The slice Burnside ring and the section Burnside ring of a finite group

Serge Bouc

Abstract : This paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets, and of Galois morphisms of G-sets, respectively. The well known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural structure of Green biset functor. The functorial structure of unit groups of these rings is also discussed.

AMS Subject Classification : 19A22, 18F30.

Keywords : Burnside ring, Green biset functor, slice, section, unit group.

1. Introduction

This paper introduces two variations on Burnside rings of finite groups, called the slice Burnside ring and the section Burnside ring. Both of them are built as Grothendieck rings of some category of morphisms of finite G-sets, instead of the category of finite G-sets used to build the usual Burnside ring. The difference between these two new Burnside rings is that the slice Burnside ring is built from arbitrary morphisms of finite G-sets, whereas the section Burnside ring uses only Galois morphisms of finite G-sets.

It turns out that most of the well known properties of the Burnside ring extend to the slice Burnside ring and to the section Burnside ring : both are commutative rings, which are free of finite rank as ℤ-module. There is an analogue of Burnside’s theorem : both of these rings embed in a product of copies of the integers, via a ghost map, and this map has a finite cokernel. After tensoring with ℚ, both rings become split semisimple ℚ-algebras, and explicit formulas for their primitive idempotents can be stated. The prime spectrum of both rings can also be described, and Dress’s characterization of solvable groups in terms of the connectedness of the spectrum of the Burnside ring can be generalized as well. Finally, both constructions have a natural biset functor structure, for which they become Green biset functors.

A major exception in this list of generalizable properties concerns unit groups : it can be shown that, unlike the case of the usual Burnside ring, the correspondence sending a finite group to the unit group of either the slice or the section Burnside ring cannot be endowed with a structure of biset functor. This is due to the lack of a suitable tensor induction for these rings.

The paper is divided in two parts, and an appendix : the first part is devoted to the slice Burnside functor. It consists of Sections 2 to 7. Section 2 recalls the basic definitions and properties on the category of morphisms of G-sets. Section 3 introduces the slice Burnside functor and its Green biset functor structure. Section 4 is devoted to the definition and main result on the ghost map. In Section 5, the explicit formulas for the primitive idempotents of the slice Burnside algebra over ℚ are stated. Section 6 gives a characterization of the image of the ghost map. Next Section 7 considers the prime spectrum of the slice Burnside ring. Finally, in Section 8, it is shown that tom Dieck’s theorem, building on Dress’s characterization of solvable groups, can be extended to the slice Burnside ring : namely, Feit-Thompson’s theorem is equivalent to the fact that the only units in the slice Burnside ring of a group of odd order are ±1. But unlike the case of the usual Burnside ring, the unit group of the slice Burnside ring cannot be endowed with a structure of biset functor.

The second part of the paper is devoted to the section Burnside ring, and it is organized similarly : Section 9 introduces Galois morphisms, and states their general properties. In particular, a left adjoint functor to the forgetful functor from Galois morphisms to arbitrary morphisms of G-sets is described, which attaches to any morphism of G-sets a canonical Galois morphism. Section 10 considers the section Burnside functor, with its Green biset functor structure. Section 11 deals with the ghost map for the section Burnside ring. Sections 12 states the formulas for the primitive idempotents of the section Burnside ℚ-algebra, and Section 13 gives a characterization of the ghost map. Section 14 considers the prime spectrum of the section Burnside ring. In Section 15, the results about unit groups of Section 8 are extended to the unit group of the section Burnside ring. In particular, it is shown that this unit group cannot be endowed with a structure of biset functor.

The appendix deals with the functorial structure of the unit group of the slice Burnside ring and the section Burnside ring : it is possible to define biset functor operations for these unit groups, but only for left inert bisets. This gives two interesting examples of somehow natural biset functors without induction. The last result of the appendix is the explicit computation of this unit group in the case of an abelian group.

I - The slice Burnside ring

2. Morphisms of G-sets

2.1. Definition : Let G be a group. If f : X → Y and f′ : X′ → Y′ are morphisms of G-sets, a morphism from f to f′ is a pair of morphisms of G-sets α : X → X′ and β : Y → Y′ such that the diagram

\xymatrix{X \ar[r]^{f} \ar[d]_{α} & Y \ar[d]^{β} \\ X′ \ar[r]^{f′} & Y′}

is commutative.

Morphisms of morphisms of G-sets can be composed in the obvious way. This composition endows the class of morphisms of G-sets with a structure of category, denoted by G-Mor.

2.2. Proposition : The disjoint union of G-sets induces a coproduct

(X →^{f} Y, X′ →^{f′} Y′) ↦ (X ⊔ X′ ⟶^{f⊔f′} Y ⊔ Y′)

in the category G-Mor.

Similarly, the direct product of G-sets, with diagonal G-action, induces a product

(X →^{f} Y, X′ →^{f′} Y′) ↦ (X × X′ ⟶^{f×f′} Y × Y′)

in the category G-Mor.

Proof : For any morphism of G-sets, the bijections

Hom_{G-Set}(X ⊔ X′, A) ≅ Hom_{G-Set}(X, A) × Hom_{G-Set}(X′, A)

induce obvious bijections between

Hom_{G-Mor}((X ⊔ X′) ⟶^{f⊔f′} (Y ⊔ Y′), A →^{α} B)

and

Hom_{G-Mor}(X →^{f} Y, A →^{α} B) × Hom_{G-Mor}(X′ →^{f′} Y′, A →^{α} B) .

These bijections are obviously functorial in G-Mor. Similarly, the bijections

Hom_{G-Set}(A, X × X′) ≅ Hom_{G-Set}(A, X) × Hom_{G-Set}(A, X′)

induce obvious bijections between

Hom_{G-Mor}(A →^{α} B, (X × X′) ⟶^{f×f′} (Y × Y′))

and

Hom_{G-Mor}(A →^{α} B, X →^{f} Y) × Hom_{G-Mor}(A →^{α} B, X′ →^{f′} Y′) .

These bijections are obviously functorial in G-Mor.

3. The slice Burnside functor

3.1. Definition and Notation : Let G be a group. A slice of G is a pair (T, S) of subgroups of G with T ≥ S. A section of G is a slice (T, S) with

S ⊴ T.

Let Π(G) denote the set of slices of G, and Σ(G) denote the set of sections

of G.

When (T, S) ∈ Π(G), denote by G/S → G/T the projection morphism.

3.2. Definition : Let G be a finite group. The slice Burnside group Ξ(G) of G is the quotient of the free abelian group on the set of isomorphism classes [X ⟶^{f} Y] of morphisms of finite G-sets, by the subgroup generated by elements of the form

[(X_{1} ⊔ X_{2}) ⟶^{f_{1}⊔f_{2}} Y] − [X_{1} →^{f_{1}} f(X_{1})] − [X_{2} →^{f_{2}} f(X_{2})] ,

whenever X →^{f} Y is a morphism of finite G-sets with a decomposition X = X_{1} ⊔ X_{2} as a disjoint union of G-sets, where f_{1} = f_{|X_{1}} and f_{2} = f_{|X_{2}} .

When f : X → Y is a morphism of finite G-sets, let π(f) denote the image in Ξ(G) of the isomorphism class of f. If S ≤ T are subgroups of G,

set ⟨T, S⟩_{G} = π(G/S → G/T).

3.3. Lemma :

1. π(∅ → ∅) = 0.

2. Let X →^{f} Y be a morphism of finite G-sets. Then

π(X →^{f} Y) = π(X →^{f} f(X)) .

3. Let X →^{f} Y and X′ →^{f′} Y′ be morphisms of finite G-sets. Then

π((X ⊔ X′) ⟶^{f⊔f′} (Y ⊔ Y′)) = π(X →^{f} Y) + π(X′ →^{f′} Y′) .

Proof : For Assertion 1, set e = π(∅ → ∅). Since the morphism ∅ ⊔ ∅ → ∅ is isomorphic to ∅ → ∅, it follows that e + e = e, hence e = 0.

For Assertion 2, writing X = X ⊔ ∅ gives

π(X →^{f} Y ) = π(X →^{f} f(X)) + π(∅ → ∅) = π(X →^{f} f(X)) .

For Assertion 3,

π((X ⊔ X′) ⟶^{f⊔f′} (Y ⊔ Y′)) = π(X →^{f} (Y ⊔ Y′)) + π(X′ →^{f′} (Y ⊔ Y′))
= π(X →^{f} f(X)) + π(X′ →^{f′} f′(X′))
= π(X →^{f} Y ) + π(X′ →^{f′} Y′) ,

where the first equality follows from the defining relations of Ξ(G), and the other ones from Assertion 2.

3.4. Lemma : Let f : X → Y be a morphism of finite G-sets. Then in the group Ξ(G)

π(X →^{f} Y) = ∑_{x∈[G\X]}⟨G_{f(x)}, G_{x}⟩_{G}

Proof : Indeed X ≅ ⊔_{x∈[G\X]} G/G_{x}, and the image f(G·x) of the G-orbit of x

is equal to the G-orbit of f(x). Moreover the morphisms f_{|G·x} : G·x → G·f(x) and G/G_{x} → G/G_{f(x)} are isomorphic. The claimed formula follows.

3.5. Corollary : The group Ξ(G) is generated by the elements ⟨T, S⟩_{G}, where (T, S) runs through a set [Π(G)] of representatives of conjugacy classes of slices of G.

Proof : Indeed, the morphisms G/^{g}S → G/^{g}T and G/S → G/T are isomorphic, for any g ∈ G, and any slice (T, S) of G.

3.6. Remark : It will be shown in Theorem 4.6 that this generating set is actually a basis of Ξ(G).

3.7. Proposition : The product of morphisms induces a commutative unital ring structure on Ξ(G). The identity element for multiplication is the image of the class [• → •], where • denotes a G-set of cardinality 1.

Proof : If we can show that the product of morphisms induces a well defined bilinear product Ξ(G) × Ξ(G) → Ξ(G), it will be clear that this product is associative, commutative, and admits [• → •] as an identity element. Hence the only point to check is that the product preserves the defining relations of Ξ(G). This is clear, since if g : Z → T is a morphism of finite G-sets, and if X_{1} ⊔ X_{2} ⟶^{f_{1}⊔f_{2}} Y is a morphism, setting X = X_{1} ⊔ X_{2}, the domain of the morphism

\xymatrix{h : Z × X \ar[r]^{g×(f_{1}⊔f_{2})} & T × Y}

has a disjoint union decomposition Z × X = (Z × X_{1}) ⊔ (Z × X_{2}), and moreover the restriction of g × f to Z × X_{1} is g × f_{1}. Thus

π(h) = π((Z × X_{1}) ⟶^{g×f_{1}} (g(Z) × f_{1}(X_{1}))) + π((Z × X_{2}) ⟶^{g×f_{2}} (g(Z) × f_{2}(X_{2})))
= π((Z × X_{1}) ⟶^{g×f_{1}} (T × f_{1}(X_{1}))) + π((Z × X_{2}) ⟶^{g×f_{2}} (T × f_{2}(X_{2})))

where the last equality follows from Lemma 3.3.

3.8. Proposition : Let (T, S) and (Y, X) be slices of G. Then in Ξ(G)

⟨T, S⟩_{G}⟨Y, X⟩_{G} = ∑_{g∈[S\G/X]}⟨T ∩^{g}Y , S ∩^{g}X⟩_{G} .

Proof : Indeed

(G/S) × (G/X) ≅ ⊔_{g∈[S\G/X]} G/(S ∩^{g}X) ,

via the map (from right to left) sending u(S ∩^{g}X) to (uS, ugX), for u ∈ G. The image of (S, gX) by the map (G/S) × (G/X) → (G/T) × (G/Y) is the pair (T, gY), whose stabilizer in G is T ∩^{g}Y. The result now follows from Lemma 3.4.

3.9. Theorem :

1. Let G and H be finite groups, and let U be a finite (H, G)-biset. The

{functor \atop (X →^{f} Y) ↦ (U ×_{G} X ⟶^{U×_{G}f} U ×_{G} Y)}

from G-Mor to H-Mor induces a group homomorphism

Ξ(U) : Ξ(G) → Ξ(H) .

2. The correspondence G ↦ Ξ(G) is a Green biset functor.

Proof : For Assertion 1, the only thing to check is that the defining relations of Ξ(G) are mapped to relations in Ξ(H). But if

X_{1} ⊔ X_{2} ⟶^{f_{1}⊔f_{2}} Y

is a morphism of finite G-sets, then

U ×_{G} (X_{1} ⊔ X_{2}) ≅ (U ×_{G} X_{1}) ⊔ (U ×_{G} X_{2}) .

Moreover the image of the map U ×_{G} f_{1} is equal to U ×_{G} f_{1}(X_{1}). It follows that the relation

[X_{1} ⊔ X_{2} ⟶^{f_{1}⊔f_{2}} Y] − [X_{1} →^{f_{1}} f_{1}(X_{1})] − [X_{2} →^{f_{2}} f_{2}(X_{2})]

in Ξ(G) is mapped to the relation

[UX_{1} ⊔ UX_{2} ⟶^{Uf_{1}⊔Uf_{2}} UY] − [UX_{1} →^{Uf_{1}} (Uf_{1})(UX_{1})] − [UX_{2} →^{Uf_{2}} (Uf_{2})(UX_{2})] ,

where U×_{G} is abbreviated to U.

It is now clear that the correspondence sending a finite group G to Ξ(G) and a finite (H, G)-biset U to Ξ(U) endows Ξ with a structure of biset functor (see [4]).

Moreover if G, G′ are finite groups, if f : X → Y is a morphism of finite G-sets and f′ : X′ → Y′ is a morphism of finite G′-sets, then

f × f′ : X × X′ → Y × Y′

is a morphism of G × G′-sets. This induces a product

Ξ(G) × Ξ(G′) → Ξ(G × G′) ,

which is associative in the obvious sense. Moreover, the morphism • → • of 1-sets is obviously an identity element for this product, up to identification G × 1 = G.

Finally, if G, G′, H, H′ are finite groups, if U is a finite (H, G)-biset, if U′ is a finite (H′, G′)-biset, it is clear that the morphisms

(U × U′) ×_{G×G′} (f × f′)

and

(U ×_{G} f) × (U′ ×_{G′} f′)

are isomorphic morphisms of (H × H′)-sets. Thus Ξ is a Green biset functor (see [4] Section 8.5).

3.10. Proposition : Let G and H be finite groups, and U be a finite (H, G)-biset. If (T, S) ∈ Π(G), then

U ×_{G} ⟨T, S⟩_{G} = ∑_{u∈[H\U/S]}⟨^{u}T ,^{u}S⟩_{H} .

(where ^{u}X = {h ∈ H | ∃x ∈ X, hu = ux}, for X ≤ G).

Proof : Indeed U ×_{G} (G/S) ≅ U/S, and the stabilizer in H of uS is equal to ^{u}S. The result follows from Lemma 3.4.

The following proposition clarifies the links existing between the usual Burnside functor and the slice Burnside functor :

3.11. Proposition :

1. Let G be a finite group. The correspondence sending the morphism X →^{f} Y of finite G-sets to the G-set X induces a unital ring homomorphism s_{G} from Ξ(G) to the Burnside ring B(G).

2. The correspondence sending the finite G-set X to the identity morphism of X induces a unital ring homomorphism i_{G} : B(G) → Ξ(G), such that

s_{G} ∘ i_{G} = Id_{B(G)}.

3. As G varies, the morphisms s_{G} and i_{G} define morphisms of Green biset functors s : Ξ → B and i : B → Ξ, such that s ∘ i = Id_{B}. In particular i is injective, and s is surjective.

Proof : This is straightforward, from the definitions.

4. Slices and ghost map

4.1. Notation : Let S ≤ T be subgroups of G. If X →^{f} Y is a morphism of finite G-sets, set

ϕ_{T,S}(X →^{f} Y) = |Hom_{G-Mor}(G/S → G/T, X →^{f} Y)| .

4.2. Notation : Define a relation ⪯ on the set Π(G) by

(T, S) ⪯ (Y, X) ⇔ (T ≤ Y and S ≤ X) .

