CHAPTER 1 Burnside rings

Serge Bouc

Université Paris 7-Denis Diderot, 2 place Jussieu, 75251 Paris Cedex 05. France

1. Introduction 1 2. Basic properties of G-sets 2 3. The ring structure 4 4. Invariants 18 5. The Mackey and Green functor structure 26 6. The Burnside ring as biset-functor 36 Bibliography 47

1. Introduction

1 Let G be a finite group. The Burnside ring B(G) of the group G is one of the fundamental representation rings of G, namely the ring of permutation representations.

It is in many ways the universal object to consider when looking at the category of G-sets. It can be viewed as an analogue of the ring ℤ of integers for this category.

It can be studied from different points of view. First B(G) is a commutative ring, and one can look at is prime spectrum and primitive idempotents. This leads to various induction theorems (Artin, Conlon, Dress): the typical statement here is that any (virtual) RG-module is a linear combination with suitable coefficients of modules induced from certain subgroups of G (cyclic, hypoelementary, or Dress subgroups).

The Burnside ring is the natural framework to study the invariants attached to structured G-sets (such as G-posets, or more generally simplicial G-sets). Those invariants are generalizations for the category of G-sets of classical notions, such as the Möbius function of a poset, or the Steinberg module of a Chevalley group. They have properties of projectivity, which lead to congruences on the values of Euler-Poincaré characteristic of some sets of subgroups of G.

The ring B(G) is also functorial with respect to G and subgroups of G, and this leads to the Mackey functor or Green functor point of view. There are close connections between the Burnside ring and the Mackey algebra. The Burnside Mackey functor is a typical example of projective Mackey functor. It is also a universal object in the category of Green functors. This leads to decomposition of the category of Mackey functors for G as a directs sum of smaller abelian categories.

Finally B(G) is also functorial with respect to bisets, and this is leads to the definition of double Burnside rings. Those rings are connected to stable homotopy theory via the Segal conjecture, and they provide tools to study the stable splittings of the classifying spaces of finite groups.

1This is a slightly corrected version [date: 16/06/2016] of the original text published in the “Handbook of Algebra”, vol 2, pp 739-804 (2000) Elsevier

2. Basic properties of G-sets

2.1. Notation. Let G be a finite group. The category of finite G-sets will be denoted by G-set. The objects of G-set are the finite sets with a left Gaction, morphisms are G-equivariant maps, and composition of morphisms is the composition of maps.

If H is a subgroup of G, and x is an element of G, the notation ^{x}H stands for x.H.x^{−1}, and the notation H^{x} for x^{−1}.H.x. The normalizer of H in G is denoted by N_{G}(H). If X is a G-set, the stabilizer in G of an element x of X is denoted by G_{x}.

The set of conjugacy classes of subgroups of G is denoted by s_{G}, and a set of representatives of s_{G} is denoted by [s_{G}]. If H and K are subgroups of G, the notation H =_{G} K (resp. H ⊆_{G} K) means that there is an element x ∈ G such that H^{x} = K (resp. H^{x} ⊆ K).

The cardinality of a set S is denoted by |S|.

If p is a prime number, the smallest normal subgroup N of G such that G/N is a p-group is denoted by O^{p}(G). It is the subgroup of G generated by the p′-elements, i.e. the elements of order coprime to p.

More generally, if π is a set of primes, the notation O^{π}(G) stands for the smallest normal subgroup N of G such that G/N is a solvable π-group. The group G is called π-perfect if O^{π}(G) = G. If G itself is a π-group, then the group O^{π}(G) is the limit of the derived series of G. In particular, if π is the set of all primes, then a group is π-perfect if and only if it is perfect, i.e. equal to its derived subgroup.

The trivial group will be denoted by 1l.

2.2. Operations on G-sets. When X is a G set, and H is a subgroup of G, one can view X as an H-set by restriction of the action. This H-set is denoted by Res_{H}^{G}X. If f : X → Y is a morphism of G-sets, let Res_{H}^{G}f denote the map f viewed as a morphism of H-sets. This defines a restriction functor Res_{H}^{G} : G-set → H-set.

Now if Z is an H-set, the induced G-set Ind_{H}^{G}Z is defined as G ×_{H} X, i.e. the quotient of the cartesian product G × X by the right action of H given by (g, x).h = (gh, h^{−1}x) for g ∈ G, h ∈ H, x ∈ X. The left action of G on G ×_{H} X is induced by its left action on G × X given by g′.(g, x) = (g′g, x), for g′, g ∈ G and x ∈ X. If f : Z → T is a morphism of H-sets, then Ind_{H}^{G}f is the morphism of G-sets from Ind_{H}^{G}Z to Ind_{H}^{G}T defined by (Ind_{H}^{G}f)((g, x)) = (g, f(x)). This defines an induction functor Ind_{H}^{G} : H-set → G-set.

Note that if Z is isomorphic to H/K for some subgroup K of H, then Ind_{H}^{G}Z is isomorphic to G/K.

The set of fixed points of H on X is denoted by X^{H}. It is viewed as a N_{G}(H)/Hset. If f : X → Y is a morphism of G-sets, then the restriction of f to X^{H} is denoted by f^{H}. It is a morphism of N_{G}(H)/H-sets from X^{H} to Y^{H}, and this defines a fixed points functor from G-set to N_{G}(H)/H-set.

When H is a normal subgroup of G, any G/H-set can be viewed as a G-set by inflation, and this operation defines an inflation functor from G/H-ens to G-ens.

Finally, let H be a subgroup of G, and x be an element of G. If Z is an H-set, then the group ^{x}H acts on Z by h.z = h^{x}z, where h ∈ ^{x}H and z ∈ Z, and h^{x}z is computed in the H-set Z. This gives an ^{x}H-set denoted by ^{x}Z. If f : Z → T is a morphism of H-sets, then c_{x,H}(f) is the map f, viewed as a morphism of ^{x}H-sets from ^{x}Z to ^{x}T. This defines a conjugation functor c_{x,H} : H-set → ^{x}H-set.

Those constructions are connected by various identities. Among them:

2. BASIC PROPERTIES OF G-SETS 3

Proposition 2.2.1. Let G be a finite group, and H and K be subgroups of G.

(1) [Mackey formula] If Z is an H-set, then there is an isomorphism of K-sets

Res_{K}^{G}Ind_{H}^{G}Z ≃ ⨆_{x∈K\G/H} Ind_{K∩^{x}H}^{K} ^{x}Res_{K^{x}∩H}^{H}Z

(2) [Frobenius identity] If X is a G-set and Z is an H-set, then there is an isomorphism of G-sets

X × Ind_{H}^{G}Z ≃ Ind_{H}^{G}((Res_{H}^{G}X).Z)

and in particular for any G-set X, there is an isomorphism of G-sets

X × (G/H) ≃ Ind_{H}^{G}Res_{H}^{G}X

(3) If Z is an H-set, then there is an isomorphism of N_{G}(H)/H-sets

(Ind_{H}^{G}Z)^{K} ≃ ⨆_{{x∈N_{G}(K)\G/H \atop K^{x}⊆H}} Ind_{N_{^{x}H}(K)/K}^{N_{G}(K)/K}(^{x}Z)^{K}

Proof. (sketch) (1) Let S be a set of representatives in G of double cosets K\G/H, and consider the map

Res_{K}^{G}Ind_{H}^{G}Z → ⨆_{x∈S} Ind_{K∩^{x}H}^{K} ^{x}Res_{K^{x}∩H}^{H}Z

sending (g, z), with g ∈ G and z ∈ Z to the element (k, hz) of the component x ∈ S, if g can be written g = kxh, for some k ∈ K and h ∈ H. This is the required isomorphism of K-sets.

(2) Consider the map

X × Ind_{H}^{G}Z → Ind_{H}^{G}((Res_{H}^{G}X) × Z)

sending the element (x, (g, z)) of the left hand side, with x ∈ X, g ∈ G, and z ∈ Z, to the element (g, (g^{−1}x, z)) of the right hand side. This is the required isomorphism of G-sets. The other isomorphism in assertion (2) is the special case Z = H/H.

(3) Note that (Ind_{H}^{G}Z)^{K} = (Res_{N_{G}(K)}^{G}Ind_{H}^{G}Z)^{K}, and use the Mackey formula. □

2.3. Characterization of G-sets.

Lemma 2.3.1. Let G be a finite group.

(1) Any G-set is a disjoint union of transitive ones. If X is a transitive G-set, and if x ∈ X, then the map

gG_{x} ∈ G/G_{x} ↦ g.x ∈ X

is an isomorphism of G-sets.

(2) If H and K are subgroups of G, then the map f ↦ f(H) is a one to one correspondence between the set of G-set homomorphisms from G/H to G/K and the set of cosets xK ∈ G/K such that H ⊆ ^{x}K. In particular, the G-sets G/H and G/K are isomorphic if and only if H and K are conjugate in G.

Proof. Both assertions are obvious. □

One can characterize a G-set up to isomorphism using the following fundamental theorem of Burnside ([15] Chapter XII Theorem I):

Theorem 2.3.2. [Burnside] Let G be a finite group, and X and Y be finite G-sets. Then the following are equivalent:

(1) The G-sets X and Y are isomorphic.

(2) For any subgroup H of G, the sets X^{H} and Y^{H} have the same cardinality.

Proof. It is clear that (1) implies (2), since any G-set isomorphism X → Y induces a bijection X^{H} → Y^{H} on the sets of fixed points by any subgroup H of G.

To show the converse, observe that it follows from Lemma 2.3.1 that any finite G-set X can be written up to isomorphism as

X = ⨆_{K∈[s_{G}]} a_{K}(X) G/K

for some a_{K}(X) ∈ ℕ, where a_{K}(X) G/K denotes the disjoint union of a_{K}(X) copies of G/K.

Now if (2) holds, for any H ∈ [s_{G}], there is an equation

∑_{K∈[s_{G}]} (a_{K}(X) − a_{K}(Y))|(G/K)^{H}| = 0

The matrix m of this system of equations is given by

m(H, K) = |(G/K)^{H}| = |{x ∈ G/K | H^{x} ⊆ K}|

for K, H ∈ [s_{G}]. In particular the entry m(H, K) is non-zero if and only if some conjugate of H is contained in K.

If the set [s_{G}] is given a total ordering ⪯ such that H ⪯ K implies |H| ≤ |K|, then the matrix m is upper triangular, with non-zero diagonal coefficient m(H, H) = |N_{G}(H) : H|. In particular m is non-singular, and it follows that a_{K}(X) = a_{K}(Y), for any K ∈ [s_{G}], and the G-sets X and Y are isomorphic. □

Definition 2.3.3. The above matrix m (or sometimes its transpose) is called the table of marks of the group G.

3. The ring structure

3.1. Definition. The following definition of the Burnside ring of the group G appears in an article of Solomon ([37]):

Definition 3.1.1. [Solomon] The Burnside ring B(G) of G is the Grothendieck group of the category G-set, for relations given by decomposition in disjoint union of G-sets. The multiplication on B(G) is induced by the direct product of G-sets.

It means that B(G) is the free ℤ-module with basis the set of equivalence classes of finite G-sets, quotiented by relations identifying the class of the disjoint union X ⊔ Y of two G-sets X and Y to the sum of the class of X and the class of Y.

The direct product of G-sets is commutative and distributive with respect to disjoint union, up to canonical isomorphisms. Hence it induces by bilinearity a commutative ring structure on B(G). The class of a set • of cardinality 1 is a unit for this ring structure.

Two finite G-sets A and B have the same image in B(G) if and only if there is a sequence of finite G-sets X_{i} and Y_{i}, for 1 ≤ i ≤ n, and an isomorphism of G-sets

A ⊔ (⨆_{i=1}^{n} X_{i}) ⊔ (⨆_{i=1}^{n} Y_{i}) ≃ B ⊔ ( ⨆_{i=1}^{n}(X_{i} ⊔ Y_{i}))

Taking fixed points of both sides shows that for any subgroup H of G, one has |A^{H}| = |B^{H}|, and Burnside’s Theorem 2.3.2 now implies that A and B are isomorphic as G-sets. In the sequel, the G-set A and its image in B(G) will be identified.

It follows from Burnside’s Theorem 2.3.2 that any finite G-set X can be written uniquely up to isomorphism as

X ≃ ⨆_{H∈[s_{G}]} a_{H}(X)G/H

Hence B(G) is a free ℤ-module, with basis indexed by elements G/H, for H ∈ [s_{G}]. In this basis, the multiplication law can be recovered by

(3.1.2)  (G/H).(G/K) = ∑_{x∈H\G/K} G/(H ∩^{x}K)

This follows from Proposition 2.2.1, since by Frobenius identities

(G/H) × (G/K) = Ind_{H}^{G}Res_{H}^{G}Ind_{K}^{G}K/K

and since by the Mackey formula

Res_{H}^{G}Ind_{K}^{G}K/K = ⨆_{x∈H\G/K} H/(H ∩^{x}K)

Finally, the operations on G-sets defined in section 2.2 all commute with disjoint unions. Hence they can be extended to the Burnside ring: the elements of B(G) can be viewed as a formal differences X − Y of two finite G-sets. If F : G-set → H-set denotes one of the functors of restriction, induction, fixed points, inflation, or conjugation, then F induces a group homomorphism still denoted by F from B(G) to B(H), defined by

F (X − Y) = F (X) − F (Y)

for any finite G-sets X and Y.

Thus for example, if H is a subgroup of G, there is a restriction homomorphism

Res_{H}^{G} : B(G) → B(H)

This homomorphism is actually a morphism of rings (with unit).

In the special case H = 1l, since B(H) ≃ ℤ, this gives an extension of the cardinality to a map X ↦ |X| = Res_{1l}^{G}X from B(G) to ℤ.

Similarly, there is an induction homomorphism

Ind_{H}^{G} : B(H) → B(G)

This morphism is not a ring homomorphism in general.

If H is a subgroup of G, there is a fixed points homomorphism X ↦ X^{H} from B(G) to B(N_{G}(H)/H), which is actually a ring homomorphism. When H is a normal subgroup of G, there is an inflation homomorphism

Inf_{G/H}^{G} : B(G/H) → B(G)

which is a ring homomorphism.

Finally, if x is an element of G, there is a conjugation homomorphism Z ↦ ^{x}Z from B(H) to B(^{x}H), which is a ring isomorphism.

The following is an obvious extension of Proposition 2.2.1:

Proposition 3.1.3. Let G be a finite group, and H and K be subgroups of G.

(1) [Mackey formula] If Z ∈ B(H), then in B(K)

Res_{K}^{G}Ind_{H}^{G}Z = ∑_{x∈K\G/H} Ind_{K∩^{x}H}^{K} ^{x}Res_{K^{x}∩H}^{H}Z

(2) [Frobenius identity] If X ∈ B(G) and Z ∈ B(H), then in B(G)

X.Ind_{H}^{G}Z = Ind_{H}^{G}((Res_{H}^{G}X).Z)

and in particular for any X ∈ B(G)

X.(G/H) = Ind_{H}^{G}Res_{H}^{G}X
(3) If Z ∈ B(H), then in B(N_{G}(H)/H)

(Ind_{H}^{G}Z)^{K} = ∑_{{x∈N_{G}(K)\G/H \atop K^{x}⊆H}} Ind_{N_{^{x}H}(K)/K}^{N_{G}(K)/K}(^{x}Z)^{K}

3.2. Fixed points as ring homomorphisms. The ring B(G) is finitely generated as ℤ-module, hence it is a noetherian ring. Burnside’s Theorem 2.3.2 can be interpreted as follows: each subgroup H of G defines a ring homomorphism ϕ_{H}^{G} : B(G) → ℤ by ϕ_{H}^{G}(X) = |X^{H}|. The kernel of ϕ_{H}^{G} is a prime ideal, since ℤ is an integral domain, and the intersection of all those kernels for subgroups H of G is zero. In particular, the ring B(G) is reduced.

Since ϕ_{H}^{G} = ϕ_{K}^{G} if H and K are conjugate in G, it follows that the product map

Φ = ∏_{H∈[s_{G}]} ϕ_{H}^{G} : B(G) → ∏_{H∈[s_{G}]} ℤ

is injective. Moreover this map Φ is a map between free ℤ-modules having the same rank. Hence the cokernel of Φ is finite.

The matrix m of Φ with respect to the basis {G/H}_{H∈[s_{G}]} and to the canonical basis {u_{H}}_{H∈[s_{G}]} of ∏_{H∈[s_{G}]} ℤ is the table of marks of the group G. Recall from Definition 2.3.3 that for H, K ∈ [s_{G}]

m(H, K) = |G/K^{H}| = |{x ∈ G/K | H^{x} ⊆ K}|

The cardinality of the cokernel of Φ is the determinant of m, hence it is equal to

|Coker(Φ)| = ∏_{H∈[s_{G}]} |N_{G}(H) : H|

This cokernel has been described by Dress ([19]):

Theorem 3.2.1. [Dress] Let G be a finite group. For H and K in [s_{G}], set

n(K, H) = |{x ∈ N_{G}(K)/K | <x, K> =_{G} H}|

Then the element y = ∑_{H∈[s_{G}]} y_{H}u_{H} of ∏_{H∈[s_{G}]} ℤ is in the image of Φ if and only if for any K ∈ [s_{G}]

∑_{H∈[s_{G}]} n(K, H)y_{H} ≡ 0 (|N_{G}(K)/K|)

Proof. First let X be a finite G-set, and let y = Φ(X). Then with the notation of the theorem, one has y_{H} = |X^{H}| for all H ∈ [s_{G}], thus for any K ∈ [s_{G}]

∑_{H∈[s_{G}]} n(K, H)y_{H} = ∑_{H∈[s_{G}]} |X^{H}||{x ∈ N_{G}(K)/K | <x, K> =_{G} H}|
= ∑_{x∈N_{G}(K)/K} |X^{<x,K>}|
= ∑_{x∈N_{G}(K)/K} |(X^{K})^{x}|

Now for any finite group L acting on a finite set Z one has

|L| × |L\Z| = |L| ∑_{z∈Z} \frac{|L_{z}|}{|L|} = |\{(l, z) ∈ L × Z | l.z = z\}|
= ∑_{l∈L} |Z^{l}|

Applying this for L = N_{G}(K)/K and Z = X^{K} gives

∑_{H∈[s_{G}]} n(K, H)y_{H} = |N_{G}(K)/K| |N_{G}(K)\X^{K}| ≡ 0 (|N_{G}(K)/K|)

By linearity, this proves the “only if” part of the theorem.

Applying this for X = G/M, for some M ∈ [s_{G}], shows that there is a matrix t indexed by [s_{G}] × [s_{G}], with entries in ℤ, such that for K ∈ [s_{G}]

∑_{H∈[s_{G}]} n(K, H)m(H, M) = |N_{G}(K)/K|t(K, M)

Now the matrix n is upper triangular, and its diagonal coefficients are equal to 1. Thus the matrix t is upper triangular, and

t(K, K) = m(K, K)/|N_{G}(K)/K| = 1

In particular t is invertible (over ℤ).

