]}), and moreover there is a scalar λ ∈ 𝔽 such that

e_{M_{ψ}^{(p)}}^{M^{[p]}×K} · u = λe_{M_{ψ}^{(p)}}^{M^{[p]}×K} ≠ 0. Hence λ ≠ 0, and e_{M_{ψ}^{(p)}}^{M^{[p]}×K} ∈ I(M^{[p]}). In other

words e_{M,ψ}^{(p)} ⊆ I, so 1 implies 2.

7.15. Notation: Let B_{K}^{(p)}-gr denote the subset of B_{K}-gr consisting of p-persistent B_{K}-groups.

As before, a subset P of B_{K}^{(p)}-gr is called closed if

∀(L, φ) ∈ P, ∀(M, ψ) ∈ B_{K}^{(p)}-gr, (M, ψ) ↠ (L, φ) ⟹ (M, ψ) ∈ P .

7.16. Theorem: Let I_{𝔽B_{K}^{(p)}} be the lattice of ideals of 𝔽B_{K}^{(p)}, ordered by inclusion of ideals, and Cl_{B_{K}^{(p)}-gr} be the lattice of closed subsets of B_{K}^{(p)}-gr, ordered by inclusion of subsets. Then the map

I ∈ I_{𝔽B_{K}^{(p)}} ↦ P_{I} = {(L, φ) ∈ B_{K}^{(p)}-gr | e_{L,φ}^{(p)} ⊆ I}

is an isomorphism of lattices from I_{𝔽B_{K}^{(p)}} to Cl_{B_{K}^{(p)}-gr}. The inverse isomorphism is the map

P ∈ Cl_{B_{K}^{(p)}-gr} ↦ I_{P} = ∑_{(L,φ)∈P} e_{L,φ}^{(p)} .

In particular I_{𝔽B_{K}^{(p)}} is completely distributive.

Proof: First the map I ∈ I_{𝔽B_{K}^{(p)}} ↦ P_{I} ∈ Cl_{B_{K}^{(p)}-gr} is well defined: indeed P_{I} ∈ Cl_{B_{K}^{(p)}-gr} by Theorem 7.10. This map is obviously order preserving. Similarly, the map P ∈ Cl_{B_{K}^{(p)}-gr} ↦ I_{P} = ∑_{(L,φ)∈P} e_{L,φ}^{(p)} is also well defined and

order preserving.

Hence all we need to show is that if I is an ideal of 𝔽B_{K}^{(p)}, then

(7.17)  I = ∑_{(L,φ)∈P_{I}} e_{L,φ}^{(p)} ,

and that if P is a closed subset of B_{K}^{(p)}-gr, and (M, ψ) ∈ B_{K}^{(p)}-gr, then

(7.18)  e_{M,ψ}^{(p)} ⊆ ∑_{(L,φ)∈P} e_{L,φ}^{(p)} ⇔ (M, ψ) ∈ P .

For 7.17, let J = ∑_{(L,φ)∈P_{I}} e_{L,φ}^{(p)}. Then J ⊆ I by definition of P_{I}. Conversely,

let G be a finite p-group, and u = ∑_{X∈E} λ_{X}e_{X}^{G×K} be an element of I(G), where λ_{X} ∈ 𝔽, and E is a set of representatives of conjugacy classes of subgroups of G × K. Then e_{X}^{G×K} · u = λ_{X}e_{X}^{G×K} ∈ I(G), for any X ∈ E. So if λ_{X} ≠ 0, then e_{X}^{G×K} ∈ I(G). Equivalently, by Theorem 7.7, e_{X_{p_{2}}^{(p)}}^{X^{[p]}×K} ∈ I(X^{[p]}), that

is e_{X,p_{2}}^{(p)} ⊆ I. Let (L, φ) be the element of B_{K}^{(p)}-gr isomorphic to β_{K}(X, p_{2}). Then e_{X,p_{2}}^{(p)} = e_{L,φ}^{(p)} by Theorem 7.10, and (L, φ) ∈ P_{I}.

