Idempotents of double Burnside algebras, L-enriched bisets,

and decomposition of p-biset functors

Serge Bouc

Abstract: Let R be a (unital) commutative ring, and G be a finite group with order invertible in R. We introduce new idempotents ϵ_{T,S}^{G} in the double Burnside algebra RB(G, G) of G over R, indexed by conjugacy classes of minimal sections (T, S) of G (i.e. sections such that S ≤ Φ(T)). These idempotents are orthogonal, and their sum is equal to the identity. It follows that for any biset functor F over R, the evaluation F (G) splits as a direct sum of specific R-modules indexed by minimal sections of G, up to conjugation. The restriction of these constructions to the biset category of p-groups, where p is a prime number invertible in R, leads to a decomposition of the category of p-biset functors over R as a direct product of categories F_{L} indexed by atoric p-groups L up to isomorphism. We next introduce the notions of L-enriched biset and L-enriched biset functor for an arbitrary finite group L, and show that for an atoric pgroup L, the category F_{L} is equivalent to the category of L-enriched biset functors defined over elementary abelian p-groups. Finally, the notion of vertex of an indecomposable p-biset functor is introduced (when p ∈ R^{×}), and when R is a field of characteristic different from p, the objects of the category F_{L} are characterized in terms of vertices of their composition factors.

AMS subject classification: 18B99, 19A22, 20J15

Keywords: Minimal sections, idempotents, double Burnside algebra, enriched biset functor, atoric

1. Introduction

Let R denote throughout a commutative ring (with identity element). For a finite group G, we consider the double Burnside algebra RB(G, G) of a G over R. In the case where the order of G is invertible in R, we introduce idempotents ϵ_{T,S}^{G} in RB(G, G), indexed by the set M(G) of minimal sections of G, i.e. the set of pairs (T, S) of subgroups of G with S ⊴ T and S ≤ Φ(T), where Φ(T) is the Frattini subgroup of G (such sections have been considered in Section 5 of [9]). The idempotent ϵ_{T,S}^{G} only depends of the conjugacy class of (T, S) in G. Moreover, the idempotents ϵ_{T,S}^{G}, where (T, S) runs through a set [M(G)] of representatives of orbits of G acting on M(G) by conjugation, are orthogonal, and their sum is equal to the identity element of RB(G, G).

The idempotents ϵ_{G,1}^{G} plays a special role in our construction, and it is denoted by φ_{1}^{G}. In particular, when F is a biset functor over R (and the order of G is invertible in R), we set δ_{Φ}F (G) = φ_{1}^{G}F (G). We show that δ_{Φ}F (G) consists of those elements u ∈ F (G) such that Res_{H}^{G}u = 0 whenever H is a proper subgroup of G, and Def_{G/N}^{G}u = 0 whenever N is a non-trivial normal subgroup of G contained in Φ(G). This yields moreover a decomposition

F (G) ≅ ( ⊕_{(T,S)∈M(G)} δ_{Φ}F (T/S))^{G} ≅ ⊕_{(T,S)∈[M(G)]} δ_{Φ}F (T/S)^{N_{G}(T,S)/T} .

Restricting these constructions to the biset category RC_{p} of p-groups with coefficients in R, where p is a prime invertible in R, we get orthogonal idempotents b_{L} in the center of RC_{p}, indexed by atoric p-groups, i.e. finite p-groups which cannot be split as a direct product C_{p} × Q, for some p-group Q. We show next that every finite p-group P admits a unique largest atoric quotient P^{@}, well defined up to isomorphism, and that there exists an elementary abelian p-subgroup E of P (non unique in general) such that P ≅ E × P^{@}. For a given atoric p-group L, we introduce a category RC_{p}^{♯L}, defined as a quotient of the subcategory of RC_{p} consisting of p-groups P such that P^{@} ≅ L. This leads to a decomposition of the category F_{p,R} of p-biset functors over R as a direct product

F_{p,R} ≅ ∏_{L∈[At_{p}]} Fun_{R}(RC_{p}^{♯L}, R-Mod)

of categories of representations of RC_{p}^{♯L} over R, where L runs through a set [At_{p}] of isomorphism classes of atoric p-groups. Similar questions on idempotents in double Burnside algebras and decomposition of biset functors categories have been considered by L. Barker ([1]), R. Boltje and S. Danz ([2], [3]), R. Boltje and B. Külshammer ([4]), and P. Webb ([16]).

In particular, via the above decomposition, to any indecomposable pbiset functor F is associated a unique atoric p-group, called the vertex of F. We show that this vertex is isomorphic to Q^{@}, for any p-group Q such that F (Q) ≠ {0} but F vanishes on any proper subquotient of Q.

Going back to arbitrary finite groups, we next introduce the notions of L-enriched biset and L-enriched biset functor, and show that when L is an atoric p-group, the abelian category Fun_{R}(RC_{p}^{♯L}, R-Mod) is equivalent to the category of L-enriched biset functors from elementary abelian p-groups to R-modules.

The paper is organized as follows: Section 2 is a review of definitions and basic results on Burnside rings and biset functors. Section 3 is concerned with the algebra E(G) obtained by “cutting” the double Burnside algebra RB(G, G) of a finite group G by the idempotent ẽ_{G}^{G} corresponding to the “top” idempotent e_{G}^{G} of the Burnside algebra RB(G). Orthogonal idempotents φ_{N}^{G} of E(G) are introduced, indexed by normal subgroups N of G contained in Φ(G). It is shown moreover that if G is nilpotent, then φ_{1}^{G} is central in E(G). In Section 4, the idempotents ϵ_{T,S}^{G} of RB(G, G) are introduced, leading in Section 5 to the corresponding direct sum decomposition of the evaluation at G of any biset functor over R. In Section 6, atoric p-groups are introduced, and their main properties are stated. In Section 7, the biset category of p-groups over R is considered, leading to a splitting of the category F_{p,R} of p-biset functors over R as a direct product of abelian categories F_{L} = Fun_{R}(RC_{p}^{♯L}, R-Mod) indexed by atoric p-groups L up to isomorphism. In Section 8, for an arbitrary finite group L, the notions of L-enriched biset and L-enriched biset functor are introduced, and it is shown that when L is an atoric p-group, the category F_{L} is equivalent to the category of L-enriched biset functors on elementary abelian p-groups. Finally, in Section 9, for a given atoric p-group L, and when p is invertible in R, the structure of the category F_{L} is considered, and the notion of vertex of an indecomposable p-biset functor over R is introduced. In particular, when R is a field of characteristic different from p, it is shown that the objects of F_{L} are those p-biset functors all composition factors of which have vertex L.

Contents

1 Introduction 1

2 Review of Burnside rings and biset functors 4

3 Idempotents in E(G) 9

4 Idempotents in RB(G, G) 17

5 Application to biset functors 21

6 Atoric p-groups 23

7 Splitting the biset category of p-groups, when p ∈ R^{×} 29

8 L-enriched bisets 41

9 The category b̂_{L}F_{p,R}, for an atoric p-group L (p ∈ R^{×}) 46

2. Review of Burnside rings and biset functors

2.1. Let G be a finite group, let s_{G} denote the set of subgroups of G, let \overline{s_{G}} denote the set of conjugacy classes of subgroups of G, and let [s_{G}] denote a set of representatives of \overline{s_{G}}.

Let B(G) denote the Burnside ring of G, i.e. the Grothendieck ring of the category of finite G-sets. It is a commutative ring, with an identity element, equal to the class of a G-set of cardinality 1. The additive group B(G) is a free abelian group on the set {[G/H] | H ∈ [s_{G}]} of isomorphism classes of transitive G-sets. 2.2. • When G and H are finite groups, and L is a subgroup of G × H, set

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

Recall that k_{i}(L) ⊴ p_{i}(L), for i ∈ {1, 2}, that (k_{1}(L) × k_{2}(L)) ⊴ L, and that there are canonical isomorphisms

p_{1}(L)/k_{1}(L) ≅ L/(k_{1}(L) × k_{2}(L)) ≅ p_{2}(L)/k_{2}(L) .

Set moreover q(L) = L/(k_{1}(L) × k_{2}(L)).

• When Z is a subgroup of G, set

Δ(Z) = {(z, z) | z ∈ Z} ≤ (G × G) .

When N is a normal subgroup of G, set

Δ_{N}(G) = {(a, b) ∈ G × G | ab^{−1} ∈ N} .

It is a subgroup of G × G.

• When G, H, and K are groups, when L ≤ (G × H) and M ≤ (H × K), set

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

It is a subgroup of (G × K). 2.3. When G and H are finite groups, a (G, H)-biset U is a set endowed with a left action of G and a right action of H which commute. In other words U is a G × H^{op}-set, where H^{op} is the opposite group of H. The opposite biset U^{op} is the (H, G)-biset equal to U as a set, with actions defined for h ∈ H, u ∈ U and g ∈ G by h · u · g (in U^{op}) = g^{−1}uh^{−1} (in U).

The Burnside group B(G, H) is the Grothendieck group of the category of finite (G, H)-bisets. It is a free abelian group on the set of isomorphism classes [(G × H)/L], for L ∈ [s_{G×H}], where the (G, H)-biset structure on (G × H)/L is given by

∀a, g ∈ G, ∀b, h ∈ H, a · (g, h)L · b = (ag, b^{−1}h)L .

When G, H, and K are finite groups, there is a unique bilinear product

×_{H} : B(G, H) × B(H, K) → B(G, K)

induced by the usual product (U, V) ↦ U ×_{H} V = (U × V)/H of bisets, where the right action of H on U × V is defined for u ∈ U, v ∈ V and h ∈ H by (u, v)·h = (uh, h^{−1}v). This product will also be denoted as a composition (α, β) ↦ α ∘ β or as a product (α, β) ↦ αβ.