Now suppose that the element y = ∑_{H∈[s_{G}]} y_{H}u_{H} ∈ ∏_{H∈[s_{G}]} ℤ satisfies all the congruences of the theorem. Since Coker(Φ) is finite, there exist rational numbers r_{M}, for M ∈ [s_{G}], such that

y = ∑_{M∈[s_{G}]} r_{M}Φ(G/M)

In other words for each H ∈ [s_{G}]

y_{H} = ∑_{M∈[s_{G}]} |G/M^{H}|r_{M}

Thus for each K ∈ [s_{G}]

∑_{H∈[s_{G}]} n(K, H)y_{H} = ∑_{H∈[s_{G}]} ∑_{M∈[s_{G}]} n(K, H)m(H, M)r_{M}
= |N_{G}(K) : K| ∑_{M∈[s_{G}]} t(K, M)r_{M}

The left hand side is a multiple of |N_{G}(K) : K| by assumption, hence there exist integers z_{K} such that for all K ∈ [s_{G}]

∑_{M∈[s_{G}]} t(K, M)r_{M} = z_{K}

Since t is invertible, it follows that r_{M} ∈ ℤ for all M, and y ∈ Im(Φ). □

3.3. Idempotents. The map Φ of the previous section is an injective map between free ℤ-modules having the same rank. Hence tensoring with ℚ gives a ℚ-algebra isomorphism

ℚ ⊗_{ℤ} Φ : ℚ ⊗_{ℤ} B(G) →^{≃} ∏_{H∈[s_{G}]} ℚ

and in particular the algebra ℚ ⊗_{ℤ} B(G) is semi-simple. It will be denoted by ℚB(G). The component ℚϕ_{H}^{G} of ℚΦ will still be written X ↦ |X^{H}|, so in general |X^{H}| will be a rational number for X ∈ ℚB(G). More generally, all the notations defined for B(G) will be extended without change to ℚB(G).

The inverse image by ℚΦ of the canonical ℚ-basis of ∏_{H∈[s_{G}]} ℚ indexed by H is denoted by e_{H}^{G}. If H′ is conjugate to H in G, one also sets e_{H′}^{G} = e_{H}^{G}. With this notation for any subgroups H and K of G, one has

|(e_{H}^{G})^{K}| = \{ \begin{matrix}1 & if H =_{G} K \\ 0 & otherwise \end{matrix}

The set of elements e_{H}^{G}, for H ∈ [s_{G}], is the set of primitive idempotents of ℚB(G). Those idempotents have been computed explicitly by Gluck ([22]), and later independently by Yoshida ([49]). This can be done using the following lemma:

Lemma 3.3.1. Let G be a finite group.

(1) Let H be a subgroup of G. Then for any X ∈ ℚB(G)

X.e_{H}^{G} = |X^{H}|e_{H}^{G}

Conversely, if Y ∈ ℚB(G) is such that X.Y = |X^{H}|Y for any X ∈ ℚB(G), then Y ∈ ℚe_{H}^{G}.

(2) Let H be a proper subgroup of G. Then Res_{H}^{G}e_{G}^{G} = 0. Conversely, if Y ∈ ℚB(G) is such that Res_{H}^{G}Y = 0 for any proper subgroup H of G, then Y ∈ ℚe_{G}^{G}.

(3) Let H be a subgroup of G. Then

e_{H}^{G} = \frac{1}{|N_{G}(H) : H|}Ind_{H}^{G}e_{H}^{H}

Proof. (1) The set of elements e_{H}^{G}, for H ∈ [s_{G}], is a ℚ-basis of ℚB(G), thus for any X ∈ ℚB(G), there are rational numbers r_{H}, for H ∈ [s_{G}], such that

X = ∑_{H∈[s_{G}]} r_{H}e_{H}^{G}

Taking fixed points of both sides by a subgroup K ∈ [s_{G}] shows that r_{K} = |X^{K}|. It follows that for all K ∈ [s_{G}]

X.e_{K}^{G} = |X^{K}|e_{K}^{G}

Now let Y be an element of ℚB(G) verifying X.Y = |X^{H}|Y for any X ∈ ℚB(G). Then in particular e_{K}^{G}.Y = 0 if K ≠_{G} H, thus Y = |Y^{H}|e_{H}^{G} is a rational multiple of e_{H}^{G}.

(2) Let H be a proper subgroup of G. Then for any subgroup of K of H

|(Res_{H}^{G}e_{G}^{G})^{K}| = |(e_{G}^{G})^{K}| = 0

since obviously |(Res_{H}^{G}X)^{K}| = |X^{K}| for any G-set X, hence for any X in ℚB(G). It shows that the restriction of e_{G}^{G} to any proper subgroup of G is zero.

Conversely, if the restriction of an element Y to any proper subgroup H of G is zero, then in particular |Y^{H}| = 0 for such a subgroup, and Y = |Y^{G}|e_{G}^{G}.

(3) Consider next the element Ind_{H}^{G}e_{H}^{H}. If X is any element of ℚB(G), then by Frobenius identity

X.Ind_{H}^{G}e_{H}^{H} = Ind_{H}^{G}((Res_{H}^{G}X).e_{H}^{H}) = Ind_{H}^{G}(|X^{H}|.e_{H}^{H}) = |X^{H}|Ind_{H}^{G}e_{H}^{H}

It follows that there is a rational number r_{H}^{G} such that

(3.3.2)  Ind_{H}^{G}e_{H}^{H} = r_{H}^{G}e_{H}^{G}

By the Mackey formula, the restriction of the left hand side to H is equal to

Res_{H}^{G}Ind_{H}^{G}e_{H}^{H} = ∑_{x∈H\G/H} Ind_{H∩^{x}H}^{H}^{x}Res_{H^{x}∩H}^{H}e_{H}^{H} = ∑_{x∈N_{G}(H)/H} ^{x}e_{H}^{H}

since the restriction of e_{H}^{H} to H^{x} ∩ H is zero if H^{x} ≠ H. Moreover if y ∈ G, then for any subgroup K of ^{y}H

|(^{y}e_{H}^{H})^{K}| = |(e_{H}^{H})^{K^{y}}| = \{ \begin{matrix}1 & if K = ^{y}H \\ 0 & otherwise \end{matrix}

Thus ^{y}e_{H}^{H} = e_{^{y}H}^{^{y}H}, and finally

Res_{H}^{G}Ind_{H}^{G}e_{H}^{H} = |N_{G}(H) : H|e_{H}^{H}

Equation 3.3.2 gives then

|N_{G}(H) : H|e_{H}^{H} = r_{H}^{G}Res_{H}^{G}e_{H}^{G}

Taking fixed points of both sides under H gives r_{H}^{G} = |N_{G}(H) : H|, and

(3.3.3)  e_{H}^{G} = \frac{1}{|N_{G}(H) : H|}Ind_{H}^{G}e_{H}^{H}

as was to be shown. As a consequence

(3.3.4)  Res_{H}^{G}e_{H}^{G} = e_{H}^{H}
□

Theorem 3.3.5. [Gluck] Let G be a finite group. If H is a subgroup of G, then

e_{H}^{G} = \frac{1}{|N_{G}(H)|} ∑_{K⊆H} |K| μ(K, H) G/K

where μ is the Möbius function of the poset of subgroups of G, and G/K denotes 1 ⊗ G/K ∈ ℚB(G).

Proof. One can write

e_{H}^{H} = ∑_{K⊆H} r(K, H) H/K

where r(K, H) is a rational number. Since ^{y}e_{H}^{H} = e_{^{y}H}^{^{y}H} for y ∈ G, one can suppose r(^{y}K,^{y}H) = r(K, H) for y ∈ G. Taking induction from H to G gives

(3.3.6)  e_{H}^{G} = \frac{1}{|N_{G}(H) : H|} ∑_{K⊆H} r(K, H) G/K

Now the sum of the elements e_{H}^{G}, for H ∈ [s_{G}], is equal to • = G/G. Summing over all subgroups H of G instead of H ∈ [s_{G}] gives

G/G = ∑_{H⊆G} \frac{|N_{G}(H)|}{|G|} \frac{1}{|N_{G}(H) : H|} ∑_{K⊆H} r(K, H) G/K

The coefficient of G/K in the right hand side is equal to the coefficient of G/K′, for any conjugate K′ of K in G, and it is equal to

∑_{K⊆H⊆G} \frac{|H|}{|G|}r(K, H)

This must be equal to zero if K ≠ G, and equal to 1 if K = G. Setting r′(K, H) = \frac{|H|}{|K|}r(K, H) for K ⊆ H, this gives

∑_{K⊆H⊆G} r′(K, H) = \{ \begin{matrix}|G : K| = 1 & if K = G \\ 0 & otherwise \end{matrix}

This shows that r′(K, H) is equal to μ(K, H), where μ is the Möbius function of the poset of subgroups of G. Thus m(K, H) = \frac{|K|}{|H|}μ(K, H), and Equation 3.3.6 gives

e_{H}^{G} = \frac{1}{|N_{G}(H)|} ∑_{K⊆H} |K| μ(K, H) G/K

as was to be shown. □

The formulae for primitive idempotents in ℚB(G) lead to a natural question: when are these idempotents actually in B(G)? More generally, if π is a set of prime numbers, let ℤ_{(π)} denote the subring of ℚ of irreducible fractions with denominator prime to all the elements of π. In other words ℤ_{(π)} is the localization of ℤ with respect to the set ℤ − ∪_{p∈π}pℤ. One may ask when the idempotents of ℚB(G) are actually in ℤ_{(π)}B(G). The answer is as follows:

Theorem 3.3.7. [Dress] Let G be a finite group, and π be a set of primes. Let F be a family of subgroups of G, closed by conjugation in G. Let [F] denote the set F ∩ [s_{G}]. Then the following conditions are equivalent:

(1) The idempotent ∑_{H∈[F]} e_{H}^{G} lies in ℤ_{(π)}B(G).

(2) Let H and K be any subgroups of G such that H is a normal subgroup of K and the quotient K/H is cyclic of prime order p ∈ π. Then H ∈ F if and only if K ∈ F.

Proof. Suppose first that (1) holds, i.e. that the idempotent e = ∑_{H∈[F]} e_{H}^{G} lies in ℤ_{(π)}B(G). This is equivalent to the existence of an integer m coprime to all the elements of π, such that me ∈ B(G). Now if H and K are subgroups of G such that H \underline{◁} K and K/H is cyclic of prime order p, then for any element X of B(G), one has |X^{H}| ≡ |X^{K}| (p). It follows that

m|e^{H}| ≡ m|e^{K}| (p)

But m|e^{H}| is equal to m if H ∈ F, and it is equal to zero otherwise. Thus if p ∈ π, it follows that assertion (2) holds, because m ≢ 0 (p).

Conversely, suppose that (2) holds. Let m be an integer such that me ∈ B(G). By Dress’s Theorem 3.2.1 this is equivalent to require that for each subgroup K of G

∑_{H∈[s_{G}]} n(K, H)m|e^{H}| = m ∑_{H∈[F]} n(K, H) ≡ 0 (|N_{G}(K) : K|)

This can also be written as

m ∑_{H∈[F]} |{x ∈ N_{G}(K)/K | <x, K> =_{G} H}| ≡ 0 (|N_{G}(K) : K|)

i.e.

(3.3.8)  m|{x ∈ N_{G}(K)/K | <x, K> ∈ F}| ≡ 0 (|N_{G}(K) : K|)

If (2) holds, then <x, K> ∈ F if and only if <x_{π′}, K> ∈ F, where x_{π′} denotes the π′-part of x. The set Γ of π′-elements y of the group W = N_{G}(K)/K such that <y, K> ∈ F is invariant under conjugation in W. Let [Γ] denote a set of representatives of W -conjugacy classes of Γ. Then:

|{x ∈ W | x_{π′} ∈ Γ}| = ∑_{γ∈Γ} |{x ∈ W | x_{π′} = γ}|
= ∑_{γ∈Γ} |C_{W} (γ)_{π}|

= ∑_{γ∈[Γ]} \frac{|W |}{|C_{W} (γ)|}|C_{W} (γ)_{π}|.

Now by a theorem of Frobenius (see Corollaire 1 of Théorème 23 of [36]), for each γ ∈ Γ, there exists an integer l_{γ} such that |C_{W} (γ)_{π}| = l_{γ}|C_{W} (γ)|_{π}. Hence

|\{x ∈ W | x_{π′} ∈ Γ\}| = ∑_{γ∈[Γ]} \frac{|W |}{|C_{W} (γ)|_{π′}} l_{γ} = |W |_{π} ∑_{γ∈[Γ]} \frac{|W |_{π′}}{|C_{W} (γ)|_{π′}} l_{γ} ≡ 0 (mod. |W |_{π}).

It follows that |W |_{π′}|{x ∈ W | x_{π′} ∈ Γ}| ≡ 0 (mod. |W |), and in particular Congruence 3.3.8 holds if m = |G|_{π′}, for any subgroup K of G. Hence me ∈ B(G), and Assertion (1) holds. □

Corollary 3.3.9. Let G be a finite group, and π be a set of primes. If J is a π-perfect subgroup of G, set

f_{J}^{G} = ∑_{{H∈[s_{G}] \atop O^{π}(H)=_{G}J}} e_{H}^{G}

Then the set of elements f_{J}^{G}, for π-perfect elements J of [s_{G}], is the set of primitive idempotents of ℤ_{(π)}B(G).

In particular, the set of primitive idempotents of B(G) is in one to one correspondence with the set of conjugacy classes of perfect subgroups of G.

The group G is solvable if and only if G/G is a primitive idempotent of B(G).

Proof. Let J be a π-perfect subgroup of G, and set

F = {H ⊆ G | O^{π}(H) =_{G} J}

Then clearly the family F is closed by conjugation in G, and satisfies condition (2) of Theorem 3.3.7. Thus f_{J}^{G} lies in ℤ_{(π)}B(G).

To prove that it is primitive, it suffices to show that the family F has no proper non-empty subfamily satisfying condition (2) of Theorem 3.3.7. But if F′ is such a non-empty subfamily of F, and if H ∈ F′, then O^{π}(H) ∈ F′ since the composition factors of H/O^{π}(H) are cyclic groups of prime order belonging to π. Thus J ∈ F′ since F′ is closed by conjugation. Now if H′ ∈ F the group O^{π}(H′) is in F′, and H′ ∈ F′ by the same argument. Thus F′ = F, and the idempotent f_{J}^{G} is primitive.

The last assertion of the corollary is the case where π is the set of all prime numbers. □

3.4. Prime spectrum. The prime spectrum of B(G) has been determined by Dress ([20]):

Theorem 3.4.1. [Dress] Let G be a finite group, let p denote a prime number or zero, and H be a subgroup of G. Let

I_{H,p}(G) = {X ∈ B(G) | |X^{H}| ≡ 0 (p)}

Then:

(1) The set I_{H,p}(G) is a prime ideal of B(G).

(2) If I is a prime ideal of B(G), then there is a subgroup H of G and an integer p equal to zero or a prime number, such that I = I_{H,p}(G).

(3) If H and K are subgroups of G, and if p and q are prime numbers or zero, then I_{H,p}(G) ⊆ I_{K,q}(G) if and only if one of the following holds:

(a) One has p = q = 0, and the subgroups H and K are conjugate in G. In this case moreover I_{H,p} = I_{K,q}.

(b) One has p = 0 and q > 0, and the groups O^{q}(H) and O^{q}(K) are conjugate in G. In this case moreover I_{H,p} ≠ I_{K,q}.

(c) One has p = q > 0, and the subgroups O^{p}(H) and O^{p}(K) are conjugate in G. In this case moreover I_{H,p} = I_{K,q}.

Proof. (1) Clearly I_{H,p} is the kernel of the ring homomorphism from B(G) to ℤ/pℤ mapping X to the class of |X^{H}|. Assertion (1) follows, since ℤ/pℤ is an integral domain.

(2) Conversely, if I is a prime ideal of B(G), then the ring R = B(G)/I is an integral domain. Let π : B(G) → R be the canonical projection. Since R ≠ {0}, there is a subgroup H of G minimal subject to the condition π(G/H) ≠ 0. Taking the image of equation 3.1.2 by π and using the minimality of H gives

π(G/H)π(G/K) = ∑_{{x∈G/K \atop H^{x}⊆K}} π(G/H)

(since moreover HxK = xK if H^{x} ⊆ K). Since R is an integral domain, and since π(G/H) ≠ 0, it follows that

π(G/K) = |{x ∈ G/K | H^{x} ⊆ K}|1_{R} = |(G/K)^{H}|1_{R}

It follows by linearity that π(X) = |X^{H}|1_{R} for any X ∈ B(G). Let p denote the characteristic of R. Then p is equal to zero or a prime number. Clearly the kernel of π, equal to I by definition, is also equal to I_{H,p}(G).

(3) Suppose that H and K are conjugate in G. Then for any X ∈ B(G), one has |X^{H}| = |X^{K}|, and in particular I_{H,0}(G) = I_{K,0}(G).

Now if p is a prime number, and if X is a finite G-set, since

X^{H} = (X^{O^{p}(H)})^{H/O^{p}(H)}

and since H/O^{p}(H) is a p-group, it follows that

|X^{H}| ≡ |X^{O^{p}(H)}| (p)

for any X ∈ B(G). In particular I_{H,0} ⊆ I_{O^{p}(H),p}, and I_{H,p}(G) = I_{K,p}(G) if O^{p}(H) and O^{p}(K) are conjugate in G. Thus I_{H,p}(G) = I_{O^{p}(H),p}(G) for any H ⊆ G.

Conversely, suppose that I_{H,p}(G) ⊆ I_{K,q}(G). There is a surjective ring homomorphism from B(G)/I_{H,p}(G) to B(G)/I_{K,q}(G). Since p is the characteristic of the ring B(G)/I_{H,p}(G), there are three possible cases:

(a) Either p = q = 0. Then the inclusion I_{H,0}(G) ⊆ I_{K,0}(G) means that if X ∈ B(G) is such that |X^{H}| = 0, then |X^{K}| = 0. Now X = |N_{G}(K)|e_{K}^{G} is in B(G) by Theorem 3.3.5, and |X^{K}| = |N_{G}(K)| ≠ 0. Thus |X^{H}| ≠ 0, hence H is conjugate to K in G. Clearly in this case I_{H,0}(G) = I_{K,0}(G).

(b) The next possible case is p = 0 and q > 0. In this case if X ∈ B(G) is such that |X^{H}| = 0, then |X^{K}| ≡ 0 (q). Consider the idempotent

f_{O^{q}(K)}^{G} = ∑_{{L∈[s_{G}] \atop O^{q}(L)=_{G}O^{q}(K)}} e_{L}^{G}

By Corollary 3.3.9, there is an integer m coprime to q such that X = mf_{O^{q}(K)}^{G} is in B(G). Now |X^{K}| ≡ |X^{O^{q}(K)}| ≡ m ≢ 0 (q). Thus |X^{H}| ≠ 0, and it follows that O^{q}(H) =_{G} O^{q}(K). The inclusion I_{H,0}(G) ⊆ I_{K,q}(G) is proper since the respective quotient rings have characteristic 0 and q.

(c) The last case is p = q > 0. Since I_{H,0}(G) ⊆ I_{H,p}(G), it follows that I_{H,0}(G) ⊆ I_{K,p}(G). Hence O^{p}(H) =_{G} O^{p}(K) by the discussion of the previous case. And in this case I_{H,p}(G) = I_{K,p}(G). □

3.5. Application to induction theorems.

Notation 3.5.1. Let G be a finite group. Denote by C(G) the set of cyclic subgroups of G. Denote by R_{ℂ}(G) the ring of complex characters of G, and set ℚR_{ℂ}(G) = ℚ ⊗_{ℤ} R_{ℂ}(G).

Theorem 3.5.2. [Artin] Let G be a finite group. Then

ℚR_{ℂ}(G) = ∑_{H∈C(G)} Ind_{H}^{G}ℚR_{ℂ}(H)

In other words, any complex character of G is a linear combination with rational coefficients of characters induced from cyclic subgroups of G.