The relation ⪯ is an order relation on Π(G).

4.3. Lemma : With this notation

ϕ_{T,S}(X →^{f} Y) = |f^{−1}(Y^{T})^{S}| .

In particular, for any A ≤ B ≤ G,

ϕ_{T,S}(G/A → G/B) = |{g ∈ G/A | (T^{g}, S^{g}) ⪯ (B, A)}| .

Proof : The morphisms of G-sets from G/S to X are in one to one correspondence with the set X^{S} of fixed points of S on X : the morphism associated to x ∈ X^{S} is defined by gS ↦ gx. Similarly, the homomorphisms of G-sets from G/T to Y are in one to one correspondence with Y^{T}. Hence the set

Hom_{G-Mor}(G/S → G/T, X →^{f} Y)

is in one to one correspondence with the set of pairs (x, y) ∈ X^{S} × Y^{T} such that f(x) = y, i.e. with the elements x of f^{−1}(Y^{T})^{S}.

4.4. Corollary : Let (T, S) ∈ Π(G) and p be a prime number. If P is a p-subgroup of N_{G}(T, S), then

ϕ_{T,S} ≡ ϕ_{P T,P S} (mod. p) .

Proof : Indeed for any morphism of finite G-sets X →^{f} Y, the set f^{−1}(Y^{T})^{S} is invariant by N_{G}(T, S), thus, as P is a p-group,

|f^{−1}(Y^{T})^{S}| ≡ |f^{−1}(Y^{T})^{P S}| (mod. p) ,

and moreover f^{−1}(Y^{T})^{P S} = f^{−1}(Y^{P T})^{P S}.

4.5. Proposition : Let S ≤ T be subgroups of G. Then the map ϕ_{T,S} induces a ring homomorphism Ξ(G) → ℤ, still denoted by ϕ_{T,S}.

Proof : Since the product of morphisms

(X →^{f} Y, X′ →^{f′} Y′) ↦ ((X × X′) ⟶^{f×f′} (Y × Y′))

is a product in the category G-Mor, it follows that

ϕ_{T,S}((X × X′) ⟶^{f×f′} (Y × Y′)) = ϕ_{T,S}(X →^{f} Y)ϕ_{T,S}(X′ →^{f′} Y′) .

Also ϕ_{T,S}(• → •) = 1. The only thing to check is that ϕ_{T,S} induces a well defined map Ξ(G) → ℤ, i.e. that the defining relations of Ξ(G) are mapped to 0 by ϕ_{T,S}. First, by Lemma 4.3, for any morphism of G-sets f : X → Y

ϕ_{T,S}(X →^{f} Y) = ϕ_{T,S}(X →^{f} f(X)) .

Now

ϕ_{T,S}((X_{1} ⊔ X_{2}) ⟶^{f_{1}⊔f_{2}} Y ) = |(f_{1} ⊔ f_{2})^{−1}(Y^{T})^{S}|
= |X_{1} ∩ (f_{1} ⊔ f_{2})^{−1}(Y^{T})^{S}| + |X_{2} ∩ (f_{1} ⊔ f_{2})^{−1}(Y^{T})^{S}|
= |f_{1}^{−1}(Y^{T})^{S}| + |f_{2}^{−1}(Y^{T})^{S}|
= ϕ_{T,S}(X_{1} →^{f_{1}} Y) + ϕ_{T,S}(X_{2} →^{f_{2}} Y)
= ϕ_{T,S}(X_{1} →^{f_{1}} f(X_{1})) + ϕ_{T,S}(X_{2} →^{f_{2}} f(X_{2})) .

This completes the proof.

4.6. Theorem : The group Ξ(G) is a free abelian group, with basis the set of elements ⟨T, S⟩_{G}, where (T, S) runs through a set [Π(G)] of representatives of conjugacy classes of slices of G. Moreover, the map (called the ghost map for Ξ(G))

Φ = ∏_{(T,S)∈[Π(G)]} ϕ_{T,S} : Ξ(G) → ∏_{(T,S)∈[Π(G)]} ℤ

is an injective ring homomorphism, with finite cokernel as morphism of abelian groups.

Proof : By Lemma 3.5, the elements ⟨T, S⟩_{G}, for (T, S) ∈ [Π(G)], generate Ξ(G). Suppose that there is a non zero linear combination in the kernel of Φ

Λ = ∑_{(T,S)∈[Π(G)]} λ_{T,S} ⟨T, S⟩_{G}

with integer coefficients λ_{T,S} ∈ ℤ, for (T, S) ∈ [Π(G)]. Extend λ to a function Π(G) → ℤ, constant on conjugacy classes. Let (Y, X) be an element of Π(G), maximal for the relation ⪯, such that λ_{Y,X} ≠ 0. Then since by Lemma 4.3

ϕ_{Y,X}(G/S → G/T) = |{g ∈ G/S | (Y^{g}, X^{g}) ⪯ (T, S)}| ,

it follows that

ϕ_{Y,X}(Λ) = ∑_{(T,S)∈[Π(G)]} λ_{T,S}ϕ_{Y,X}(G/S → G/T)
= λ_{Y,X}ϕ_{Y,X}(G/X → G/Y)
= λ_{Y,X}|N_{G}(X, Y)/X| = 0 .

Hence λ_{Y,X} = 0, and this contradiction shows that Φ is injective. In particular, the elements ⟨T, S⟩_{G}, for (T, S) ∈ [Π(G)], form a ℤ-basis of Ξ(G). Thus Φ is an injective morphism between free abelian groups with the same finite rank, hence it has finite cokernel.

4.7. Corollary : Set ℚΞ(G) = ℚ ⊗_{ℤ} Ξ(G), and ℚΦ = ℚ ⊗_{ℤ} Φ. Then

ℚΦ : ℚΞ(G) → ∏_{(T,S)∈[Π(G)]} ℚ

is an isomorphism of ℚ-algebras.

5. Slices and idempotents

By Corollary 4.7, the commutative ℚ-algebra ℚΞ(G) is split semisimple. Its primitive idempotents are indexed by slices of G, up to conjugation : they are the inverse images under ℚΦ of the primitive idempotents of the algebra

∏_{(T,S)∈[Π(G)]} ℚ :

5.1. Notation : If (T, S) ∈ Π(G), denote by ξ_{T,S}^{G} the unique element of ℚ ⊗_{ℤ} Ξ(G) such that

∀(Y, X) ∈ Π(G), ℚϕ_{Y,X}(ξ_{T,S}^{G}) = \{ \begin{matrix}1 & if (Y, X) =_{G} (T, S) \\ 0 & otherwise \end{matrix}

The set of elements ξ_{T,S}^{G}, for (T, S) ∈ [Π(G)], is the set of primitive idempotents of ℚΞ(G).

5.2. Theorem : Let (T, S) ∈ Π(G). Then

ξ_{T,S}^{G} = \frac{1}{|N_{G}(T, S)|} ∑_{(V,U)⪯(T,S)} |U|μ_{Π}((V, U), (T, S))⟨V, U⟩_{G} ,

where μ_{Π} is the Möbius function of the poset (Π(G), ⪯).

Proof^{1} : Denote by \overline{Π}(G) the set of orbits of G for its conjugacy action on Π(G). Thus \overline{Π}(G) is in one to one correspondence with [Π(G)], and the map ℚΦ can also be viewed a ℚ-algebra isomorphism from ℚΞ(G) to ℚ^{\overline{Π}(G)}. The ℚ-vector space ℚΞ(G) has a basis consisting of the elements ⟨V, U⟩_{G}, for (V, U) ∈ [Π(G)]. Let ℚ^{Π(G)} denote the ℚ-vector space with basis Π(G), and let p : ℚ^{Π(G)} → ℚΞ(G) denote the ℚ-linear map sending (V, U) ∈ Π(G) to ⟨V, U⟩_{G}.

Let β_{T,S}^{G} denote the vector of the canonical basis of ℚ^{\overline{Π}(G)} indexed by the G-orbit of (T, S) ∈ Π(G), and let q : ℚ^{Π(G)} → ℚ^{\overline{Π}(G)} denote the ℚ-linear map sending (T, S) ∈ Π(G) to β_{T,S}^{G}.

With this notation, Lemma 4.3 shows that for any (V, U) ∈ Π(G)

Φ ∘ p((V, U)) = ∑_{(T,S)∈[Π(G)]} ϕ_{T,S}(⟨V, U⟩_{G})β_{T,S}^{G}

= ∑_{(T,S)∈[Π(G)]} \frac{1}{|U|}|\{g ∈ G | (T, S) ⪯ (^{g}V ,^{g}U)\}| β_{T,S}^{G}

1As suggested by the referee, a direct computation using the formula for ξ_{T,S}^{G} in Theorem 5.2 shows that if (Y, X) is a slice of G, then ℚϕ_{Y,X}(ξ_{T,S}^{G}) is equal to 0 if (Y, X) and (T, S) are not G-conjugate, and to 1 otherwise. This gives a short proof of Theorem 5.2.

The proof given here is longer, but it may have the advantage of explaining the origin of the formula for ξ_{T,S}^{G}, in particular why Möbius functions appear in this expression.

Φ ∘ p((V, U)) = ∑_{(T,S)∈Π(G)} \frac{|N_{G}(T, S)|}{|G||U|}|\{g ∈ G | (T^{g}, S^{g}) ⪯ (V, U)\}| β_{T,S}^{G}

= ∑_{(T,S)∈Π(G) ; {g∈G \atop (T^{g},S^{g})⪯(V,U)}} \frac{|N_{G}(T^{g}, S^{g})|}{|G||U|} β_{T^{g},S^{g}}^{G}

But |N_{G}(T^{g}, S^{g})| = |N_{G}(T, S)| and β_{T^{g},S^{g}}^{G} = β_{T,S}^{G}, for any g ∈ G. Thus

Φ∘p((V, U)) = ∑_{{(T,S)∈Π(G) \atop (T,S)⪯(V,U)}} \frac{|N_{G}(T, S)|}{|U|} β_{T,S}^{G} = ∑_{{(T,S)∈Π(G) \atop (T,S)⪯(V,U)}} \frac{|N_{G}(T, S)|}{|U|} q((T, S)) .

This shows that if Φ̃ is the ℚ-endomorphism of ℚ^{Π(G)} defined by

Φ̃((V, U)) = ∑_{{(T,S)∈Π(G) \atop (T,S)⪯(V,U)}} \frac{|N_{G}(T, S)|}{|U|}(T, S) ,

then Φ∘p = q ∘Φ̃. The matrix of the map Φ̃ is equal to the product E ·J ·D, where D is a diagonal matrix with diagonal coefficients (|N_{G}(T, S)|)_{(T,S)∈Π(G)}, where E is a diagonal matrix with diagonal coefficients (\frac{1}{|U|})_{(V,U)∈Π(G)}, and J is the incidence matrix of the order relation ⪯ on Π(G). It follows that Φ̃ is invertible, with inverse equal to D^{−1} · J^{−1} · E^{−1}. Now the entries of the matrix J^{−1} are precisely the values of the Möbius function μ_{Π} of the poset (Π(G), ⪯ ). It follows that

ξ_{T,S}^{G} = Φ^{−1}(β_{T,S}^{G})
= Φ^{−1} ∘ q((T, S)) = p ∘ Φ̃^{−1}((T, S))
= \frac{1}{|N_{G}(T, S)|} ∑_{{(V,U)∈Π(G) \atop (V,U)⪯(T,S)}} |U|μ_{Π}((V, U), (T, S)) ⟨V, U⟩_{G} ,

which completes the proof.

5.3. Proposition : Let (X , ≤) be a finite poset. Let Π(X) denote the set of pairs (y, x) of elements of X such that x ≤ y. Define a partial order ⪯ on

{Π(X) by \atop ∀(y, x), (t, z) ∈ Π(X), (y, x) ⪯ (t, z) ⇔ y ≤ t and x ≤ z .}

Then the Möbius function μ_{Π} of the poset (Π(X), ⪯) can be computed as follows, for any (y, x), (t, z) ∈ Π(X) :

(5.4)  μ_{Π}((y, x), (t, z)) = \{ \begin{matrix}μ_{X} (x, z)μ_{X} (y, t) & if x ≤ z ≤ y ≤ t \\ 0 & otherwise \end{matrix} ,

where μ_{X} is the Möbius function of the poset (X , ≤).

Proof : Let m((y, x), (t, z)) denote the expression defined by the right hand side of Equation 5.4. Then if (y, x) ⪯ (t, z), i.e. if y ≤ t and x ≤ z,

∑_{{(v,u)∈Π(X) \atop (y,x)⪯(v,u)⪯(t,z)}} m((y, x), (v, u)) = ∑_{y≤v≤t ; {x≤u≤z \atop u≤v, u≤y}} μ_{X}(x, u)μ_{X}(y, v)

= ∑_{y≤v≤t ; {x≤u≤z \atop u≤y}} μ_{X}(x, u)μ_{X}(y, v)

= ( ∑_{y≤v≤t} μ_{X} (y, v))( ∑_{{x≤u≤z \atop u≤y}} μ_{X} (x, u)) .

The first factor ∑_{y≤v≤t} μ_{X} (y, v) is equal to 0 if y ≠ t, and to 1 if y = t. In

this case, the second factor ∑_{{x≤u≤z \atop u≤y}} μ_{X} (x, u) is equal to ∑_{x≤u≤z} μ_{X} (x, u). This is

equal to zero if x ≠ z, and to 1 if x = z.

It follows that ∑_{{(v,u)∈Π(X) \atop (y,x)⪯(v,u)⪯(t,z)}} m((y, x), (v, u)) is equal to 0 if (y, x) ≠ (t, z)

and to 1 otherwise. The proposition follows.

Applying Proposition 5.3 to the poset of subgroups of G, ordered by inclusion of subgroups, gives the following :

5.5. Corollary : Let (V, U) and (T, S) be slices of G. Then

(5.6) μ_{Π}((V, U), (T, S)) = \{ \begin{matrix}μ(U, S)μ(V, T) & if U ≤ S ≤ V ≤ T \\ 0 & otherwise \end{matrix} ,

where μ is the Möbius function of the poset of subgroups of G. In particular

{in Ξ(G) \atop ξ_{T,S}^{G} = \frac{1}{|N_{G}(T, S)|} ∑_{U≤S≤V ≤T} |U|μ(U, S)μ(V, T) ⟨V, U⟩_{G} .}

6. The image of the ghost map

The following characterization of the image of the ghost map is the analogue for the slice Burnside ring of a theorem of Dress ([6]) on the ordinary Burnside ring :

6.1. Theorem : Let G be a finite group, and let m = (m_{T,S})_{(T,S)∈Π(G)} be a sequence of integers indexed by Π(G), constant on G-conjugacy classes of slices. Then the sequence [m] = (m_{T,S})_{(T,S)∈[Π(G)]} of representatives lies in the image of the ghost map Φ if and only if, for any slice (T, S) of G

∑_{g∈N_{G}(T,S)/S} m_{<gT >,<gS>} ≡ 0 (mod. |N_{G}(T, S)/S|) .

Proof : Saying that [m] lies in the image of Φ is equivalent to saying that the element

s_{m} = ∑_{(T,S)∈[Π(G)]} m_{T,S} ξ_{T,S}^{G}

of ℚΞ(G) lies in Ξ(G), i.e. that it is a linear combination with integer coefficients of the elements ⟨V, U⟩_{G}, for (V, U) ∈ [Π(G)]. Now by Theorem 5.2

s_{m} = ∑_{(T,S)∈Π(G)} \frac{|N_{G}(T, S)|}{|G|}m_{T,S} ξ_{T,S}^{G}

= \frac{1}{|G|} ∑_{(T,S)∈Π(G)} m_{T,S} ∑_{(V,U)⪯(T,S)} |U|μ_{Π}((V, U), (T, S)) ⟨V, U⟩_{G}

= \frac{1}{|G|} ∑_{(V,U)∈Π(G)} |U|( ∑_{(V,U)⪯(T,S)} μ_{Π}((V, U), (T, S))m_{T,S})⟨V, U⟩_{G}

= ∑_{(V,U)∈[Π(G)]} \frac{1}{|N_{G}(V, U)/U|}( ∑_{(V,U)⪯(T,S)} μ_{Π}((V, U), (T, S))m_{T,S})⟨V, U⟩_{G} .