This leads to the following definitions:

2.4. Definition: The biset category of finite groups C is defined as follows:

• The objects of C are the finite groups.

• When G and H are finite groups,

Hom_{C}(G, H) = B(H, G) .

• When G, H, and K are finite groups, the composition

∘ : Hom_{C}(H, K) × Hom_{C}(G, H) → Hom_{C}(G, K)

is the product

×_{H} : B(K, H) × B(H, G) → B(K, G) .

• The identity morphism of the group G is the class of the set G, viewed as a (G, G)-biset by left and right multiplication.

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

When R is a commutative (unital) ring, the category RC is defined similarly by extending coefficients to R, i.e. by setting

Hom_{RC}(G, H) = R ⊗_{ℤ} B(H, G) ,

which will be simply denoted by RB(H, G). A biset functor over R is an Rlinear functor from RC to the category R-Mod of R-modules. The category of biset functors over R (where morphisms are natural transformations of functors) is denoted by F_{R}.

The correspondence sending a (G, H)-biset U to its opposite U^{op} extends to an isomorphism of R-modules RB(G, H) → RB(H, G). These isomorphisms give an equivalence of R-linear categories from RC to its opposite category, which is the identity on objects. 2.5. Let G and H be finite groups, and F be a biset functor (with values in R-Mod). For any finite (H, G)-biset U, the isomorphism class [U] of U belongs to B(H, G), and it yields an R-linear map F ([U]) : F (G) → F (H), simply denoted by F (U), or even f ∈ F (G) ↦ U(f) ∈ F (H). In particular:

• When H is a subgroup of G, denote by Ind_{H}^{G} the set G, viewed as a (G, H)-biset for left and right multiplication, and by Res_{H}^{G} the same set, viewed as an (H, G)-biset. This gives a map Ind_{H}^{G} : F (H) → F (G), called induction, and a map Res_{H}^{G} : F (G) → F (H), called restriction.

• When N is a normal subgroup of G, let Inf_{G/N}^{G} denote the set G/N, viewed as a (G, G/N)-biset for the left action of G, and right action of G/N by multiplication. Also let Def_{G/N}^{G} denote the set G/N, viewed as a (G/N, G)-biset. This gives a map Inf_{G/N}^{G} : F (G/N) → F (G), called inflation, and a map Def_{G/N}^{G} : F (G) → F (G/N), called deflation.

• Finally, when f : G → G′ is a group isomorphism, denote by Iso(f) the set G′, viewed as a (G′, G)-biset for left multiplication in G′, and right action of G given by multiplication by the image under f. This gives a map Iso(f) : F (G) → F (G′), called transport by isomorphism.

When G and H are finite groups, any (G, H)-biset is a disjoint union of transitive ones. It follows that any element of B(G, H) is a linear combination of morphisms of the form [(G×H)/L], where L ∈ s_{G×H}. Moreover, any such morphism factors as

(2.6) [(G×H)/L] = Ind_{p_{1}(L)}^{G}∘Inf_{p_{1}(L)/k_{1}(L)}^{p_{1}(L)}∘Iso(f)∘Def_{p_{2}(L)/k_{2}(L)}^{p_{2}(L)}∘Res_{p_{2}(L)}^{H} ,

where f : p_{2}(L)/k_{2}(L) → p_{1}(L)/k_{1}(L) is the canonical group isomorphism. In particular, for N ⊴ G,

(2.7)  [(G × G)/Δ_{N}(G)] = Inf_{G/N}^{G} ∘ Def_{G/N}^{G} .

For finite groups G, H, K, for L ≤ (G × H) and M ≤ (H × K), one has that

(2.8) [(G × H)/L] ×_{H} [(H × K)/M] = ∑_{h∈p_{2}(L)\H/p_{1}(M)}[(G × K)/(L ∗^{(h,1)}M)

in B(G, K). 2.9. When G is a finite group, the group B(G, G) is the ring of endomorphisms of G in the category C. This ring is called the double Burnside ring of G. It is a non-commutative ring (if G is non trivial), with identity element equal to the class of the set G, viewed as a (G, G)-biset for left and right multiplication.

There is a unitary ring homomorphism α ↦ α̃ from B(G) to B(G, G), induced by the functor X ↦ X̃ from G-sets to (G, G)-bisets, where X̃ = G × X, with (G, G)-biset structure given by

∀a, b, g ∈ G, ∀x ∈ X, a(g, x)b = (agb, ax) .

This construction has in particular the following properties ([7], Corollary 2.5.12):

2.10. Lemma: Let G be a finite group.

1. If H is a subgroup of G, and X is a finite G-set, then there is an isomorphism of (G, H)-bisets

X̃ ×_{G} Ind_{H}^{G} ≅ Ind_{H}^{G} ×_{H} { \atop Res_{H}^{G}X} ,

and an isomorphism of (H, G)-bisets

Res_{H}^{G} ×_{G} X̃ ≅ { \atop Res_{H}^{G}X} ×_{H} Res_{H}^{G} .

2. If H is a subgroup of G, and Y is a finite H-set, then there is an isomorphism of (G, G)-bisets

Ind_{H}^{G} ×_{H} Ỹ ×_{H} Res_{H}^{G} ≅ { \atop Ind_{H}^{G}Y} .

3. If N is a normal subgroup of G, and X is a finite G/N-set, then there is an isomorphism of (G/N, G)-bisets

X̃ ×_{G/N} Def_{G/N}^{G} ≅ Def_{G/N}^{G} ×_{G} { \atop Inf_{G/N}^{G}X} .

4. If N is a normal subgroup of G, and X is a finite G-set, then there is an isomorphism of (G/N, G/N)-bisets

Def_{G/N}^{G} ×_{G} X ×_{G} Inf_{G/N}^{G} ≅ { \atop Def_{G/N}^{G}X} .

2.11. Let RB(G) denote the R-algebra R ⊗_{ℤ} B(G). Assume moreover that the order of G is invertible in R. Then for H ≤ G, let e_{H}^{G} ∈ RB(G) be defined by

(2.12)  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. The elements e_{H}^{G}, for H ∈ [s_{G}], are orthogonal idempotents of RB(G), and their sum is equal to the identity element of RB(G). It follows that the elements ẽ_{H}^{G}, for H ∈ [s_{G}], are orthogonal idempotents of the R-algebra RB(G, G) = R ⊗_{ℤ} B(G, G), and the sum of these idempotents is equal to the identity element of RB(G, G). The idempotents ẽ_{G}^{G} play a special role, due to the following lemma:

2.13. Lemma: Let R be a commutative ring, and G be a finite group with order invertible in R.

1. Let H be a proper subgroup of G. Then

Res_{H}^{G} ∘ ẽ_{G}^{G} = 0 and ẽ_{G}^{G} ∘ Ind_{H}^{G} = 0 .

2. When N ⊴ G, let m_{G,N} ∈ R be defined by

m_{G,N} = \frac{1}{|G|} ∑_{{X∈s_{G} \atop XN=G}} |X|μ(X, G) .

Then

Def_{G/N}^{G} ∘ ẽ_{G}^{G} ∘ Inf_{G/N}^{G} = m_{G,N} ẽ_{G/N}^{G/N} .

3. Let N ⊴ G, and suppose that N is contained in the Frattini subgroup

Φ(G) of G. Then

ẽ_{G/N}^{G/N} ∘ Def_{G/N}^{G} = Def_{G/N}^{G} ∘ ẽ_{G}^{G} and Inf_{G/N}^{G} ∘ ẽ_{G/N}^{G/N} = ẽ_{G}^{G} ∘ Inf_{G/N}^{G} .

Proof : Assertion 1 follows from Lemma 2.10 and Assertion 1 of Theorem 5.2.4. of [7].

Assertion 2 follows from Lemma 2.10 and Assertion 4 of Theorem 5.2.4. of [7].

Finally the first part of Assertion 3 follows from Lemma 2.10 and Assertion 3 of Theorem 5.2.4. of [7]: indeed Inf_{G/N}^{G}e_{G/N}^{G/N} is equal to the sum of the different idempotents e_{X}^{G} of RB(G) indexed by subgroups X such that XN = G. If N ≤ Φ(G), then XN = G implies XΦ(G) = G, hence X = G. The second part of Assertion 3 follows by taking opposite bisets, since ẽ_{G}^{G} and

{ \atop e_{G/N}^{G/N}} are equal to their opposite bisets, and since (Def_{G/N}^{G})^{op} ≅ Inf_{G/N}^{G}.

2.14. Remark: For the same reason, if N ≤ Φ(G), then m_{G,N} = 1.

2.15. Remark: It follows from Assertion 1 and Remark 2.6 that if G and H are finite groups and if L ≤ (G × H), then ẽ_{G}^{G}[(G × H)/L] = 0 if p_{1}(L) ≠ G, and [(G × H)/L]ẽ_{H}^{H} = 0 if p_{2}(L) ≠ H.

3. Idempotents in E(G)

3.1. Notation: When G is a finite group with order invertible in R, denote by E(G) the R-algebra defined by

E(G) = ẽ_{G}^{G}RB(G, G)ẽ_{G}^{G} .

{Set \atop Σ(G, G) = \{M ∈ s_{G×G} | p_{1}(L) = p_{2}(L) = G\} ,}

and for L ∈ s_{G×G}, set

Y_{L} = ẽ_{G}^{G} [(G × G)/L] ẽ_{G}^{G} ∈ E(G) .