Proof. There is a natural homorphism from the Burnside ring B(G) of G to the ring R_{ℂ}(G), which maps a G-set X to the associated ℂG-module ℂX. This extends to a map of vector spaces ℚB(G) ↦ ℚR_{ℂ}(G). Now the value of the character of the permutation module ℂX at the element s of G is the trace of s on ℂX, which is equal to the number of fixed points |X^{s}| of s on X.

Let H be a subgroup of G. Since |(e_{H}^{G})^{s}| is equal to 0 if H is not conjugate in G to the subgroup generated by s, it follows that the image of e_{H}^{G} in ℚR_{ℂ}(G) is zero unless H is cyclic. And if H is cyclic, the image of e_{H}^{G} is a linear combination with rational coefficients of permutations characters Ind_{K}^{G}1, for subgroups K of G. Taking the image in ℚR_{ℂ}(G) of the decomposition

G/G = ∑_{H∈[s_{G}]} e_{H}^{G}

shows that the trivial character is a linear combination with rational coefficients of such characters Ind_{K}^{G}1, which are induced from cyclic subgroups K of G. Then if χ is any character of G

χInd_{H}^{G}1 = Ind_{H}^{G}(Res_{H}^{G}χ)

and the theorem follows. □

Notation 3.5.3. Let p be a prime number, and O be a complete local noetherian commutative ring with maximal ideal 𝔭 and residue field k of characteristic p. If G is a finite group, an OG-lattice is a finitely generated O-free OG-module. Also denote by O the trivial OG-lattice of O-rank 1.

The Green ring A_{O}(G) is the Grothendieck group of the category of OG- lattices, for relations given by direct sums decompositions.

As an additive group, it is the quotient of the free abelian group with basis the set of isomorphism classes of OG-lattices, by the subgroup generated by elements [M ⊕ N] − [M] − [N], for OG-lattices M and N, where [M] denotes the class of M. Since Krull-Schmidt theorem holds for OG-lattices, the group A_{O}(G) is free, with basis the set of indecomposable OG-lattices. The ring structure on A_{O}(G) is induced by tensor product (over O) of OG-lattices.

There is a natural ring homomorphism π_{O} from B(G) to A_{O}(G), mapping the (class of the) finite G-set X to the (class of the) permutation lattice OX. It is natural to look at the kernel I_{O}(G) of this morphism.

Since A_{O}(G) is torsion-free, the sequence

0 → I_{O}(G) → B(G) →^{π_{O}} A_{O}(G)

is a split exact sequence of abelian groups.

Hence there is an exact sequence

0 → ℚI_{O}(G) → ℚB(G) →^{ℚπ_{O}} ℚA_{O}(G)

It follows that there exists a set N_{O}(G) of subgroups of G, closed by conjugation, such that

ℚI_{O}(G) = ∑_{H∈N_{O}(G)/G} ℚe_{H}^{G}

or equivalently

I_{O}(G) = {X ∈ B(G) | ∀H ⊆ G, H ∉ N_{O}(G) ⇒ |X^{H}| = 0}

Clearly the morphism π_{O} commutes with induction and restriction. Now formulae 3.3.3 and 3.3.4 show that e_{H}^{G} ∈ ℚI_{O}(G) if and only if e_{H}^{H} ∈ ℚI_{O}(H). It follows that there exists a family of finite groups N_{O} such that

N_{O}(G) = {H ⊆ G | H ∈ N_{O}}

The main result of this section is a characterization of the family N_{O}. First a definition:

Definition 3.5.4. Let p be a prime number. If G is a finite group, let O_{p}(G) denote the largest normal p-subgroup of G. The group G is called p-hypoelementary, or cyclic modulo p, if the quotient group G/O_{p}(G) is cyclic. The family of such finite groups is denoted by Z_{p}. The set of p-hypoelementary subgroups of a finite group H is denoted by Z_{p}(G).

It turns out that the family N_{O} depends only on p, and is equal to the complement of Z_{p}, by the following theorem (see [17]):

Theorem 3.5.5. [Conlon] Let G be a finite group, and X, Y be finite G-sets. The following conditions are equivalent:

(1) The OG-lattices OX and OY are isomorphic.

(2) For any p-hypoelementary subgroup H of G, the sets X^{H} and Y^{H} have the same cardinality.

Proof. Recall that if M is an OG-lattice, then a vertex of M is a subgroup P of G, minimal such that M is a direct summand of Ind_{P}^{G}Res_{P}^{G}M. The deep argument for theorem 3.5.5 relies on the Green correspondence, which, for any p-subgroup P of G, is a bijection between the set of isomorphism classes of indecomposable OG-lattices with vertex P and the set of isomorphism classes of indecomposable ON_{G}(P)-lattices with vertex P (see [1] Chapter 3.12).

In the case of permutation modules, or more generally of their direct summands (or p-permutation modules), there is a more elementary approach, due to Broué ([12]). It relies on the Brauer construction: if M is an OG-module, and P is a subgroup of G, then the Brauer quotient M[P] of M at P is defined by

M[P] = M^{P} /(𝔭M^{P} + ∑_{Q⊂P} T r_{Q}^{P}(M^{Q}))

where the sum runs over proper subgroups Q of P, and T r_{Q}^{P} : M^{Q} → M^{P} is the relative trace map or transfer map, defined by

∀m ∈ M^{Q}, T r_{Q}^{P}(m) = ∑_{x∈P/Q} xm

If S is a Sylow p-subgroup of P, and if m ∈ M^{P}, then m = T r_{S}^{P} (\frac{1}{|P:S|}m), thus M[P] = 0 if P is not a p-subgroup of G. When P is a p-subgroup of G and M = OX is a permutation lattice associated to a G-set X, the image of the set X^{P} in M[P] is a k-basis of M[P].

In this case moreover, the value of the Brauer character of M at the element s of G is equal to the number of fixed points of s on X. Hence if s ∈ N_{G}(P), the value of the Brauer character of the kN_{G}(P)/P -module M[P] at the element sP ∈ N_{G}(P)/P is equal to |X^{H_{P,s}}|, where H_{P,s} = P <s> is the subgroup of G generated by s and P. Note that H_{P,s} is cyclic modulo p, and that conversely, if H is a p-hypoelementary subgroup of G, there is a p-subgroup P of G and an s ∈ N_{G}(P) such that H = H_{P,s}.

It follows that if OX and OY are isomorphic, then for any p-hypoelementary subgroup H of G, one has |X^{H}| = |Y^{H}|.

Conversely, if |X^{H}| = |Y^{H}| for any p-hypoelementary subgroup H of G, then for any p-subgroup P of G, the modules kX^{P} ≃ OX[P] and kY^{P} ≃ OY [P] have the same Brauer character. The next proposition shows that OX and OY are isomorphic, and this completes the proof of Theorem 3.5.5. □

Proposition 3.5.6. Let G be a finite group, and let M and N be p-permutation OG-lattices. Then the following are equivalent:

(1) The OG-lattices M and N are isomorphic.

(2) For any p-subgroup P of G, the kN_{G}(P)/P -modules M[P] and N[P] have the same Brauer character.

Proof. It is clear that (1) implies (2). The proof of the converse is by induction on the cardinality of the set

S(M) = {P ⊆ G | M[P] ≠ 0}

Note that if (2) holds, then S(M) = S(N).

If S(M) = ∅, then M[1] = M/𝔭M = N/𝔭N = 0, hence M = N = 0. If S(M) ≠ ∅, choose a maximal element P of S(M). Write M = M_{P} ⊕ M′ (resp. N = N_{P} ⊕ N′), where all direct summands of M_{P} (resp. N_{P}) have non-zero Brauer quotient at P, and where M′[P] = 0 (resp. N′[P] = 0). Theorem 3.5.7 below shows that M[P] = M_{P} [P] and N[P] = N_{P} [P] are projective kN_{G}(P)/P - modules. By assumption, they have the same Brauer character, hence they are isomorphic, and then Theorem 3.5.7 shows that M_{P} and N_{P} are isomorphic. Now (2) holds for M′ and N′, and moreover |S(M′)| < |S(M)|, since S(M′) ⊆ S(M) and P ∈ S(M)−S(M′). Hence M′ ≃ N′ by induction hypothesis, and M ≃ N. □

Theorem 3.5.7. [Broué] Let G be a finite group, and P be a p-subgroup of G.

(1) Let M be an indecomposable p-permutation OG-module with vertex P. Then for any subgroup Q of G, the module M[Q] is non-zero if and only if Q is conjugate to a subgroup of P.

(2) The correspondence M ↦ M[P] induces a bijection between the set of isomorphism classes of indecomposable p-permutation OG-lattices with vertex P and the set of isomorphism classes of indecomposable projective kN_{G}(P)/P -modules.

Proof. (1) Recall (Higman criterion) that if H is a subgroup of G and if M is an OG-module, then M is a direct summand of Ind_{H}^{G}Res_{H}^{G}M if and only if there exists an OH-endomorphism ϕ of M such that Id_{M} = T r_{H}^{G}(ϕ), where T r_{H}^{G}(ϕ) = ∑_{g∈G/H} gϕg^{−1}. If S is a Sylow p-subgroup of H, then |H : S| is invertible in O, and Id_{M} = T r_{S}^{G}(ϕ/|H : S|). In particular, any vertex of M is a p-group.

Recall also that if P is a p-group, and Q is a subgroup of P, then the permutation OP -lattice Ind_{Q}^{P}O is indecomposable, since its reduction modulo 𝔭 is isomorphic to Ind_{Q}^{P}k, which has simple socle k. Hence any p-permutation OP -lattice is a permutation OP -lattice.

Let M be an indecomposable p-permutation OG-lattice. Then M is a direct summand of a permutation lattice OX, for some finite G-set X. Let P be a psubgroup of G such that M is a direct summand of Ind_{P}^{G}Res_{P}^{G}M. Then Res_{P}^{G}M is a direct summand of ORes_{P}^{G}X, hence it is a permutation OP -lattice, and there is a set S of subgroups of P such that

Res_{P}^{G}M ≃ ⊕_{Q∈S} Ind_{Q}^{P}O

Thus M is a direct summand of some module Ind_{Q}^{G}O, for Q ∈ S. Since Ind_{Q}^{P}O is a direct summand of Res_{P}^{G}M, and since O is a direct summand of Res_{Q}^{P}Ind_{Q}^{P}O, it follows that O is a direct summand of Res_{Q}^{G}M, and that Ind_{Q}^{G}O is a direct summand of Ind_{Q}^{G}Res_{Q}^{G}M.

Now if P is a vertex of M, then Q = P ∈ S. Hence M is a direct summand of Ind_{P}^{G}O, and O is a direct summand of Res_{P}^{G}M (in other words the module M has trivial source). It follows that M[Q] ≠ 0 for any subgroup of Q. Conversely, if M[Q] ≠ 0 for some subgroup Q of G, then (Ind_{P}^{G}O)[Q] ≠ 0, thus Q has a fixed point on the set G/P, i.e. Q ⊆_{G} P. Assertion (1) follows. It shows in particular that all the vertices of M are conjugate in G.

(2) Since Ind_{P}^{G}O ≃ Ind_{N_{G}(P)}^{G}ON_{G}(P)/P, and since the ON_{G}(P)/P is a direct sum of indecomposable projective ON_{G}(P)/P -lattices, it follows from (1) that if M is an indecomposable p-permutation OG-lattice with vertex P, then there is an indecomposable projective ON_{G}(P)/P -lattice E such that M is a direct summand of Ind_{N_{G}(P)}^{G}E. It is easy to check by Mackey formula that

(Ind_{N_{G}(P)}^{G}E)[P] ≃ ∑_{{x∈G/N_{G}(P) \atop P^{x}⊆N_{G}(P)}} E[P^{x}P/P] = E/𝔭E

since E[Q/P] = 0 for any non-trivial subgroup Q/P of N_{G}(P)/P.

Now E/𝔭E is an indecomposable projective kN_{G}(P)/P -module, having M[P] as a non-zero direct summand. It follows that M[P] ≃ E/𝔭E is an indecomposable projective kN_{G}(P)/P -module, and the correspondence of assertion (2) is well defined.

Let E be an indecomposable projective ON_{G}(P)/P -lattice. Write

Ind_{N_{G}(P)}^{G}E ≃ L ⊕ L′

where all the direct summands of L have vertex conjugate to P in G, and no direct summands of L′ have vertex conjugate to P. Since L′ is a direct summand of Ind_{P}^{G}O, all the indecomposable direct summands of L′ have vertex strictly contained in P (up to G-conjugation). Hence L′[P] = 0, and L[P] ≃ E/𝔭E is indecomposable. It follows that L is indecomposable, and that it is the only indecomposable direct summand of Ind_{N_{G}(P)}^{G}E with vertex P, up to isomorphism.

Thus if M is an indecomposable p-permutation module with vertex P, and if E is the only indecomposable projective ON_{G}(P)/P -lattice such that M[P] ≃ E/𝔭E, then M is isomorphic to the only indecomposable direct summand of Ind_{N_{G}(P)}^{G}E with vertex P. This shows that the correspondence M ↦ M[P] of assertion (2) is injective.

Conversely, if \overline{E} is an indecomposable projective kN_{G}(P)/P -module, then there is a projective ON_{G}(P)/P -lattice E such that \overline{E} ≃ E/𝔭E. In particular E is indecomposable. Now Ind_{N_{G}(P)}^{G}E has a unique indecomposable direct summand M with vertex P, and M[P] ≃ \overline{E}. This shows that the correspondence M ↦ M[P] of assertion (2) is surjective, and completes the proof of Theorem 3.5.7. □

Corollary 3.5.8. Let G be a finite group. Then

ℚA_{O}(G) = ∑_{H∈Z_{p}(G)} Ind_{H}^{G}ℚA_{O}(H)

Two OG-lattices M and N are isomorphic if and only if for any p-hypoelementary subgroup H of G, the restrictions Res_{H}^{G}M and Res_{H}^{G}N are isomorphic.

Proof. The image by ℚπ_{O}(e_{H}^{G}) of the idempotent e_{H}^{G} of ℚB(G) in the ring ℚA_{O}(G) is zero if H is not p-hypoelementary. Thus

O = ℚπ_{O}(G/G) = ∑_{H∈Z_{p}(G)} ℚπ_{O}(e_{H}^{G})

Now e_{H}^{G} is a linear combination of elements G/K, for subgroups K of H. Since subgroups of p-hypoelementary groups are p-hypoelementary, it follows that there exists rational numbers r_{K} such that

O = ∑_{K∈Z_{p}(G)} r_{K}Ind_{K}^{G}O

since moreover ℚπ_{O}(G/K) is the (class of) the permutation lattice Ind_{K}^{G}O. Tensoring this identity with M over O, and using Frobenius identity, it follows that

M = ∑_{K∈Z_{p}(G)} r_{K}Ind_{K}^{G}Res_{K}^{G}M

This proves both assertions of the corollary. □

Remark 3.5.9. A different proof of Theorem 3.5.5 and Corollary 3.5.8 has been given by Dress ([21]). It is exposed in Curtis-Reiner ([18] Chapter 11.80D).

Definition 3.5.10. Let p and q be (non-necessarily distinct) prime numbers. A finite group H is called a (p, q)-Dress group if the group O^{q}(H) is p-hypoelementary. A (p, q)-Dress subgroup H of a finite group G is a subgroup of G which is a (p, q)- Dress group. The set of (p, q)-Dress subgroups of a finite group H is denoted by D_{p,q}(G).

Theorem 3.5.11. [Dress] Let G be a finite group. Then

A_{O}(G) = ∑_{{H∈D_{p,q}(G) \atop any q}} Ind_{H}^{G}A_{O}(H)

Proof. It follows from the expression of the idempotents of the Burnside ring B(G): if q is a prime and J is a q-perfect subgroup of G, then the idempotent f_{J}^{G} of Corollary 3.3.9 is mapped to zero by ℤ_{(q)}π_{O} if J is not p-hypoelementary.

And if J is p-hypoelementary, the idempotent f_{J}^{G} is a linear combination with coefficients in ℤ_{(q)} of elements G/K, where K runs through the (p, q)-Dress subgroups of G. This shows that there is an integer m_{q} coprime to q and integers n_{K} such that

m_{q}O = ∑_{K∈D_{p,q}(G)} n_{K}Ind_{K}^{G}O  in  A_{O}(G)

Setting

A′_{O}(G) = ∑_{{H∈D_{p,q}(G) \atop any q}} Ind_{H}^{G}A_{O}(H)

it follows that the quotient A_{O}(G)/A′_{O}(G) is a torsion group, with finite exponent coprime to q. Since this holds for any prime q, the groups A_{O}(G) and A′_{O}(G) are equal. □

3.6. Further results and references. The group of units of the Burnside ring has been studied by Matsuda ([29]), Matsuda-Miyata ([30]), and Yoshida ([50]).

Examples of non-isomorphic groups having isomorphic Burnside rings have been given by Thévenaz ([40]).

The Burnside ring of a compact Lie group has been defined and studied by tom Dieck ([44], [45], [46]) and Schwänzl ([34], [35]).

General exposition of the properties of Burnside rings can be found in Benson ([1] Chapter 5.4), Curtis-Reiner ([18] Chapter 11), Karpilovsky ([24] Chapter 15), tom Dieck ([46]).

4. Invariants

The Burnside ring is an analogue for finite G-sets of the ring ℤ for finite sets (and ℤ is actually isomorphic to the Burnside ring of the trivial group). One can attach various invariants to structured G-sets, such as G-posets or G-simplicial complexes.

This section is a self-contained algebraic exposition of the properties of those invariants. The original definitions and methods of Quillen ([32], [33]) are used throughout, avoiding however the topological part of this material. Thus for example no use will be made of the geometric realization of a poset, and the accent will be put on acyclic posets rather than contractible ones.

In other words, in order to define and state properties of the invariants attached to finite G-posets in the Burnside ring, one can forget about the fundamental group of those posets, and consider only homology groups.

4.1. Homology of posets. Let (X, ≤) be a partially ordered set (poset for short). As usual, if x, x′ are in X, the notation x < x′ means x ≤ x′ and x ≠ x′. The notation [x, x′]_{X} (resp. [x, x′[_{X}, ]x, x′]_{X}, ]x, x′[_{X}) stands for the set of elements z ∈ X with x ≤ z ≤ x′ (resp. x ≤ z < x′, x < z ≤ x′, x < z < x′). The notation [x, .[_{X} (resp. ]x, .[_{X}, ]., x]_{X}, ]., x[_{X}) stands for the set of elements z ∈ X with x ≤ z (resp. x < z, z ≤ x, z < x).

If n ∈ ℕ, let Sd_{n}(X) denote the set of chains x_{0} < … < x_{n} of elements of X of cardinality n + 1. The chain complex C_{∗}(X, ℤ) is the complex of ℤ-modules defined as follows: for n ∈ ℕ, the module C_{n}(X, ℤ) is the free ℤ-module with basis Sd_{n}(X). The differential d_{n} : C_{n}(X, ℤ) → C_{n−1}(X, ℤ) is given by

d_{n}(x_{0}, …, x_{n}) = ∑_{i=0}^{n}(−1)^{i}(x_{0}, …, x̂_{i}, …, x_{n})

where (x_{0}, …, x̂_{i}, …, x_{n}) denotes the chain (x_{0}, …, x_{n}) − {x_{i}}.