Moreover e_{X_{p_{2}}^{(p)}}^{X^{[p]}×K} ∈ e_{L,φ}^{(p)}(X^{[p]}), or equivalently e_{X}^{G×K} ∈ e_{L,φ}^{(p)}(G) ⊆ J(G). As this holds for any X ∈ E such that λ_{X} ≠ 0, we have also u ∈ J(G), so J(G) = I(G), as u was arbitrary in I(G), and J = I, as G was an arbitrary finite p-group. This completes the proof of 7.17.

As for 7.18, clearly if (M, ψ) ∈ P, then e_{M,ψ}^{(p)} ⊆ ∑_{(L,φ)∈P} e_{L,φ}^{(p)}. Conversely if

e_{M,ψ}^{(p)} ⊆ ∑_{(L,φ)∈P} e_{L,φ}^{(p)}, then by Lemma 7.14, there exists (L, φ) ∈ P such that

e_{M,ψ}^{(p)} ⊆ e_{L,φ}^{(p)}. Hence (M, ψ) ↠ (L, φ), by Corollary 7.13. Since (L, φ) ∈ P and P is closed, we get that (M, ψ) ∈ P, as was to be shown.

7.19. Theorem: Let (L, φ) be a p-persistent B_{K}-group. Let [s_{K}] be a set of representatives of conjugacy classes of subgroups of K. Let H be the unique element of [s_{K}] conjugate to φ(L), and j_{H} : H ↪ K be the inclusion map. Then one and one only of the following holds:

1. Ker φ = 1, and (L, φ) ≅ (H, j_{H}) in grp_{⇓K}.

2. Kerφ ≅ C_{p}, the group H^{[p]} is cyclic and non trivial, and (L, φ) ≅ (C_{p} × H, j_{H} ∘ π_{H}) in grp_{⇓K}, where π_{H} : C_{p} × H → K is the projection

onto H.

3. Kerφ ≅ C_{p} × C_{p}, the group H^{[p]} is trivial - in other words H is a pperfect subgroup of K - and (L, φ) ≅ (C_{p} × C_{p} × H, j_{H} ∘ π_{H}) in grp_{⇓K}, where π_{H} : C_{p} × C_{p} × H → K is the projection onto H.

Proof: Since O^{p}(L) ∩ Ker φ = 1 by Corollary 7.5, the group Ker φ embeds into L^{[p]}, so it is a p-group. Let F denote the Frattini subgroup of Ker φ. Then F is a normal subgroup of L. Moreover if X is a subgroup of L such that XF = L, then F ≤ Ker φ ≤ XF, so Ker φ = (Ker φ ∩ X)F, hence Ker φ ∩ X = Ker φ, and then XF = X = L since F ≤ Ker φ ≤ X. It follows that m_{L,F} = 1, thus F = 1 as (L, φ) is a B_{K}-group. This shows that Ker φ is elementary abelian.

Let now N = ∩_{P ∈M} P, where M is the set of normal subgroups of L which

are contained in Ker φ, and maximal for these conditions (in other words the factor group Ker φ/P is a simple 𝔽_{p}L-module). If X is a subgroup of L such that XN = L, then N ≤ Ker φ ≤ XN, so Ker φ = (Ker φ ∩ X)N. But Ker φ ∩ X is normalized by X and Ker φ, so it is normal in L. If Ker φ ∩ X < Ker φ, then there is P ∈ M such that Ker φ ∩ X ≤ P. Then N ≤ P also, and Ker φ = (Ker φ ∩ X)N ≤ P, contradicting P < Ker φ. It follows that Ker φ ≤ X, and XN = L implies X = L. Thus m_{L,N} = 1 and N = 1.

But then the product of the projection maps Ker φ → ∏_{P ∈M} Ker φ/P is injective, and the latter is a semisimple 𝔽_{p}L-module. Hence Ker φ is also a semisimple 𝔽_{p}L-module. Now since O^{p}(L) and Ker φ are normal subgroups of L with trivial intersection, they centralize each other. In other words Ker φ is a module for the factor group L^{[p]} = L/O^{p}(L). Then Ker φ is a semisimple 𝔽_{p}L^{[p]}-module. As L^{[p]} is a p-group, the action of L^{[p]} on Ker φ has to be trivial. Hence Ker φ is central in L.