The R-algebra E(G) has been considered in [5], Section 9, in the case R is a field of characteristic 0. The extension of the results proved there to the case where R is a commutative ring in which the order of G is invertible is straightforward. In particular:

3.2. Proposition: Let G be a finite group with order invertible in R.

1. If L ∈ s_{G×G} − Σ(G, G), then Y_{L} = 0.

2. The elements Y_{L}, for L in a set of representatives of (G×G)-conjugacy classes on Σ(G, G), form a R-basis of E(G).

{3. For L, M ∈ Σ(G, G) \atop Y_{L}Y_{M} = \frac{m_{G,k_{2}(L)∩k_{1}(M)}}{|G|} ∑_{Z≤G ; {Zk_{2}(L)=Zk_{1}(M)=G \atop Z≥k_{2}(L)∩k_{1}(M)}} |Z|μ(Z, G) Y_{L∗Δ(Z)∗M}}

in E(G).

3.3. Corollary: Let L, M ∈ Σ(G, G). If one of the groups k_{2}(L) or k_{1}(M) is contained in Φ(G), then

Y_{L}Y_{M} = Y_{L∗M} .

Proof : Indeed if k_{2}(L) ≤ Φ(G), then Zk_{2}(L) = G implies ZΦ(G) = G, hence Z = G. Similarly, if k_{1}(M) ≤ Φ(G), then Zk_{1}(M) = G implies Z = G. In each case, Proposition 3.2 then gives

Y_{L}Y_{M} = m_{G,k_{2}(L)∩k_{1}(M)}Y_{L∗M} ,

and moreover m_{G,k_{2}(L)∩k_{1}(M)} = 1 since k_{2}(L) ∩ k_{1}(M) ≤ Φ(G), by Remark 2.14.

3.4. Notation: For a normal subgroup N of G such that N ≤ Φ(G), set

φ_{N}^{G} = ∑_{{M ⊴ G \atop N≤M≤Φ(G)}} μ_{⊴ G}(N, M)Y_{Δ_{M} (G)} ,

where μ_{⊴ G} is the Möbius function of the poset of normal subgroups of G.

3.5. Proposition: Let N ⊴ G with N ≤ Φ(G). Then

φ_{N}^{G} = Inf_{G/N}^{G}φ_{1}^{G/N}Def_{G/N}^{G} .

Proof : Indeed if N ≤ M ⊴ G, then μ_{⊴ G}(N, M) = μ_{⊴ G/N}(1, M/N). Since moreover N ≤ Φ(G), setting \overline{G} = G/N and \overline{M} = M/N, we have by Lemma 2.13

Inf_{G/N}^{G}Y_{Δ_{\overline{G}}(\overline{M})}Def_{G/N}^{G} = Inf_{G/N}^{G} ∘ ẽ\overline{G} ; \overline{G}((\overline{G} × \overline{G})/Δ_{\overline{G}}(\overline{M}))ẽ\overline{G} ; \overline{G} ∘ Def_{G/N}^{G}
= ẽ_{G}^{G} ∘ Inf_{G/N}^{G}((\overline{G} × \overline{G})/Δ_{\overline{G}}(\overline{M}))Def_{G/N}^{G} ∘ ẽ_{G}^{G}
= ẽ_{G}^{G}((G × G)/Δ_{M}(G))ẽ_{G}^{G}
= Y_{Δ_{M} (G)} ,

since Inf_{G/N}^{G}((\overline{G} × \overline{G}))/Δ_{\overline{G}}(\overline{M}))Def_{G/N}^{G} = (G × G)/Δ_{M}(G).

3.6. Proposition:

1. Let N ⊴ G, with N ≤ Φ(G). Then

φ_{N}^{G} = ẽ_{G}^{G} ×_{G} ( ∑_{{M ⊴ G \atop N≤M≤Φ(G)}} μ_{⊴ G}(N, M)[(G × G)/Δ_{M}(G)])

= ( ∑_{{M ⊴ G \atop N≤M≤Φ(G)}} μ_{⊴ G}(N, M)[(G × G)/Δ_{M}(G)]) ×_{G} ẽ_{G}^{G} .

2. In particular

φ_{1}^{G} = \frac{1}{|G|} ∑_{{X≤G,M ⊴ G \atop M≤Φ(G)≤X≤G}} |X|μ(X, G)μ_{⊴ G}(1, M) Indinf_{X/M}^{G} ∘ Defres_{X/M}^{G} .

Proof : For Assertion 1, by definition

φ_{N}^{G} = ∑_{{M ⊴ G \atop N≤M≤Φ(G)}} μ_{⊴ G}(N, M)ẽ_{G}^{G}[(G×G)/Δ_{M}(G)]×_{G} ∑_{X≤G} \frac{|X|}{|G|}μ(X, G)[(G×G)/Δ(X)].

Moreover [(G×G)/Δ_{M}(G)]×_{G}[(G×G)/Δ(X)] = [(G×G)/(Δ_{M}(G)∗Δ(X))], by (2.8), and Δ_{M}(G) ∗ Δ(X) = {(xm, x) | x ∈ X, m ∈ M}. The first projection of this group is equal to XM, hence it is equal to G if and only if X = G, since M ≤ Φ(G). The first equality of Assertion 1 follows, by Remark 2.15. The second one follows by taking opposite bisets, since ẽ_{G}^{G} and [(G × G)/Δ_{M}(G)] are equal to their opposite.

Assertion 2 follows in the special case where N = 1, expanding ẽ_{G}^{G} as

ẽ_{G}^{G} = \frac{1}{|G|} ∑_{X≤G} |X|μ(X, G)[(G × G)/Δ(X)] ,

observing that μ(X, G) = 0 unless X ≥ Φ(G), and that if X ≥ Φ(G) ≥ M, then

[(G × G)/Δ(X)] ∘ [(G × G)/Δ_{M}(G)] = [(G × G)/Δ_{M}(X)] ,

which is equal to Indinf_{X/M}^{G} ∘ Defres_{X/M}^{G}.

3.7. Corollary:

1. Let H < G. Then Res_{H}^{G}φ_{N}^{G} = 0 and φ_{N}^{G}Ind_{H}^{G} = 0.

2. Let M ⊴ G. If M ∩Φ(G) ≰ N, then Def_{G/M}^{G}φ_{N}^{G} = 0 and φ_{N}^{G}Inf_{G/M}^{G} = 0.

Proof : The first part of Assertion 1 follows from Lemma 2.13, since

Res_{H}^{G}φ_{N}^{G} = Res_{H}^{G} ẽ_{G}^{G}φ_{N}^{G} = 0 .

The second part follows by taking opposite bisets.

For Assertion 2, let P = M ∩ Φ(G). Since Def_{G/M}^{G} = Def_{G/M}^{G/P} ∘ Def_{G/P}^{G}, it suffices to consider the case M = P, i.e. the case where M ≤ Φ(G). Then, since [(G × G)/Δ_{M}(G)] = Inf_{G/M}^{G}Def_{G/M}^{G} for any M ⊴ G, by 2.7, and since Def_{G/M}^{G}Inf_{G/Q}^{G} = Inf_{G/MQ}^{G/M}Def_{G/MQ}^{G/Q} for any M, Q ⊴ G,

Def_{G/M}^{G}φ_{N}^{G} = Def_{G/M}^{G} ∑_{{Q ⊴ G \atop N≤Q≤Φ(G)}} μ_{⊴ G}(N, Q)Inf_{G/Q}^{G}Def_{G/Q}^{G}ẽ_{G}^{G}

= ∑_{{Q ⊴ G \atop N≤Q≤Φ(G)}} μ_{⊴ G}(N, Q)Inf_{G/MQ}^{G}Def_{G/MQ}^{G}ẽ_{G}^{G}

= ∑_{{P ⊴ G \atop N≤P ≤Φ(G)}} ( ∑_{Q ⊴ G ; {N≤Q≤Φ(G) \atop QM=P}} μ_{⊴ G}(N, Q))Inf_{G/P}^{G} Def_{G/P}^{G} ẽ_{G}^{G} .

Now for a given P ⊴ G with P ⊆ Φ(G), the sum ∑_{Q ⊴ G ; {N≤Q≤Φ(G) \atop QM=P}} μ_{⊴ G}(N, Q) is

equal to zero unless NM = N, that is M ≤ N, by classical properties of the Möbius function ([15] Corollary 3.9.3). This proves the first part of Assertion 2, and the second part follows by taking opposite bisets.

3.8. Theorem: Let G be a finite group with order invertible in R.

1. The elements φ_{N}^{G}, for N ⊴ G with N ≤ Φ(G), form a set of orthogonal idempotents in the algebra E(G), and their sum is equal to the identity element ẽ_{G}^{G} of E(G).

2. Let N ⊴ G with N ≤ Φ(G), and let H be a finite group.

(a) If L ≤ (G × H), then φ_{N}^{G} ×_{G} [(G × H)/L] = 0 unless p_{1}(L) = G

and k_{1}(L) ∩ Φ(G) ≤ N.

(b) If L′ ≤ (H × G), then [(H × G)/L′] ×_{G} φ_{N}^{G} = 0 unless p_{2}(L′) = G

and k_{2}(L′) ∩ Φ(G) ≤ N.

Proof : For N ⊴ G, set u_{N}^{G} = Y_{Δ_{N} (G)}. Since Δ_{N}(G) ∗ Δ_{M}(G) = Δ_{NM}(G) for any normal subgroups N and M of G, it follows from Corollary 3.3 that if either N or M is contained in Φ(G), then u_{N}^{G}u_{M}^{G} = u_{NM}^{G}.