The reduced chain complex C̃_{∗}(X, ℤ) is the augmented complex obtained by setting C_{−1}(X, ℤ) = ℤ, the augmentation map d_{0} : C_{0}(X, ℤ) → C_{−1}(X, ℤ) sending each x_{0} ∈ X to 1 ∈ ℤ.

The n^{th} homology group of the complex C_{∗}(X, ℤ) (resp. C̃_{∗}(X, ℤ)) is denoted by H_{n}(X, ℤ), and called the n^{th} homology group (resp. reduced homology group) of X. A poset X is acyclic if all its reduced homology groups are equal to zero.

More generally, if K is a ring, the n^{th} homology group of X with coefficients in K is the n^{th} homology group of the complex K ⊗_{ℤ} C_{∗}(X, ℤ). When K is a field, one has H_{n}(X, K) = K ⊗ H_{n}(X, ℤ).

The Euler-Poincaré characteristic χ(X) of a finite poset X is defined by

χ(X) = ∑_{n≥0}(−1)^{n}rank_{ℤ}C_{n}(X, ℤ) = ∑_{n≥0}(−1)^{n}|Sd_{n}(X)|

Similarly, the reduced Euler-Poincaré characteristic χ̃(X) is defined by

χ̃(X) = ∑_{n≥−1}(−1)^{n}rank_{ℤ}C̃_{n}(X, ℤ) = χ(X) − 1

If K is a field, then setting k_{n} = dim_{K} Ker(K ⊗_{ℤ} d_{n}) for any n ∈ ℕ

dim_{K} H_{n}(X, K) = dim_{K} K ⊗_{ℤ} H_{n}(X, ℤ) = k_{n} − dim_{K} Im(K ⊗ d_{n+1})
= k_{n} + k_{n+1} − dim_{K} K ⊗_{ℤ} C_{n+1}(X, ℤ)

It follows that

χ(X) = ∑_{n≥0}(−1)^{n} dim_{K} H_{n}(X, K)  χ̃(X) = ∑_{n≥−1}(−1)^{n} dim_{K} H̃_{n}(X, K)

In particular, if X is acyclic, then χ̃(X) = 0. Similarly, if X and Y are finite posets, and if there is an homotopy equivalence f_{∗} from the complex C_{∗}(X, ℤ) to the complex C_{∗}(Y, ℤ), then χ(X) = χ(Y), since f induces a group isomorphism from H_{n}(X, ℤ) to H_{n}(Y, ℤ), for any n ∈ ℕ.

If X and Y are posets, a map of posets f : X → Y is a map from X to Y such that f(x) ≤ f(x′) whenever x and x′ are elements of X such that x ≤ x′. If f is such a map, there is an induced map of chain complexes C_{∗}(f, ℤ) : C_{∗}(X, ℤ) → C_{∗}(Y, ℤ) defined for n ∈ ℕ by

C_{n}(f, ℤ)(x_{0}, …, x_{n}) = \{ \begin{matrix}(f(x_{0}), …, f(x_{n})) & if f(x_{0}) < … < f(x_{n}) \\ 0 & otherwise \end{matrix}

One also defines a reduced map C̃_{∗}(f, ℤ) : C̃_{∗}(X, ℤ) → C̃_{∗}(Y, ℤ) by C̃_{n}(f, ℤ) = C_{n}(f, ℤ) if n ≤ 0, and C̃_{−1} = Id_{ℤ}.

If f and g are maps of posets from X to Y, the notation f ≤ g means that f(x) ≤ g(x) for any x ∈ X. The maps f and g are said to be comparable if either f ≤ g or g ≤ f.

Lemma 4.1.1. Let f and g be maps of posets from X to Y. If f and g are comparable, then the maps of complexes C_{∗}(f, ℤ) and C_{∗}(g, ℤ) are homotopic, as well as the maps C̃_{∗}(f, ℤ) and C̃_{∗}(g, ℤ).

Proof. Suppose for instance that f ≤ g. Consider the map h_{n} : C_{n}(X, ℤ) → C_{n+1}(Y, ℤ) defined for n ∈ ℕ by

h_{n}(x_{0}, …, x_{n}) = ∑_{i=0}^{n}(−1)^{i}(f(x_{0}), …, f(x_{i}), g(x_{i}), …, g(x_{n}))

where the sequence (f(x_{0}), …, f(x_{i}), g(x_{i}), …, g(x_{n})) is replaced by 0 if it is not strictly increasing. It is easy to check that

d_{n+1} ∘ h_{n} + h_{n−1} ∘ d_{n} = C_{n}(g, ℤ) − C_{n}(f, ℤ)

thus the maps C_{n}(f, ℤ) and C_{n}(g, ℤ) are homotopic. A similar argument can be used for the augmented complexes, the map h_{−1} being the zero map. □

Corollary 4.1.2. [Quillen]

(1) Let X and Y be posets, and let f : X → Y and g : Y → X be maps of posets. If g∘f is comparable to Id_{X} and if f ∘g is comparable to Id_{Y}, then the maps of complexes C̃_{∗}(f, ℤ) and C̃_{∗}(g, ℤ) are mutual inverse homotopy equivalences between C̃_{∗}(X, ℤ) and C̃_{∗}(Y, ℤ).

(2) If the poset X has a biggest element, or a smallest element, the chain complex C̃_{∗}(X, ℤ) is contractible.

Proof. The first assertion is a direct consequence of the previous lemma. For the second one, denote by m the biggest (or the smallest) element of X, and let • denote a poset of cardinality one. Apply assertion (1) to the unique map f : X → • and to the map g : • → X sending the unique element of • to m. The result follows, since the complex C̃_{∗}(•, ℤ) is clearly contractible. □

4.2. Invariants attached to finite G-posets. The following definition of the Lefschetz invariants is due to Thévenaz ([39]):

Definition 4.2.1. Let G be a finite group. A G-poset X is a G-set equipped with an order relation ≤ compatible to the G-action: if x ≤ x′ are elements of X and if g ∈ G, then gx ≤ gx′.

If X and Y are G-posets, a map of G-posets f : X → Y is a map such that f(gx) = gf(x) if g ∈ G and x ∈ X, and such that f(x) ≤ f(x′) in Y, whenever x ≤ x′ in X. If y ∈ Y, then

f^{y} = {x ∈ X | f(x) ≤ y} f_{y} = {x ∈ X | f(x) ≥ y}

Those sets are sub-G_{y}-posets of (the restriction of) X to the stabilizer G_{y} of y in G.

If x ≤ y are elements of X, the set ]x, y[_{X} is a G_{x,y}-poset, where G_{x,y} is the stabilizer G_{x,y} of the pair {x, y}. Similarly, the sets ]x, .[_{X} and ]., x[_{X} are G_{x}-posets.

If X is a G-poset, then for n ∈ ℕ, the set Sd_{n}(X) is a G-set. When X is finite, the Lefschetz invariant Λ_{X} of X is the element of B(G) defined by

Λ_{X} = ∑_{n≥0}(−1)^{n}Sd_{n}(X)

The reduced Lefschetz invariant Λ̃_{X} is the element of B(G) defined by

Λ̃_{X} = Λ_{X} − G/G

If x < y are elements of X, the Möbius invariant μ_{X}(x, y) is defined as the Lefschetz invariant of the poset ]x, y[_{X}. It is an element of the Burnside ring B(G_{x,y}). By convention, the Möbius invariant μ_{X}(x, x) is equal to G_{x}/G_{x}.

It follows from those definitions that |Λ_{X}| is equal to the Euler-Poincaré characteristic of X. One can say more:

Lemma 4.2.2. Let G be a finite group.

(1) If X is a finite G-poset, then for any subgroup H of G

(Λ_{X})^{H} = Λ_{X^{H}}

in B(N_{G}(H)/H). In particular |(Λ_{X})^{H}| = χ(X^{H}).

(2) If X and Y are finite G-posets, then Λ_{X} = Λ_{Y} in B(G) if and only if χ(X^{H}) = χ(Y^{H}) for any subgroup H of G.

Proof. The first assertion is obvious, since Sd_{n}(X)^{H} = Sd_{n}(X^{H}) for all n ∈ ℕ. The second one follows from Burnside’s Theorem 2.3.2. □

Definition 4.2.3. A G-poset X is called G-acyclic if the poset X^{H} is acyclic for any subgroup H of G.

The following is a direct consequence of this definition:

Lemma 4.2.4. Let G be a finite group, and X be a finite G-poset. If X is G-acyclic, then Λ̃_{X} = 0 in B(G).

Proof. This follows from Burnside’s theorem, since |(Λ̃_{X})^{H}| = χ̃(X^{H}) = 0 for any subgroup H of G. □

Proposition 4.2.5. Let G be a finite group.

(1) Let X and Y be finite G-posets, and let f : X → Y and g : Y → X be maps of G-posets. If g ∘ f is comparable to Id_{X} and if f ∘ g is comparable to Id_{Y}, then Λ̃_{X} = Λ̃_{Y} in B(G).

(2) If a G-poset X has a biggest element, or a smallest element, then it is G-acyclic.

Proof. This is a direct consequence of Corollary 4.1.2: for any subgroup H of G, the restrictions of f and g to the posets X^{H} and Y^{H} verify the hypotheses of Corollary 4.1.2. Hence χ̃(X^{H}) = χ̃(Y^{H}), and the first assertion follows. For the second, note that the biggest (resp. smallest) element of X is also a biggest (resp. smallest) element of X^{H}, for any subgroup H of G. □

Example 4.2.6. Let f : X → Y be a maps of finite G-posets. Denote by X ∗_{f} Y the G-poset defined as follows: the underlying G-set is the disjoint union X ⊔ Y of X and Y. The ordering is defined for z and z′ in X ∗_{f} Y by

z ≤ z′ ⇔ \{ \begin{matrix}z, z′ ∈ X & and z ≤ z′ in X \\ z, z′ ∈ Y & and z ≤ z′ in Y \\ z ∈ X, z′ ∈ Y, & and f(z) ≤ z′ in Y \end{matrix}

Let f denote the injection from Y to X ∗_{f} Y, and g denote the map from X ∗_{f} Y to Y defined by

g(z) = \{ \begin{matrix}f(z) & if z ∈ X \\ z & if z ∈ Y \end{matrix}

Then f and g are maps of G-posets, such that g ∘ f = Id_{Y} and Id_{X∗_{f} Y} ≤ f ∘ g. It follows that Λ_{X∗_{f} Y} = Λ_{Y}.

The consequence is the following relation between Λ̃_{X} and Λ̃_{Y}:

Proposition 4.2.7. Let f : X → Y be a map of finite G-posets. Then in B(G)

Λ̃_{Y} = Λ̃_{X} + ∑_{y∈G\Y} Ind_{G_{y}}^{G}(Λ̃_{f^{y}} Λ̃_{]y,.[_{Y}})

Λ̃_{Y} = Λ̃_{X} + ∑_{y∈G\Y} Ind_{G_{y}}^{G}(Λ̃_{f_{y}} Λ̃_{].,y[_{Y}})

In particular, if Λ̃_{f^{y}} = 0 for all y ∈ Y (for instance if f^{y} is G_{y}-acyclic), then Λ̃_{X} = Λ̃_{Y} in B(G).

Proof. Let n ∈ ℕ. The set Sd_{n}(X ∗_{f} Y) is the disjoint union of Sd_{n}(X) and of the set of sequences z_{0} < … < z_{n} for which z_{n} ∈ Y. Such a sequence has a smallest element y = z_{i} in Y, hence it can be written as

x_{0} < … < x_{i−1} < y < y_{0} < … < y_{n−i−1}

where x_{0} < … < x_{i−1} is in Sd_{i−1}(f^{y}) (for i = 0, the convention is that Sd_{−1}(f^{y}) is a set of cardinality one), and y_{0} < … < y_{n−i−1} is in Sd_{n−i−1}(]y, .[_{Y}) (for i = n, the convention is that Sd_{−1}(]y, .[_{Y}) has cardinality one).

Keeping track of the action of G, this leads to the following isomorphism of G-sets

Sd_{n}(X ∗_{f} Y) = Sd_{n}(X) ⊔ ⨆_{y∈G\Y} Ind_{G_{y}}^{G} (⨆_{i=0}^{n} (Sd_{i−1}(f^{y}) × Sd_{n−i−1}(]y, .[_{Y})))

Taking alternating sums gives the first equality of the proposition. The second one follows, by considering the map f : X^{op} → Y^{op} between the opposite posets of X and Y, since clearly Λ̃_{X^{op}} = Λ̃_{X} for any finite group G and any finite G-poset X. □

Corollary 4.2.8. Let X be a finite G-poset.

(1) The reduced Lefschetz invariant of X is equal to

Λ̃_{X} = −G/G − ∑_{x∈G\X} Ind_{G_{x}}^{G} Λ̃_{]x,.[_{X}}

(2) If x ≤ y in X, then

∑_{z∈G_{x,y}\[x,y]} Ind_{G_{x,y,z}}^{G_{x,y}}Res_{G_{x,y,z}}^{G_{y,z}}μ_{X}(z, y) = \{ \begin{matrix}0 & if x < y \\ G_{x}/G_{x} & if x = y \end{matrix}

(3) If f : X → Y is a map of finite G-posets, then

Λ_{X} = − ∑_{y∈G\Y} Ind_{G_{y}}^{G}Λ_{f^{y}} Λ̃_{]y,.[_{Y}} = − ∑_{y∈G\Y} Ind_{G_{y}}^{G}Λ_{f_{y}} Λ̃_{].,y[_{Y}}

Assertion 2) is the reason for the name of the Möbius invariant.

Proof. Assertion (1) follows from the previous proposition, applied to the inclusion ∅ → X, since Λ̃_{∅} = −G/G. Assertion (2) follows from assertion (1), applied to the G_{x,y}-poset [x, y[_{X}, which has a smallest element x if x < y.

Assertion (3) follows from assertion (1) and Proposition 4.2.7:

Λ̃_{Y} = Λ̃_{X} + ∑_{y∈G\Y} Ind_{G_{y}}^{G}((Λ_{f^{y}} − G_{y}/G_{y})Λ̃_{]y,.[_{Y}} )
= Λ_{X} − G/G + ∑_{y∈G\Y} Ind_{G_{y}}^{G}(Λ_{f^{y}} Λ̃_{]y,.[_{Y}}) − ∑_{y∈G\Y} Ind_{G_{y}}^{G} Λ̃_{]y,.[_{Y}}
= Λ_{X} + ∑_{y∈G\Y} Ind_{G_{y}}^{G}(Λ_{f^{y}} Λ̃_{]y,.[_{Y}}) + Λ̃_{Y}

The second equality in assertion 3) is similar. □

Corollary 4.2.9. Let G be a finite group and X be a finite poset. Denote by X_{♯} (resp. X^{♯}) the set of elements x ∈ X such that Λ̃_{].,x[_{X}} ≠ 0 (resp. Λ̃_{]x,.[_{X}} ≠ 0) in B(G_{x}). If Y is a sub-G-poset of X such that X_{♯} ⊆ Y ⊆ X (resp. X^{♯} ⊆ Y ⊆ X), then Λ̃_{Y} = Λ̃_{X} in B(G).

Proof. By induction on the cardinality of X: if X = ∅, then X = X_{♯} = X^{♯} and there is nothing to prove. For the inductive step, consider the inclusion map i : X_{♯} → Y. It is a map of G-posets. Moreover if y ∈ X_{♯}, then i^{y} has a biggest element y, hence it is G_{y}-acyclic, and Λ̃_{G_{y}} = 0 in B(G_{y}). Now if y ∉ X_{♯}, then Λ̃_{].,y[_{X}} = 0 in B(G_{y}) by definition of X_{♯}. Moreover in this case

i^{y} =]., y[_{X}∩X_{♯} = {z ∈ X | z < y, Λ̃_{].,z[_{X}} ≠ 0 in B(G_{z})}
⊇ {z ∈ X | z < y, Λ̃_{].,z[_{X}} ≠ 0 in B(G_{z} ∩ G_{y})} = (]., y[_{X})_{♯}

It follows that there are inclusions of G_{y}-posets

(]., y[_{X})_{♯} ⊆ i^{y} ⊆]., y[_{X}

Moreover |]., y[_{X}| < |X|. By induction hypothesis, it follows that Λ̃_{i^{y}} = Λ̃_{].,y[_{X}} = 0. Now the corollary follows from Proposition 4.2.7. □

4.3. Steinberg invariants.

Definition 4.3.1. Let G be a finite group, and p be a prime number. The Steinberg invariant St_{p}(G) of G at p is the reduced Lefschetz invariant of the poset s_{p}(G) of non-trivial p-subgroups of G, on which G acts by conjugation.

The reason for this terminology is that if G is a finite simple Chevalley group in characteristic p, then the (virtual) permutation character associated to St_{p}(G) is equal up to a sign to the Steinberg character of G.

Proposition 4.3.2. Let G be a finite group, and p be a prime number. Then St_{p}(G) = 0 in B(G) if and only if G has a non-trivial normal p-subgroup.

Proof. If St_{p}(G) = 0, then in particular χ̃(s_{p}(G)^{G}) = 0. Thus s_{p}(G)^{G} is non-empty, and G has a non-trivial normal p-subgroup.

Conversely, suppose that R ≠ 1l is a non-trivial normal p-subgroup of G. Let f be the map from s_{p}(G) to [R, .[_{s_{p}(G)} defined by f(Q) = Q.R, and let g denote the inclusion map from [R, .[_{s_{p}(G)} to s_{p}(G). Then f and g are maps of G-posets, and moreover Id ≤ g ∘ f and f ∘ g = Id. Since [R, .[_{s_{p}(G)} has a smallest element, it follows from Proposition 4.2.5 that St_{p}(G) = 0 in B(G). □

Remark 4.3.3. Quillen has conjectured that s_{p}(G) is contractible if and only if O_{p}(G) ≠ 1l. The above proof shows actually that s_{p}(G) is G-contractible if and only if O_{p}(G) ≠ 1l (see Thévenaz-Webb [42]).

Proposition 4.3.4. Let G be a finite group, and p be a prime number. Let a_{p}(G) denote the sub-G-poset of s_{p}(G) consisting of non-trivial elementary abelian p-subgroups of G, and b_{p}(G) denote the sub-G-poset of s_{p}(G) consisting of nontrivial p-subgroups P of G such that P = O_{p}(N_{G}(P)). Then

St_{p}(G) = Λ̃_{a_{p}(G)} = Λ̃_{b_{p}(G)} in B(G)

Proof. Let P be a non-trivial p-subgroup of G. Suppose P is not elementary abelian, and denote by Φ(P) the Frattini subgroup of P. Let f :]., P [_{s_{p}(G)}→ [Φ(P), P [_{s_{p}(G)} be the map defined by f(Q) = Q.Φ(P), and let g be the inclusion map from [Φ(P), P [_{s_{p}(G)} to ]., P [_{s_{p}(G)}. Then f and g are maps of N_{G}(P)-posets. Moreover f ∘ g = Id, and Id ≤ g ∘ f. Now [Φ(P), P [_{s_{p}(G)} has a smallest element, thus Λ̃_{].,P [_{sp(G)}} = 0 in B(N_{G}(P)) by Proposition 4.2.5. In other words

(s_{p}(G))_{♯} ⊆ a_{p}(G) ⊆ s_{p}(G)

and Corollary 4.2.9 shows that Λ̃_{s_{p}(G)} = Λ̃_{a_{p}(G)}.