Let Z be any subgroup of order p of Ker φ. Then 0 = m_{L,Z} = 1 − \frac{k_{L}(Z)}{p}, by Proposition 5.6.4 of [2], where k_{L}(Z) denotes the number of complements of Z in L. It follows that k_{L}(Z) = p, so in particular there is a subgroup H of L such that L = Z × H. Then the complements of Z in L are the groups of the form {(f(h), h) | h ∈ H}, where f : H → Z is any group homomorphism. It follows that there are exactly p homomorphisms from H to Z ≅ C_{p}. Equivalently, there are exactly p homomorphisms from the p-group H^{[p]} to C_{p}, so H^{[p]} is cyclic and non trivial. Since Ker φ embeds in L^{[p]} ≅ Z × H^{[p]}, the rank of Ker φ is at most 2.

We now observe that if (M, ψ) ↠ (L, φ) is a surjective morphism of groups over K - in particular if it is an isomorphism -, then ψ(M) and φ(L) are conjugate in K. Then there are three disjoint cases:

1. Ker φ = 1. In this case, denoting by π_{H} the inclusion map H ↪ K and by φ^{0} : L → H the isomorphism induced by φ, we have i ∘ φ = π_{H} ∘ φ^{0} for some inner automorphism i of K which conjugates φ(L) to H. So φ^{0} is an isomorphism from (L, φ) to (H, π_{H}) in grp_{⇓K}, and we are in Case 1 of Theorem 7.19.

2. Ker φ = Z ≅ C_{p}. Then we have seen that L = Z × H_{1}, where H_{1} is a subgroup of L such that H_{1}^{[p]} is cyclic and non trivial. In this case φ induces an isomorphism φ^{0} : H_{1} → H = φ(L), and Id_{Z} × φ^{0} is an isomorphism from (L, φ) to (Z × H, j_{H} ∘ π_{H}), where π_{H} : Z × H → K is the projection onto H. Hence we are in Case 2 of Theorem 7.19.

3. Ker φ ≅ C_{p} × C_{p}. Then let Z be a subgroup of order p of Ker φ. Then we have seen that L = Z × H_{1}, where H_{1} is a subgroup of L such that H_{1}^{[p]} is cyclic and non trivial. In this case Z_{1} = Ker φ ∩ H_{1} has order p, and m_{L,Z_{1}} = 0 since (L, φ) is a B_{K}-group. It follows that Z_{1} must have also exactly p complements in L. In particular, there is a subgroup J of L such that L = Z_{1} × J. But then Z_{1} ≤ H_{1} ≤ Z_{1}J implies that H_{1} = Z_{1} × H_{2}, where H_{2} = H_{1} ∩ J. Hence L = Z × Z_{1} × H_{2}, and moreover H_{2}^{[p]} = 1 since H_{1}^{[p]} ≅ Z_{1} × H_{2}^{[p]} is cyclic. Then φ induces an isomorphism φ^{0} : H_{2} → H = φ(L), and Id_{Z×Z_{1}} × φ^{0} is an isomorphism from (L, φ) to (Z × Z_{1} × H, j_{H} ∘ π_{H}), where π_{H} : Z × Z_{1} × H → K is the projection onto H. Hence we are in Case 3 of Theorem 7.19.

This completes the proof of Theorem 7.19.

7.20. Corollary: Let \underline{1} = \{0, 1\} and \underline{2} = \{0, 1, 2\} be totally ordered lattices of cardinality 2 and 3, respectively. Let c_{K} (resp. nc_{K}) be the number of conjugacy classes of subgroups H of K such that H^{[p]} is cyclic (resp. non cyclic). Then the lattice I_{𝔽B_{K}^{(p)}} of ideals of 𝔽B_{K}^{(p)} is isomorphic to the direct product of c_{K} copies of \underline{2} and nc_{K} copies of \underline{1}. In particular it is a finite distributive lattice.