Now Assertion 1 follows from the following general procedure for building orthogonal idempotents (see [13] Theorem 10.1 for details): we have a finite lattice P (here P is the lattice of normal subgroups of G contained in Φ(G)), and a set of elements g_{x} of a ring A, for x ∈ P (here A = E(G) and g_{N} = u_{N}^{G}), with the property that g_{x}g_{y} = g_{x∨y} for any x, y ∈ P, and g_{0} = 1, where 0 is the smallest element of P (here this element is the trivial subgroup of G, and u_{1}^{G} = Y_{Δ_{1}(G)} = ẽ_{G}^{G}). The the elements f_{x} defined for x ∈ P by

f_{x} = ∑_{{y∈P \atop x≤y}} μ(x, y)g_{y} ,

where μ is the Möbius function of P, are orthogonal idempotents of A, and their sum is equal to the identity element of A. This is exactly Assertion 1 (since f_{x} = φ_{N}^{G} here, for x = N ∈ P).

Let L ≤ (G × H), then by 2.6

φ_{N}^{G} ×_{G} [(G × H)/L] = φ_{N}^{G} ∘ Ind_{p_{1}(L)}^{G} ∘ [(p_{1}(L) × H)/L] = 0

unless p_{1}(L) = G, by Corollary 3.7. And if p_{1}(L) = G, then by 2.6

φ_{N}^{G} ×_{G} [(G × H)/L] = φ_{N}^{G} ∘ Inf_{G/k_{1}(L)}^{G} ∘ [(G/k_{1}(L) × H)/L_{1} ,

for some subgroup L_{1} of (G/k_{1}(L) × H). Again, by Corollary 3.7 this is equal to 0 unless k_{1}(L) ∩ Φ(G) ≤ N. The proof of Assertion (b) is similar. Alternatively, one can take opposite bisets in (a).

3.9. Proposition: Let G be a finite group with order invertible in R.

1. Let L ∈ Σ(G, G). Then

φ_{1}^{G}Y_{L} = ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)Y_{(N×1)L} .

This is non zero if and only if k_{1}(L) ∩ Φ(G) = 1. Similarly

Y_{L}φ_{1}^{G} = ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)Y_{L(1×N)} ,

and Y_{L}φ_{1}^{G} ≠ 0 if and only if k_{2}(L) ∩ Φ(G) = 1.

2. The elements φ_{1}^{G}Y_{L} (resp. Y_{L}φ_{1}^{G}), when L runs through a set of representatives of conjugacy classes of elements of Σ(G, G) such that k_{1}(L) ∩ Φ(G) = 1 (resp k_{2}(L) ∩ Φ(G) = 1), form an R-basis of the right ideal φ_{1}^{G}E(G) (resp. the left ideal E(G)φ_{1}^{G}) of E(G).

Proof : Let L ∈ Σ(G, G). By Proposition 3.8, we have

φ_{1}^{G}Y_{L} = ẽ_{G}^{G} ×_{G} ( ∑_{{N ⊴ G \atop N≤Φ(G)}}μ_{⊴ G}(1, N)[(G × G)/Δ_{N}(G)]) ×_{G} [(G × G)/L] ×_{G} ẽ_{G}^{G}

= ẽ_{G}^{G} ×_{G} ( ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)[(G × G)/Δ_{N}(G) ∗ L]) ×_{G} ẽ_{G}^{G}

= ẽ_{G}^{G} ×_{G} ( ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)[(G × G)/(N × 1)L]) ×_{G} ẽ_{G}^{G} .

= ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)Y_{(N×1)L} .

Set M = k_{1}(L) ∩ Φ(G). Then M ⊴ G, and (N × 1)L = (NM × 1)L for any normal subgroup N of G contained in φ(G). Thus

(3.10)  φ_{1}^{G}Y_{L} = ∑_{{P ⊴ G \atop M≤P ≤Φ(G)}} ( ∑_{{N ⊴ G \atop NM=P}} μ_{⊴ G}(1, N))Y_{(P ×1)L} .

If M ≠ 1, then ( ∑_{{N ⊴ G \atop NM=P}} μ_{⊴ G}(1, N)) = 0 for any P ⊴ G with M ≤ P ≤ Φ(G).

Hence φ_{1}^{G}Y_{L} = 0 in this case. And if M = 1, Equation (3.10) reads

φ_{1}^{G}Y_{L} = ∑_{{P ⊴ G \atop P ≤Φ(G)}} μ_{⊴ G}(1, P)Y_{(P ×1)L} .

The element Y_{(P ×1)L} is equal to Y_{L} if and only if (P × 1)L is conjugate to L. This implies that k_{1}((P × 1)L) is conjugate to (hence equal to) k_{1}(L). Thus P ≤ k_{1}((P × 1)L) ≤ k_{1}(L) ∩ Φ(G), hence P = 1. So the coefficient of Y_{L} in φ_{1}^{G}Y_{L} is equal to 1, hence φ_{1}^{G}Y_{L} ≠ 0. The remaining statements of Assertion 1 follow by taking opposite bisets.

Assertion 2 follows from Proposition 3.2, and from the fact that the coefficient of Y_{L} in φ_{1}^{G}Y_{L} is equal to 1 when k_{1}(L) ∩ Φ(G) = 1.

3.11. Corollary: Let G be a finite group of order invertible in R. If every minimal (non-trivial) normal subgroup of G is contained in Φ(G), then φ_{1}^{G} is central in E(G), and the algebra φ_{1}^{G}E(G) is isomorphic to ROut(G).

Proof : Indeed if L ∈ Σ(L, L) and φ_{1}^{G}Y_{L} ≠ 0, then k_{1}(L) ∩ Φ(G) = 1. It follows that k_{1}(L) contains no minimal normal subgroup of G, and then k_{1}(L) = 1. Equivalently q(L) ≅ p_{1}(L)/k_{1}(L) ≅ G ≅ p_{2}(L)/k_{2}(L), i.e. k_{2}(L) = G also, or equivalently k_{2}(L) ∩ Φ(G) = 1. Hence φ_{1}^{G}Y_{L} ≠ 0 if and only if Y_{L}φ_{1}^{G} ≠ 0, and in this case, there exists an automorphism θ of G such that

L = Δ_{θ}(G) = {(θ(x), x) | x ∈ G} .

In this case for any normal subgroup N of G contained in Φ(G)

(N × 1)L = {(a, b) ∈ G × G | aθ(b)^{−1} ∈ N}
= {(a, b) ∈ G × G | a^{−1}θ(b) ∈ N}
= L(1 × θ^{−1}(N)) .

Now N ↦ θ^{−1}(N) is a permutation of the set of normal subgroups of G contained in Φ(G). Moreover μ_{⊴ G}(1, N) = μ_{⊴ G}(1, θ^{−1}(N)).

It follows that φ_{1}^{G}Y_{L} = Y_{L}φ_{1}^{G}, so φ_{1}^{G} is central in E(G). Moreover the map θ ∈ Aut(G) ↦ φ_{1}^{G}Y_{Δ_{θ}(G)} clearly induces an algebra isomorphism ROut(G) → φ_{1}^{G}E(G).

3.12. Theorem: Let G be a finite group with order invertible in R. If G is nilpotent, then φ_{1}^{G} is a central idempotent of E(G).

Proof : Let L ∈ Σ(G, G). Setting Q = q(L), there are two surjective group homomorphisms s, t : G → Q such that L = {(x, y) ∈ G × G | s(x) = t(y)}. Then k_{1}(L) = Ker s and k_{2}(L) = Ker t. Now by Proposition 3.9

φ_{1}^{G}Y_{L} = ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)Y_{(N×1)L} ,

and this is non zero if and only if Ker s ∩ Φ(G) = 1. Now s(Φ(G)) is equal to Φ(Q) since G is nilpotent: indeed G = ∏_{p} G_{p} (resp. Q = ∏_{p} Q_{p}) is the direct product of its p-Sylow subgroups G_{p} (resp. Q_{p}), and s induces a surjective group homomorphism G_{p} → Q_{p}, for any prime p. Moreover Φ(G) = ∏_{p} Φ(G_{p}) (resp. Φ(Q) = ∏_{p} Φ(Q_{p})). Finally Φ(G_{p}) is the subgroup of G_{p} generated by commutators and p-powers of elements of G_{p}, hence it maps by s onto the subgroup of Q_{p} generated by commutators and p-powers of elements of Q_{p}, that is Φ(Q_{p}). Similarly t(Φ(G)) = Φ(Q).

If Ker s ∩ Φ(G) = 1, it follows that s induces an isomorphism from Φ(G) to Φ(Q). Then the surjective homomorphism Φ(G) → Φ(Q) induced by t is also an isomorphism, and in particular Ker t ∩ Φ(G) = 1.

Let D = L∩(Φ(G)×Φ(G)). Then k_{1}(D) ⊆ k_{1}(L)∩Φ(G) = Ker s∩Φ(G), hence k_{1}(D) = 1. Similarly k_{2}(L) ⊆ k_{2}(L) ∩ Φ(G) = Ker t ∩ Φ(G) = 1, hence k_{2}(D) = 1. Since s(Φ(G)) = Φ(Q) = t(Φ(G)), we have moreover p_{1}(D) = Φ(G) = p_{2}(D). It follows that there is an automorphism α of Φ(G) such that D = {(x, α(x)) | x ∈ Φ(G)}.

Moreover for any y ∈ G, there exists z ∈ G such that (y, z) ∈ L. It follows that (x^{y}, α(x)^{z}) ∈ D for any x ∈ Φ(G), that is α(x^{y}) = α(x)^{z}. In particular if N is a normal subgroup of G contained in Φ(G), then so is α(N). Hence α induces an automorphism of the poset of normal subgroups of G contained in Φ(G). In particular μ_{⊴ G}(1, N) = μ_{⊴ G}(1, α(N)).