The other equality is similar: let P be a non-trivial p-subgroup of G. Set R = O_{p}(N_{G}(P)/P ), and suppose R ≠ P. Let f :]P, .[_{s_{p}(G)}→ s_{p}(N_{G}(P)/P ) defined by f(Q) = N_{Q}(P)/P, and let g : s_{p}(N_{G}(P)/P ) →]P, .[_{s_{p}(G)} defined by g(Q/P) = Q. Then f ∘ g = Id, and g ∘ f ≤ Id. By Proposition 4.2.5, it follows that

Λ̃_{]P,.[_{sp(G)}} = Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{p}(N_{G}(P)/P )

and this is zero since O_{p}(N_{G}(P)/P ) = R/P ≠ 1l. Hence Λ̃_{]P,.[_{sp(G)}} = 0 in B(N_{G}(P)/P ). In other words

(s_{p}(G))^{♯} ⊆ b_{p}(G) ⊆ s_{p}(G)

and the proposition follows from Corollary 4.2.9. □

Notation 4.3.5. For the remainder of this section, the group G will be a finite group, and \underline{F} will denote a family of subgroups of G such that:

(1) 1l ∈ \underline{F}.

(2) \underline{F} is closed by conjugation.

(3) \underline{F} is closed by products, i.e. if P and Q are elements of \underline{F} such that P

normalizes Q, then P.Q ∈ \underline{F}.

If H is a subgroup of G, then \underline{F}(H) is the set of subgroups of H which are in \underline{F}. It is a family of subgroups of H having the above three properties.

Denote by F the family \underline{F} with the trivial group removed. Then \underline{F} and F are ordered by inclusion of subgroups, and they are G-posets. When H is a subgroup of G, denote similarly by F(H) the set of non-trivial subgroups of H which are in F.

Definition 4.3.6. The Steinberg invariant St_{F}(G) of G with respect to F is the reduced Lefschetz invariant of the G-poset F.

Thus if F = s_{p}(G), then St_{F}(G) = St_{p}(G).

Lemma 4.3.7. Let G and F as in 4.3.5. If P ∈ F, then St_{F}(G)^{P} = 0 in B(N_{G}(P)/P ).

Proof. Let a denote the inclusion map from [P, .[_{F} to F^{P}, and b denote the map from F^{P} to [P, .[_{F} defined by b(P) = F P. Then b ∘ a = Id_{[P,.[_{F}}, and Id_{F^{P}} ≤ a ∘ b. Thus St_{F}(G)^{P} = Λ̃_{F^{P}} = Λ̃_{[P,.[_{F}} = 0 since [P, .[_{F} has a smallest element P. □

Theorem 4.3.8. [Bouc] Let G and F as in 4.3.5. If X ∈ B(G), set

St_{F}(G, X) = ∑_{P ∈G\\underline{F}} Ind_{N_{G}(P)}^{G}(μ\underline{F}(1l, P) Inf_{N_{G}(P)/P}^{N_{G}(P)} X^{P} )

Then the map X ↦ St_{F}(G, X) is an idempotent group endomorphism of B(G), and its image is the set of elements X of B(G) such that X^{P} =0 in B(N_{G}(P)/P ) for all P ∈ F.

Proof. Clearly if X^{P} = 0 for P ∈ F, then St_{F}(G, X) = X. Hence the only thing to check is that if P ∈ F, then St_{F}(G, X)^{P} = 0.

By linearity, one can suppose that X is a G-set, viewed as a G-poset for the discrete ordering. Let Z denote the subposet of X × F consisting of pairs (x, P) such that P ⊆ G_{x}. Let a : Z → X defined by a((x, P)) = x. Then a is a map of G-posets, and a^{x} is isomorphic to F(G_{x}).

Now obviously Λ_{X} = X for any dicrete G-poset X. Moreover all the elements of X are maximal in X, thus Λ̃_{]x,.[_{X}} = −G_{x}/G_{x} for all x ∈ X. It follows from Proposition 4.2.7 that

(4.3.9)  Λ_{X} = X = Λ_{Z} − ∑_{x∈G\X} Ind_{G_{x}}^{G}St_{F(G_{x})}(G_{x})

Now let b : Z → F be the map defined by b((x, P)) = P. Then b is a map of G-posets, and for Q ∈ F

g_{Q} = {(x, P) ∈ X × F | Q ⊆ P ⊆ G_{x}}

The maps c : q_{Q} → X^{Q} and d : X^{Q} → g_{Q} defined by c((x, P)) = x and d(x) = (x, Q) are maps of N_{G}(Q)-posets, such that d ∘ c ≤ Id_{g_{Q}} and c ∘ d = Id_{X^{Q}}. Thus by Corollary 4.2.8

Λ_{Z} = − ∑_{Q∈G\F} Ind_{N_{G}(Q)}^{G}(μ\underline{F}(1l, Q)X^{Q})

Hence equation 4.3.9 gives

X = − ∑_{Q∈G\F} Ind_{N_{G}(Q)}^{G}(μ\underline{F}(1l, Q)X^{Q}) − ∑_{x∈G\X} Ind_{G_{x}}^{G}St_{F(G_{x})}(G_{x})

and finally

St_{F}(G, X) = − ∑_{x∈G\X} Ind_{G_{x}}^{G}St_{F(G_{x})}(G_{x})

Now for any x ∈ X and any P ∈ F, if P ⊆ G_{x}, then St_{F(G_{x})}(G_{x})^{P} = 0 by Lemma 4.3.7. Theorem 4.3.8 follows from assertion (3) of Proposition 2.2.1. □

Remark 4.3.10. Another proof of Theorem 4.3.8 can be found in [5], where decompositions of B(G) associated to F are constructed.

Corollary 4.3.11. Let G and \underline{F} as in 4.3.5, let p be a prime number, and let O be a complete local noetherian commutative ring with residue field of characteristic p. Suppose that F contains the set s_{p}(G) of non-trivial p-subgroups of G. Then for any finite G-poset X the image of St_{F}(G, X) in A_{O}(G) is a linear combination of projective OG-lattices.

Proof. Indeed, in this case, if Y is an element of B(G) such that Y^{P} = 0 for P ∈ F, then in particular Y^{P} = 0 for any non-trivial p-subgroup P of G. It follows that |Y^{H}| = 0 whenever H is a subgroup of G with O_{p}(H) ≠ 1l.

Now Y = ∑_{H} |Y^{H}|e_{H}^{G} in ℚB(G), and by Theorem 3.5.5, the idempotent e_{H}^{G} maps to zero in ℚA_{O}(G) if H is not p-hypoelementary. If H is hypoelementary and if O_{p}(H) = 1l, then H is a cyclic p′-group, and the idempotent e_{H}^{G} is a linear combination of elements G/K, for subgroups K of H.

Those remarks show that the image of Y in A_{O}(G) is a linear combination of permutation lattices Ind_{K}^{G}O, for cyclic p′-subgroups K of G. The corollary follows, since those lattices are OG-projective. □

Remark 4.3.12. If X = G/G, then St_{F}(G, X) = −St_{F}(G). Hence the image of St_{F}(G) in A_{O}(G) is a linear combination of projective OG-lattices. In fact, it has been shown by Webb ([47]) that the chain complex C̃_{∗}(F, ℤ_{p}) is the direct sum of a complex of projective ℤ_{p}G-lattices, and of a split acyclic augmented subcomplex.

Corollary 4.3.13. Let G and \underline{F} as in 4.3.5, and suppose moreover that \underline{F} is closed by taking subgroups. Let X be a finite G-poset.

(1) If P ∈ \underline{F}, then there exists an integer m_{P} such that

Res_{P}^{G}St_{F}(G, X) = m_{P} P/1l

(2) Denote by |G|_{F} the l.c.m of the orders of elements of F. Then

χ̃(X) + ∑_{P ∈F} μ\underline{F}(1l, P) χ̃(X^{P}) ≡ 0 (|G|_{F})

and in particular

χ̃(F) ≡ 0 (|G|_{F})

Proof. Let Y be an element of B(G) such that Y^{P} = 0 for all P ∈ F. Consider the restriction of Y to an element Q of F. Then

Res_{Q}^{G}Y = ∑_{R} |Y^{R}|e_{R}^{Q}

where R runs through a set of representatives of conjugacy classes of subgroups of Q. Since \underline{F} is closed by taking subgroups, the only non-zero term in this sum is obtained for R = 1l. Hence

Res_{Q}^{G}Y = |Y |e_{1l}^{Q} = \frac{|Y |}{|Q|} Q/1l

This shows that |Y | ≡ 0 (|Q|), and Res_{Q}^{G}Y is an integer multiple of Q/1l. The first assertion follows, for Y = St_{F}(G, X). Taking cardinalities gives the first congruence. The second one follows from Lemma 4.3.7 in the case X = F. □

Remark 4.3.14. When F is the poset s_{p}(G) of non-trivial p-subgroups of G, the second congruence is χ̃(s_{p}(G)) ≡ 0 (|G|_{p}), and it is due to Brown ([13]). Similar congruences have been stated by Thévenaz ([38]) and Brown-Thévenaz ([14]).

The computation of the Steinberg invariants of the symmetric groups, or more generally of wreath products of a finite group with symmetric groups, can be found in Bouc ([6]). It requires the use of a ring of formal power series with coefficients in Burnside rings.

5. The Mackey and Green functor structure

The notion of Mackey functor is a formal generalization of the properties of induction, restriction, and conjugation exposed in section 2.2. The notion of Green functor keeps track moreover of the ring structure. Both have several equivalent definitions, that are quickly recalled hereafter. The Burnside functor is a universal object in this framework also.

5.1. Mackey functors and subgroups. Let G be a finite group, and R be a ring. Let R-Mod denote the category of R-modules. The first definition of Mackey functors is due to Green ([23]):

Definition 5.1.1. [Green] A Mackey functor M for the group G over R (or with values in R-Mod) consists of the following data:

• For each subgroup H of G, an R-module M(H).

• Whenever H ⊆ K are subgroups of G with H ⊆ K, a map of R-modules t_{H}^{K} : M(H) → M(K), called transfer or induction, and a map of Rmodules r_{H}^{K} : M(K) → M(H) called restriction.

• For each subgroup H of G and each element x ∈ G, a map of R-modules

c_{x,H} : M(H) → M(^{x}H), called conjugation.

Those maps are subject to four types of conditions:

• (Triviality conditions) For any subgroup H of G, and any h ∈ H, the maps t_{H}^{H}, r_{H}^{H}, and c_{h,H} are equal to the identity map of M(H).

• (Transitivity conditions) If H ⊆ K ⊆ L are subgroups of G, then t_{K}^{L} ∘t_{H}^{K} = t_{H}^{L} and r_{H}^{K} ∘ r_{K}^{L} = r_{H}^{L}. If x, y ∈ G, then c_{y,^{x}H} ∘ c_{x,H} = c_{yx,H}.

• (Compatibility conditions) If H ⊆ K are subgroups of G and x ∈ G, then c_{x,K} ∘ t_{H}^{K} = t_{^{x}H}^{^{x}K} ∘ c_{x,H} and c_{x,H} ∘ r_{H}^{K} = r_{^{x}H}^{^{x}K} ∘ c_{x,K}.

• (Mackey axiom) If H ⊆ K ⊇ L are subgroups of G, then

r_{H}^{K} ∘ t_{L}^{K} = ∑_{x∈H\K/L} t_{H∩^{x}L}^{H} ∘ c_{x,H^{x}∩L} ∘ r_{H^{x}∩L}^{L}

If M and N are Mackey functors for G over R, then a morphism of Mackey functors f : M → N is a collection of morphisms of R-modules f_{H} : M(H) → N(H), which commute to the maps t_{H}^{K}, r_{H}^{K} and c_{x,H}.

The category of Mackey functors for G over R is denoted by Mack_{R}(G).

Example 5.1.2. The Burnside Mackey functor B is the Mackey functor with values in ℤ-Mod which value at H is the Burnside ring B(H). If H ⊆ K are subgroups of G, then t_{H}^{K} = Ind_{H}^{K} and r_{H}^{K} = Res_{H}^{K}. The conjugation maps c_{x,H} are defined by c_{x,H}(Z) = ^{x}Z for Z ∈ B(H) and x ∈ G.

More generally, if R is a ring, the Burnside functor RB is defined by “tensoring everything with R”, i.e. setting RB(H) = R ⊗_{ℤ} B(H), and extending the maps t_{H}^{K}, r_{H}^{K} and c_{x,H} in the obvious way.

Another example of Mackey functor is the Green ring functor A_{O}, which value at H is the Green ring A_{O}(H) of OH-lattices defined in section 3.5. The transfer is given by induction of lattices, and the operations of restriction and conjugation are the obvious ones. It is clear that the morphism π_{O} actually defines a morphism of Mackey functors from the Burnside functor to the Green ring functor.

5.2. Mackey functors and G-sets. If M is a Mackey functor for G over R, if H and K are subgroups of G, and if x ∈ G is such that H^{x} ⊆ K, then there are maps of R-modules

a_{x} = t_{H^{x}}^{K} ∘ c_{x^{−1},H} : M(H) → M(K) b_{x} = c_{x,H^{x}} ∘ r_{H^{x}}^{K} : M(K) → M(H)

Moreover, if k ∈ K, then since M is a Mackey functor

a_{xk} =t_{H^{xk}}^{K} ∘ c_{(xk)^{−1},H} =c_{k^{−1}x^{−1},K} ∘ t_{H}^{^{x}K} =c_{k^{−1},K} ∘ c_{x^{−1},^{x}K} ∘ t_{H}^{^{x}K} =c_{x^{−1},^{x}K} ∘ t_{H}^{^{x}K} =a_{x}

Similarly b_{xk} = b_{x}. Hence a_{x} and b_{x} only depend on the class xK. The crucial observation is that the set of classes xK such that H^{x} ⊆ K is in one to one correspondence with the set of G-sets homomorphisms from G/H to G/K. This leads to the second definition of Mackey functors, due to Dress ([21]).

Recall that a bivariant functor M from a category C to a category D is a pair of functors M = (M_{∗}, M^{∗}), where M_{∗} is a functor from C to D and M^{∗} is a functor from C to D^{op} (or a cofoncteur from C to D), which coincide on objects, i.e. M_{∗}(C) = M^{∗}(C) for all objects C of C. This common value is simply denoted by M(C).

Definition 5.2.1. [Dress] A Mackey functor M for the group G with values in the category R-Mod of R-modules is a bivariant functor M = (M_{∗}, M^{∗}) from the category of finite G-sets to R-Mod, with the following two properties:

(1) Let X and Y be any finite G-sets, and let i_{X} (resp. i_{Y}) denote the canonical injection from X (resp. Y) into X ⊔ Y. Then the morphisms

(M_{∗}(i_{X}), M_{∗}(i_{Y})) : M(X) ⊕ M(Y) → M(X ⊔ Y)

\binom{M^{∗}(i_{X})}{M^{∗}(i_{Y})} : M(X ⊔ Y) → M(X) ⊕ M(Y)

are mutually inverse isomorphisms.

(2) Let

X \underrightarrow{a} Y
b ↓  ↓ c
Z \overrightarrow{d} T

be any cartesian (i.e. pull-back) square of finite G-sets. Then

M_{∗}(b) ∘ M^{∗}(a) = M^{∗}(d) ∘ M_{∗}(c)

A morphism of Mackey functors f : M → N is a natural transformation of bivariant functors, i.e. a collection of morphisms of R-modules f_{X} : M(X) → N(X), for finite G-sets X, which commute to the maps M_{∗}(a) and M^{∗}(a) for any morphism of finite G-sets a : X → Y.

If M is a Mackey functor for the first definition, then M yields a Mackey functor M̂ for this definition by choosing a set of representatives [G\X], for each finite G-set X, and then setting

M̂(X) = ⊕_{x∈[G\X]} M(G_{x})

Let f : X → Y be a morphism of G-sets. If x ∈ [G\X] and y ∈ [G\Y] are such that f(x) ∈ Gy, then there exists g ∈ G such that f(x) = gy. In this case G_{x} ⊆ ^{g}G_{y}, and one can define two maps

α_{y,x} = t_{^{g}G_{x}}^{G_{y}} ∘ c_{g,G_{x}} : M(G_{x}) → M(G_{y})

β_{x,y} = c_{g^{−1},^{g}G_{x}} ∘ r_{^{g}G_{x}}^{G_{y}} : M(G_{y}) → M(G_{x})

Those maps depend only on x and y, and do not depend on the chosen element g. Define moreover α_{y,x} = β_{x,y} = 0 for x ∈ [G\X] and y ∈ [G\Y] if f(x) ∉ Gy.

Then the map M̂_{∗}(f) is defined by the block matrix (α_{y,x})_{y∈[G\Y],x∈[G\X]}, and the map M̂^{∗}(f) is defined by the block matrix (β_{x,y})_{x∈[G\X],y∈[G\Y]}. One can check that M̂ is a Mackey functor for the second definition.

Conversely, a Mackey functor M̂ for the second definition yields a Mackey functor M for the first definition, by setting

M(H) = M̂(G/H)  t_{H}^{K} = M̂_{∗}(π_{H}^{K})  r_{H}^{K} = M̂^{∗}(π_{H}^{K})  c_{x,H} = M̂_{∗}(γ_{x,H})

where the morphism π_{H}^{K} is the canonical projection G/H → G/K for H ⊆ K, and γ_{x,H} is the isomorphism yH ↦ yx^{−1}^{x}H from G/H to G/^{x}H, for x ∈ G.

Example 5.2.2. One can show (see for instance [9] Chapter 2.4) that the Mackey functor associated to the Burnside functor B (still denoted by B) can be described as follows: if X is a finite G-set, let G-set↓_{X} denote the category which objects are the finite G-sets over X, i.e. the pairs (Y, f), where Y is a finite G-set and f : Y → X is a map of G-sets. A morphism in G-set↓_{X} from (Y, f) to (Z, g) is a map of G-sets h : Y → Z such that g ∘ h = f. The composition of morphisms is the composition of maps.

Then B(X) is the Grothendieck group of the category G-set↓_{X}, for relation given by decomposition in disjoint union. If ϕ : X → X′ is a morphism of G-sets, then the map B_{∗}(ϕ) sends (Y, f) to (Y, ϕ ∘ f), and the map M^{∗}(ϕ) sends (Z, g) to the G-set over X defined by the pull-back square

Y ⟶ Z
↓  ↓ g
X \overrightarrow{ϕ} X′

5.3. Mackey functors as modules. The third definition of Mackey functors is due to Thévenaz and Webb ([43]), who defined the following algebra:

Definition 5.3.1. [Thévenaz-Webb] Let G be a finite group, and R be a commutative ring. The Mackey algebra μ_{R}(G) is the R-algebra with generators t_{H}^{K}, r_{H}^{K}, and c_{x,H}, for subgroups H ⊆ K of G, and x ∈ G, subject to the following relations:

• If H is a subgroup of G, and if h ∈ H, then t_{H}^{H} = r_{H}^{H} = c_{h,H}. Moreover ∑_{H⊆G} t_{H}^{H} = 1_{μ_{R}(G)}, and if H ≠ K are subgroups of G, then t_{H}^{H}t_{K}^{K} = 0.

• If H ⊆ K ⊆ L are subgroups of G, then t_{K}^{L} ∘ t_{H}^{K} = t_{H}^{L} and r_{H}^{K} ∘ r_{K}^{L} = r_{H}^{L}. If x, y ∈ G, then c_{y,^{x}H} ∘ c_{x,H} = c_{yx,H}.

• If H ⊆ K are subgroups of G and x ∈ G, then c_{x,K} ∘ t_{H}^{K} = t_{^{x}H}^{^{x}K} ∘ c_{x,H} and c_{x,H} ∘ r_{H}^{K} = r_{^{x}H}^{^{x}K} ∘ c_{x,K}.