Hence s_{m} ∈ Im Φ if and only if the number

β_{V,U} = \frac{1}{|N_{G}(V, U)/U|}( ∑_{(V,U)⪯(T,S)} μ_{Π}((V, U), (T, S))m_{T,S})

is an integer, for any slice (V, U) of G.

With this notation, for any (Y, X) ∈ Π(G)

m_{Y,X} = ∑_{{(V,U)∈Π(G) \atop (Y,X)⪯(V,U)}} β_{V,U}|N_{G}(V, U)/U| .

Hence, setting σ_{Y,X} = ∑_{g∈N_{G}(Y,X)/X} m_{<gY >,<gX>} :

σ_{Y,X} = ∑_{g∈N_{G}(Y,X)/X} ∑_{{(V,U)∈Π(G) \atop (<gY >,<gX>)⪯(V,U)}} β_{V,U}|N_{G}(V, U)/U|

= ∑_{g∈N_{G}(Y,X) ; {(V,U)∈Π(G) \atop (<gY >,<gX>)⪯(V,U)}} \frac{1}{|X|}β_{V,U}|N_{G}(V, U)/U|

= ∑_{(V,U)∈Π(G) ; {(Y,X)⪯(V,U) \atop g∈U∩N_{G}(Y,X)}} \frac{1}{|X|}β_{V,U}|N_{G}(V, U)/U|

σ_{Y,X}= ∑_{{(V,U)∈[N_{G}(Y,X)\Π(G)] \atop (Y,X)⪯(V,U)}} \frac{|N_{G}(Y, X)/X|}{|N_{G}(Y, X, V, U)|}β_{V,U}|N_{G}(V, U)/U||U ∩ N_{G}(Y, X)|

= |N_{G}(Y, X)/X| ∑_{{(V,U)∈[N_{G}(Y,X)\Π(G)] \atop (Y,X)⪯(V,U)}} β_{V,U} \frac{|N_{G}(V, U)||U ∩ N_{G}(Y, X)|}{|U||N_{G}(Y, X, V, U)|}

= |N_{G}(Y, X)/X| ∑_{{(V,U)∈[N_{G}(Y,X)\Π(G)] \atop (Y,X)⪯(V,U)}} β_{V,U}|N_{G}(V, U) : U·N_{G}(Y, X, V, U)| .

Setting α_{Y,X} = σ_{Y,X}/|N_{G}(Y, X)/X|, it follows that

α_{Y,X} = ∑_{{(V,U)∈N_{G}(Y,X)\Π(G) \atop (Y,X)⪯(V,U)}} β_{V,U}|N_{G}(V, U) : U·N_{G}(Y, X, V, U)| .

Since |N_{G}(V, U) : U·N_{G}(Y, X, V, U)| = 1 if (Y, X) = (V, U), it follows that the transition matrix from the β_{V,U}’s to the α_{Y,X}’s is triangular, with integer coefficients, and 1’s on the diagonal. Hence it is invertible over ℤ, and the numbers β_{V,U} are all integers, for (V, U) ∈ Π(G), if and only if the numbers α_{Y,X} are all integers, for (Y, X) ∈ Π(G). This completes the proof.

7. Prime spectrum

7.1. Notation : Let p denote either 0 or a prime number.

• If (T, S) ∈ Π(G), let I_{T,S,p} be the prime ideal of Ξ(G) defined as the kernel of the ring homomorphism

Ξ(G) ⟶^{ϕ_{T,S}} ℤ → ℤ/pℤ ,

where the right hand side map is the projection.

• Let Θ(G) denote the set of triples (T, S, p), where (T, S) ∈ Π(G) is such that |N_{G}(T, S)/S| ≢ 0 (mod. p).

The group G acts on Θ(G), by ^{g}(T, S, p) = (^{g}T ,^{g}S, p), for g ∈ G, and the ideal I_{T,S,p} only depends on the G-orbit of (T, S, p). Conversely :

7.2. Proposition : Let I be a prime ideal of Ξ(G), and R = Ξ(G)/I. Denote by ϕ : Ξ(G) → R the projection map, and denote by p ≥ 0 the characteristic of R. Then R ≅ ℤ/pℤ and :

1. If p = 0, there exists a slice (T, S) of G such that ϕ = ϕ_{T,S}, and (T, S) is unique up to G-conjugation, with this property.

2. If p > 0, there exists a slice (T, S) of G such that ϕ is the reduction modulo p of ϕ_{T,S} and N_{G}(T, S)/S is a p′-group, and (T, S) is unique up to G-conjugation, with these properties.

In particular, there exists a unique (T, S, p) ∈ Θ(G), up to conjugation, such

that I = I_{T,S,p}.

Proof : Let (T, S) be a slice of G, minimal for the relation ⪯, such that ⟨T, S⟩_{G} ∉ I. Then by Proposition 3.8, for any (Y, X) ∈ Π(G)

⟨T, S⟩_{G}⟨Y, X⟩_{G} = ∑_{g∈[S\G/X]}⟨T ∩^{g}Y , S ∩^{g}X⟩_{G}
≡ ∑_{g∈G/X ; {S≤^{g}X \atop T ≤^{g}Y}} ⟨T, S⟩_{G} (mod. I)

= ϕ_{T,S}(⟨Y, X⟩_{G})⟨T, S⟩_{G} .

Since I is prime, it follows that ⟨Y, X⟩_{G} − ϕ_{T,S}(⟨Y, X⟩_{G})1_{Ξ(G)} ∈ I. In particular R = Ξ(G)/I is generated by the image of 1_{Ξ(G)}, hence R ≅ ℤ/pℤ, where p is the characteristic of R. Since R is an integral domain, the number p is either 0 or a prime.

1. If p = 0, then R = ℤ, and ϕ = ϕ_{T,S}. And if (T′, S′) ∈ Π(G) is such that ϕ_{T,S} = ϕ_{T′,S′}, then both ϕ_{T,S}(⟨T′, S′⟩_{G}) and ϕ_{T′,S′}(⟨T, S⟩_{G}) are non zero. Then there exist elements g, g′ ∈ G such that (T^{g}, S^{g}) ⪯ (T′, S′) and (T^{′g′}, S^{′g′}) ⪯ (T, S), so (T, S) and (T′, S′) are conjugate in G.

2. If p > 0, then R = ℤ/pℤ, and ϕ is equal to the reduction of ϕ_{T,S} modulo p. Since ϕ(⟨T, S⟩_{G}) = |N_{G}(T, S)/S| is non zero in R, it follows that N_{G}(T, S)/S is a p′-group. If (T′, S′) is another slice of G such that ϕ is the reduction modulo p of ϕ_{T′,S′}, and N_{G}(T′, S′)/S′ is a p′-group, then

|N_{G}(T, S)/S| = ϕ_{T,S}(⟨T, S⟩_{G}) ≡ ϕ_{T′,S′}(⟨T, S⟩_{G}) (mod. p) .

This is non zero. Similarly |N_{G}(T, S)/S| ≡ ϕ_{T,S}(⟨T′, S′⟩_{G}) (mod. p) is non zero. In particular ϕ_{T′,S′}(⟨T, S⟩_{G}) and ϕ_{T,S}(⟨T′, S′⟩_{G}) are both non zero, and it follows as above that (T, S) and (T′, S′) are conjugate in G.

7.3. Notation : Let p be a prime number.

• Let Π_{p}(G) denote the subset of Π(G) consisting of the slices (T, S) such that N_{G}(T, S)/S is a p′-group.

• For any (T, S) ∈ Π(G), let (T, S)p̂ denote the unique element (V, U) of Π_{p}(G), up to conjugation, such that I_{T,S,p} = I_{V,U,p}.

7.4. Proposition : Let p be a prime number. If (T, S) is a slice of G, let (T, S)_{p}^{+} denote a slice of the form (P T, P S) of G, where P is a Sylow p-subgroup of N_{G}(T, S).

Define inductively an increasing sequence (T_{n}, S_{n}) in (Π(G), ⪯) by (T_{0}, S_{0}) = (T, S), and (T_{n+1}, S_{n+1}) = (T_{n}, S_{n})_{p}^{+}, for n ∈ ℕ. Then (T, S)p̂ is conjugate to the largest term (T_{∞}, S_{∞}) of the sequence (T_{n}, S_{n}).

Proof : The proof of Proposition 7.2 shows that (T, S)p̂ is a minimal element (V, U) of the poset (Π(G), ⪯) such that

ϕ_{T,S}(V, U) = |{g ∈ G/U | (T^{g}, S^{g}) ⪯ (V, U)}| ≢ 0 (mod.p) .

Thus one can assume that (T, S) ⪯ (V, U). But ϕ_{T,S} ≡ ϕ_{P T,P S} (mod. p) by Corollary 4.4, for any p-subgroup P of N_{G}(T, S), hence one can also assume that (T, S)_{p}^{+} ⪯ (V, U), and by induction, that (T_{∞}, S_{∞}) ⪯ (V, U). Moreover ϕ_{T_{∞},S_{∞}} ≡ ϕ_{V,U} (mod. p). As N_{G}(T_{∞}, S_{∞})/S_{∞} is a p′-group, it follows that (T_{∞}, S_{∞}) = (V, U), as was to be shown.

7.5. Remark : Let (T, S) ∈ Π(G), and (V, U) ∈ Π_{p}(G). It is easy to check, by induction on the integer n such that (T_{n}, S_{n}) = (T_{∞}, S_{∞}), that (T, S)p̂ is equal to (V, U) if and only if T is a subnormal subgroup of V, if S is a subnormal subgroup of U, if |U : S| is a power of p, and if the set T ·U is equal to V.

7.6. Proposition : Let (T, S, p), (T′, S′, p′) be elements of Θ(G). Then I_{T′,S′,p′} ⊆ I_{T,S,p} if and only if

• either p′ = p and the slices (T′, S′) and (T, S) are conjugate in G.

• or p′ = 0 and p > 0, and the slices (T′, S′)p̂ and (T, S) are conjugate

in G.

Proof : Set I = I_{T,S,p} and I′ = I_{T′,S′,p′}. Then Ξ(G)/I′ ≅ ℤ/p′ℤ maps surjectively to Ξ(G)/I ≅ ℤ/pℤ. Thus if p = p′, this projection map is an isomorphism, hence I = I′ and the slices (T, S) and (T′, S′) are conjugate in G. And if p ≠ p′, then p′ = 0 and p > 0. The morphism ϕ_{T,S} is equal to the reduction modulo p of the morphism ϕ_{T′,S′}. In other words I_{T′,S′,p} = I_{T,S,p}, hence (T, S) is conjugate to (T′, S′)p̂.

7.7. Corollary : Let p be a prime number, and let ℤ_{(p)} be the localization of ℤ at the set ℤ − pℤ. Let Θ_{p}(G) denote the subset of Θ(G) consisting of triples (T, S, 0), for (T, S) ∈ Π(G), and (T, S, p), for (T, S) ∈ Π_{p}(G). Then :

1. The prime ideals of the ring ℤ_{(p)}Ξ(G) are the ideals ℤ_{(p)}I_{T,S,q}, for

(T, S, q) ∈ Θ_{p}(G).

2. If (T, S, q), (T′, S′, q′) ∈ Θ_{p}(G), then ℤ_{(p)}I_{T′,S′,q′} ⊆ ℤ_{(p)}I_{T,S,q} if and only if :

• either q = q′, and the slices (T, S) and (T′, S′) are conjugate in G.

• or q′ = 0, q = p, and the slices (T′, S′)p̂ and (T, S) are conjugate

in G.

3. The connected components of the spectrum of ℤ_{(p)}Ξ(G) are indexed by the conjugacy classes of Π_{p}(G). The component indexed by (T, S) ∈ Π_{p}(G) consists of a unique maximal element ℤ_{(p)}I_{T,S,p}, and of the ideals ℤ_{(p)}I_{T′,S′,0}, where (T′, S′) ∈ Π(G) is such that (T′, S′)p̂ is conjugate to

(T, S) in G.

Proof : The prime ideals of ℤ_{(p)}Ξ(G) are of the form ℤ_{(p)}I, where I is a prime ideal of Ξ(G) such that I ∩(ℤ−pℤ) = ∅. Equivalently I = I_{T,S,0} or I = I_{T,S,p}. This proves Assertion 1. Now Assertion 2 follows from Proposition 7.6, and Assertion 3 follows from Assertion 2.

7.8. Corollary :

1. The primitive idempotents of the ring ℤ_{(p)}Ξ(G) are indexed by the conjugacy classes of Π_{p}(G). The primitive idempotent η_{V,U}^{G} indexed by (V, U) ∈ Π_{p}(G) is equal to

η_{V,U}^{G} = ∑_{{(T,S)∈[Π(G)] \atop (T,S)p̂=_{G}(T,S)}} ξ_{T,S}^{G} .

2. Let π be a set of prime numbers, and ℤ_{(π)} be the localization of ℤ relative to ℤ − ∪_{p∈π}pℤ. Let F be a set of slices of G, invariant by Gconjugation, and [F] be a set of representatives of G-conjugacy classes of F. Then the following conditions are equivalent :

(a) The idempotent

ξ_{F}^{G} = ∑_{(T,S)∈[F]} ξ_{T,S}^{G}

of ℚΞ(G) lies in ℤ_{(π)}Ξ(G).

(b) Let (T, S) ∈ Π(G), and let P be a p-subgroup of N_{G}(T, S), for

some p ∈ π. Then (T, S) ∈ F if and only if (P T, P S) ∈ F.

Proof : Let F be a set of slices of G, invariant by G-conjugation, and [F] be a set of representatives of G-conjugacy classes of F. The idempotent

ξ_{F}^{G} = ∑_{(T,S)∈[F]} ξ_{T,S}^{G}

of ℚΞ(G) lies in ℤ_{(p)}Ξ(G), for some prime p, if and only if there exists an integer m, not divisible by p, such that u = mξ_{F}^{G} ∈ Ξ(G). Let (T, S) ∈ Π(G), and let P be a p-subgroup of N_{G}(T, S). The integer ϕ_{T,S}(u) is equal to m if (T, S) ∈ F, and to 0 otherwise. Hence it is coprime to p if and only if (T, S) ∈ F. Since ϕ_{T,S} and ϕ_{P T,P S} are congruent modulo p, it follows that (T, S) ∈ F if and only if (P T, P S) ∈ F.

Hence if (T, S) and (T′, S′) are slices of G such that (T, S)p̂ =_{G} (T′, S′)p̂, then (T, S) ∈ F if and only if (T′, S′) ∈ F. Thus F is a disjoint union of sets of the form

E_{V,U} = {(T, S) ∈ Π(G) | (T, S)p̂ =_{G} (V, U)} ,

for some slices (V, U) ∈ Π_{p}(G). In other words the idempotent ξ_{F}^{G} is a sum of some idempotents η_{V,U}^{G}, for (V, U) ∈ Π_{p}(G).

But the primitive idempotents of the ring ℤ_{(p)}Ξ(G) are in one to one correspondence with the connected components of its spectrum, which precisely are indexed by the conjugacy classes of Π_{p}(G). It follows that ξ_{F}^{G} = η_{V,U}^{G} is equal to the idempotent corresponding to the component indexed by (V, U), for any (V, U) ∈ Π_{p}(G). This proves Assertion 1. This also proves Assertion 2 in the case where π consists of a single prime number.

For the general case, observe that ξ_{F}^{G} lies in ℤ_{(π)}Ξ(G) if and only if it lies in ℤ_{(p)}Ξ(G), for any p ∈ π.