Proof: By Theorem 7.16, the lattice I_{𝔽B_{K}^{(p)}} is distributive, isomorphic to the

lattice Cl_{B_{K}^{(p)}-gr} of closed subsets of B_{K}^{(p)}-gr. Moreover, the join-irreducible elements of I_{𝔽B_{K}^{(p)}} are the ideals e_{L,φ}, for (L, φ) ∈ I_{𝔽B_{K}^{(p)}}. By Theorem 7.19,

the set B_{K}^{(p)}-gr is finite, and contains three types of elements:

1. the elements (H, j_{H}) of the first type, for H ∈ [s_{K}].

2. the elements (C_{p} × H, j_{H} ∘ π_{H}) of the second type, for H ∈ [s_{K}] such that H^{[p]} is cyclic and non-trivial.

3. the elements (C_{p} × C_{p} × H, j_{H} ∘ π_{H}) of the third type, for H ∈ [s_{K}] such that H^{[p]} is trivial.

The only possible surjective morphisms between elements of B_{K}^{(p)}-gr are of the following form:

• (C_{p} × H, j_{H} ∘ π_{H}) ↠ (H, j_{H}), where H ∈ [s_{K}] is such that H^{[p]} is cyclic and non trivial.

• (C_{p}×C_{p}×H, j_{H} ∘π_{H}) ↠ (H, j_{H}), where H ∈ [s_{K}] is such that H^{[p]} = 1.

It follows that the poset B_{K}^{(p)}-gr has as many connected components as conjugacy classes of subgroups of K. The connected components corresponding to subgroups H for which H^{[p]} is cyclic - trivial or not - are isomorphic to a totally ordered poset of size 2, and the other ones are posets with one element. Hence B_{K}^{(p)}-gr is a disjoint union of c_{K}-components which are totally ordered of size 2, and nc_{K} isolated points. The lattice of closed subsets of a totally ordered poset of size n is a totally ordered lattice of size n + 1, and the lattice of closed subsets of a disjoint union of posets is the direct product of the lattices of closed subsets of the pieces. This completes the proof.

7.21. Remark: As in Remark 5.6, it follows from Section 5.2.2 of [3] that the category 𝔽B_{K}^{(p)}-Mod splits as a product

𝔽B_{K}^{(p)}-Mod ≅ ∏_{H} e_{H}^{K}𝔽B_{K}^{(p)}-Mod ,

of categories of modules over smaller Green biset functors e_{H}^{K}𝔽B_{K}^{(p)}, where H ∈ [s_{K}]. The above connected components correspond to this decomposition. In particular, when H is a subgroup of K such that H^{[p]} is non cyclic, then the (commutative) Green functor e_{H}^{K}𝔽B_{K}^{(p)} has no non zero proper ideals. It might therefore be called a Green field.

References

[1] S. Bouc. Foncteurs d’ensembles munis d’une double action. J. of Algebra, 183(0238):664–736, 1996.

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

[3] S. Bouc and N. Romero. The center of a Green biset functor. Pacific J. Math., 303 (2019) 459-490.

[4] D. Gluck. Idempotent formula for the Burnside ring with applications to the p-subgroup simplicial complex. Illinois J. Math., 25:63–67, 1981.

[5] S. Mac Lane. Categories for the working mathematician, volume 5 of Graduate texts in Mathematics. Springer, 1971.

[6] N. Romero. On fibred biset functors with fibres of order prime and four. Journal of Algebra, 387:185–194, 2013.

[7] I. Tounkara. The ideals of the slice Burnside p-biset functor. J. Algebra, 495:81–113, 2018.

[8] T. Yoshida. Idempotents of Burnside rings and Dress induction theorem. J. of Algebra, 80:90–105, 1983.

Serge Bouc - CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens Cedex 01 - France.

email :  serge.bouc@u-picardie.fr
web   :  http://www.lamfa.u-picardie.fr/bouc/