• If H ⊆ K ⊇ L are subgroups of G, then

r_{H}^{K} ∘ t_{L}^{K} = ∑_{x∈H\K/L} t_{H∩^{x}L}^{H} ∘ c_{x,H^{x}∩L} ∘ r_{H^{x}∩L}^{L}

A Mackey functor for G over R is a just a μ_{R}(G)-module, and a morphism of Mackey functors is a morphism of μ_{R}(G)-modules.

If M is a Mackey functor over R for the first definition, set

M̃ = ⊕_{H⊆G} M(H)

This is endowed with an obvious structure of μ_{R}(G)-module: for instance, the action of the generator t_{H}^{K} of μ_{R}(G) is zero on the component M(H′) of M̃, if H′ ≠ H, and it is equal to the map t_{H}^{K} : M(H) → M(K) on the component M(H).

Conversely, being given a μ_{R}(G)-module M̃, one recovers a Mackey functor for the first definition by setting M(H) = t_{H}^{H} M̃. The transfer maps, restriction maps, and conjugation maps for M are obtained by multiplication by the generators of μ_{R}(G) with the same name: for instance, the relations of the Mackey algebra show that t_{H}^{K}M(H) ⊆ M(K).

Proposition 5.3.2. [Thévenaz-Webb] Let G be a finite group and R be a commutative ring. Then the algebra μ_{R}(G) is a finitely generated free R-module, with basis the set of elements t_{K}^{H}c_{x,K^{x}}r_{K^{x}}^{L}, for subgroups H and L of G, for x ∈ H\G/L, and a subgroup K of H ∩^{x}L up to H ∩^{x}L-conjugacy.

Proof. See [43] Propositions (3.2) and (3.3), or [9] Chapter 4.4. □

Remark 5.3.3. It follows in particular that the rank over R of μ_{R}(G) does not depend on R. In other words, the algebra μ_{R}(G) is isomorphic to R ⊗_{ℤ} μ_{ℤ}(G). It is sometimes convenient to define μ_{R}(G) = R⊗_{ℤ} μ_{ℤ}(G) for any ring R (not necessarily commutative). The case R = μ_{S}(G) for a commutative ring S is of interest (see [9] Chapter 1.2). Not that if R is not commutative, then μ_{R}(G) is not strictly speaking an R-algebra.

The next corollary states a link between the Burnside ring and the Mackey algebra:

Corollary 5.3.4. Let G be a finite group, and R be a commutative ring. Define a G-set Ω_{G} by Ω_{G} = ⨆_{H⊆G} G/H. If H and L are subgroups of G, if x ∈ G and K is a subgroup of H ∩^{x}L, denote by π_{H,x,L} the map of G-sets defined by

π_{H,x,L} : yK ∈ G/K ↦ (yH, y^{x}L) ∈ Ω_{G}^{2}

Then the R-linear map defined by

t_{K}^{H}c_{x,K^{x}}r_{K^{x}}^{L} ∈ μ_{R}(G) ↦ (G/K, π_{H,x,L}) ∈ RB(Ω_{G}^{2})

is an isomorphism of R-modules.

Proof. It is easy to check that the images of the basis elements of μ_{R}(G) form an R-basis of RB(Ω_{G}^{2}). □

Remark 5.3.5. Of course, the isomorphism of Corollary 5.3.4 is not an algebra isomorphism, since μ_{R}(G) is not commutative in general. One can show (see [9] Chapter 4.5.1) that the multiplication law it induces on RB(Ω_{G}^{2}) is given by

(V, W) ∈ B(Ω_{G}^{2}) ↦ V ∘_{Y} W = B_{∗} \binom{xyz}{xz} B^{∗} \binom{xyz}{xyyz} (V × W)

where \binom{xyz}{xz} is the map (x, y, z) ∈ Ω_{G}^{3} ↦ (x, z) ∈ Ω_{G}^{2}, and \binom{xyz}{xyyz} is the map

(x, y, z) ∈ Ω_{G}^{3} ↦ (x, y, y, z) ∈ Ω_{G}^{4}, and V × W is the product for the Green functor structure, to be defined in the next section, in Example 5.4.4.

5.4. Green functors. Roughly speaking, a Green functor for the finite group G over the commutative ring R is a “Mackey functor with a compatible ring structure”. More precisely, there are two equivalent definitions of Green functors. The first one is due to Green ([23]):

Definition 5.4.1. A Green functor A for G over R is a Mackey functor for G over R, such that for any subgroup H of G, the R-module A(H) has a structure of R-algebra (associative, with unit), with the following properties:

(1) If K is a subgroup of G and x ∈ G, then the map c_{x,H} is a morphism of rings (with unit) from A(K) to A(^{x}K).

(2) If H ⊆ K are subgroups of G, then the map r_{H}^{K} is a morphism of rings (with unit) from A(K) to A(H).

(3) [Frobenius identities] In the same conditions, if a ∈ A(K) and b ∈ A(H), then

a.t_{H}^{K}(b) = t_{H}^{K}(r_{H}^{K}(a).b) t_{H}^{K}(b).a = t_{H}^{K}(b.r_{H}^{K}(a))

If A and A′ are Green functors for G over R, a morphism of Green functors f : A → A′ is a morphism of Mackey functors such that for each subgroup H of G, the map f_{H} : A(H) → A′(H) is a ring homomorphism. The morphism is unitary if moreover all the maps f_{H} are unitary, or equivalently, if f_{G} is unitary.

Example 5.4.2. The Burnside Mackey functor RB is a Green functor, if the product on RB(H) for H ⊆ G is defined by linearity from the direct product of H-sets. The Frobenius identities follow from Proposition 2.2.1.

The second definition of Green functors is in terms of G-sets, and is detailed in [9] Chapter 2.2:

Definition 5.4.3. A Green functor A for G over R is a Mackey functor for G over R, endowed for any G-sets X and Y with R-bilinear maps A(X) × A(Y) → A(X × Y) with the following properties:

• (Bifunctoriality) If f : X → X′ and g : Y → Y′ are morphisms of G-sets, then the squares

A(X) × A(Y) \underrightarrow{×} A(X × Y)
A_{∗}(f) × A_{∗}(g) ↓  ↓ A_{∗}(f × g)
A(X′) × A(Y′) \overrightarrow{×} A(X′ × Y′)
A(X) × A(Y) \underrightarrow{×} A(X × Y)
A^{∗}(f) × A^{∗}(g) ↑  ↑ A^{∗}(f × g)
A(X′) × A(Y′) \overrightarrow{×} A(X′ × Y′)

are commutative.

• (Associativity) If X, Y and Z are G-sets, then the square

A(X) × A(Y) × A(Z) \underrightarrow{Id_{A(X)} × (×)} A(X) × A(Y × Z)
(×) × Id_{A(Z)} ↓  ↓ ×
A(X × Y) × A(Z)  \overrightarrow{×}  A(X × Y × Z)

is commutative, up to identifications (X × Y) × Z ≃ X × Y × Z ≃ X × (Y × Z).

• (Unitarity) If • denotes the trivial G-set G/G, there exists an element ε_{A} ∈ A(•), called the unit of A, such that for any G-set X and for any a ∈ A(X)

A_{∗}(p_{X})(a × ε_{A}) = a = A_{∗}(q_{X})(ε_{A} × a)

denoting by p_{X} (resp. q_{X}) the (bijective) projection from X × • (resp. from • × X) to X.

If A and A′ are Green functors for G over R, a morphism of Green functors f : A → A′ is a morphism of Mackey functors such that for any G-sets X and Y, the square

A(X) × A(Y) \underrightarrow{×} A(X × Y)
f_{X} × f_{Y} ↓  ↓ f_{X×Y}
A′(X) × A′(Y) \overrightarrow{×} A′(X × Y)

is commutative. The morphism f is unitary if moreover f_{•}(ε_{A}) = ε_{A′}.

Example 5.4.4. If X and Y are finite G-sets, then the product B(X)×B(Y) → B(X × Y) is defined by linearity from the map sending the finite G-set U over X and the finite G-set V over Y to the cartesian product U × V, which is a G-set over X × Y. The element ε_{B} ∈ B(•) is the image of the trivial G-set • = G/G.

Remark 5.4.5. There is an obvious notion of module over a Green functor: a (left) module M over the Green functor A is a Mackey functor M endowed for any G-sets X and Y with bilinear maps A(X) × M(Y) → M(X × Y) which are bifunctorial, associative, and unitary. With this definition, a Mackey functor for G over R is just a module over the Green functor RB. This can be viewed as a generalization of the identification of abelian groups with ℤ-modules.

5.5. Induction, Restriction, Inflation. The second definition of Mackey functors leads to a natural notion of induction, restriction, and inflation for Mackey functors:

Definition 5.5.1. Let G be a finite group, and R be a commutative ring.

• If H is a subgroup of G, and M is a Mackey functor for G over R, then the restriction Res_{H}^{G}M is the Mackey functor for H over R obtained by composition of the functor M : G-set → R-Mod with the induction functor Ind_{H}^{G} : H-set → G-set. If A is a Green functor for G over R, then Res_{H}^{G}A has a natural structure of Green functor for H over R.

• If H is a subgroup of G, and N is a Mackey functor for H over R, then the induced Mackey functor Ind_{H}^{G}M is the Mackey functor for G over R obtained by composition of the functor M : H-set → R-Mod with the restriction functor Res_{H}^{G} : G-set → H-set. If A is a Green functor for H over R, then Ind_{H}^{G}A has a natural structure of Green functor for H over R.

• If K is a normal subgroup of G, and N is a Mackey functor for G/K, then the inflated Mackey functor Inf_{G/K}^{G}N is the Mackey functor for G over R obtained by composition of the functor M : G/K-set → R-Mod with the fixed points functor H-set → G-set. If A is a Green functor for G/K over R, then Inf_{H}^{G}A has a natural structure of Green functor for G over R.

Remark 5.5.2. If M is a Mackey functor for G, and if K ⊆ H are subgroups of G, then

(Res_{H}^{G}M)(K) = (Res_{H}^{G}M)(H/K) = M(Ind_{H}^{G}H/K) = M(G/K) = M(K)

so the above definition of Res_{H}^{G}M coincides with the naive one. In particular, the restriction of the Burnside functor for G to the subgroup H is isomorphic to the Burnside functor for H.

Remark 5.5.3. The constructions of Definition 5.5.1 are examples of functors between categories of Mackey functors, obtained by composition with functors between the corresponding categories of G-sets. A uniform description of this kind of functors is given in [7], using the formalism of bisets. In [9] Chapter 8, it is shown that those constructions also apply to Green functors.

Another fundamental example of this kind of functors is the following:

Definition 5.5.4. [Dress] If X is a finite G-set, and if M is is a Mackey functor for G over R, then the Mackey functor M_{X} is the Mackey functor for G over R obtained by composition of the functor M : G-set → R-Mod with the endofunctor Y ↦ Y × X of G-set. If A is a Green functor for G over R, then A_{X} has a natural structure of Green functor for G over R.

The endofunctor M ↦ M_{X} on Mack_{R}(G) is denoted by I_{X}.

Remark 5.5.5. In the case of a transitive G-set X = G/H, the isomorphism

G/H × Y ≃ Ind_{H}^{G}Res_{H}^{G}Y

of Proposition 2.2.1 leads to an isomorphism of Mackey functors

M_{G/H} ≃ Ind_{H}^{G}Res_{H}^{G}M

Induction and restriction are mutual left and right adjoints:

Proposition 5.5.6. [Thévenaz-Webb] Let G be a finite group, let H be a subgroup of G, and R be a commutative ring.

(1) The functors

Ind_{H}^{G} : Mack_{R}(H) → Mack_{R}(G) and Res_{H}^{G} : Mack_{R}(G) → Mack_{R}(H)

are mutual left and right adjoint.

(2) For any finite G-set X, the endofunctor I_{X} is self adjoint.

Proof. For assertion (1), see [41] Proposition 4.2. Assertion (2) follows trivially. □

5.6. The Burnside functor as projective Mackey functor. The third definition of Mackey functors shows that the category Mack_{R}(G) is an abelian category, with enough projective objects. A sequence of Mackey functors

0 → L → M → N → 0

is exact if and only if for each subgroup H of G, the sequence

0 → L(H) → M(H) → N(H) → 0

is an exact sequence of R-modules.

The Burnside functor RB provides the first example of projective Mackey functor:

Proposition 5.6.1. Let G be a finite group, and R be a commutative ring. Set • = G/G.

(1) If M is a Mackey functor for G over R, then the map

Θ_{M} : f ∈ Hom_{Mack_{R}(G)}(RB, M) ↦ f_{•}(•) ≃ M(•) = M(G)

is an isomorphism of R-modules.

(2) The Burnside functor RB is a projective object in Mack_{R}(G).

(3) The map Θ_{RB} is an isomorphism of rings (with unit)

Θ_{RB} : End_{Mack_{R}(G)}(RB) →^{≃} RB(G)

Proof. Let X be a finite G-set, let (Y, a) be a G-set over X, and denote also by (Y, a) its image in RB(X) = R ⊗_{ℤ} B(X). Then clearly (Y, a) = RB_{∗}(a)(Y, Id_{Y}). Denote by p_{Y} : Y → • be the unique morphism of G-sets. Then the square

Y \underrightarrow{p_{Y}} •
Id_{Y} ↓  ↓ Id_{•}
Y \overrightarrow{p_{Y}} •

is cartesian. Hence denoting by ε_{R} the element (•, Id_{•}) of RB(•), it follows that

(Y, a) = RB_{∗}(a)RB^{∗}(p_{Y})(ε_{R})

Now a morphism f : RB → M is entirely determined by the element u = f_{•}(ε_{R}) of M(•) = M(G), since the map f_{X} must verify

f_{X}((Y, a)) = M_{∗}(a)M^{∗}(p_{Y})(u)

Conversely, if u ∈ M(•) is given, this equality defines a map of R-modules f_{X} from RB(X) to M(X). If g : X → X′ is a morphism of G-sets, then obviously

M_{∗}(g)f_{X}((Y, a)) = M_{∗}(ga)M^{∗}(p_{Y})(u) = f_{X′}((Y, ga))
= f_{X′}RB_{∗}(g)((Y, a))

Similarly, if the square

Y \underrightarrow{h} Y′
a ↓  ↓ a′
X \overrightarrow{g} X′

is cartesian, then

M^{∗}(g)f_{X′}((Y′, a′)) = M^{∗}(g)M_{∗}(a′)M^{∗}(p_{Y′})(u) = M_{∗}(a)M^{∗}(h)M^{∗}(p_{Y′})(u)
= M_{∗}(a)M^{∗}(p_{Y})(u) = f_{X}((Y, a))
= f_{X}RB^{∗}((Y′, a′))

This proves that f is a morphism of Mackey functors, and assertion (1) follows. Assertion (2) is now clear, since the evaluation functor M ↦ M(G) from Mack_{R}(G) to R-Mod is exact.

Now let f and g be two endomorphisms of RB, and suppose that f_{•}(•) (resp. g_{•}(•)) is equal to (U, p_{U}) (resp. (V, p_{V})) for some G-set U (resp. V). Then

(g ∘ f)_{•}(•) = g_{•}((U, p_{U})) = RB_{∗}(p_{U})RB^{∗}(p_{U})((V, p_{V}))
= RB_{∗}(p_{U})((U × V, p_{1})) = (U × V, p_{U×V})

where p_{1} : U × V → U is the first projection, since the square

U × V ⟶ V
p_{1} ↓  ↓ p_{V}
U  \overrightarrow{p_{U}} •

is cartesian.

By linearity, this shows that Θ_{RB} is a ring homomorphism. Clearly Θ_{RB} maps the identity endomorphism to the trivial G-set •, which is the unit of RB(G). Assertion (3) follows. □

Colollary 5.3.4 states an isomorphism of R-modules

μ_{R}(G) ≃ RB(Ω_{G}^{2})

Thus setting M = RB_{Ω_{G}}, the R-module μ_{R}(G) is isomorphic to the module M̃ = ⊕_{H⊆G}M(H). The next proposition shows that this is actually an isomorphism of μ_{R}(G)-modules:

Proposition 5.6.2. [Thévenaz-Webb] Let G be a finite group, and R be a commutative ring.

(1) There is an isomorphism of Mackey functors

μ_{R}(G) ≃ RB_{Ω_{G}} ≃ ⊕_{H⊆G} Ind_{H}^{G}Res_{H}^{G}RB

(2) Any Mackey functor is a quotient of a direct sum of induced Burnside Mackey functors.

Proof. See [43] Corollary 8.4. □

5.7. The Burnside functor as universal Green functor. The following proposition shows that the Burnside functor is an initial object in the category of Green functors:

Proposition 5.7.1. Let G be a finite group, and R be a commutative ring. If A is a Green functor for G over R, then there is a unique unitary morphism of Green functor from RB to A.

Proof. By Proposition 5.6.1, there is a unique morphism f of Mackey functors from RB to A such that f_{•}(•) = ε_{A}. It is easy to check that f is a morphism of Green functors (see [9] Proposition 2.4.4 for details). □

Remark 5.7.2. For any Green functor A for G over R, there is a natural algebra structure on A(Ω_{G}^{2}), defined as in Remark 5.3.5 (see [9] Chapter 4). Moreover (see [9] Chapter 12.2), one can build a Green functor ζ_{A} for G over R such that ζ_{A}(•) = Z(A(Ω_{G}^{2})): the evaluation of ζ_{A} at the finite G-set X is the set of natural transformation from the identity endofunctor I = I_{•} of Mack_{R}(G) to the endofunctor I_{X}. The previous proposition shows that for any Green functor A, there is a ring homomorphism from RB(G) to Z(A(Ω_{G}^{2})).

Hence there should exist a ring homomorphism from RB(G) to the center of the Mackey algebra. But the center of ring S is isomorphic to the endomorphism algebra of the identity functor on S-Mod. Thus, for any Mackey functor M, there should be a morphism from RB(G) to the endomorphism algebra of M, with functorial properties in M.

It can be defined as follows: if X is a finite G-set, then there is a natural transformation a_{X} : M → M_{X} defined for the finite G-set Z by

a_{X,Z} = M^{∗} \binom{zx}{z} : M(Z) → M(Z × X)

where \binom{zx}{z} denotes the projection map Z × X → Z. Similarly, there is a natural transformation b_{X} : M_{X} → M defined by

b_{X,Z} : M_{∗} \binom{zx}{z} : M(Z × X) → M(Z)

This defines an endomorphism z(X)_{M} of M by z(X)_{M,Z} = b_{X,Z} ∘ a_{X,Z}. It is easy to check that for finite G-sets X and Y

z(X ⊔ Y)_{M} = z(X)_{M} + z(Y)_{M} z(X)_{M} ∘ z(Y)_{M} = z(X × Y)_{M}

This leads by linearity to an R-algebra homomorphism

X ∈ RB(G) ↦ z(X)_{M} ∈ End_{Mack_{R}(G)}(M)

for any Mackey functor M for G over R.

Moreover, if f : M → N is a morphism of Mackey functors, then the square

M \underrightarrow{z(X)_{M}} M
f ↓  ↓ f
N \overrightarrow{z(X)_{N}} N

is commutative.