7.9. Theorem : Let G be a finite group.

1. Let ∼ denote the finest equivalence relation on the set Π(G) such that

{for any (T, S), (T′, S′) ∈ Π(G), \atop ∃p, (T, S)p̂ =_{G} (T′, S′)p̂ ⟹ (T, S) ∼ (T′, S′) .}

Then the primitive idempotents of Ξ(G) are indexed by the equivalence classes of Π(G) for the relation ∼. The idempotent ξ_{C}^{G} indexed by the component C is equal to

ξ_{C}^{G} = ∑_{(T,S)∈[C]} ξ_{T,S}^{G} ,

where [C] is a set of representatives of G-conjugacy classes in C.

2. The prime spectrum of Ξ(G) is connected if and only if G is solvable.

Proof : Assertion 1 follows from Corollary 7.8, applied to the set π of all primes.

For Assertion 2, observe that the spectrum of Ξ(G) is connected if and only if 1 is a primitive idempotent of Ξ(G). This means that for any two slices (T, S) and (T′, S′) of G, there exists a sequence (T_{i}, S_{i}) of slices, for i ∈ {0, …, n}, and a sequence p_{i} of prime numbers, for i ∈ {0, …, n − 1}, such that

(T, S) = (T_{0}, S_{0}) ∼^{p_{0}} (T_{1}, S_{1}) ∼^{p_{1}} … ∼^{p_{n−2}} (T_{n−1}, S_{n−1}) ∼^{p_{n−1}} (T_{n}, S_{n}) = (T′, S′) ,

where the notation (Y, X) ∼^{p} (Y′, X′) means that (Y, X)p̂ =_{G} (Y′, X′)p̂.

But clearly, if (Y, X) ∼^{p} (Y′, X′), then the slices (O^{p}(Y), O^{p}(X)) and (O^{p}(Y′), O^{p}(X′)) are conjugate in G. Hence the slices (D^{∞}(Y), D^{∞}(X)) and (D^{∞}(Y′), D^{∞}(X′)) are conjugate in G, where D^{∞}(H) denotes the last term in the derived series of the group H.

If the spectrum of G is connected, taking (T, S) = (G, G) and (T′, S′) = (1, 1), it follows that D^{∞}(G) = 1, i.e. that G is solvable.

Conversely, if G is solvable, let (T, S) be a slice of G. Then S is solvable, and there exists a prime p such that O^{p}(S) < S. Let P be a Sylow psubgroup of S. Then P ≤ N_{G}(T, S), and (T, S) = (P T, P O^{p}(S)). Thus (T, S) ∼^{p} (T, O^{p}(S)). By induction, there is a sequence of prime numbers p_{i}, for i ∈ {1, …, k}, such that

(T, S) = (T, S_{0}) ∼^{p_{0}} (T, S_{1}) ∼^{p_{1}} … ∼^{p_{k−1}} (T, S_{k}) ∼^{p_{k}} (T, 1) ,

where S_{i+1} = O^{p_{i}}(S_{i}).

Now T is solvable, so there exists a prime q such that O^{q}(T) < T. If Q is a Sylow q-subgroup of T, then Q ≤ N_{G}(T, 1), and (T, 1) ∼^{q} (O^{q}(T), 1). Hence there exists a sequence of primes q_{j}, for j ∈ {0, …, l}, such that

(T, 1) = (T_{0}, 1) ∼^{q_{0}} (T_{1}, 1) ∼^{q_{1}} … ∼^{q_{l−1}} (T_{l}, 1) ∼^{q_{l}} (1, 1) ,

where T_{j+1} = O^{q_{j}}(T_{j}). This shows that the spectrum of Ξ(G) is connected.

8. Unit group

8.1. When G is a finite group, denote by Ξ(G)^{×} the group of invertible elements of the ring Ξ(G). It follows from Theorem 4.6 that the restricted ghost map yields an injective group homomorphism

Φ^{×} : Ξ(G)^{×} ↪ ∏_{(T,S)∈Π(G)} ℤ^{×} .

The following lemma is a straightforward consequence of the existence of this injective group homomorphism :

8.2. Lemma : Let G be a finite group, and let u ∈ Ξ(G). The following conditions are equivalent :

1. u ∈ Ξ(G)^{×}.

2. ϕ_{T,S}(u) ∈ {±1}, for any (T, S) ∈ Π(G).

3. u^{2} = 1.

In particular Ξ(G)^{×} is a finite elementary abelian 2-group.

The main motivation in considering the group Ξ(G)^{×} lies in the following proposition, which extends a theorem of tom Dieck ([9] Proposition 1.5.1) about the unit group of the usual Burnside ring :

8.3. Proposition : Feit-Thompson’s theorem is equivalent to the statement that, if G has odd order, then Ξ(G)^{×} = {±1}.

Proof : The first observation is that for any finite group G, by Lemma 8.2, the correspondences u ↦ \frac{1−u}{2} and e ↦ 1 − 2e are mutually inverse bijections between Ξ(G)^{×} and the set of idempotents e ∈ ℚΞ(G) such that 2e ∈ Ξ(G).

Now Theorem 5.2 shows that |G|e ∈ Ξ(G), for any idempotent e of ℚΞ(G). Hence if G has odd order, and if e is an idempotent of Ξ(G) such that 2e ∈ Ξ(G), then (2, |G|)e = e ∈ Ξ(G). Thus if |G| is odd, the set Ξ(G)^{×} is in one to one correspondence with the set of idempotents of the ring Ξ(G). By a standard argument from commutative ring theory, this set is in one to one correspondence with the set of connected components of the spectrum of Ξ(G). It follows that if G has odd order, the spectrum of G is connected if and only if Ξ(G)^{×} = {±1}. By Theorem 7.9, this is equivalent to saying that G is solvable.

The following theorem is an analogue of Yoshida’s characterization ([10]) of the unit group of the usual Burnside ring :

8.4. Theorem : Let G be a finite group, and let m = (m_{T,S})_{(T,S)∈Π(G)} be a sequence of integers in {±1} indexed by Π(G), constant on G-conjugacy classes of slices. Then the sequence [m] = (m_{T,S})_{(T,S)∈[Π(G)]} of representatives lies in the image of the restricted ghost map Φ^{×} if and only if for any (T, S) ∈

{Π(G), the map \atop g ∈ N_{G}(T, S)/S ↦ m_{<gT >,<gS>}/m_{T,S} ∈ \{±1\}}

is a group homomorphism.

Proof : Let X →^{f} Y be a morphism of finite G-sets. It follows from Lemma 4.3 that for any (T, S) ∈ Π(G), the monoid of endomorphisms of G/S → G/T in the category G-Mor is actually a group, isomorphic to N_{G}(T, S)/S. This group acts on the set of morphisms from G/S → G/T to X →^{f} Y, by pre-composition : if

\xymatrix{G/S \ar[r] \ar[d]^{α} & G/T \ar[d]^{β} \\ X \ar[r]^{f} & Y}

is a morphism in G-Mor, and if g ∈ N_{G}(T, S), the morphism ^{g}(α, β) = (^{g}α,^{g}β) is defined by (^{g}α)(xS) = xgS and (^{g}β)(xT) = xgT, for any g ∈ T. The morphism (α, β) is invariant under g ∈ N_{G}(T, S) if and only if α(gS) = α(S), i.e. if (α, β) factors as

\xymatrix{G/S \ar[r] \ar[d] & G/gT \ar[d] \\ G/<gS> \ar[r] \ar[d]^{\overline{α}} & G/<gT > \ar[d]^{\overline{β}} \\ X \ar[r]^{f} & Y}

It follows that the number of fixed points of g ∈ N_{G}(T, S)/S on the set of homomorphisms from G/S → G/T to X →^{f} Y, i.e. the value at gS of the corresponding permutation character θ_{T,S} of N_{G}(T, S)/S, is equal to ϕ_{<gT >,<gS>}(X →^{f} Y). This shows more generally that for any u ∈ Ξ(G), the correspondence

θ_{T,S} : g ∈ N_{G}(T, S)/S ↦ ϕ_{<gT >,<gS>}(u)

is a generalized character of the group N_{G}(T, S)/S.

Now if u ∈ Ξ(G)^{×}, this generalized character has all its values in {±1}. It follows that ⟨θ_{T,S}, θ_{T,S}⟩_{G} = 1, hence θ_{T,S} is up to a sign equal to an irreducible character of N_{G}(T, S)/S, of degree 1. Hence θ_{T,S}/θ_{T,S}(1) is a group homomorphism from N_{G}(T, S)/S to {±1}. Thus if m ∈ Im(Φ^{×}), then the map g ∈ N_{G}(T, S)/S ↦ m_{<gT >,<gS>}/m_{T,S} ∈ {±1} is a group homomorphism.

Conversely, if m = (m_{T,S})_{(T,S)∈Π(G)} is a G-invariant sequence with values in {±1}, such that for any (T, S) ∈ Π(G), the map g ∈ N_{G}(T, S)/S ↦ m_{<gT >,<gS>}/m_{T,S} ∈ {±1} is a group homomorphism θ_{T,S}, then

∑_{g∈N_{G}(T,S)/S} m_{<gT >,<gS>} = m_{T,S}|N_{G}(T, S)/S| ⟨θ_{T,S}, 1⟩_{G}

is an integer multiple of |N_{G}(T, S)/S|. By Theorem 6.1, it follows that [m] = Φ(u), for some u ∈ Ξ(G). Then u ∈ Ξ(G)^{×}, by Lemma 8.2, and this completes the proof.

8.5. In the case of the usual Burnside ring B(G), the correspondence sending a finite group G to the unit group B(G)^{×} can be endowed with a structure of biset functor, using tensor induction (see e.g. [3], Proposition 5.5). One may ask whether a similar structure exists for the group of units of the section Burnside ring :

8.6. Proposition : The correspondence sending a finite group G to Ξ(G)^{×} cannot be endowed with a structure of biset functor.

Proof : If such a structure exists, one may view it as a biset functor Ξ^{×} with values in 𝔽_{2}-vector spaces. Obviously Ξ^{×}(1) = 𝔽_{2}. This shows that there are two subfunctors F_{2} ⊂ F_{1} of Ξ^{×} such that F_{1}/F_{2} is isomorphic to the simple functor S_{1,𝔽_{2}}.

An easy computation (a special case of Theorem 16.13 in the Appendix) shows moreover that Ξ^{×}(C_{2}) ≅ (𝔽_{2})^{3}, where C_{2} is a group of order 2. As S_{1,𝔽_{2}}(C_{2}) ≅ (𝔽_{2})^{2} (see e.g. [4] Proposition 4.4.6), this shows that either (F/F_{1})(C_{2}) ≅ 𝔽_{2} and F_{2}(C_{2}) = {0}, or F (C_{2}) = F_{1}(C_{2}) and F_{2}(C_{2}) ≅ 𝔽_{2}. In the first case, there are subfunctors F_{3} and F_{4} of F with F_{1} ⊆ F_{4} ⊂ F_{3} ⊆ F and F_{3}/F_{4} ≅ S_{C_{2},𝔽_{2}}. In the latter case, there are subfunctors F_{4} ⊂ F_{3} of F_{2} such that F_{3}/F_{4} ≅ S_{C_{2},𝔽_{2}}. In any case

dim_{𝔽_{2}} Ξ^{×}((C_{2})^{2}) ≥ dim_{𝔽_{2}} S_{1,𝔽_{2}}((C_{2})^{2}) + dim_{𝔽_{2}} S_{C_{2},𝔽_{2}}((C_{2})^{2}) .

Now dim_{𝔽_{2}} S_{1,𝔽_{2}}((C_{2})^{2}) = 4 by [4] Proposition 4.4.6 or Corollary 10.5.6, and dim_{𝔽_{2}} S_{C_{2},𝔽_{2}}((C_{2})^{2}) = 5 by inspection, using [4] Proposition 4.4.6. Thus dim_{𝔽_{2}} Ξ^{×}((C_{2})^{2}) ≥ 9.

When G is a finite abelian group, the unit group Ξ(G)^{×} can be determined explicitly (see Theorem 16.13). In particular

(8.7)  Ξ((C_{2})^{2})^{×} ≅ (𝔽_{2})^{7} .

This contradiction shows that the biset functor Ξ^{×} cannot exist.

8.8. Remark : Boltje and Pfeiffer ([1]) have described an efficient algorithm to compute the unit group of the ordinary Burnside ring. A straightforward adaptation of this algorithm to the ring Ξ(G) allows for a quick computation of the group Ξ(G)^{×}, for not too large finite groups G, using GAP4 ([7]). These computations agree in particular with 8.7.

II - The section Burnside ring

9. Galois morphisms of G-sets

9.1. Definition : Let G be a group. A morphism f : X → Y of G-sets is a Galois morphism if for any x, x′ ∈ X such that f(x) = f(x′), there exists φ ∈ Aut_{G}(X) such that f ∘ φ = f and φ(x) = x′.

9.2. Example : Any injective morphism of G-sets is a Galois morphism, for trivial reasons.

9.3. Proposition : Let f : X → Y be a morphism of G-sets. The following conditions are equivalent:

1. f is a Galois morphism.

2. ∀x, x′ ∈ X, f(x) = f(x′) ⟹ G_{x} = G_{x′}.

Proof : Suppose first that f is a Galois morphism. Then if f(x) = f(x′), there exists a G-automorphism φ of X such that φ(x) = x′. This implies G_{x} ≤ G_{x′}, hence G_{x} = G_{x′} by symmetry. Thus Condition 1 implies Condition 2.

Conversely, suppose that Condition 2 holds, and let x, x′ ∈ X with f(x) = f(x′). There are two cases :

• Either x and x′ are in the same G-orbit ω. Let Z = X − ω. Define φ : X → X by φ(t) = t if t ∈ Z and φ(ux) = ux′ if u ∈ G. This is a well defined G-automorphism of X, since G_{x} = G_{x′}. Clearly φ(x) = x′, and f ∘ φ(t) = f(t) if t ∈ Z. Moreover

f ∘ φ(ux) = f(ux′) = uf(x′) = uf(x) = f(ux)

for any u ∈ G. Hence f ∘ φ = f.

• If x and x′ are in different G-orbits ω and ω′, respectively, let Z = X − (ω ⊔ ω′). Define φ : X → X by φ(t) = t if t ∈ Z, and φ(ux) = ux′ and φ(ux′) = ux, for any u ∈ G. This is a well defined G-automorphism of X, since G_{x} = G_{x′}. Clearly φ(x) = x′, and f ∘ φ(t) = f(t) if t ∈ Z. Moreover

f ∘ φ(ux) = f(ux′) = uf(x′) = uf(x) = f(ux)

for any u ∈ G. Also

f ∘ φ(ux′) = f(ux) = uf(x) = uf(x′) = f(ux′) ,

for any u ∈ G. Hence f ∘ φ = f.

It follows that f is a Galois morphism, and Condition 2 implies Condition 1.

9.4. Corollary : Let f : X → Y be a morphism of G-sets. Then f is a Galois morphism if and only if for any y ∈ f(X), there exists a normal subgroup N_{y} of G_{y} such that G_{x} = N_{y} for any x ∈ f^{−1}(y).

Proof : Suppose first that f is a Galois morphism. If x ∈ X, set y = f(x), and choose g ∈ G_{y}. Then f(gx) = gy = y = f(x), thus G_{gx} = gG_{x} = G_{x}, thus G_{x} ⊴ G_{y}. Moreover G_{x} does not depend on x ∈ f^{−1}(y), by Proposition 9.3.

Conversely, suppose that for any y ∈ f(X), there exists N_{y} ⊴ G_{y} such that G_{x} = N_{y} for any x ∈ f^{−1}(y). Then obviously G_{x} = G_{x′} = N_{y} if f(x) = f(x′) = y, so f is a Galois morphism, by Proposition 9.3.

9.5. Remark : In particular, when (T, S) is a slice of G, the projection morphism G/S → G/T is a Galois morphism of G-sets if and only if S ⊴ T, i.e. if (T, S) is a section of G.

9.6. Corollary : Let f : X → Y be a Galois morphism of G-sets. If X_{1} is a G-subset of X, the restricted morphism f_{|X_{1}} : X_{1} → f(X_{1}) is a Galois morphism of G-sets.