Moreover for n ∈ N and (y, z) ∈ L, we have

(n, 1)(y, z) = (y, z)(n^{y}, 1) = (y, z)(n^{y}, α(n^{y}))(1, α(n^{y})^{−1}) .

Since (n^{y}, α(n^{y})) ∈ D ≤ L, we have (N × 1)L = L(1 × α(N)). It follows that

φ_{1}^{G}Y_{L} = ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)Y_{(N×1)L} = ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)Y_{L(1×α(N))}

= ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, α(N))Y_{L(1×α(N))} = ∑_{{N ⊴ G \atop N≤Φ(G)}} μ_{⊴ G}(1, N)Y_{L(1×N)}

= Y_{L}φ_{1}^{G} ,

as was to be shown.

3.13. Remark: When G is not nilpotent, it is not true in general that φ_{1}^{G} is central in E(G). This is because t(Φ(G)) need not be equal to Φ(Q) for a surjective group homomorphism t : G → Q. For example, there is a surjection t from the group G = C_{4} × (C_{5} ⋊ C_{4}) to Q = C_{4} with kernel C_{4} × C_{5} containing Φ(G) = C_{2} × 1, and another surjection s : G → Q with kernel 1 × (C_{5} ⋊ C_{4}) intersecting trivially Φ(G). In this case, the group L = {(x, y) ∈ G × G | s(x) = t(y)} is in Σ(G, G), and k_{1}(L) ∩ Φ(G) = 1, but k_{2}(L) ∩ Φ(G) = Φ(G) ≠ 1. By Proposition 3.9, we have φ_{1}^{G}Y_{L} ≠ 0 and Y_{L}φ_{1}^{G} = 0, so φ_{1}^{G} is not central in E(G).

4. Idempotents in RB(G, G)

4.1. Definition: When G is a finite group, a section (T, S) of G is a pair of subgroups of G such that S ⊴ T.

A section (T, S) is called minimal (cf. [9]) if S ≤ Φ(T). Let M(G) denote the set of minimal sections of G.

A group H is called a subquotient of G (notation H ⊑ G) if there exists a section (T, S) of G such that T/S ≅ H.

A section (T, S) is minimal if and only if the only subgroup H of T such that H/(H ∩ S) ≅ T/S is T itself.

4.2. Notation: Let G be a finite group, and let (T, S) be a section of G.

1. Let Indinf_{T/S}^{G} ∈ B(G, T/S) denote (the isomorphism class of) the (G, T/S)-biset G/S, and let Defres_{T/S}^{G} ∈ B(T/S, G) denote (the isomorphism class of) the (T/S, G)-biset S\G.

2. Let R be a commutative ring in which the order of G is invertible. Let u_{T,S}^{G} ∈ RB(G, T/S) be defined by

u_{T,S}^{G} = Indinf_{T/S}^{G}φ_{1}^{T/S} ,

and let v_{T,S}^{G} ∈ RB(T/S, G) be defined by

v_{T,S}^{G} = φ_{1}^{T/S}Defres_{T/S}^{G} .

4.3. Remark: Observe that v_{T,S}^{G} = (u_{T,S}^{G})^{op}: indeed (G/S)^{op} ≅ S\G, and (φ_{1}^{T/S})^{op} = φ_{1}^{T/S}.

4.4. Theorem: Let G be a finite group with order invertible in R. 1. If (T, S) and (T′, S′) are minimal sections of G, then

v_{T′,S′}^{G}u_{T,S}^{G} = 0

unless (T, S) and (T′, S′) are conjugate in G.

2. If (T, S) is a minimal section of G, then

v_{T,S}^{G}u_{T,S}^{G} = φ_{1}^{T/S}( ∑_{g∈N_{G}(T,S)/T} Iso(c_{g})) ,

where N_{G}(T, S) = N_{G}(T)∩N_{G}(S), and c_{g} is the automorphism of T/S induced by conjugation by g.

Proof : Indeed (S′\G) ×_{G} (G/S) ≅ S′\G/S as a (T′/S′, T/S)-biset. Hence

v_{T′,S′}^{G}u_{T,S}^{G} = φ_{1}^{T′/S′}( ∑_{g∈T′\G/T} S′\T′gT/S)φ_{1}^{T/S} .

For any g ∈ G, the (T′/S′, T/S)-biset S′\T′gT/S is transitive, isomorphic to ((T′/S′) × (T/S))/L_{g}, where

L_{g} = {(t′S′, tS) ∈ (T′/S′) × (T/S) | t′gt^{−1} ∈ S′gS} .

Then t′S′ ∈ p_{1}(L_{g}) if and only if t′ ∈ S′ · gT g^{−1} ∩ T′. Hence

p_{1}(L_{g}) = (^{g}T ∩ T′)S′/S′ .

Similarly p_{2}(L_{g}) = (T^{′g} ∩ T)S/S. In particular p_{1}(L_{g}) = T′/S′ if and only if (^{g}T ∩ T′)S′ = T′, i.e. ^{g}T ∩ T′ = T′, since S′ ≤ Φ(T′). Thus p_{1}(L_{g}) = T′/S′ if and only if T′ ≤ ^{g}T. Similarly p_{2}(L_{g}) = T/S if and only if T ≤ T^{′g}. By Theorem 3.8, it follows that φ_{1}^{T′/S′}(S′\T′gT/S)φ_{1}^{T/S} = 0 unless T′ = ^{g}T.

Assume now that T′ = ^{g}T. Then t′S′ ∈ k_{1}(L_{G}) if and only if t′ lies in S′ · gSg^{−1} ∩ T′. Hence

k_{1}(L_{g}) = (^{g}S ∩ T′)S′/S′ ,

and similarly k_{2}(L_{g}) = (S^{′g} ∩ T)S/S. But since S ≤ Φ(T) and S ⊴ T, it follows that ^{g}S ⊴^{g}T = T′ and ^{g}S ≤ ^{g}Φ(T) = Φ(T′). Hence ^{g}S · S′/S′ is contained in k_{1}(L_{g}) ∩ Φ(T′)/S′. Moreover Φ(T′)/S′ = Φ(T′/S′), as

Φ(T′/S′) = ⋂_{S′≤M′<T′}(M′/S′) = ⋂_{M′<T′}(M′/S′) = ( ⋂_{M′<T′} M′)/S′ = Φ(T′)/S′ ,

where M′ runs through maximal subgroups of T′, which all contain S′ since S′ ≤ Φ(T′).

It follows that if k_{1}(L_{g})∩Φ(T′/S′) = 1, then ^{g}S ·S′ = S′, that is ^{g}S ≤ S′. Similarly if k_{2}(L_{g}) ∩ Φ(T/S) = 1, then S^{′g} ≤ S. By Theorem 3.8, it follows that φ_{1}^{T′/S′}(S′\T′gT/S)φ_{1}^{T/S} = 0 unless T′ = ^{g}T and S′ = ^{g}S. This proves Assertion 1.

For Assertion 2, the same computation shows that

v_{T,S}^{G}u_{T,S}^{G} = ∑_{g∈N_{G}(T,S)/T} φ_{1}^{T/S}(S\T gT/S)φ_{1}^{T/S} .

But S\T gT/S = gT/S if g ∈ N_{G}(T, S), and this (T/S, T/S)-biset is isomorphic to Iso(c_{g}). Assertion 2 follows, since moreover φ_{1}^{T/S} commutes with any biset of the form Iso(θ), where θ is an automorphism of T/S.

4.5. Notation: For a minimal section (T, S) of the group G, set

ϵ_{T,S}^{G} = \frac{1}{|N_{G}(T,S):T |}u_{T,S}^{G}v_{T,S}^{G} = \frac{1}{|N_{G}(T,S):T |}Indinf_{T/S}^{G}φ_{1}^{G}Defres_{T/S}^{G} ∈ RB(G, G) .

Note that ϵ_{T,S}^{G} = ϵ_{^{g}T,^{g}S}^{G} for any g ∈ G, and that ϵ_{G,N}^{G} = φ_{N}^{G} when N ⊴ G and N ≤ Φ(G), by Proposition 3.5.

4.6. Proposition: Let (T, S) be a minimal section of G. Then

ϵ_{T,S}^{G} = \frac{1}{|N_{G}(T, S)|} ∑_{{X≤T,M ⊴ T \atop S≤M≤Φ(T)≤X≤T}} |X|μ(X, T)μ_{⊴ T} (S, M) Indinf_{X/M}^{G}∘Defres_{X/M}^{G} .

Proof : This is a straightforward consequence of the above definition of ϵ_{T,S}^{G}, and from Assertion 2 of Proposition 3.6.

4.7. Theorem: Let G be a finite group with order invertible in R, let [M(G)] be a set of representatives of conjugacy classes of minimal sections of G. Then the elements ϵ_{T,S}^{G}, for (T, S) ∈ [M(G)], are orthogonal idempotents of RB(G, G), and their sum is equal to the identity element of

RB(G, G).