The ring homomorphism from RB(G) to Z(μ_{R}(G)) can be described concretely as follows: if H is a subgroup of G, then the relations of the Mackey algebra show that there is a (non-unitary ) ring homomorphism β_{H} from RB(H) to μ_{R}(G) defined by β_{H}(H/K) = t_{K}^{H}r_{K}^{H}. With this notation

Proposition 5.7.3. Let G be a finite group, and R be a commutative ring. Then the map z : RB(G) → μ_{R}(G) defined by

z(X) = ∑_{H⊆G} β_{H}(Res_{H}^{G}X)

is an injective unitary ring homomorphism from RB(G) to Z(μ_{R}(G)).

Proof. By inspection, or see [43] Proposition 9.2. □

Taking the image by z of a set of primitive idempotents of RB(G) will give a decomposition of the unit of μ_{R}(G) into a sum of orthogonal central idempotents. This in turn will give a decomposition of the category of Mackey functors as a direct sum of abelian categories.

Notation 5.7.4. If G is a finite group and R is a commutative ring, denote by π = π_{R}(G) be the set of prime factors of |G| which are not invertible in R. If J is a π-perfect subgroup of G, denote by Mack_{R}(G, J) the subcategory of Mack_{R}(G) on which the idempotent z(f_{J}^{G}) acts trivially. If M is a Mackey functor for G over R, denote by f_{J}^{G} × M the direct summand of M on which z(f_{J}^{G}) acts trivially.

Theorem 5.7.5. [Thévenaz-Webb] Let G be a finite group, and R be a commutative ring. Set π = π_{R}(G).

(1) If J is a π-perfect subgroup of G, then the direct summand f_{J}^{G}×B of B has a natural structure of Green functor. Moreover, there are isomorphisms of Green functors

f_{J}^{G} × B ≃ Ind_{N_{G}(H)}^{G}Inf_{N_{G}(H)/H}^{G}(f_{1l}^{N_{G}(H)/H} × B)

RB ≃ ⨁_{{J∈[s_{G}] \atop J=O^{π}(J)}} f_{J}^{G} × RB

(2) The functor

M ↦ Ind_{N_{G}(J)}^{G}Inf_{N_{G}(J)/J}^{N_{G}(J)}M

is an equivalence of categories from Mack_{R}(N_{G}(J)/J, 1l) to the category Mack_{R}(G, J).

Proof. See [43] Section 10 (or [9] Proposition 12.1.11 for a generalization to arbitrary Green functors). □

Corollary 5.7.6. Let G be a finite group, and π be a set of prime numbers. If J is a π-perfect subgroup of G, then there is a ring isomorphism

f_{J}^{G}B_{(π)}(G) ≃ f_{1l}^{N_{G}(J)/J}B_{(π)}(N_{G}(J)/J)

Proof. By evaluation at the trivial G-set of the first isomorphism of Green functors in Theorem 5.7.5, in the case R = ℤ_{(π)}. Of course, there is also an elementary proof, using the ring homomorphism X ↦ X^{J} from B(G) to B(N_{G}(J)/J). □

Remark 5.7.7. By definition, the category Mack_{R}(G, 1l) is the category of modules over the algebra μ_{R}(G, 1l) = z(f_{1l}^{G})μ_{R}(G). One can show that μ_{R}(G, 1l) is equal to the R-submodule of μ_{R}(G) generated by the elements t_{P}^{H}c_{x,P^{x}}r_{P^{x}}^{K}, where H and K are subgroups of G, where x ∈ G, and P is a π-solvable subgroup of H ∩^{x}K.

One can also show (see [10] Lemme 2.2 for the case |π| = 1) that this algebra is Morita-equivalent to its subalgebra μ_{R,1l}(G) generated by the elements t_{P}^{H}c_{x,P^{x}}r_{P^{x}}^{K}, where x ∈ G and H and K are π-solvable subgroups of G. The μ_{R,1l}(G)-modules can be viewed as “Mackey functors defined only on π-solvable subgroups”.

Recall (see [47] Section 3.2) that a Mackey functor M is said to be projective relative to a finite G-set X if the morphism b_{X} : M_{X} → M is split surjective. If X is a set of subgroups of G, the functor M is said to be projective relative to X if it is projective relative to the G-set X = ⊔_{H∈X} G/H. With these definitions, one can show that the Mackey functors in Mack_{R}(G, 1l) are exactly the Mackey functors which are projective relative to π-solvable subgroups of G, for π = π_{R}(G).

Remark 5.7.8. If R is a field k of characteristic p, there is a nice interpretation of the indecomposable projective objects in Mack_{k}(G, 1l), in terms of p-permutation modules:

Theorem 5.7.9. [Thévenaz-Webb] Let G be a finite group, and k be a field of characteristic p > 0. Then evaluation at the trivial subgroup M ↦ M(1l) induces a one to one correspondence between the set of isomorphism classes of indecomposable projective objects in Mack_{k}(G, 1l) and the set of isomorphism classes of indecomposable p-permutation kG-modules.

Proof. See [43] Theorem 12.7. □

6. The Burnside ring as biset-functor

6.1. Bisets. In the previous section, the Burnside functor was defined on the subgroups of a given finite group G. The three operations relating the Burnside rings of subgroups of G were induction, restriction, and conjugation.

But there is at least another natural operation on the Burnside ring, which has not yet been considered, namely inflation: if N is a normal subgroup of the group G, then any G/N-set can be viewed as a G-set by inflation. This induces an inflation functor Inf_{G/N}^{G} : G/N-set → G-set and a ring homomorphism Inf_{G/N}^{G} : B(G/N) → B(G).

There is also a deflation functor Def_{G/N}^{G} : G-set → G/N-set mapping a G-set X to the set of orbits N\X of N on X, inducing an operation Def_{G/N}^{G} : B(G) → B(G/N).

The common point about all those operations is the following: in all cases, there are two finite groups G and H, and a functor F from G-set to H-set, preserving disjoint unions, i.e. such that F (X ⊔ Y) ≃ F (X) ⊔ F (Y). I all cases moreover there exists a finite set U with a left H-action and a right G-action, such that

F (X) ≃ U ×_{G} X

where U ×_{G} X is the quotient of U × X by the right action of G given for (u, x) ∈ U × X and g ∈ G by

(u, x)g = (ug, g^{−1}x)

The set U × X is an H-set for the action defined by

h(u, x) = (hu, x)

for (u, x) ∈ U × X and h ∈ H. If the actions of G and H on U commute, i.e. if (hu)g = h(ug) for all u ∈ U, h ∈ H, and g ∈ G, then this action passes down to a well defined left action of H on U ×_{G} X. This leads to the following definition:

Definition 6.1.1. Let G and H be groups. An H-set-G, or a biset for short, is an (H × G^{op})-set, i.e. a set U with a left H-action and an right G-action which commute.

If K is another group, if V is a K-set-H, then the product V ×_{H}U is the quotient of the product V × U by the right action of H given by (v, u)h = (vh, h^{−1}u) for v ∈ V, u ∈ U, and h ∈ H. The class of (v, u) in V ×_{H} U is denoted by (v,_{H} u).

The set V ×_{H} U is a K-set-G for the action given by

k(v,_{H} u)g = (kv,_{H} ug)

for k ∈ K, g ∈ G, u ∈ U, and v ∈ V.

Example 6.1.2. For the induction functor Ind_{H}^{G} from a subgroup H of G to G, the set U is the set G, with its natural left G-action and right H-action by multiplication.

For the restriction functor Res_{H}^{G}, the set U is the set G, for its left H-action and right G-action.

For the inflation functor Inf_{G/N}^{G}, the set U is the set G/N, for its left Gaction given by projection and multiplication, and its right G/N-action given by multiplication.

For the deflation functor Def_{G/N}^{G}, the set U is the set G/N, for its left G/Naction and right G-action.

There is still another operation, involving the case of conjugation. It is associated to a group isomorphism f : G → H. Any G-set X can be viewed as an H-set on which h ∈ H acts as f^{−1}(h) ∈ G. Here the set U is the set G, with right G-action by multiplication, and left H-action of h ∈ H by left multiplication by f^{−1}(h). This operation is denoted by Iso_{G}^{H} (without reference to f, which is generally clear from the context).

Notation 6.1.3. Let G and H be finite groups, and L be a subgroup of H ×G. Set

p_{1}(L) = {h ∈ H | ∃g ∈ G, (h, g) ∈ L} p_{2}(L) = {g ∈ G | ∃h ∈ H, (h, g) ∈ L}

k_{1}(L) = {h ∈ H | (h, 1) ∈ L} k_{2}(L) = {g ∈ G | (1, g) ∈ L}

Then k_{i}(L) is a normal subgroup of p_{i}(L), for i = 1, 2. The quotient group is denoted by q_{i}(L), for i = 1, 2. There is a canonical group isomorphism q_{1}(L) → q_{2}(L).

The quotient set (H × G)/L is viewed as an H-set-G by

h.(x, y)L.g = (hx, g^{−1}y)L ∀h, x ∈ H, ∀y, g ∈ G

This biset is transitive, i.e. it is a transitive (H×G^{op})-set. Conversely, any transitive H-set-G is isomorphic to a biset (H × G)/L, for some subgroup L of H × G. Any H-set-G is a disjoint union of transitive bisets.

If G, H, and K are finite groups, if L is a subgroup of H × G and M is a subgroup of K × H, set

M ∗ L = {(k, g) ∈ K × G | ∃h ∈ H, (k, h) ∈ M and (h, g) ∈ L}

It is a subgroup of K × G.

Proposition 6.1.4. (1) Let G and H be finite groups, and L be a subgroup of H × G. Then there is an isomorphism of bisets

(H × G)/L ≃ H ×_{p_{1}(L)} q_{1}(L) ×_{q_{1}(L)} q_{1}(L) ×_{q_{2}(L)} q_{2}(L) ×_{p_{2}(L)} G

(2) Let G, H, and K be finite groups, let L be a subgroup of H × G and M be a subgroup of K × H. Then there is an isomorphism of K-sets-G

((K × H)/M) ×_{H} ((H × G)/L) ≃ ⨆_{x∈p_{2}(M)\H/p_{1}(L)}(K × G)/(M ∗^{(x,1)}L)

Proof. See [8] Lemme 3 and Proposition 1. □

Remark 6.1.5. Assertion (1) shows that any transitive biset is a product of bisets associated to a restriction, a deflation, an isomorphism, an inflation, and an induction. Assertion (2) can be viewed as a biset version of Mackey formula (see Proposition 2.2.1).

6.2. Bisets and functors. If G and H are finite groups, then any finite Hset-G U induces a functor I_{U} : G-set → H-set defined for a finite G-set X by I_{U}(X) = U ×_{G} X, and for a morphism of finite G-sets f : X → Y by

I_{U}(f)((u,_{G}x)) = (u,_{G}f(x)) : I_{U}(X) → I_{U}(Y)

If K is another finite group, if V is a K-set-H, and if X is a finite G-set, there is a natural isomorphism of K-sets

V ×_{H} (U ×_{G} X) → (V ×_{H} U) ×_{G} X

mapping (v,_{H} (u,_{G}x)) to ((v,_{H} u),_{G}x). This induces an isomorphism of functors I_{V} ∘ I_{U} ≃ I_{V ×_{H}U}.

The functor I_{U} preserves disjoint unions, hence it induces a morphism 𝔹(U) from B(G) to B(H) (this morphism should not be confused with the value B(U) of the Burnside functor at the (H × G^{op})-set U!). If U′ is an H-set-G, and if U′ is isomorphic to U (as biset), then the functors I_{U} and I_{U′} are clearly isomorphic. Hence 𝔹(U) = 𝔹(U′).

With the same notation, one has 𝔹(V) ∘ 𝔹(U) = 𝔹(V ×_{H} U). Moreover, if the biset U is a disjoint union U_{1} ⊔ U_{2} of two H-sets-G U_{1} and U_{2}, then the functor I_{U} is clearly isomorphic to the disjoint union of the functors I_{U_{1}} and I_{U_{2}}. It follows that 𝔹(U) = 𝔹(U_{1}) + 𝔹(U_{2}).

Those remarks show that there is a well defined morphism from the Burnside group B(H ×G^{op}) to the group Hom_{ℤ}(B(G), B(H)) of group homomorphisms from B(G) to B(H).

If G = H, then viewing G as a G-set-G by left and right multiplication, it is clear that the functor I_{G} is isomorphic to the identity functor. Hence 𝔹(G) is the identity map of B(G). The biset G will be called the identity biset for G.

There are many other situations where similar transformations can be associated to bisets: for instance, a finite H-set-G U induces a morphism R(U) : R_{ℚ}(G) → R_{ℚ}(H) between the corresponding Grothendieck groups of rational representations, defined for a finite dimensional ℚG-module M by

R(U)(M) = ℚU ⊗_{ℚG} M

where ℚU is the permutation bimodule with basis U.

More generally, if k is a field, one can define a morphism R(U) : R_{k}(G) → R_{k}(H) by R(U)(M) = kU ⊗_{kG} M for a finitely generated kG-module M. However, this map is well defined only if tensoring with kU preserves exact sequences, i.e. if kU is projective as right kG-module. If k has characteristic p, this is equivalent to requiring that for each u ∈ U, the stabilizer in G of u is a p′-group.

This shows that in some natural situations, not all the bisets are allowed, and leads to the following definition:

Definition 6.2.1. Let P and Q be classes of finite groups closed under taking sections and extensions. This means that if H is a subgroup or a quotient of an element of P, then H in in P, and conversely that if N is a normal subgroup of H such that N and H/N are in P, then H ∈ P.

If G and H are finite groups, then an H-set-G U is said to be P-free-Q if for any u ∈ U, the stabilizer of u in H is in P, and the stabilizer of u in G is in Q.

One denotes by B_{P,Q}(H, G) the subgroup of B(H × G^{op}) generated by the Hsets-G which are P-free-Q. If R is a commutative ring, then RB_{P,Q}(H, G) denotes R ⊗_{R} B_{P,Q}(H, G).

Lemma 6.2.2. Let G, H, and K be finite groups. Let U be an H-set-G, and V be a K-set-H. If U and V are P-free-Q, then so does V ×_{H} U.

Proof. See [8] Lemme 4. □

It follows that if G, H, and K are finite groups, there is a bilinear product

×_{H} : B_{P,Q}(K, H) × B_{P,Q}(H, G) → B_{P,Q}(K, G)

extending the product (X, Y) ↦ X ×_{H} Y. It should be noted moreover that the identity bisets are left and right free, hence they are P-free-Q, for any families P and Q.

Definition 6.2.3. Denote by C_{P,Q} the category defined as follows: • The objects of C_{P,Q} are the finite groups.

• If G and H are finite groups, then

Hom_{C_{P,Q}}(G, H) = B_{P,Q}(H, G)

• The composition of morphisms in C_{P,Q} is defined for finite groups G, H, and K by

v ∘ u = v ×_{H} u ∀u ∈ B_{P,Q}(H, G), ∀u ∈ B_{P,Q}(K, H)

The category C_{P,Q} is preadditive (see [27] Chapter 1.8), i.e. the sets of morphisms in C_{P,Q} are abelian groups, and moreover the composition of morphisms is left and right distributive with respect to addition of morphisms.

More generally, if R is a commutative ring, then the category RC_{P,Q} is the category obtained from C_{P,Q} by tensoring with R: the objects are the finite groups, but

Hom_{RC_{P,Q}}(G, H) = R ⊗_{ℤ} Hom_{C_{P,Q}}(G, H)

Denote by F_{P,Q} the category which objects are additive functors from C_{P,Q} to ℤ-Mod, morphisms are natural transformations of functors, and composition of morphisms is composition of natural transformations. More generally, denote by RF_{P,Q} the category of R-linear functors from RC_{P,Q} to R-Mod.

The category RF_{P,Q} is abelian. A sequence

0 → L → M → N → 0

of objects and morphisms in RF_{P,Q} is exact if and only if its evaluation at any finite group is exact.

It is now clear that the correspondence sending a finite group G to its Burnside group B(G), and a finite P-free-Q H-set-G U, for a finite groups G and H, to the map 𝔹(U), induces an additive functor from C_{P,Q} to ℤ-Mod. More generally, the correspondence sending the finite group G to RB(G) and the biset U to the map R𝔹(U) : RB(G) → RB(H) induced by 𝔹(U), is an object of the category RF_{P,Q}. It is called the Burnside biset-functor with coefficients in R.

For any finite group G, the set Hom_{RC_{P,Q}}(1l, G) identifies with the R-submodule RB_{P}(G) of RB(G) generated by finite P-free G-sets, or by equivalently by sets G/P, for P ∈ P. This correspondence G ↦ RB_{P}(G) is clearly a subfunctor of the biset functor RB.

The analogue of Proposition 5.6.1 is the following:

Proposition 6.2.4. Let M be any object of RF_{P,Q}. Then the map

f ∈ Hom_{RF_{P,Q}}(RB_{P}, M) ↦ f_{1l}(•) ∈ M(1l)

is an isomorphism of R-modules. In particular, the functor RB_{P} is a projective object in RF_{P,Q}, and End_{RF_{P,Q}}(RB_{P}) ≃ R.

Proof. This is essentially Yoneda’s Lemma: let G be any finite group, and X be any finite P-free G-set. Then X is also a G-set-1l, for the right (trivial!) action of the trivial group, and moreover the G-set X is isomorphic to X ×_{1l} •, where • is the trivial set for the trivial group.

Hence if f : RB_{P} → M is a natural transformation, the map f_{G} : RB_{P}(G) → M(G) is the map of R-modules sending the G-set X to M(X)(f_{1l}(•)). Conversely, if m ∈ M(1l) is given, one can define a map f_{G} from RB_{P}(G) to M(G) by setting

f_{G}(X) = M(X)(m)

for a finite P-free G-set X, and extending f_{G} by linearity. It is easy to check that this defines a natural transformation from RB_{P} to M. □

When the ring R is a field, and P and Q are both equal to the family All of all finite groups, one can say a little more (see [8] Lemme 1 and Proposition 8):

Theorem 6.2.5. [Bouc] Let K be a field. Then the functor KB has a unique maximal proper subfunctor J in KF_{All,All}, defined for a finite group G by

J(G) = {X ∈ KB(G) | ∀Y ∈ KB(G), |G\(Y × X)| = 0_{K}}

If K has characteristic 0, the quotient functor KB/J is isomorphic to the functor KR_{ℚ}. It is a simple object in KF_{All,All}.

Proof. If L is a subfunctor of KB, then L(1l) is a K-subspace of KB(1l) = K. Thus either L(1l) = K. In this case • ∈ L(1l), thus for any finite group G and any finite G-set X, the G-set X = X ×_{1l} • = L(X)(•) is in L(G). Hence L = RB in this case.

The other case is L(1) = 0, and then for any finite group G, any X ∈ L(G), and any morphism Y from G to 1l in KC_{P,Q}, the image L(Y)(X) has to be zero. But clearly Hom_{KC_{P,Q}}(G, 1l) ≃ KB(G), and with this identification, the element L(Y)(X) of B(1l) identifies with |G\(Y × X)|. Thus L ⊆ J, and moreover J is clearly a subfunctor of KB.

This shows in particular that the dimension over K of KB(G)/J(G) is equal to the rank of the bilinear form on KB(G) with values in K defined by

(X, Y) ∈ (KB(G))^{2} ↦ <X, Y > = |G\(Y × X)|

Now if K ⊇ ℚ, the K-basis {e_{H}^{G}}_{H∈[s_{G}]} of KB(G) is orthogonal for this bilinear form, and moreover

<e_{H}^{G}, e_{H}^{G}> = |G\e_{H}^{G}| = \frac{1}{|N_{G}(H)|} ∑_{K⊆H} |K|μ(K, H)

= \frac{1}{|N_{G}(H)|} ∑_{x∈H} ∑_{<x>⊆K⊆H} μ(K, H) = \frac{ϕ_{1}(H)}{|N_{G}(H)|}

where ϕ_{1}(H) is the number of elements x ∈ H such that <x> = H. This is non-zero if and only if H is cyclic.