Proof : This is a straightforward consequence of Proposition 9.3.

9.7. Lemma : Let f : X → Y be a morphism of G-sets, and j : Y ↪ Z be an injective morphism of G-sets. Then f is a Galois morphism if and only if j ∘ f is a Galois morphism.

Proof : This is straightforward.

9.8. Lemma : Let

\xymatrix{X \ar[r]^{a} \ar[d]_{b} & Y \ar[d]^{c} \\ Z \ar[r]_{d} & T}

be a cartesian square of G-sets. If c is a Galois morphism, then b is a Galois morphism.

Proof : Suppose that x, x′ ∈ X are such that b(x) = b(x′). Then

ca(x) = db(x) = db(x′) = ca(x′) .

If c is a Galois morphism, it follows that G_{a(x)} = G_{a(x′)}. Let g ∈ G_{x}. Then g ∈ G_{a(x)} = G_{a(x′)}, and g ∈ G_{b(x)} = G_{b(x′)}, thus

b(x′) = gb(x′) = b(gx′)
a(x′) = ga(x′) = a(gx′) .

It follows that x′ = gx′, so G_{x} ≤ G_{x′}, and G_{x} = G_{x′} by symmetry. Hence b is a Galois morphism, by Proposition 9.3.

9.9. Remark : In particular, any morphism isomorphic to a Galois morphism (that is, when a and d are isomorphisms) is a Galois morphism.

9.10. Proposition : A morphism f : X → Y_{1} ⊔ Y_{2} is a Galois morphism if and only if the restricted morphisms f^{−1}(Y_{1}) → Y_{1} and f^{−1}(Y_{2}) → Y_{2} are Galois morphisms.

Proof : Set X_{i} = f^{−1}(Y_{i}), and denote by f_{i} : X_{i} → Y_{i} the restriction of f, for i = 1, 2. Assume first that f is a Galois morphism. If x, x′ ∈ X_{1} have the same image under f_{1}, then they have the same image under f, and G_{x} = G_{x′}. So f_{1} is a Galois morphism, and f_{2} is also a Galois morphism, by symmetry.

Conversely, suppose that f_{1} and f_{2} are Galois morphisms. If x, x′ ∈ X are such that f(x) = f(x′) ∈ Y_{1}, then x, x′ ∈ X_{1}, and f_{1}(x) = f_{1}(x′). Thus G_{x} = G_{x′}, as f_{1} is a Galois morphism. If f(x) = f(x′) ∈ Y_{2}, the argument is similar, with f_{1} replaced by f_{2}. In any case G_{x} = G_{x′}, and f is a Galois morphism, by Proposition 9.3.

Let G and H be groups. A morphism f : U → U′ of (H, G)-bisets is called a Galois morphism if it is a Galois morphism of (H ×G^{op})-sets. Then :

9.11. Proposition : Let G, H and K be groups. If f : U → U′ is a Galois morphism of (H, G)-bisets, and g : V → V′ is a Galois morphism of (K, H)-bisets, then g ×_{H} f : V ×_{H} U → V′ ×_{H} U′ is a Galois morphism of

(K, G)-bisets.

Proof : Let (v,_{H} u) and (v′,_{H} u′) be elements of V ×_{H} U with the same image under g ×_{H} f. It means that there exists h ∈ H such that

g(v′) = g(v)h = g(vh) and f(u′) = h^{−1}f(u) = f(h^{−1}u) .

As g is a Galois morphism, the stabilizers of v′ and vh in K × H^{op} are equal. Similarly, as f is a Galois morphism, the stabilizers of u′ and h^{−1}u in H ×G^{op} are equal.

Now let (k, g) ∈ K × G^{op} such that k(v′,_{H} u′)g^{−1} = (v′,_{H} u′). It means that there exists a ∈ H such that kv′a^{−1} = v′ and au′g^{−1} = u′. Hence

kvha^{−1} = vh and ah^{−1}ug^{−1} = h^{−1}u .

It follows that

k(v,_{H} u)g^{−1} = (kv,_{H} ug^{−1}) = (vhah^{−1},_{H} ug^{−1})
= (v,_{H} hah^{−1}ug^{−1}) = (v,_{H} hh^{−1}u) = (v,_{H} u) .

By symmetry, the stabilizers of (v,_{H} u) and (v′,_{H} u′) in K × G^{op} are equal. Thus (g ×_{H} f) is a Galois morphism of (K, G)-bisets.

9.12. Corollary : Let G and H be finite groups, and U be an (H, G)-biset. If f : X → Y is a Galois morphism of G-sets, then

U ×_{G} f : U ×_{G} X → U ×_{G} Y and U ∘_{G} f : U ∘_{G} X → U ∘_{G} Y

are Galois morphisms of H-sets.

Proof : The assertion on U ×_{G} f is the special case of Proposition 9.11, with the following changes in the notation : the group G becomes the trivial group, the group H becomes G, the group K becomes H, the biset U becomes X, the biset V becomes U, and the map g becomes the identity map of U.

Now recall from [2] that U ∘_{G} X is the H-subset of U ×_{G} X defined by

U ∘_{G} X = {(u,_{G} x) ∈ U ×_{G} X | ∀g ∈ G, ug = u ⟹ gx = x} ,

and that the map U ∘_{G}f is the restriction of U ×_{G}f to U ∘_{G}X. By Lemma 9.7 and Corollary 9.6, the morphism U ∘_{G}f is also a Galois morphism of H-sets.

9.13. Corollary : Let f : X → Y be a Galois morphism of G-sets. If H is a subgroup of G, the restriction Res_{H}^{G}f : Res_{H}^{G}X → Res_{H}^{G}Y is a Galois morphism of H-sets.

Proof : In Corollary 9.12, set U = G, viewed as and (H, G)-biset for left and right multiplication.

9.14. Corollary : Let G and H be groups. If f : X → Y is a Galois morphism of G sets and g : Z → T is a Galois morphism of H-sets, then f × g : X × Z → Y × T is a Galois morphism of (G × H)-sets.

Proof : Consider f as a morphism of (G, 1)-bisets, and g as a morphism of (1, H^{op})-bisets.

9.15. Notation : Let G-Mor^{Gal} denote the full subcategory of G-Mor consisting of Galois morphisms of G-sets.

9.16. Notation : Let X →^{f} Y be a morphism of G-sets. For x ∈ X, set

G_{x}^{f} = <G_{z} | ∀z ∈ X, f(z) = f(x)> .

Let ∼_{f} be the relation on X defined by

x ∼_{f} x′ ⇔ ∃g ∈ G_{x}^{f}, gx = x′ .

9.17. Lemma : With this notation :

1. The relation ∼_{f} is an equivalence relation on X. Let X_{f}^{Gal} denote the set of equivalence classes, and let γ_{X,f} : X → X_{f}^{Gal} denote the projection map.

2. If x, x′ ∈ X and x ∼_{f} x′, then gx ∼_{f} gx′ for any g ∈ G. Hence there exists a unique structure of G-set on X_{f}^{Gal} such that γ_{X,f} is a morphism

of G-sets.

3. There is a unique map f^{Gal} : X_{f}^{Gal} → Y such that the diagram

\xymatrix{X \ar[r]^{f} \ar[d]_{γ_{X,f}} & Y \\ X_{f}^{Gal} \ar@{-->}[ur]_{f^{Gal}}}

is commutative.

Proof : For Assertion 1, the relation ∼_{f} is clearly reflexive, since G_{x} ≤ G_{x}^{f}. Now if x, x′ ∈ X and x ∼_{f} x′, let g ∈ G_{x}^{f} such that gx = x′. There exist r ∈ ℕ, elements z_{1}, …, z_{r} in X and elements g_{1}, …, g_{r} of G such that g_{i}z_{i} = z_{i} and f(z_{i}) = f(x), for i = 1, …, r, and such that g = g_{1} ⋯ g_{r}. It follows that

f(x′) = g_{1} ⋯ g_{r}f(x) = g_{1} ⋯ g_{r}f(z_{r}) = g_{1} ⋯ g_{r−1}f(g_{r}z_{r}) = g_{1} ⋯ g_{r−1}f(z_{r})
= g_{1} ⋯ g_{r−1}f(x) ,

thus f(x′) = f(x), by induction on r. It follows that G_{x′}^{f} = G_{x}^{f}, hence that x′ ∼_{f} x, since x = g^{−1}x′. So the relation ∼_{f} is symmetric.

Finally ∼_{f} is transitive : if x, x′, x′′ ∈ X, if x ∼_{f} x′ and x′ ∼_{f} x′′, there exist g ∈ G_{x}^{f} and g′ ∈ G_{x′}^{f} such that gx = x′ and g′x′ = x′′. But the previous argument shows that G_{x}^{f} = G_{x′}^{f} = G_{x′′}^{f}. Hence g′g ∈ G_{x}^{f}, and x ∼_{f} x′′, since g′gx = x′′.

Assertion 2 follows from the straightforward fact that ^{g}(G_{x}^{f}) = G_{gx}^{f} for any x ∈ X and any g ∈ G. Assertion 3 follows as well, since x ∼_{f} x′ implies f(x) = f(x′).

9.18. Proposition : Let G be a finite group.

1. Let X →^{f} Y be a morphism of G-sets. Then the morphism X_{f}^{Gal} ⟶^{f^{Gal}} Y is a Galois morphism of G-sets.

[figure: 2. If; X; f; α; Y; β; A; a; B]

is a morphism in the category G-Mor, and if A →^{a} B is a Galois morphism of G-sets, then there exists a unique morphism of G-sets α̃ : X_{f}^{Gal} → A such that the diagram

[figure: X; f; γ_{X,f}; α; Y; X_{f}^{Gal}; α̃; f^{Gal}; Y; β; A; a; B]

is commutative.

3. The correspondence sending X →^{f} Y to X_{f}^{Gal} ⟶^{f^{Gal}} Y is a functor from G-Mor to G-Mor^{Gal}, and this functor is left adjoint to the forgetful functor G-Mor^{Gal} → G-Mor.

Proof : When x ∈ X, let x̃ = γ_{X,f}(x) ∈ X_{f}^{Gal} denote its equivalence class for the relation ∼_{f}. Let g ∈ G. Then gx̃ = x̃ if and only if gx ∼_{f} x, i.e. if there exists h ∈ G_{x}^{f} such that gx = hx, or equivalently h^{−1}g ∈ G_{x}. Hence the stabilizer of x̃ in G is equal to G_{x}^{f} · G_{x} = G_{x}^{f}, since G_{x} ≤ G_{x}^{f}. So if x, x′ ∈ X are such that f^{Gal}(x̃) = f^{Gal}(x̃′), i.e. if f(x) = f(x′), then the stabilizers G_{x}^{f} of x̃ and G_{x′}^{f} of x̃′ are equal, since G_{x}^{f} depends only on f(x). Assertion 1 follows, by Proposition 9.3.

Let x, x′ ∈ X such that x ∼_{f} x′. Then there exist r ∈ ℕ, elements z_{1}, …, z_{r} in X and elements g_{1}, …, g_{r} of G such that g_{i}z_{i} = z_{i} and f(z_{i}) = f(x), for i = 1, …, r, and such that g = g_{1} ⋯ g_{r}. It follows that

βf(z_{i}) = aα(z_{i}) = βf(x) = aα(x) ,

for i = 1, …, r. By Proposition 9.3, since A →^{a} B is a Galois morphism, this implies G_{α(z_{i})} = G_{α(x)}. Moreover G_{z} ≤ G_{α(z)} for any z ∈ X. Thus g_{i} ∈ G_{α(z_{i})} = G_{α(x)} for i = 1, …, r. It follows that g = g_{1} ⋯ g_{r} ∈ G_{α(x)}, hence α(x′) = gα(x) = α(x). This shows the existence of a map α̃ : X_{f}^{Gal} → A, sending the equivalence class of x ∈ X for ∼_{f} to α(x). Such a map is obviously unique, and it is a morphism of G-sets. This proves Assertion 2.

For Assertion 3, suppose that

\xymatrix{X \ar[r]^{f} \ar[d]_{α} & Y \ar[d]^{β} \\ X′ \ar[r]^{f′} & Y′}

is a morphism from X →^{f} Y to X′ →^{f′} Y′ in the category G-Mor. Then one can compose this morphism with the morphism

[figure: X′; f′; γ_{X′,f′}; Y′; _{X′{Gal \atop f′}}; _{f′_{Gal}}; Y′ .]

This yields a morphism from X →^{f} Y to X′_{f′}^{Gal} ⟶^{f′^{Gal}} Y′, which is a Galois morphism by Assertion 1. By Assertion 2, this composition factors in a unique way through the morphism X_{f}^{Gal} ⟶^{f^{Gal}} Y. In other words there is a unique morphism of G-sets α̃ : X_{f}^{Gal} → X′_{f}^{Gal} such that the diagram

[figure: X; f; α; γ_{X,f}; Y; β; X′; f′; γ_{X′,f′}; Y′; X_{f}^{Gal}; f^{Gal}; α̃; Y; β; _{X′{Gal \atop f′}}; _{f′_{Gal}}; Y′]

is commutative. Clearly, the map (α, β) ↦ (α̃, β) endows the correspondence

(X →^{f} Y ) ↦ (X_{f}^{Gal} ⟶^{f^{Gal}} Y )

with a structure of functor from G-Mor to G-Mor^{Gal}. Moreover, for any Galois morphism A →^{a} B, Assertion 2 yields a bijection

Hom_{G-Mor}(X →^{f} Y, A →^{a} B) ≅ Hom_{G-Mor^{Gal}}(X_{f}^{Gal} ⟶^{f^{Gal}} Y, A →^{a} B) ,

and this bijection is clearly functorial with respect to X →^{f} Y and A →^{a} B. Assertion 3 follows.

9.19. Example : Let (T, S) be a slice of G, and f be the projection map from X = G/S to Y = G/T. Then for x = S ∈ X, the group G_{x}^{f} is the group generated by the stabilizers G_{tS}, for t ∈ T. As G_{tS} = ^{t}S, it follows that G_{x}^{f} is equal to the normal closure S^{⊴ T} of S in T. In this case moreover, the G-set X_{f}^{Gal} is isomorphic to G/(S^{⊴ T}), and the map f^{Gal} is the projection to G/T.

10. The section Burnside functor

10.1. Proposition : If X →^{f} Y and X′ →^{f′} Y′ are Galois morphisms of G-sets, then X ⊔X′ ⟶^{f⊔f′} Y ⊔Y′ and X ×X′ ⟶^{f×f′} Y ×Y′ are Galois morphisms

of G-sets.

Proof : The case of disjoint union follows from Proposition 9.10 and Corollary 9.6. Moreover Proposition 9.14 shows that if f : X → Y and f′ : X′ → Y′ are Galois morphisms of G-sets, then

f × f′ : X × X′ → Y × Y′

is a Galois morphism of (G × G)-sets. By Corollary 9.13, the restriction of this morphism to the diagonal G ≤ G × G is a Galois morphism of G-sets.

10.2. Definition : Let G be a finite group. The section Burnside group Γ(G) of G is the subgroup of the slice Burnside ring Ξ(G) generated by the classes of Galois morphisms of G-sets.

By Proposition 10.1, the group Γ(G) is actually a subring of Ξ(G), called the section Burnside ring of G.

10.3. Lemma : Let f : X → Y be a Galois morphism of finite G-sets. Then in the group Γ(G)

π(X →^{f} Y) = ∑_{x∈[G\X]}⟨G_{f(x)}, G_{x}⟩_{G}
= ∑_{y∈[G\Y]} |G_{y}\f^{−1}(y)| ⟨G_{y}, G_{x_{y}}⟩_{G} ,

where x_{y} is chosen in f^{−1}(y), for each y ∈ [G\f(X)].

Proof : The first formula follows from Lemma 3.4. For the second one, write

π(X →^{f} Y) = ∑_{y∈[G\Y]} ∑_{x∈[G_{y}\f^{−1}(Y)]} π(G/G_{x} → G/G_{y}) ,

and observe that G_{x} = G_{x′} if f(x) = f(x′).