Proof : Let (T, S) and (T′, S′) be distinct elements of [M(G)]. Then

ϵ_{T′,S′}^{G}ϵ_{T,S}^{G} = \frac{1}{|N_{G}(T′,S′):T′|} \frac{1}{|N_{G}(T,S):T |}u_{T′,S′}^{G}v_{T′,S′}^{G}u_{T,S}^{G}v_{T,S}^{G} = 0 ,

since v_{T′,S′}^{G}u_{T,S}^{G} = 0 by Theorem 4.4. Moreover:

∑_{(T,S)∈[M(G)]} ϵ_{T,S}^{G} = ∑_{(T,S)∈[M(G)]} \frac{1}{|N_{G}(T,S):T |}u_{T,S}^{G}v_{T,S}^{G}
= ∑_{(T,S)∈M(G)} \frac{1}{|G:T |}u_{T,S}^{G}v_{T,S}^{G}
= ∑_{(T,S)∈M(G)} \frac{1}{|G:T |}Indinf_{T/S}^{G}φ_{1}^{T/S}Defres_{T/S}^{G}

Now φ_{1}^{T/S} = ẽ_{T/S}^{T/S}f_{T/S} by Proposition 3.6, where

f_{T/S} = ∑_{{N/S ⊴(T/S) \atop N/S≤Φ(T/S)}} μ_{⊴ G}(1, N/S)[((T/S) × (T/S))/Δ_{N/S}(T/S)] .

Hence φ_{1}^{T/S} = ẽ_{T/S}^{T/S}Def_{T/S}^{T}Inf_{T/S}^{T}f_{T/S}, and

∑_{(T,S)∈[M(G)]} ϵ_{T,S}^{G} = ∑_{(T,S)∈M(G)} \frac{1}{|G:T |}Ind_{T}^{G}Inf_{T/S}^{T}ẽ_{T/S}^{T/S}Def_{T/S}^{T}Inf_{T/S}^{T}f_{T/S}Def_{T/S}^{T}Res_{T}^{G} .

Now Inf_{T/S}^{T}ẽ_{T/S}^{T/S}Def_{T/S}^{T} = { \atop Inf_{T/S}^{T}e_{T/S}^{T/S}}, and Inf_{T/S}^{T}e_{T/S}^{T/S} is equal to the sum over subgroups X or T such that XS = T, up to conjugation, of the idempotents e_{X}^{T}. Since S ≤ Φ(T), the only subgroup X of T such that XS = T is T itself. Hence

Inf_{T/S}^{T}ẽ_{T/S}^{T/S}Def_{T/S}^{T} = ẽ_{T}^{T} .

On the other hand

Inf_{T/S}^{T}[((T/S) × (T/S))/Δ_{N/S}(T/S)]Def_{T/S}^{T} = [(T × T)/Δ_{N}(T)] .

It follows that the sum Σ = ∑_{(T,S)∈[M(G)]} ϵ_{T,S}^{G} is equal to

Σ = ∑_{(T,S)∈M(G)} \frac{1}{|G:T |}Ind_{T}^{G}ẽ_{T}^{T} ∑_{{N ⊴ T \atop S≤N≤Φ(T)}} μ_{⊴ T} (S, N)[(T × T)/Δ_{N}(T)]Res_{T}^{G}

= ∑_{(T,S)∈M(G)} \frac{1}{|G:T |}Ind_{T}^{G}ẽ_{T}^{T}φ_{S}^{T}Res_{T}^{G} [by definition of φ_{S}^{T}]
= ∑_{T ≤G} \frac{1}{|G:T |}Ind_{T}^{G}ẽ_{T}^{T} ∑_{{S ⊴ T \atop S≤Φ(T)}} φ_{S}^{T}Res_{T}^{G}

= ∑_{T ≤G} \frac{1}{|G:T |}Ind_{T}^{G}ẽ_{T}^{T}Res_{T}^{G} [by Theorem 3.8]

= ∑_{T ≤G} \frac{1}{|G:T |} { \atop Ind_{T}^{G}e_{T}^{T}} [by Lemma 2.10]
= ∑_{T ≤G} \frac{1}{|G:N_{G}(T)|} ẽ_{T}^{G} [by (2.12)]
= ∑_{T ∈[s_{G}]} ẽ_{T}^{G} = { \atop G/G} = [(G × G)/Δ(G)] .

So the sum Σ is equal to the identity of RB(G, G). Since ϵ_{T,S}^{G}ϵ_{T′,S′}^{G} = 0 if (T, S) and (T′, S′) are distinct elements of [M(G)], it follows that for any (T, S) ∈ [M(G)]

ϵ_{T,S}^{G} = ϵ_{T,S}^{G}Σ = (ϵ_{T,S}^{G})^{2} ,

which completes the proof of the theorem.

5. Application to biset functors

5.1. Notation: Let F be a biset functor over R. When G is a finite group with order invertible in R, we set

δ_{Φ}F (G) = φ_{1}^{G}F (G)

5.2. Proposition: Let F be a biset functor over R. Then for any finite group G with order invertible in R, the R-submodule δ_{Φ}F (G) of F (G) is the set of elements u ∈ F (G) such that

\{ \begin{matrix}Res_{H}^{G}u = 0 & ∀H < G \\ Def_{G/N}^{G}u = 0 & ∀N ⊴ G, N ∩ Φ(G) ≠ 1 \end{matrix} .

Proof : If u ∈ δ_{Φ}F (G) = φ_{1}^{G}F (G), then Res_{H}^{G}u = 0 for any proper subgroup H of G, and Def_{G/N}^{G}u = 0 for any N ⊴ G such that N ∩ Φ(G) ≠ 1, by Corollary 3.7.

Conversely, if u ∈ F (G) fulfills the two conditions of the proposition, then ẽ_{G}^{G}u = u, because ẽ_{G}^{G} is equal to the identity element [(G × G)/Δ(G)] of RB(G, G), plus a linear combination of elements of the form [(G×G)/Δ(H)] = Ind_{H}^{G} ∘ Res_{H}^{G}, for proper subgroups H of G. Similarly Inf_{G/N}^{G}Def_{G/N}^{G}u = 0 for any non-trivial normal subgroup of G contained in Φ(G), thus φ_{1}^{G}u = u.

5.3. Remark: Since Def_{G/N}^{G} = Def_{G/N}^{G/M} ∘ Def_{G/M}^{G}, where M = N ∩ Φ(G), saying that Def_{G/N}^{G}u = 0 for any N ⊴ G with N ∩ Φ(G) ≠ 1 is equivalent to saying that Def_{G/N}^{G}u = 0 for any non trivial normal subgroup N of G contained in Φ(G).

5.4. Theorem: Let F be a biset functor over R. Then for any finite group G with order invertible in R, the maps

[figure: F (G); ⊕_{(T,S)∈[M(G)]} (δ_{Φ}F (T/S))^{N_{G}(T,S)/T}; w; V; ⊕_{(T,S)} \frac{1}{|N_{G}(T,S):T |}v_{T,S}^{G}w; ∑_{(T,S)} u_{T,S}^{G}w_{T,S}; ⊕_{(T,S)} w_{T,S}; U]

are well defined isomorphisms of R-modules, inverse to one other.

Proof : We have first to check that if w ∈ F (G), then the element v_{T,S}^{G}w of φ_{1}^{T/S}F (T/S) = δ_{Φ}F (T/S) is invariant under the action of N_{G}(T, S)/T. But for any g ∈ N_{G}(T/S)

Iso(c_{g})v_{T,S}^{G} = v_{^{g}T,^{g}S}^{G}Iso(c_{g}) = v_{T,S}^{G}Iso(c_{g}) ,

where Iso(c_{g}) : F (G) → F (G) on the right hand side is conjugation by g, that is an inner automorphism, hence the identity map, for g ∈ G.

Now for w ∈ F (G)

UV (w) = ∑_{(T,S)∈[M(G)]} \frac{1}{|N_{G}(T,S):T |}u_{T,S}^{G}v_{T,S}^{G}w
= ∑_{(T,S)∈[M(G)]} ϵ_{T,S}^{G}w = w ,

so UV is the identity map of F (G).

Conversely, if w_{T,S} ∈ (δ_{Φ}F (T/S))^{N_{G}(T,S)/T}, for (T, S) ∈ [M(G)], then

V U( ⊕_{(T,S)∈[M(G)]} w_{T,S}) = ⊕_{(T,S)∈[M(G)]} ∑_{(T′,S′)∈[M(G)]} \frac{1}{|N_{G}(T,S):T |}v_{T,S}^{G}u_{T′,S′}^{G}w_{T′,S′}
= ⊕_{(T,S)∈[M(G)]} \frac{1}{|N_{G}(T,S):T |}v_{T,S}^{G}u_{T,S}^{G}w_{T,S}
= ⊕_{(T,S)∈[M(G)]} \frac{1}{|N_{G}(T,S):T |} ∑_{g∈N_{G}(T,S)/T} Iso(c_{g})w_{T,S}
= ⊕_{(T,S)∈[M(G)]} w_{T,S} ,

so V U is also equal to the identity map.

6. Atoric p-groups

For the remainder of the paper, we denote by p a (fixed) prime number.

6.1. Notation and Definition:

• If P is a finite p-group, let Ω_{1}P denote the subgroup of P generated by the elements of order p.

• A finite p-group P is called atoric if it does not admit any decomposition P = E × Q, where E is a non-trivial elementary abelian p-group. Let At_{p} denote the class of atoric p-groups, and let [At_{p}] denote a set of representatives of isomorphism classes in At_{p}.

The terminology “atoric” is inspired by [14], where elementary abelian p-groups are called p-tori. Atoric p-groups have been considered (without naming them) in [6], Example 5.8.

6.2. Lemma: Let P be a finite p-group, and N be a normal subgroup of P. The following conditions are equivalent:

1. N ∩ Φ(P) = 1

2. N is elementary abelian and central in P, and admits a complement

in P.

3. N is elementary abelian and there exists a subgroup Q of P such that

P = N × Q.