Hence the dimension of KB(G)/J(G) is equal to the number c(G) of conjugacy classes of cyclic subgroups of G.

Now the natural morphism KB(G) → KR_{ℚ}(G) mapping a G-set to its permutation module induces clearly a natural transformation of functors

(6.2.6)  χ : KB → KR_{ℚ}

since for any finite group H and any H-set-G U, there is an isomorphism of ℚGmodules

ℚ(U ×_{G} X) ≃ ℚU ⊗_{ℚG} ℚX

Since χ is non-zero, the kernel of χ is contained in J. Hence the image of KB(G) in KR_{ℚ}(G) has dimension at least equal to c(G). But it is well known (see [36] Chapitre 13 Théorème 29 Corollaire 1) that the dimension of KR_{ℚ}(G) is equal to c(G). The proposition follows. □

Remark 6.2.7. ([8] Proposition 2) The set of isomorphism classes of simple objects in RF_{P,Q} is in one to one correspondence with the set of isomorphism classes of pairs (H, V), where H is a finite group, and V is a simple ROut(H)- module, where Out(H) is the group of outer automorphisms of H. The simple functor associated to such a pair (H, V) is denoted by S_{H,V}.

For example, when K is a field of characteristic zero, and P = Q = All, the functor KR_{ℚ} of Proposition 6.2.5 is the functor S_{1l,K}. In this case, the subfunctors of KB are studied in [8]. In particular, the composition factors (i.e. the simple sections) of KB are simple functors S_{H,K} associated to the trivial KOut(H)-module, for a class of finite group called b-groups ([8] Proposition 10).

Remark 6.2.8. In the case where P and Q are reduced to the trivial group, the objects of RF_{P,Q} are called globally defined Mackey functors. They have been studied by Webb ([48]). The reason for this denomination is that the operations associated to bisets which are both left and right free only involve induction, restriction and group isomorphisms.

In this paper, Webb also considers the inflation functors, which correspond to the case where Q is reduced to the trivial group, and P is the class of all finite groups.

Remark 6.2.9. It is also of interest to consider subcategories of the above categories RC_{P,Q}. For example, let p be a prime number, and D_{p} denote the full subcategory of ℚC_{All,All} consisting of finite p-groups. Denote by ℚB and ℚR_{ℚ} the restriction to D_{p} of the functors ℚB and ℚR_{ℚ} on ℚC_{All,All}.

It has been shown recently (Bouc-Thévenaz [11]) that the (restriction to D_{p} of the) kernel of the morphism χ defined in 6.2.6 above is isomorphic to the torsionfree Dade functor ℚD, which value at a p-group is equal to ℚ⊗_{ℤ} D(P), where D(P) is the Dade group of endo-permutation kP -modules, for a field k of characteristic p. This functor ℚD is simple, isomorphic to (the restriction to D_{p} of) S_{E_{2},K}, where E_{2} is an elementary abelian p-group of order p^{2}.

6.3. Double Burnside rings. The endomorphism ring of a finite group G in the category RC_{P,Q} is called the double Burnside ring of G with coefficients in R, for the classes P and Q.

An essential tool to study those rings is the following lemma:

Lemma 6.3.1. Let G be a finite group. If X is a finite G-set, denote by X̃ the set G × X, with its G-set-G structure defined by

g(a, x)g′ = (gag′, gx) ∀g, a, g′ ∈ G and x ∈ X

Then the correspondence X ↦ X̃ induces a morphism of rings (with unit) from B(G) to B_{P,Q}(G, G).

Proof. See [8] Lemme 13. □

In particular, this lemma gives a way to carry the idempotents of the Burnside ring to the double Burnside ring, giving information on the projective and simple modules for those rings. This was used intensively in [8].

Remark 6.3.2. If K is a field of characteristic 0, then the Burnside algebra KB(G) is semi-simple for any finite group G. It is natural to ask if the double Burnside algebra KB_{P,Q}(G) is. The answer is no in general. One can show for instance that if P = Q = All is the class of all finite groups, then the algebra KB_{P,Q}(G) is semi-simple if and only if the group G is cyclic.

6.4. Stable maps between classifying spaces. Double Burnside rings have been studied intensively in the case where P = {1l} is reduced to the trivial group, and Q = All is the class of all finite groups, because they are an essential tool to describe the stable splitting of the classifying spaces of finite groups. The origin of this theory is the Segal conjecture, proved by Carlsson ([16]), which states an isomorphism between the stable cohomotopy groups of the classifying space BG of a finite group G and the completion of the Burnside ring at the augmentation ideal.

Recall first some notation and definitions (see [2] Chapter 2.8):

Notation 6.4.1. If X is a pointed CW -complex, denote by SX its (reduced) suspension, and by ΩX its loop space. If X and Y are pointed CW -complexes, denote by [X; Y] the set of homotopy classes of pointed continuous maps from X to Y. If m ∈ ℕ, there is a map

Ω^{m}S^{m}X = [S^{m}; S^{m}X] → [S^{m+1}; S^{m+1}X] = Ω^{m+1}S^{m+1}X

defined by suspension, and one can set

Ω^{∞}S^{∞}X = lim ; ⟶_{m→∞} Ω^{m}S^{m}X

If X and Y are pointed CW -complexes, then the set {X; Y } of stable maps from X to Y is defined by

{X; Y } = [X; Ω^{∞}S^{∞}Y]

It is an abelian group. If X, Y, and Z are pointed CW -complexes, there is a bilinear map

{Y; Z} × {X; Y } → {X; Z}

The stable cohomotopy groups π_{s}^{r}(X) of a pointed CW -complex X are defined for r ∈ ℤ by

π_{s}^{r}(X) = {X_{+}; S^{r}} = [X; lim ; ⟶_{n→∞} Ω^{n}S^{n+r}]

where X_{+} is the space X with a disjoint basepoint. Note that this last expression makes sense even for r < 0.

The other definition required in the Segal conjecture is the notion of completion. Observe that another consequence of Lemma 6.3.1 is that if G and H are finite groups, then the natural structure of B_{P,Q}(H, H)-module-B_{P,Q}(G, G) on B_{P,Q}(H, G) gives by restriction a structure of B(H)-module-B(G).

It is easy to check (see [8] Lemme 15) that if X is an H-set-G and Y is a G-set, then X ×_{G} Ỹ identifies with the cartesian product X × Y, for the double action given by

h(x, y)g = (hxg, g^{−1}y) ∀h ∈ H, ∀g ∈ G, ∀x ∈ X, ∀y ∈ Y

Notation 6.4.2. Let G be a finite group. Denote by I_{G} the prime ideal I_{1l,0}(G) of B(G) defined in section 3.4. It is called the augmentation ideal. The completion of B(G) at the ideal I_{G} is the inverse limit

B(G)ˆ = lim ; ⟵_{n∈ℕ} B(G)/I_{G}^{n}

More generally, if H and G are finite groups, the completion B_{{1l},All}(H, G)ˆ of B_{{1l},All}(H, G) is defined as the inverse limit

B_{{1l},All}(H, G)ˆ = lim ; ⟵_{n∈ℕ} B_{{1l},All}(H, G)/B_{{1l},All}(H, G)I_{G}^{n}

where B_{{1l},All}(H, G)I_{G}^{n} denotes B_{{1l},All}(H, G) ×_{G} Ĩ_{G}^{n}.

If Y is a pointed CW -complex, then the functors X ↦ {X; S^{r}Y } form a generalized comology theory (see [2] Chapter 2.5). In particular, if H is a subgroup of a finite group G, there is a transfer map from {BH_{+}; Y } to {BG_{+}; Y }. Taking Y = BH_{+}, the image of the identity of {BH_{+}; BH_{+}} is an element T r_{H}^{G} of {BG_{+}; BH_{+}}, also called transfer. This element can be composed with the stable map from BH_{+} to S^{0} obtained by identifying BH to a point. This gives an element τ_{H} of {BG_{+}, S^{0}} = π_{s}^{0}(BG).

The precise statement of Segal’s conjecture is now the following:

Theorem 6.4.3. [Carlsson] Let G be a finite group. Then the map H ↦ τ_{H} defined above induces a group isomorphism

B(G)ˆ ≃ π_{s}^{0}(BG)

Furthermore π_{s}^{r}(BG) = 0 for r > 0.

This has been generalized by Lewis, May and McClure ([25]): suppose H and G are finite groups. If Q is a subgroup of G and ϕ : Q → H is a group homomorphism, then the stable map T r_{Q}^{G} ∈ {BG_{+}; BQ_{+}} can be composed with the element of {BQ_{+}; BH_{+}} deduced from ϕ, to get an element τ_{Q,ϕ} ∈ {BG_{+}; BH_{+}}.

This element depends only on the conjugacy class of the subgroup Δ_{ϕ}(Q) of H × G consisting of the elements (ϕ(l), l), for l ∈ Q. Moreover, the biset (H × G)/Δ_{ϕ}(Q) is transitive, and free on the left. Conversely, any left-free transitive H-set-G is isomorphic to a biset (H × G)/Δ_{ϕ}(Q), for some subgroup Q of G and some morphism ϕ from Q to H.

This construction gives all the stable maps from BG_{+} to BH_{+}. More precisely:

Theorem 6.4.4. [Lewis-May-McClure] Let G and H be finite groups. The above correspondence sending the left-free transitive H-set-G (H × G)/Δ_{ϕ}(Q) to τ_{Q,ϕ} ∈ {BG_{+}; BH_{+}} induces an isomorphism

B_{{1l},All}(H, G)ˆ → {BG_{+}; BH_{+}}

This theorem was at the origin of the study of stable splittings of the classifying spaces of finite groups. It provides indeed an algebraic translation from the stable endomorphisms of BG_{+} for a finite group G, in terms of the completion of the double Burnside ring B_{{1l},All}(G, G). For details see the survey paper of Benson ([3]), or the articles of Benson and Feschbach ([4]), Martino and Priddy ([28], Priddy ([31]), Webb ([48]).

Those two deep theorems on stable homotopy raise some natural questions on the algebraic side. For example, being given three finite groups G, H, and K, the composition of stable maps

{BH_{+}; BK_{+}} × {BG_{+}; BH_{+}} → {BG_{+}; BK_{+}}

shows that there is a map

B_{{1l},All}(K, H) × B_{{1l},All}(H, G) → B_{{1l},All}(K, G)

induced by the product (X, Y) ↦ X ×_{H} Y. Without Theorem 6.4.4, the existence of such a map is not obvious a priori, and seems to require the following lemma:

Lemma 6.4.5. Let G and H be finite groups. Then for any integer m ∈ ℕ, there exists an integer n ∈ ℕ such that

I_{H}^{n}B_{{1l},All}(H, G) ⊆ B_{{1l},All}(H, G)I_{G}^{m}

where I_{H}^{n}B_{{1l},All}(H, G) denotes Ĩ_{H}^{n} ×_{H} B_{{1l},All}(H, G).

Proof. Let X be a finite H-set, and Y be an H-set-G. Then it is easy to see that the biset Z = X̃ ×_{H} Y identifies with the cartesian product X × Y, for the double action given by

h(x, y)g = (hx, hyg) ∀h ∈ H, ∀g ∈ G, ∀x ∈ X, ∀y ∈ Y

In particular for any subgroup L of H × G, the fixed points of L on Z, i.e. the set of z ∈ Z such that az = zb whenever (a, b) ∈ L, identifies with X^{p_{1}(L)} × Y^{L}. It follows that for any X ∈ B(H) and any Y ∈ B(H × G^{op})

|(X̃ ×_{H} Y)^{L}| = |X^{p_{1}(L)}||Y^{L}|

In particular if X ∈ I_{G}^{n}, for an integer n ≥ 1, then X is a sum of terms X_{1}×…×X_{n}, with X_{1}, …, X_{n} ∈ I_{G}. Hence |X| = 0, and moreover |X^{P} | ≡ 0 (p^{n}) for any prime p and any p-subgroup P of H, since |X_{i}^{P} | ≡ |X_{i}| = 0 (p), for i = 1, …, n. Thus

I_{H}^{n}B_{{1l},All}(H, G) ⊆ J_{n}(H, G)

where J_{n}(H, G) is the subset of B(H, G) = B_{{1l},All}(H, G) defined by

J_{n}(H, G) = {Z ∈ B(H, G) | |Z| = 0, |Z^{L}| ≡ 0 (p^{n}) ∀L ⊆ H × G, p_{1}(L) p-group}

For each prime number p, set m_{p} = |H|_{p′}|G|_{p′}. Then for any p-perfect subgroup K of G, the idempotent

f_{p,K}^{G} = ∑_{{M∈[s_{G}] \atop O^{p}(M)=_{G}K}} e_{M}^{G}

of ℚB(G) lies in ℤ_{(p)}B(G), and m_{p}f_{K}^{G} ∈ B(G). Moreover |e_{M}^{G}| = 0 unless M = 1l. Thus m_{p}f_{p,K}^{G} ∈ I_{G} unless K = 1l.

It follows that for any integer n, the element m_{p}^{n}f_{p,K}^{G} = (m_{p}f_{p,K}^{G})^{n} is in I_{G}^{n} if K ≠ 1l. Now if Z ∈ B(H, G)

m_{p}^{n}Z = m_{p}^{n}Zf_{p,1l}^{G} + ∑_{{K∈[s_{G}] \atop O^{p}(K)=K}} m_{p}^{n}Zf_{p,K}^{G} ≡ m_{p}^{n}Zf_{p,1l}^{G} (B(H, G)I_{G}^{n})

Moreover, the integers m_{p}, for p dividing |H||G|, are relatively prime, and there exist integers a_{p,n}, for p prime (equal to 0 for p not dividing |H||G|), such that ∑_{p} a_{p,n}m_{p}^{n} = 1. It follows that

Z ≡ ∑_{p} a_{p,n}Y_{p} (B(H, G)I_{G}^{n})

where Y_{p} = m_{p}^{n}Zf_{p,1l}^{G}.

Let L be a subgroup of H × G, and M be a subgroup of G. Then in B(H × G)

e_{L}^{H×G} × e_{M}^{G} = |(e_{M}^{G})^{L}|e_{L}^{H×G} = |(e_{M}^{G})^{p_{2}(L)}|e_{L}^{H×G} = \{ \begin{matrix}e_{L}^{H×G} & if p_{2}(L) =_{G} M \\ 0 & otherwise \end{matrix}

Now for each prime p, the element Y_{p} is equal to

Y_{p} = Y_{p}f_{p,1l}^{G} = ∑_{L∈[s_{H×G}]} |Y_{p}^{L}|e_{L}^{H×G}

and e_{L}^{H×G}f_{p,1l}^{G} = 0 unless p_{2}(L) is a p-group. Thus

Y_{p} = ∑_{{L∈[s_{[}H×G] \atop O^{p}(p_{2}(L))=1l}} m_{p}^{n}|Z^{L}|e_{L}^{H×G}

Moreover, for each subgroup L of H × G, one has |L| = |k_{1}(L)||p_{2}(L)|, and if k_{1}(L) ≠ 1l, then |Z^{L}| = 0 since Z is a linear combination of left-free bisets. It follows that if |Z^{L}| ≠ 0, and if p_{2}(L) is a p-group, then then L is a p-group, equal to Δ_{ϕ}(Q) for some p-subgroup Q of G, and some morphism ϕ : Q → H. Hence p_{1}(L) = ϕ(Q) is a p-group. Hence

Y_{p} = ∑_{L} m_{p}^{n}|Z^{L}|e_{L}^{H×G}

where L runs through those subgroups Δ_{ϕ}(Q), up to conjugation by H ×G. Finally, for such a subgroup L = Δ_{ϕ}(Q)

m_{p}^{n}|Z^{L}|e_{L}^{H×G} = |H|_{p′}^{n}|G|_{p′}^{n}|Z^{L}|e_{L}^{H×G}e_{Q}^{G}
= |H|_{p′}^{n−1}|G|_{p′}^{n−m−1} \frac{|Z^{L}|}{|H|_{p}|G|_{p}^{m+1}} (|H||G|e_{L}^{H×G})(|G|^{m}e_{Q}^{G})

This is zero if Q = 1l and Z ∈ J_{n}(H, G), because |Z^{L}| = |Z| = 0 in this case. And if Q ≠ 1l, the element (|H||G|e_{L}^{H×G})(|G|^{m}e_{Q}^{G}) is in B(H, G)I_{G}^{m}, and if n is big

enough, the quotient \frac{|Z^{L}|}{|H|_{p}|G|_{p}^{m+1}} is an integer for Z ∈ J_{n}(H, G), since p_{1}(L) = ϕ(Q) is a p-group. This completes the proof of the lemma. □

Proposition 6.4.6. Let G, H and K be finite groups. Then the product

(X, Y) ∈ B_{{1l},All}(K, H) × B_{{1l},All}(H, G) ↦ X ×_{H} Y ∈ B_{{1l},All}(K, G)

induces a well defined product

B_{{1l},All}(K, H)ˆ × B_{{1l},All}(H, G)ˆ → B_{{1l},All}(K, G)ˆ

Proof. This is clear, since by the previous lemma, the product

(X, Y) ∈ B_{{1l},All}(K, H) × B_{{1l},All}(H, G) ↦ X ×_{H} Y ∈ B_{{1l},All}(K, G)

is continuous for the I_{H}-adic and I_{G}-adic topologies. □

Proposition 6.4.7. Let G be a finite group. Then the correspondence mapping a subgroup H of G to B(H)ˆ has a natural structure of Green functor.

Proof. Indeed, for any finite group H, there is an isomorphism

B(H)ˆ ≃ B_{{1l},All}(1l, H)ˆ

Thus for two finite groups H and K, there are maps

B(H)ˆ × B_{{1l},All}(H, K) → B(K)ˆ

Now if H ⊆ K are subgroups of G, this gives a transfer map from B(H)ˆ to B(K)ˆ , using K as a left-free H-set-K, and a restriction map from B(K)ˆ to B(H)ˆ , using K as a left-free K-set-H. Similarly, for x ∈ G, there is a conjugation map from B(H)ˆ to B(^{x}H)ˆ , induced by the biset H with its obvious structure of H-set-^{x}H. The axioms of Mackey functors follow from the properties of product of bisets.

In other words, the problem to define a Mackey functor structure on Bˆ is that the correspondence H ↦ I_{H}^{n} is not clearly a sub-Mackey functor of B. However if H is a subgroup of G, and n a positive integer, let

J_{n}(H) = {X ∈ B(H) | |X| = 0, ∀p prime , ∀P ⊆ H, P p-group, |X^{P} | ≡ 0 (p^{n})}

Then J_{n}(H) is an ideal of B(G), and one can show as in Lemma 6.4.5 that the topology defined by the ideals J_{n}(H) is equivalent to the topology defined by the ideals I_{H}^{n}. It is easy to see moreover that the correspondence H ↦ J_{n}(H) is a sub-Mackey functor of B. Hence the completion B(H)ˆ is isomorphic to

B(H)ˆ ≃ lim ; ⟵_{n→∞} B(H)/J_{n}(H)

and in this form, it has a natural structure of Green functor. □

Remark 6.4.8. For further results on completion of Green functors, see Luca ([26]).

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