10.4. Corollary : The elements ⟨T, S⟩_{G}, where (T, S) runs through a set [Σ(G)] of representatives of conjugacy classes of sections of G, form a basis

of Γ(G).

Proof : Indeed, these elements generate Γ(G), by Proposition 10.3, and they are linearly independent, by 4.6.

10.5. Remark : This also shows that Γ(G) is the quotient of the free abelian group on the set of isomorphism classes [X ⟶^{f} Y] of Galois morphisms of finite G-sets, by the subgroup generated by elements of the form

[(X_{1} ⊔ X_{2}) ⟶^{f_{1}⊔f_{2}} Y] − [X_{1} →^{f_{1}} f(X_{1})] − [X_{2} →^{f_{2}} f(X_{2})] ,

whenever X →^{f} Y is a Galois morphism of finite G-sets with a decomposition X = X_{1} ⊔ X_{2} as a disjoint union of G-sets, where f_{1} = f_{|X_{1}} and f_{2} = f_{|X_{2}} .

10.6. Remark : Since Γ(G) is a subring of Ξ(G), the product formula of Proposition 3.8

⟨T, S⟩_{G}⟨Y, X⟩_{G} = ∑_{g∈[S\G/X]}⟨T ∩^{g}Y , S ∩^{g}X⟩_{G} ,

also expresses the product of two sections ⟨T, S⟩_{G} and ⟨Y, X⟩_{G} of G. Other ring structures with the same additive group Γ(G), but different multiplications, have been considered by O. Coşkun in [5] (Definition 2.2 and Remark 2.3), as well as their functorial properties.

10.7. Theorem :

1. Let G and H be finite groups, and let U be a finite (H, G)-biset. The

{functor \atop (X →^{f} Y) ↦ (U ×_{G} X ⟶^{U×_{G}f} U ×_{G} Y)}

from G-Mor^{Gal} to H-Mor^{Gal} induces a group homomorphism

Γ(U) : Γ(G) → Γ(H) .

2. The correspondence G ↦ Γ(G) is a Green biset functor.

Proof : This follows from Theorem 3.9 and Corollary 9.12.

10.8. Remark : It follows from Remark 9.2 that the image of the morphism i : B → Ξ of Proposition 3.11 is actually contained in Γ. Thus i is a morphism of Green biset functors from B to Γ.

11. Sections and ghost map

11.1. Notation : If (T, S) is a slice of G, denote by ψ_{T,S} : Γ(G) → ℤ the restriction of the ring homomorphism ϕ_{T,S} : Ξ(G) → ℤ.

11.2. Lemma : Let (T, S), (T′, S′) ∈ Π(G). Then ψ_{T,S} = ψ_{T′,S′} if and only if the sections (T, S^{⊴ T}) and (T′, S^{′ ⊴ T′}) are conjugate in G (recall that S^{⊴ T} denotes the normal closure of S in T). In particular ψ_{T,S} = ψ_{T,S^{⊴ T}}.

Proof : This follows from Proposition 9.18 and Remark 9.19, but the following is a short direct proof : by Lemma 4.3, ψ_{T,S} = ψ_{T′,S′} if and only if for any section (V, U) of G

|{g ∈ G/U | (T, S)^{g} ⪯ (V, U)}| = |{g ∈ G/U | (T′, S′)^{g} ⪯ (V, U)}| .

Taking (V, U) = (T, S^{⊴ T}) shows that there exists g ∈ G such that (T′, S′)^{g} ≤ (T, S^{⊴ T}). This implies (T′, S^{′ ⊴ T′})^{g} ⪯ (T, S^{⊴ T}). Taking now (V, U) = (T′, S^{′ ⊴ T′}) shows that (T′, S^{′ ⊴ T′})^{g′} ⪯ (T, S^{⊴ T}), for some g′ ∈ G. It follows that (T′, S^{′ ⊴ T′})^{g} = (T, S^{⊴ T}).

11.3. Theorem : The map (called the ghost map for Γ(G))

Ψ = ∏_{(T,S)∈[Σ(G)]} ψ_{T,S} : Γ(G) → ∏_{(T,S)∈[Σ(G)]} ℤ

is an injective ring homomorphism, with finite cokernel as morphism of abelian groups.

Proof : The proof is exactly the same as for Theorem 4.6 : by Lemma 3.5, the elements ⟨T, S⟩_{G}, for (T, S) ∈ [Σ(G)], generate Γ(G). Suppose that there is a non zero linear combination in the kernel of Ψ

Λ = ∑_{(T,S)∈[Σ(G)]} λ_{T,S} ⟨T, S⟩_{G}

with integer coefficients λ_{T,S} ∈ ℤ, for (T, S) ∈ [Σ(G)]. Extend λ to a function Σ(G) → ℤ, constant on conjugacy classes. Let (Y, X) be an element of Σ(G), maximal for the relation ⪯, such that λ_{Y,X} ≠ 0. Then since by Lemma 4.3

ψ_{Y,X}(G/S → G/T) = |{g ∈ G/S | (Y^{g}, X^{g}) ⪯ (T, S)}| ,

it follows that

ψ_{Y,X}(Λ) = ∑_{(T,S)∈[Σ(G)]} λ_{T,S}ψ_{Y,X}(G/S → G/T)
= λ_{Y,X}ψ_{Y,X}(G/X → G/Y)
= λ_{Y,X}|N_{G}(X, Y)/X| = 0 .

Hence λ_{Y,X} = 0, and this contradiction shows that Ψ is injective (and in particular, we recover the fact that the elements ⟨T, S⟩_{G}, for (T, S) ∈ [Σ(G)], form a ℤ-basis of Γ(G)). Thus Ψ is an injective morphism between free abelian groups with the same finite rank, hence it has finite cokernel.

11.4. Corollary : Set ℚΓ(G) = ℚ ⊗_{ℤ} Γ(G), and ℚΨ = ℚ ⊗_{ℤ} Ψ. Then

ℚΨ : ℚΓ(G) → ∏_{(T,S)∈[Σ(G)]} ℚ

is an isomorphism of ℚ-algebras.

12. Sections and idempotents

Corollary 11.4 shows that ℚΓ(G) is a split semisimple commutative algebra. Its primitive idempotents are indexed by sections of G, up to conjugation : they are the inverse images under ℚΨ of the primitive idempotents of the

algebra ∏_{(T,S)∈[Σ(G)]} ℚ :

12.1. Notation : If (T, S) ∈ Σ(G), denote by γ_{T,S}^{G} the unique element of ℚ ⊗_{ℤ} Γ(G) such that

∀(Y, X) ∈ Σ(G), ℚψ_{Y,X}(γ_{T,S}^{G}) = \{ \begin{matrix}1 & if (Y, X) =_{G} (T, S) \\ 0 & otherwise \end{matrix}

The elements γ_{T,S}^{G}, for (T, S) ∈ Σ(G), are the primitive idempotents of ℚΓ(G).

12.2. Theorem : Let ⪯ denote the restriction of the relation ⪯ to Σ(G). Then for (T, S) ∈ Σ(G)

γ_{T,S}^{G} = \frac{1}{|N_{G}(T, S)|} ∑_{(V,U)⪯(T,S)} |U|μ_{Σ}((V, U), (T, S))⟨V, U⟩_{G} ,

where μ_{Σ} is the Möbius function of the poset (Σ(G), ⪯).

Proof : The proof is the same as the proof of Theorem 5.2.

12.3. Proposition : Let (X , ≤) be a finite poset. Let φ : X → X be a map of posets such that φ ∘ φ = φ and x ≤ φ(x) for any x ∈ X. Then the Möbius function μ_{Y} of the subposet Y = φ(X) of X is given by

(12.4)  ∀y, z ∈ Y, μ_{Y}(y, z) = ∑_{{y≤u∈X \atop φ(u)=z}} μ_{X} (y, u) ,

where μ_{X} is the Möbius function of (X , ≤).

Proof : For y, z ∈ Y, denote by m(y, z) the right hand side of Equation 12.4. Then for y, t ∈ Y

∑_{{z∈Y \atop y≤z≤t}} m(y, z) = ∑_{{z∈Y \atop y≤z≤t}} ∑_{{y≤u∈X \atop φ(u)=z}} μ_{X} (y, u)

= ∑_{(z,u)∈P} μ_{X} (y, u) ,

where

P = {(z, u) | z ∈ Y, u ∈ X , y ≤ u ≤ φ(u) = z ≤ t} .

Set Q = {u ∈ X | y ≤ u ≤ t}. If (z, u) ∈ P, then clearly u ∈ Q. Conversely, if u ∈ Q, then (φ(u), u) ∈ P : indeed φ(u) ∈ Y = φ(X), and moreover φ(u) ≤ φ(t) = t, since t ∈ φ(X) and φ ∘ φ = φ. Now the maps (z, u) ∈ P ↦ u ∈ Q and u ∈ Q ↦ (φ(u), u) ∈ P are mutual inverse bijections. It follows that

∑_{{z∈Y \atop y≤z≤t}} m(y, z) = ∑_{{u∈X \atop y≤u≤t}} μ_{X} (y, u) .

This is equal to 1 if y = t, and to zero otherwise. The proposition follows.

12.5. Corollary : Let (V, U) and (T, S) be sections of G. Then

(12.6)  μ_{Σ(G)}((V, U), (T, S)) = μ(V, T)( ∑_{{U≤X≤V \atop X^{⊴ T} =S}} μ(U, X)) .

In particular in ℚΓ(G)

γ_{T,S}^{G} = \frac{1}{|N_{G}(T, S)|} ∑_{U ⊴ V ≤T ; {U≤X≤V \atop X^{⊴ T} =S}} |U|μ(U, X)μ(V, T) ⟨V, U⟩_{G} .

Proof : For Equation 12.6, apply Proposition 12.3 to the poset X = (Π(G), ⪯ ), and to the map φ : X → X defined by

φ((Y, X)) = (Y, X^{⊴ Y}) ,

and then use Equation 5.6. Then substitute the value of μ_{Σ}((V, U), (T, S)) in the formula of Theorem 12.2.

13. The image of the ghost map

The following is a characterization of the image of the ghost map for the section Burnside ring, similar to Theorem 6.1 :

13.1. Theorem : Let G be a finite group. Let m = (m(T, S))_{(T,S)∈Σ(G)} be a sequence of integers indexed by Σ(G), constant on G-conjugacy classes of sections. Then the sequence [m] = (m(T, S))_{(T,S)∈[Σ(G)]} of representatives lies in the image of the ghost map Ψ if and only if, for any section (T, S)

of G ; ∑_{g∈N_{G}(T,S)/S} m(<gT >, <gS>^{⊴ <gT >}) ≡ 0 (mod. |N_{G}(T, S)/S|) .

Proof : The proof is very similar to the proof of Theorem 6.1, with two differences : the first one is that the poset Π(G) of slices of G has to be replaced by the poset Σ(G) of sections of G. The second one is that the slice (<gT >, <gS>) has to be changed to the corresponding section (<gT >, <gS>^{⊴ gT} ) of G. Apart from these two differences, the proof goes through without changes.

14. Prime spectrum

14.1. Notation : Let p denote either 0 or a prime number.

• If (T, S) ∈ Σ(G), denote by J_{T,S,p} the prime ideal I_{T,S,p} ∩Γ(G) of Γ(G). In other words J_{T,S,p} is the kernel of the ring homomorphism

Γ(G) ⟶^{ψ_{T,S}} ℤ → ℤ/pℤ ,

where the right hand side map is the projection.

• Let Ω(G) denote the set of triples (T, S, p), where (T, S) ∈ Σ(G) is such that |N_{G}(T, S)/S| ≢ 0 (mod. p).

The group G acts on Ω(G), by ^{g}(T, S, p) = (^{g}T ,^{g}S, p), for g ∈ G, and the ideal J_{T,S,p} only depends on the G-orbit of (T, S, p). Conversely :

14.2. Proposition : Let I be a prime ideal of Γ(G), and R = Γ(G)/I. Denote by ψ : Γ(G) → R the projection map, and denote by p ≥ 0 the characteristic of R. Then R ≅ ℤ/pℤ and :

1. If p = 0, there exists a section (T, S) of G such that ψ = ψ_{T,S}, and (T, S) is unique up to G-conjugation, with this property.

2. If p > 0, there exists a section (T, S) of G such that ψ is the reduction modulo p of ψ_{T,S} and N_{G}(T, S)/S is a p′-group, and (T, S) is unique up to G-conjugation, with these properties.

In particular, there exists a unique (T, S, p) ∈ Ω(G), up to conjugation, such

that I = J_{T,S,p}.

Proof : The proof is exactly the same as the proof of Proposition 7.2, with slices replaced by sections : let (T, S) be a section of G, minimal for the relation ⪯, such that ⟨T, S⟩_{G} ∉ I. Then by Proposition 3.8, for any (Y, X) ∈ Σ(G)

⟨T, S⟩_{G}⟨Y, X⟩_{G} = ∑_{g∈[S\G/X]}⟨T ∩^{g}Y , S ∩^{g}X⟩_{G}
≡ ∑_{g∈G/X ; {S≤^{g}X \atop T ≤^{g}Y}} ⟨T, S⟩_{G} (mod. I)

= ψ_{T,S}(⟨Y, X⟩_{G})⟨T, S⟩_{G} .

Since I is prime, it follows that ⟨Y, X⟩_{G} − ψ_{T,S}(⟨Y, X⟩_{G})1_{Γ(G)} ∈ I. In particular R = Γ(G)/I is generated by the image of 1_{Γ(G)}, hence R ≅ ℤ/pℤ, where p is the characteristic of R. Since R is an integral domain, the number p is either 0 or a prime.

1. If p = 0, then R = ℤ, and ψ = ψ_{T,S}. And if (T′, S′) ∈ Σ(G) is such that ψ_{T,S} = ψ_{T′,S′}, then both ψ_{T,S}(⟨T′, S′⟩_{G}) and ψ_{T′,S′}(⟨T, S⟩_{G}) are non zero. Then there exist elements g, g′ ∈ G such that (T^{g}, S^{g}) ⪯ (T′, S′) and (T^{′g′}, S^{′g′}) ⪯ (T, S), so (T, S) and (T′, S′) are conjugate in G.

2. If p > 0, then R = ℤ/pℤ, and ψ is equal to the reduction of ψ_{T,S} modulo p. Since ψ(⟨T, S⟩_{G}) = |N_{G}(T, S)/S| is non zero in R, it follows that N_{G}(T, S)/S is a p′-group. If (T′, S′) is another section of G such that ψ is the reduction modulo p of ψ_{T′,S′}, and N_{G}(T′, S′)/S′ is a p′- group, then

|N_{G}(T, S)/S| = ψ_{T,S}(⟨T, S⟩_{G}) ≡ ψ_{T′,S′}(⟨T, S⟩_{G}) (mod. p) .

This is non zero. Similarly |N_{G}(T, S)/S| ≡ ψ_{T,S}(⟨T′, S′⟩_{G}) (mod. p) is non zero. In particular ψ_{T′,S′}(⟨T, S⟩_{G}) and ψ_{T,S}(⟨T′, S′⟩_{G}) are both non zero, and it follows as above that (T, S) and (T′, S′) are conjugate in G.

14.3. Notation : Let p be a prime number.

• Let Σ_{p}(G) denote the set of sections (T, S) of G such that N_{G}(T, S)/S is a p′-group. In other words Σ_{p}(G) = Σ(G) ∩ Π_{p}(G).

• For any (T, S) ∈ Σ(G), let (T, S)_{p̂} denote the unique element (V, U) of Σ_{p}(G), up to conjugation, such that J_{T,S,p} = J_{V,U,p}.

14.4. Proposition : Let p be a prime number. If (T, S) is a section of G, let (T, S)_{+}_{p} denote a section of the form (P T, (P S)^{⊴ P T} ) of G, where P is a Sylow p-subgroup of N_{G}(T, S).