Proof :

1 ⇒ 3 Let N ⊴ P with N ∩ Φ(P) = 1. Then N maps injectively in the elementary abelian p-group P/Φ(P), so N is elementary abelian. Let Q/Φ(P) be a complement of NΦ(P)/Φ(P) in P/Φ(P). Then Q ≥ Φ(P) ≥ [P, P], so Q is normal in P. Moreover Q·N = P and Q∩NΦ(P) = (Q∩N)Φ(P) = Φ(P), thus Q ∩ N ≤ Φ(P) ∩ N = 1. Now N and Q are normal subgroups of P which intersect trivially, hence they centralize each other. It follows that P = N × Q.

3 ⇒ 2 This is clear.

2 ⇒ 1 If P = N · Q for some subgroup Q of P, and if N is central in P, then P = N × Q. Thus Φ(P) = 1 × Φ(Q), as N is elementary abelian. Then N ∩ Φ(P) ≤ N ∩ Q = 1.

6.3. Lemma: Let P be a finite p-group. The following conditions are equivalent:

1. P is atoric.

2. If N ⊴ P and N ∩ Φ(P) = 1, then N = 1.

3. Ω_{1}Z(P) ≤ Φ(P).

Proof :

1 ⇒ 2 Suppose that P is atoric. Let N ⊴ P with N ∩ Φ(P) = 1. Then by Lemma 6.2, the group N is elementary abelian and there exists a subgroup Q of P such that P = N × Q. Hence N = 1.

2 ⇒ 3 Suppose now that Assertion 2 holds. If x is a central element of order p of P, then the subgroup N of P generated by x is normal in P, and non trivial. Then N ∩ Φ(P) ≠ 1, hence N ≤ Φ(P) since N has order p, thus x ∈ Φ(P).

3 ⇒ 1 Finally, if Assertion 3 holds, and if P = E × Q for some subgroups E and Q of P with E elementary abelian, then Φ(P) = 1 × Φ(Q). Moreover E ≤ Ω_{1}Z(P) ≤ Φ(P) ≤ Q, so E = E ∩ Q = 1, and P is atoric.

6.4. Proposition: Let P be a finite p-group, and N be a maximal normal subgroup of P such that N ∩ Φ(P) = 1. Then:

1. The group N is elementary abelian and there exists a subgroup T of P such that P = N × T.

2. The group P/N ≅ T is atoric.

3. If Q is an atoric p-group and s : P ↠ Q is a surjective group homomorphism, then s(T) = Q. In particular Q is isomorphic to a quotient

of T.

Proof : (1) This follows from Lemma 6.2.

(2) By (1), there exists T ≤ P such that P = N ×T. In particular P/N ≅ T. Now if T = E × S, for some subgroups E and S of T with E elementary abelian, then P ≅ P_{1} = (N ×E)×S, and N ×E is an elementary abelian normal subgroup of P_{1} which intersects trivially Φ(P_{1}) = Φ(S). By maximality of N, it follows that E = 1, so T ≅ P/N is atoric.

(3) Let s : P ↠ Q be a surjective group homomorphism, where Q is atoric. By (1), the group N is elementary abelian, and there exists a subgroup T of P such that P = N × T. Then T ≅ P^{@}, and Φ(P) = Φ(T). Moreover s(Φ(P)) = Φ(Q) as P is a p-group, and s(Z(P)) ≤ Z(Q) as s is surjective. It follows that s(N) is an elementary abelian central subgroup of Q, so s(N) ≤ Φ(Q) since Q is atoric, by Lemma 6.3. Now s(P) = Q = s(N)s(T), thus Q = Φ(Q)s(T), and s(T) = Q, as was to be shown.

6.5. Notation: When P is a finite p-group, and N is a maximal normal subgroup of P such that N ∩ Φ(P) = 1, we set P^{@} = P/N.

By Proposition 6.4, the group P^{@} does not depend on the choice of N, up to isomorphism: it is the greatest atoric quotient of P, in the sense that any atoric quotient of P is isomorphic to a quotient of P^{@}. In particular P^{@} is trivial if and only if P is elementary abelian.

6.6. Proposition: Let s : P ↠ Q be a surjective group homomorphism. Then P^{@} ≅ Q^{@} if and only if Ker(s) ∩ Φ(P) = 1.

Proof : Let E be a maximal normal subgroup of P such that E ∩ Φ(P) = 1, and T be a subgroup of P such that P = E × T. Then E is elementary abelian, and Φ(P) = Φ(T). Let π : Q → Q^{@} be the canonical projection. By definition, we have T ≅ P^{@}, and by Proposition 6.4, we have π ∘ s(T) = Q^{@}.

Hence Q^{@} is a quotient of P^{@}, and P^{@} ≅ Q^{@} if and only if the map π ∘ s induces an isomorphism from T to Q^{@}, that is if Ker(π ∘ s) ∩ T = 1. This implies Ker(s) ∩ T = 1, hence Ker(s) ∩ Φ(P) = 1.

Conversely, if Ker(s) ∩ Φ(P) = 1, then Ker(s) ∩ Φ(T) = 1. Now the group M = Ker(s) ∩ T is a normal subgroup of T such that M ∩ Φ(T) = 1. Since T is atoric, it follows from Lemma 6.3 that M = 1, hence s(T) ≅ T. Now Q = s(E)s(T), and s(E) is a central elementary abelian subgroup of Q, since s is surjective. Let F be a complement of G = s(E) ∩ s(T) in s(E). Then Q = (F · G)s(T) = F · s(T), thus Q = F × s(T) since F is central in Q. It follows that s(T) is a quotient of Q. Since s(T) ≅ T ≅ P^{@} is atoric, the group P^{@} is isomorphic to a quotient of Q^{@}, thus P^{@} ≅ Q^{@}.

6.7. Proposition: Let P be a finite p-group, and Q be a subquotient of P. Then Q^{@} is a subquotient of P^{@}.

Proof : Let (V, U) be a section of P such that V/U ≅ Q. Then Q^{@} is isomorphic to a quotient of V^{@}, by Lemma 6.4. Hence it suffices to prove that V^{@} is a subquotient of P^{@}.

Let E be a maximal normal subgroup of P such that E ∩ Φ(P) = 1, and T be a subgroup of P such that P = E × T. Then V ≤ E × T, so there exist a subgroup F of E, a subgroup X of T, a group Y, and surjective group homomorphisms α : F → Y and β : X → Y such that

V = {(f, x) ∈ F × X | α(f) = β(x)} .

Now F ≤ E is elementary abelian. If (f, x), (f′, x′) ∈ V, then [(f, x), (f′, x′)] = (1, [x, x′]), so [V, V] ≤ 1 × [X, X]. Conversely if x, x′ ∈ X, then there exist f, f′ ∈ F such that α(f) = β(x) and α(f′) = β(x′), i.e. (f, x), (f′, x′) ∈ V. Then [(f, x), (f′, x′)] = (1, [x, x′]), and it follows that [V, V] = 1 × [X, X]. Similarly, if (f, x) ∈ V, then (f, x)^{p} = (1, x^{p}). Conversely, if x ∈ X, then there exists f ∈ F such that α(f) = β(x), i.e. (f, x) ∈ V, and (1, x^{p}) = (f, x)^{p}. It follows that Φ(V) = 1 × Φ(X).

Now N = Ker(α) × 1 is a normal subgroup of V, and N ∩ Φ(V) = 1. By Proposition 6.6, it follows that V^{@} ≅ (V/N)^{@}. Moreover the group homomorphism (f, x) ∈ V ↦ x ∈ X is surjective with kernel N, hence V/N ≅ X. It follows that V^{@} ≅ X^{@} is a isomorphic to a quotient of the subgroup X of T ≅ P^{@}. Hence V^{@} is a subquotient of P^{@}, as was to be shown.

6.8. Proposition: Let P be a finite p-group, let N be a normal subgroup of P such that P/N ≅ P^{@}, and let Q be a subgroup of P. The following are equivalent:

1. Q^{@} ≅ P^{@}.

2. QN = P.

3. There exists a central elementary abelian subgroup E of P such that

P = EQ.

4. There exists an elementary abelian subgroup E of P such that P =

E × Q.

Proof : 1 ⇒ 2 Suppose Q^{@} ≅ P^{@}. We have N ∩ Φ(T) = 1, by Proposition 6.6. Moreover Φ(Q) ≤ Φ(P), as P is a p-group. Setting M = N ∩ Q, we have M ∩ Φ(Q) = 1, so (Q/M)^{@} ≅ Q^{@} ≅ P^{@}. But \overline{Q} = Q/M ≅ QN/N is a subgroup of P/N ≅ P^{@}, and moreover there exists an elementary abelian subgroup E of \overline{Q} such that \overline{Q} ≅ E × \overline{Q}^{@} ≅ E × P^{@}. Hence E = 1 and \overline{Q} ≅ QN/N ≅ P/N, so QN = P, as was to be shown.

2 ⇒ 3 We have N ∩ Φ(P) = 1, by Proposition 6.6. Hence N is elementary abelian, and central in P, and 2 implies 3.

2 ⇒ 3 Let E be an elementary abelian central subgroup of P such that P = EQ. Let F be a complement of E ∩ Q in E. Then F is elementary abelian and central in P. Moreover QF = QE = P, and Q ∩ F = 1. Hence P = F × Q.

4 ⇒ 1 If P = E × Q and E is elementary abelian, then Φ(P) = 1 × Φ(Q). Thus E ∩ Φ(P) = 1, so (P/E)^{@} ≅ P^{@} by Proposition 6.6, and Q^{@} ≅ P^{@}.