Define inductively an increasing sequence (T_{n}, S_{n}) in (Σ(G), ⪯) by

(T_{0}, S_{0}) = (T, S), and (T_{n+1}, S_{n+1}) = (T_{n}, S_{n})_{+}_{p}, for n ∈ ℕ. Then (T, S)_{p̂}

is conjugate to the largest term (T_{∞}, S_{∞}) of the sequence (T_{n}, S_{n}).

Proof : Again, the proof is the same as the proof of Proposition 7.4 : the section (T, S)_{p̂} is a minimal element (V, U) of the poset (Σ(G), ⪯) such that

ψ_{T,S}(V, U) = |{g ∈ G/U | (T^{g}, S^{g}) ⪯ (V, U)}| ≢ 0 (mod.p) .

Thus one can assume that (T, S) ⪯ (V, U). But ψ_{T,S} ≡ ψ_{P T,(P S)^{^{⊴} ^{P T}}} (mod. p) by Corollary 4.4 and Lemma 11.2, for any p-subgroup P of N_{G}(T, S), hence one can also assume that (T, S)_{+}_{p} ⪯ (V, U), hence that (T_{∞}, S_{∞}) ⪯ (V, U), by induction. Moreover ψ_{T_{∞},S_{∞}} ≡ ψ_{V,U} (mod. p). As N_{G}(T_{∞}, S_{∞})/S_{∞} is a p′-group, it follows that (T_{∞}, S_{∞}) = (V, U), as was to be shown.

14.5. Proposition : Let (T, S, p), (T′, S′, p′) be elements of Ω(G). Then J_{T′,S′,p′} ⊆ J_{T,S,p} if and only if • either p′ = p and the sections (T′, S′) and (T, S) are conjugate in G.

• or p′ = 0 and p > 0, and the sections (T′, S′)_{p̂} and (T, S) are conjugate

in G.

Proof : (see Proposition 7.6) Set J = J_{T,S,p} and J′ = J_{T′,S′,p′}. Then Γ(G)/J′ ≅ ℤ/p′ℤ maps surjectively to Γ(G)/J ≅ ℤ/pℤ. Thus if p = p′, this projection map is an isomorphism, hence J = J′ and the sections (T, S) and (T′, S′) are conjugate in G. And if p ≠ p′, then p′ = 0 and p > 0. The morphism ψ_{T,S} is equal to the reduction modulo p of the morphism ψ_{T′,S′}. In other words J_{T′,S′,p} = J_{T,S,p}, hence (T, S) is conjugate to (T′, S′)_{p̂}.

14.6. Corollary : Let p be a prime number, and let ℤ_{(p)} be the localization of ℤ at the set ℤ − pℤ. Let Ω_{p}(G) denote the subset of Ω(G) consisting of triples (T, S, 0), for (T, S) ∈ Σ(G), and (T, S, p), for (T, S) ∈ Σ_{p}(G). Then :

1. The prime ideals of the ring ℤ_{(p)}Γ(G) are the ideals ℤ_{(p)}J_{T,S,q}, for

(T, S, q) ∈ Ω_{p}(G).

2. If (T, S, q), (T′, S′, q′) ∈ Θ_{p}(G), then ℤ_{(p)}J_{T′,S′,q′} ⊆ ℤ_{(p)}J_{T,S,q} if and only if :

• either q = q′, and the sections (T, S) and (T′, S′) are conjugate

in G.

• or q′ = 0, q = p, and the sections (T′, S′)_{p̂} and (T, S) are conjugate

in G.

3. The connected components of the spectrum of ℤ_{(p)}Γ(G) are indexed by the conjugacy classes of Σ_{p}(G). The component indexed by (T, S) ∈ Σ_{p}(G) consists of a unique maximal element ℤ_{(p)}J_{T,S,p}, and of the ideals ℤ_{(p)}J_{T′,S′,0}, where (T′, S′) ∈ Σ(G) is such that (T′, S′)_{p̂} is conjugate to

(T, S) in G.

Proof : (See Corollary 7.7) The prime ideals of ℤ_{(p)}Γ(G) are of the form ℤ_{(p)}I, where I is a prime ideal of Γ(G) such that I ∩ (ℤ − pℤ) = ∅. Equivalently I = J_{T,S,0} or I = J_{T,S,p}. This proves Assertion 1. Now Assertion 2 follows from Proposition 14.5, and Assertion 3 follows from Assertion 2.

14.7. Corollary :

1. The primitive idempotents of the ring ℤ_{(p)}Γ(G) are indexed by the conjugacy classes of Σ_{p}(G). The primitive idempotent ε_{V,U}^{G} indexed by

(V, U) ∈ Σ_{p}(G) is equal to

ε_{V,U}^{G} = ∑_{{(T,S)∈[Σ(G)] \atop (T,S)_{p̂}=_{G}(T,S)}} γ_{T,S}^{G} .

2. Let π be a set of prime numbers, and ℤ_{(π)} be the localization of ℤ relative to ℤ−∪_{p∈π}pℤ. Let F be a set of sections of G, invariant by Gconjugation, and [F] be a set of representatives of G-conjugacy classes of F. Then the following conditions are equivalent :

(a) The idempotent

γ_{F}^{G} = ∑_{(T,S)∈[F]} γ_{T,S}^{G}

of ℚΓ(G) lies in ℤ_{(π)}Γ(G).

(b) Let (T, S) ∈ Σ(G), and let P be a p-subgroup of N_{G}(T, S), for some p ∈ π. Then (T, S) ∈ F if and only if (P T, (P S)^{⊴ P T} ) ∈ F.

Proof : The proof is almost identical to the proof of Corollary 7.8 : let F be a set of sections of G, invariant by G-conjugation, and [F] be a set of representatives of G-conjugacy classes of F. The idempotent

γ_{F}^{G} = ∑_{(T,S)∈[F]} γ_{T,S}^{G}

of ℚΓ(G) lies in ℤ_{(p)}Γ(G), for some prime p, if and only if there exists an integer m, not divisible by p, such that u = mγ_{F}^{G} ∈ Γ(G). Let (T, S) ∈ Σ(G), and let P be a p-subgroup of N_{G}(T, S). The integer ψ_{T,S}(u) is equal to m if (T, S) ∈ F, and to 0 otherwise. Hence it is coprime to p if and only if (T, S) ∈ F. Since ψ_{T,S} and ψ_{P T,(P S)^{⊴ P T}} are congruent modulo p, it follows that (T, S) ∈ F if and only if (P T, (P S)^{⊴ P T} ) ∈ F.

Hence if (T, S) and (T′, S′) are sections of G such that (T, S)_{p̂} =_{G} (T′, S′)_{p̂}, then (T, S) ∈ F if and only if (T′, S′) ∈ F. Thus F is a disjoint union of sets of the form

E_{V,U} = {(T, S) ∈ Σ(G) | (T, S)_{p̂} =_{G} (V, U)} ,

for some sections (V, U) ∈ Σ_{p}(G). In other words the idempotent γ_{F}^{G} is a sum of some idempotents ε_{V,U}^{G}, for (V, U) ∈ Σ_{p}(G).

But the primitive idempotents of the ring ℤ_{(p)}Γ(G) are in one to one correspondence with the connected components of its spectrum, which precisely are indexed by the conjugacy classes of Σ_{p}(G). It follows that γ_{F}^{G} = ε_{V,U}^{G} is equal to the idempotent corresponding to the component indexed by (V, U), for any (V, U) ∈ Σ_{p}(G). This proves Assertion 1. This also proves Assertion 2 in the case where π consists of a single prime number.

For the general case, observe that γ_{F}^{G} lies in ℤ_{(π)}Γ(G) if and only if it lies in ℤ_{(p)}Γ(G), for any p ∈ π.

The following proposition is the analogue of Theorem 7.9, for the ring Γ(G). Its statement is a bit simpler :

14.8. Theorem : Let G be a finite group.

1. The primitive idempotents of Γ(G) are indexed by the conjugacy classes of perfect subgroups of G. The idempotent γ_{H}^{G} indexed by the perfect subgroup H is equal to

γ_{H}^{G} = ∑_{{(T,S)∈[Σ(G)] \atop D^{∞}(T)=_{G}H}} γ_{T,S}^{G} ,

where D^{∞}(T) denotes the last term in the derived series of T.

2. The prime spectrum of Γ(G) is connected if and only if G is solvable.

Proof : As in the proof of Theorem 7.9, let ∼ denote the finest equivalence relation on the set Σ(G) such that for any (T, S), (T′, S′) ∈ Σ(G)

∃p, (T, S)_{p̂} =_{G} (T′, S′)_{p̂} ⟹ (T, S) ∼ (T′, S′) .

If we can show that

(T, S) ∼ (T′, S′) ⇔ D^{∞}(T) =_{G} D^{∞}(T′) ,

then Assertion 1 follows from Corollary 14.7, applied to the set π of all primes.

Clearly, if there exists a p-subgroup P ≤ N_{G}(T, S) such that (T′, S′) is conjugate to (P T, (P S)^{⊴ P T} ), then T′ is conjugate to P T, and D^{∞}(T′) is conjugate to D^{∞}(T). By transitivity, for any (T, S), (T′, S′) ∈ Σ(G), if (T, S) ∼ (T′, S′), then D^{∞}(T) =_{G} D^{∞}(T′).

To show the converse, is it enough to show that (T, S) ∼ (D^{∞}(T), D^{∞}(T)), for any (T, S) ∈ Σ(G). Let p be any prime, and P be a Sylow p-subgroup of T. Then (T, S)_{p̂} = (T, (P S)^{⊴ P T} )_{p̂}, hence (T, S) ∼ (T, S′), where S′ = (P S)^{⊴ P T} is a normal subgroup of T, containing S, and of p′-index in T. Since p was arbitrary, it follows by induction that (T, S) ∼ (T, T). Now for any prime p, if P is a Sylow p-subgroup of T, then P ≤ N_{G}(O^{p}(T)), and T = P O^{p}(T). It follows that (T, T)_{p̂} = (O^{p}(T), O^{p}(T))_{p̂}. Again, since p is arbitrary, it follows that (T, T) ∼ (D^{∞}(T), D^{∞}(T)), and this completes the proof of Assertion 1.

Assertion 2 follows easily, since by Assertion 1, the spectrum of Γ(G) is connected if and only if the trivial group is the only perfect subgroup of G, i.e. if G is solvable.

14.9. Corollary : The images of the primitive idempotents of B(G) by the morphism i_{G} : B(G) → Γ(G) are the primitive idempotents of Γ(G). In other words, if H is a perfect subgroup of G, then

γ_{H}^{G} = \frac{1}{|N_{G}(H)|} ∑_{K≤H} |K|μ(K, H)⟨K, K⟩_{G} .

Proof : Indeed by a theorem of Dress ([6]), the primitive idempotents of B(G) are indexed by the conjugacy classes of perfect subgroups of G. As the morphism i_{G} : B(G) → Γ(G) is an injective unital ring homomorphism (see Remark 10.8 and Proposition 3.11), it follows that the primitive idempotents of B(G) are mapped to primitive idempotents in Γ(G), and that every primitive idempotent of Γ(G) is in the image of B(G).

15. Unit group

All the results of Section 8 about the unit group of the slice Burnside ring have an analogue for the group Γ(G)^{×} of the section Burnside ring of a finite group G. Namely :

• The restricted ghost map yields an injective group homomorphism

Ψ^{×} : Γ(G)^{×} ↪ ∏_{(T,S)∈Σ(G)} ℤ^{×} .

The following lemma follows :

15.1. Lemma : Let G be a finite group, and let u ∈ Γ(G). The following conditions are equivalent :

1. u ∈ Γ(G)^{×}.

2. ψ_{T,S}(u) ∈ {±1}, for any (T, S) ∈ Σ(G).

3. u^{2} = 1.

In particular Γ(G)^{×} is a finite elementary abelian 2-group.

• The following is an analogue of Proposition 8.3 :

15.2. Proposition : Feit-Thompson’s theorem is equivalent to the statement that, if G has odd order, then Γ(G)^{×} = {±1}.

Proof : The proof is the same as the proof of Proposition 8.3 : the argument uses only the formulas for the primitive idempotents of ℚΓ(G) (Corollary 12.5), and the characterization of solvable groups by the connectedness of the prime spectrum of Γ(G) (Proposition 14.8).

• The following theorem is an analogue of Yoshida’s characterization ([10]) of the unit group of the usual Burnside ring :

15.3. Theorem : Let G be a finite group, and let m = (m(T, S))_{(T,S)∈Σ(G)} be a sequence of integers in {±1} indexed by Σ(G), constant on G-conjugacy classes of sections. Then the sequence [m] = (m(T, S))_{(T,S)∈[Σ(G)]} of representatives lies in the image of the restricted ghost map Ψ^{×} if and only if for any (T, S) ∈ Σ(G), the map

g ∈ N_{G}(T, S)/S ↦ m(<gT >, <gS>^{⊴ <gT >})/m(T, S) ∈ {±1}

is a group homomorphism.

Proof : Again, the proof is the same as for Theorem 8.4 : it only requires Theorem 13.1 and Lemma 11.2.

• Finally, the correspondence G ↦ Γ(G)^{×} is not a biset functor :

15.4. Proposition : The correspondence sending a finite group G to Γ(G)^{×} cannot be endowed with a structure of biset functor.

Proof : The proof of Proposition 8.6 goes through without change here : the argument uses computations for the trivial group, the group C_{2}, and the group (C_{2})^{2}. All these groups are abelian, hence sections and slices coincide.

Appendix

16.1. Let G and H be finite groups, and U be a finite (H, G)-biset. Let U^{op} denote the opposite biset, i.e. the (G, H)-biset equal to U as a set, with actions reversed by taking inverses (i.e. g · u · h [in U^{op}] = h^{−1}ug^{−1} [in U], for g ∈ G, u ∈ U, and h ∈ H).

When X is a finite G-set, the set Hom_{G-set}(U^{op}, X) is a finite H-set : if φ : U^{op} → X is a morphism of G-sets, and if h ∈ H, then the morphism hφ : U^{op} → X is defined by (hφ)(u) = φ(h^{−1}u), for u ∈ U. This correspondence

T_{U} : X ↦ Hom_{G-set}(U^{op}, X)

is actually a functor from the category G-set of finite G-sets to the category H-set. One can show (see e.g. Section 11.2.13 of [4]) that this functor induces a map t_{U} : B(G) → B(H) between the usual Burnside rings of G and H, called the generalized tensor induction with respect to U. This induction is not additive, but multiplicative (i.e. t_{U}(ab) = t_{U}(a)t_{U}(b), for any a, b ∈ B(G)). It yields a biset functor structure on the unit group of the usual Burnside ring.

A natural question is to know whether this construction can be extended to the rings Ξ(G) and Γ(G) : indeed, if X →^{f} Y is a morphism of finite G-sets, then the morphism T_{U}(X →^{f} Y)

\xymatrix{Hom_{G-set}(U^{op}, X) \ar[r]^{Hom_{G-set}(U^{op},f)} & Hom_{G-set}(U^{op}, Y)}

is a morphism of finite H-sets. Does this correspondence induce a map Ξ(G) → Ξ(H) or a map Γ(G) → Γ(H) ?

In other words :

(Q1) Are the defining relations of Ξ(G) preserved by T_{U} ?

(Q2) Does T_{U} map a Galois morphism to a Galois morphism ?

The answer to these two questions has to be no in general, for otherwise T_{U} would yield a biset functor structure on the unit groups of the slice Burnside ring and the section Burnside ring, contradicting Proposition 8.6 and Proposition 15.4. But the answer is yes with the additional assumption that the biset U be left inert, according to the following definition :

16.2. Definition : An (H, G)-biset U is called left inert if Hu ⊆ uG, for any u ∈ U, in other words if H acts trivially on the set of orbits U/G.

16.3. Example :

• If U is transitive as a right G-set, then U is left inert, since |U/G| = 1. In particular, the identity (G, G)-biset G is left inert. Conversely, if U is left inert, then each right orbit uG, for u ∈ U, is an (H, G)-biset, so U is a disjoint union of (H, G)-bisets which are transitive as right G-sets.

• The disjoint union of two left inert (H, G)-bisets is left inert. Any sub-biset of a left inert biset if left inert.

• Left inert bisets can be composed : if G, H, and K are groups, if U is a left inert (H, G)-biset, and if V is a left inert (K, H)-biset, then V ×_{H} U is a left inert (K, G)-biset : indeed, for u ∈ U and v ∈ V

K(v,_{H} u) = (Kv,_{H} u) ⊆ (vH,_{H} u) = (v,_{H} Hu) ⊆ (v,_{H} uG) = (v,_{H} u)G ,

where (v,_{H} u) denotes the image of (v, u) in V ×_{H} U.

16.4. The following proposition deals with Question (Q2) above, in the case of a left inert biset :

16.5. Proposition : Let X →^{f} Y be a Galois morphism of G-sets. If U is a left inert (H, G)-biset, then the morphism T_{U}(X →^{f} Y) is a Galois morphism

of H-sets.

Proof : Let φ, ψ ∈ T_{U}(X) = Hom_{G-set}(U^{op}, X) having the same image in T_{U}(Y). This means that f ∘ φ = f ∘ ψ, i.e. that f(φ(u)) = f(ψ(u)), for any u ∈ U. Since f is a Galois morphism, it follows that G_{φ(u)} = G_{ψ(u)}, for any u ∈ U.

Suppose that h ∈ H stabilizes φ. It means that φ(h^{−1}u) = φ(u), for any u ∈ U. But since Hu ⊆ uG, there exists g ∈ G (depending on u) such that hu = ug, i.e. h^{−1}u = ug^{−1}. Then

φ(u) = φ(h^{−1}u) = φ(ug^{−1}) = gφ(u) ,

hence g ∈ G_{φ(u)}. It follows that

ψ(u) = gψ(u) = ψ(ug^{−1}) = ψ(h^{−1}u) = (hψ)(u) .

Hence h stabilizes ψ, and by symmetry φ and ψ have the same stabilizer in H. This shows that T_{U}(X →^{f} Y) is a Galois morphism of H-sets.

16.6. Remark : The following example shows that Lemma 16.5 is no longer true without the hypothesis that U is left inert : suppose that G = 1, and that H is non trivial. Let U = H, viewed as an (H, G)-biset. Then U is not left inert. Let X be a set of cardinality 2, let Y be a set of cardinality 1, and let f : X → Y be the unique map. Then f is a Galois morphism of G-sets, for trivial reasons.

In this case Hom_{G}(U^{op}, X) is isomorphic to the set 2^{H} of subsets of H, on which H acts by left translation. The set Hom_{G}(U^{op}, Y) is a set • of cardinality 1, and the map T_{U}(f) is the only possible map 2^{H} → •. In particular, all the elements of 2^{H} have the same image. But the subset {1} of H has a trivial stabilizer, whereas the subset H of H has stabilizer equal to H. Thus T_{U}(X →^{f} Y) is not a Galois morphism of H-sets, by Proposition 9.3.

16.7. Remark : The example given in Remark 16.6 also shows that the answer to Question (Q1) is no, in general : indeed, keeping the same notation, the image of the morphism X →^{f} Y in Ξ(G) is equal to the sum of two copies of the image of Y →^{Id} Y. In other words, in Ξ(G) ≅ ℤ,

π(X →^{f} Y) = 2 = π(X →^{Id} X) .

But T_{U}(X →^{f} Y) is isomorphic to 2^{H} → •, hence

(16.8)  π(T_{U}(X →^{f} Y)) = ∑_{{A⊆H \atop A mod.H}}⟨H, H_{A}⟩_{H}

by Lemma 3.4, where the summation runs over a set of representatives of subsets of H, up to translation by H, and H_{A} denotes the stabilizer in H of such a subset A. In particular H_{A} = H if and only if A = H or A = ∅, hence if K is a subgroup of H, the only term of the form ⟨K, K⟩_{H} in the right hand side of Equation 16.8 is ⟨H, H⟩_{H}, with coefficient 2.

On the other hand, by Lemma 3.4 again

π(X →^{Id} X) = ∑_{{A⊆H \atop A mod.H}}⟨H_{A}, H_{A}⟩_{H} .

As there are proper non empty subsets A of H, there are some terms of the form ⟨K, K⟩_{H} with non zero coefficient in this summation, for some subgroups K < H. Hence π(X →^{f} Y) ≠ π(X →^{Id} X), thus T_{U} does not preserve the defining relations of Ξ(G).

16.9. The following proposition answers Question (Q1) in the case of left inert bisets :

16.10. Proposition :

1. Let G and H be finite groups, and U be a finite left inert (H, G)-biset. Then T_{U} induces a well defined map t_{U} : Ξ(G) → Ξ(H), such that t_{U}(Γ(G)) ⊆ Γ(H). Moreover t_{U} only depends on the isomorphism class of the biset U.

2. The map t_{U} is multiplicative, i.e. t_{U}(ab) = t_{U}(a)t_{U}(b), for any a, b ∈ Ξ(G). Moreover t_{U}(1_{Ξ(G)}) = 1_{Ξ(H)}. In particular t_{U} restricts to group homomorphisms Ξ(G)^{×} → Ξ(H)^{×} and Γ(G)^{×} → Γ(H)^{×}.

3. If U and U′ are finite left inert (H, G)-bisets, then t_{U⊔U′} = t_{U}t_{U′}.

4. If G, H, and K are finite groups, if U is a finite left inert (H, G)-biset and V is a finite left inert (K, H)-biset, then t_{V} ∘ t_{U} = t_{V ×_{H}U}.

5. If U is the identity (G, G)-biset, then t_{U} is the identity map.

Proof : Since U is left inert, Example 16.3 shows that U splits as a disjoint union of (H, G)-bisets

U ≅ ⨆_{u∈[U/G]} uG .

It follows that the functor T_{U} = Hom_{G-set}(U, −) is isomorphic to the direct product of the functors T_{uG}, for u ∈ [U/G]. Hence to prove Assertion 1, it suffices to consider the case where U is transitive as a right G-set.

In this case, the functor T_{U} induces actually a group homomorphism from Ξ(G) to Ξ(H) : to see this, it suffices to check that the defining relations of Ξ(G) are mapped to relations in Ξ(H). Fix u ∈ U, and denote by G_{u} its stabilizer in G. So U = uG ≅ G/G_{u} as G-set, and for any G-set X, there is a bijection

Hom_{G-set}(U^{op}, X) ≅ X^{G_{u}} .

This is actually an isomorphism of H-sets if the left action of H on X^{G_{u}} is defined as follows : for h ∈ H, there is some g ∈ G such that hu = ug, and the action of h on X^{G_{u}} is defined by hx = g^{−1}x, for x ∈ X^{G_{u}}.

Now let X →^{f} Y be a morphism of finite G-sets such that X splits as a disjoint union X = X_{1} ⊔ X_{2} of two G-sets. The image π(X →^{f} Y) of this morphism in Ξ(G) is equal to the sum of the images of the morphisms

X_{1} →^{f_{1}} f(X_{1}) and X_{2} →^{f_{2}} f(X_{2}) ,

where f_{1} and f_{2} are the restrictions of f to X_{1} and X_{2}, respectively.

On the other hand T_{U}(X →^{f} Y) is isomorphic to X_{1}^{G_{u}} ⊔ X_{2}^{G_{u}} ⟶^{f^{Gu}} Y^{G_{u}} in the category H-Mor, where f^{G_{u}} is the restriction of f to X^{G_{u}}. The image π(T_{U}(X →^{f} Y)) of this morphism in Ξ(H) is equal to the sum of the images of the morphisms

X_{1}^{G_{u}} ⟶^{f_{1}^{Gu}} f(X_{1}^{G_{u}}) and X_{2}^{G_{u}} ⟶^{f_{2}^{Gu}} f(X_{2}^{G_{u}}) .

By Lemma 3.3, since f(X_{1}^{G_{u}}) ⊆ f(X_{1})^{G_{u}}

π(X_{1}^{G_{u}} ⟶^{f_{1}^{Gu}} f(X_{1}^{G_{u}})) = π(X_{1}^{G_{u}} ⟶^{f_{1}^{Gu}} f(X_{1})^{G_{u}}) ,

which is equal to π (T_{U}(X_{1} →^{f_{1}} f(X_{1}))). It follows that in Ξ(H)

π(T_{U}(X →^{f} Y)) = π (T_{U}(X_{1} →^{f_{1}} f(X_{1}))) + π (T_{U}(X_{2} →^{f_{2}} f(X_{2}))) ,

hence T_{U} induces a group homomorphism t_{U} : Ξ(G) → Ξ(H). As the functor T_{U} maps direct product of G-sets to direct product of H-sets, and the trivial G-set to the trivial H-set, the morphism t_{U} is actually a unital ring homomorphism from Ξ(G) to Ξ(H) (recall that U is assumed transitive as a right G-set, here).

If U is an arbitrary finite left inert (H, G)-biset, then T_{U} induces the map

t_{U} = ∏_{u∈[U/G]} t_{uG} : Ξ(G) → Ξ(H) .

It follows from Proposition 16.5 that t_{U}(Γ(G)) ⊆ Γ(H).

Now Assertion 2 follows, since t_{U} is equal to a product of unital ring homomorphisms. Assertions 3, 4, and 5 are straightforward consequences of the properties of the functor T_{U}.

16.11. Remark : It follows that the correspondences G ↦ Ξ(G)^{×} and G ↦ Γ(G)^{×} are biset functors for which the biset operations are only defined for left inert bisets. Equivalently, these correspondences are biset functors without induction : the usual basic operations for biset functors are defined for these correspondences, namely restriction to a subgroup, deflation from G to a factor group G/N (this is induced by taking fixed points by N on G-sets), transport by isomorphism, and inflation from a factor group. But there is no induction from a subgroup.

16.12. The last result of this Appendix is the computation of the group Ξ(G)^{×}, when G is abelian :

16.13. Theorem : Let G be a finite abelian group. Then Ξ(G)^{×}(= Γ(G)^{×}) has an 𝔽_{2}-basis consisting of the elements

[ \begin{matrix}−⟨G, G⟩_{G} \\ \begin{matrix}⟨G, G⟩_{G} − ⟨S, S⟩_{G} \\ ⟨G, G⟩_{G} − ⟨G, S⟩_{G} \end{matrix} \} for |G : S| = 2 . \end{matrix}

In particular

Ξ(G)^{×} ≅ (𝔽_{2})^{2r+1} ,

where r is the number of subgroups of index 2 in G.

Proof : By Theorem 8.4, the group Ξ(G)^{×} is isomorphic to the group of sequences (u_{T,S})_{(T,S)∈Π(G)} with values in {±1}, such that for any (T, S) ∈ Π(G), the map

g ∈ G/S ↦ u_{<gT >,<gS>}/u_{T,S}

is a group homomorphism. Switching to an additive notation, the group Ξ(G)^{×} is isomorphic to the 𝔽_{2}-vector spaces of sequences (λ_{T,S})_{(T,S)∈Π(G)}, with values in 𝔽_{2}, such that for any (T, S) ∈ Π(G) and any (g, h) ∈ G

λ_{<ghT >,<ghS>} + λ_{<gT >,<gS>} + λ_{<hT >,<hS>} + λ_{T,S} = 0 .

If none of g, h, and gh are in S, this yields an expression

λ_{T,S} = λ_{<ghT >,<ghS>} + λ_{<gT >,<gS>} + λ_{<hT >,<hS>}

of λ_{T,S} as a linear combination of λ_{T′,S′}, for S′ > S. If |G : S| > 2, it is always possible to find such elements g and h. It follows that the sequence (λ_{T,S})_{(T,S)∈Π(G)} is entirely determined by the values λ_{T,S}, for |G : S| ≤ 2, i.e. the values λ_{G,G}, λ_{S,S} and λ_{G,S}, where S runs through the set of subgroups of index 2 in G. If there are r such subgroups, this gives 2r + 1 such values, so dim_{𝔽_{2}} Ξ(G)^{×} ≤ 2r + 1.

To prove the converse, and actually the more precise statement in the theorem, it suffices to show that the elements −1_{Ξ(G)}, u_{S}^{G} = ⟨G, G⟩_{G}−⟨S, S⟩_{G} and v_{S}^{G} = ⟨G, G⟩_{G}−⟨G, S⟩_{G}, for |G : S| = 2 are linearly independent elements of the 𝔽_{2}-vector space Ξ(G)^{×}.

First as ⟨G, G⟩_{G} = 1_{Ξ(G)}, and as ⟨S, S⟩_{G}^{2} = 2⟨S, S⟩ if |G : S| = 2, by Proposition 3.8, it follows that (u_{S}^{G})^{2} = (v_{S}^{G})^{2} = 1, hence {u_{S}^{G}, v_{S}^{G}} ⊆ Ξ(G)^{×}.

Observe now that for |G : S| = 2, both elements u_{S}^{G} and v_{S}^{G} are obtained by inflation from G/S to G of the corresponding elements u_{1}^{G/S} and v_{1}^{G/S}.

The group C = G/S has order 2. The additive group Ξ(C) has a ℤ-basis consisting of ⟨C, C⟩_{C}, a = ⟨C, 1⟩_{C} and b = ⟨1, 1⟩_{C}. The element ⟨C, C⟩_{C} is the identity element 1_{Ξ(C)} of Ξ(C), and by Proposition 3.8, the products of the other basis elements are given by a^{2} = 2a, b^{2} = ab = ba = 2b. The group of units Ξ(C)^{×} has an 𝔽_{2}-basis consisting of −1_{Ξ(C)}, x = 1_{Ξ(C)} − a and y = 1_{Ξ(C)} − b. Moreover, the set {x, y} is an 𝔽_{2}-basis of the kernel ∂Ξ(G)^{×} of the deflation map Def_{C/C}^{C} : Ξ(C)^{×} → Ξ(1)^{×} = {±1_{Ξ(1)}} (recall that deflation in this case consists in taking fixed points under C).

Now assume that in Ξ(G)^{×}, there is a linear relation (with an additive notation) of the form

λ(−1_{Ξ(G)}) + ∑_{|G:S|=2}(α_{S}u_{S}^{G} + β_{S}v_{S}^{G}) = 0 ,

for some coefficients λ, α_{S}, β_{S} in 𝔽_{2}. Fix a subgroup X of index 2 in G, and apply Def_{G/X}^{G} to this relation. Observe that

Def_{G/X}^{G}u_{S}^{G} = Def_{G/X}^{G}Inf_{G/S}^{G}u_{1}^{G/S} = Inf_{G/SX}^{G/X}Def_{G/SX}^{G/S}u_{1}^{G/S}

is equal to 0 if S ≠ X, and to u_{1}^{G/S} if S = X. Similarly Def_{G/X}^{G}v_{S}^{G} is equal to 0 if S ≠ X, and to v_{1}^{G/S} if S = X. It follows that

λ(−1_{Ξ(G/X)}) + α_{X}u_{1}^{G/X} + β_{X}v_{1}^{G/X} = 0 ,

hence λ = α_{X} = β_{X} = 0. This completes the proof, since this holds for any X of index 2 in G.

16.14. Remark : Using the inclusion i_{G} : B(G) → Ξ(G) of Proposition 3.11, one can identify B(G) with a subring of Ξ(G). Then it is easy to see that B(G)^{×} is the subgroup of Ξ(G)^{×} generated by the elements −⟨G, G⟩_{G} and ⟨G, G⟩_{G} −⟨S, S⟩_{G}, for |G : S| = 2 : this gives another proof of Matsuda’s theorem ([8]), saying that the unit group B(G)^{×} of the usual Burnside ring of a finite abelian group G is isomorphic to (𝔽_{2})^{r+1}, where r is the number of subgroups of index 2 in G.

Acknowledgements : I am grateful to Geoffrey Powell for pointing out a false argument in the proof of Corollary 9.6, in an early version of this work. I also thank the referee for his/her careful reading of this paper, and helpful comments about it (in particular, the fact that Corollary 9.12 is indeed a corollary of Proposition 9.11).

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

email :  serge.bouc@u-picardie.fr
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