6.9. Proposition:

1. Let L be an atoric p-group, let P = E × L and Q = F × L, where E and F are elementary abelian p-groups, and let s : P → Q be a group homomorphism. Then s is surjective if and only if there exist a surjective group homomorphism a : E → F, group homomorphisms b : L → F and c : E → Ω_{1}Z(L), and an automorphism d of L such

that ; ∀(e, l) ∈ E × L, s(e, l) = (a(e)b(l), c(e)d(l)) .

Moreover in this case b ∘ c(e) = 1 for any e ∈ E, and s is an isomorphism if and only if a is an isomorphism.

2. Let P be a finite p-group. For a group homomorphism

λ : P → Ω_{1}Z(P) ∩ Φ(P) ,

let α_{λ} : P → P be defined by α(x) = xλ(x), for x ∈ P. Then α_{λ} is an automorphism of P.

3. Let P be a finite p-group, and let P = E×Q, where Q is atoric and E is elementary abelian. Then the correspondence λ ↦ α_{λ}(E) is a bijection from the set of group homomorphisms λ : P → Ω_{1}Z(P) ∩ Φ(P) such that Q ≤ Ker λ to the set of subgroups N of P such that P = N × Q.

Proof : (1) If s is surjective, then s(E) is central in Q, so s(E) ≤ Ω_{1}Z(Q) = F × Ω_{1}Z(L). Hence there exists group homomorphisms a : E → F and c : E → Ω_{1}Z(L) such that s(e, 1) = (a(e), c(e)), for any e ∈ E. Let b : L → F and d : L → L be the group homomorphisms defined by s(1, l) = (b(l), d(l)), for l ∈ L. Then s(e, l) = s(e, 1)s(1, l) = (a(e)b(l), c(e)d(l)) for all (e, l) ∈ P. Moreover b ∘ c(e) = 1 for any e ∈ E, since c(E) ≤ Ω_{1}Z(L) ≤ Φ(L), as L is atoric, and Φ(L) ≤ Ker b, as F is elementary abelian.

Now the composition of s with the projection F × L → L is surjective, hence s(1 × L) = L by Proposition 6.4. In other words d is surjective, hence it is an automorphism of L.

Since s is surjective, for any (f, y) ∈ Q, there exists (e, x) ∈ P such that a(e)b(x) = f and c(e)d(x) = y. The latter gives x = d^{−1}(c(e)^{−1}y). Then b(x) = bd^{−1}(c(e)^{−1})bd^{−1}(y), and bd^{−1}(c(e)^{−1}) = 1 since d^{−1}(c(e)^{−1}) ∈ d^{−1}Ω_{1}Z(L) = Ω_{1}Z(L), and Ω_{1}Z(L) ≤ Φ(L) ≤ Ker b. Then b(x) = bd^{−1}(y), and f = a(e)bd^{−1}(y). In particular, taking y = 1, we get that for any f ∈ L, there exists e ∈ E such that f = a(e). In other words a is surjective.

Conversely, given a surjective group homomorphism a : E → F, a group homomorphism b : L → F, a group homomorphism c : E → Ω_{1}Z(L), and an automorphism d of L, we can define s : P → Q by s(e, x) = (a(e)b(x), c(e)d(x)), for (e, x) ∈ P. This is clearly a group homomorphism, as F is abelian, and the image of c is central in L. We have again Ω_{1}Z(L) ≤ Φ(L) ≤ Ker b, since F is elementary abelian. If (f, y) ∈ Q, we can choose an element e ∈ E such that f = a(e)bd^{−1}(y), and then set x = d^{−1}(c(e)^{−1}y), i.e. c(e)d(x) = y. We also have b(x) = bd^{−1}(y), since d^{−1}(c(e)) ∈ Ω_{1}Z(L), so f = a(e)b(x). Hence s(e, x) = (f, y), and s is surjective.

Finally if s is an isomorphism, then E ≅ F, and then the surjection a is an isomorphism. Conversely, if a is an isomorphism, then E ≅ F, so P ≅ Q, and the surjection s is an isomorphism.

(2) Clearly α_{λ} is a group homomorphism, since λ(P) ≤ Z(P). Moreover if x ∈ Ker α_{λ}, then λ(x) = x, so x ∈ Ω_{1}Z(P) ∩ Φ(P) ≤ Φ(P) ≤ Ker λ, since Ω_{1}Z(P)∩Φ(P) is elementary abelian. Thus x = 1, and α_{λ} is injective. Hence it is an automorphism.

(3) Since P = E ×Q, we have Ω_{1}Z(P) = E ×Ω_{1}Z(Q), and Φ(P) = 1×Φ(Q).

So if λ is a group homomorphism from P to Ω_{1}Z(P) ∩ Φ(P) with Q ≤ Ker λ, we have λ(e, l) = (1, β(e)) for some group homomorphism β : E → Ω_{1}Z(Q). Then the group N = α_{λ}(E) = {(e, β(e)) | e ∈ E} is central in P. Moreover N ∩ Q = 1, and NQ = P, so P = N × Q. Note that N determines the homomorphism β, hence also the homomorphism λ, so the map λ ↦ α_{λ}(E) is injective.

It is moreover surjective: indeed, if N is a subgroup of P = E × Q such that P = N × Q, then N ≅ P/Q ≅ E is elementary abelian, hence central in P. Since NQ = P, for any e ∈ E, there exists (a, b) ∈ N and q ∈ Q such that (e, 1) = (a, b)(1, q), that is e = a and q = b^{−1}. In other words p_{1}(N) = E. Moreover N ∩ Q = 1, so k_{2}(N) = 1. So for e ∈ E, there exists a unique x ∈ Q such that (e, x) ∈ N. Setting x = β(e), we get a group homomorphism β : E → Q, such that N = {(e, β(e)) | e ∈ E}. Since N is central in P, the image of β is contained in Ω_{1}Z(Q) ≤ Φ(Q). Moreover Ω_{1}Z(P) = E × Ω_{1}Z(Q), and Φ(P) = 1 × Φ(Q), so (1 × β(E)) ≤ Ω_{1}Z(P) ∩ Φ(P). Setting λ(e, l) = (1, β(e)), we get a group homomorphism from P to Ω_{1}Z(P) ∩ Φ(P), such that Q ≤ Ker λ, and N = α_{λ}(E).

7. Splitting the biset category of p-groups, when p ∈ R^{×}

7.1. Notation and Definition: Let RC_{p} denote the full subcategory of the biset category RC consisting of finite p-groups. A p-biset functor over R is an R-linear functor from RC_{p} to the category of R-modules. Let F_{p,R} denote the full subcategory of F_{R} consisting of p-biset functors over R.

In the statements below, we indicate by [ p ∈ R^{×}] the assumption that p is invertible in R.

7.2. Theorem: [ p ∈ R^{×}] Let P and Q be finite p-groups, let (T, S) be a minimal section of P, and (V, U) be a minimal section of Q. Then

ϵ_{V,U}^{Q} RB(Q, P) ϵ_{T,S}^{P} ≠ {0} ⟹ (V/U)^{@} ≅ (T/S)^{@} .

Proof : If ϵ_{V,U}^{Q}RB(Q, P)ϵ_{T,S}^{P}, there exists a ∈ RB(Q, P) such that

ϵ_{V,U}^{Q} a ϵ_{T,S}^{P} = Indinf_{V/U}^{Q}φ_{1}^{V/U}Defres_{V/U}^{Q} a Indinf_{T/S}^{P}φ_{1}^{T/S}Defres_{T/S}^{P} ≠ 0 ,

and in particular the element b = Defres_{V/U}^{Q} a Indinf_{T/S}^{P} of RB(V/U, T/S)

is such that φ_{1}^{V/U} b φ_{1}^{T/S} ≠ 0. It follows that there is a subgroup L of the product (V/U) × (T/S) such that

φ_{1}^{V/U}[((V/U) × (T/S))/L]φ_{1}^{T/S} ≠ 0 .

Then Theorem 3.8 implies that p_{1}(L) = V/U, k_{1}(L) ∩ Φ(V/U) = 1, p_{2}(L) = T/S, and k_{2}(L) ∩ Φ(T/S) = 1. By Proposition 6.6, it follows that

(V/U)^{@} ≅ (p_{1}(L)/k_{1}(L))^{@} ≅ (p_{2}(L)/k_{2}(L))^{@} ≅ (T/S)^{@} ,

as was to be shown.

7.3. Notation: [ p ∈ R^{×}] Let L be an atoric p-group. If P is a finite p-group, we set

b_{L}^{P} = ∑_{{(T,S)∈[M(G)] \atop (T/S)^{@}≅L}} ϵ_{T,S}^{P} .

7.4. Theorem: [ p ∈ R^{×}]

1. Let L be an atoric p-group, and P be a finite p-group. Then b_{L}^{P} ≠ 0 if and only if L ⊑ P^{@}.

2. Let L and M be atoric p-groups, and let P and Q be finite p-groups. If

b_{M}^{Q}RB(Q, P)b_{L}^{P} ≠ {0}, then M ≅ L.

3. Let L be an atoric p-group, and let P and Q be finite p-groups. Then

{for any a ∈ RB(Q, P) \atop b_{L}^{Q} a = a b_{L}^{P} .}

4. The family of elements b_{L}^{P} ∈ RB(P, P), for finite p-groups P, is an idempotent endomorphism b_{L} of the identity functor of the category RC_{p} (i.e. an idempotent of the center of RC_{p}). The idempotents b_{L}, for L ∈ [At_{p}], are orthogonal, and their sum is equal to the identity element of the center of RC_{p}.

5. For a given finite p-group P, the elements b_{L}^{P}, for L ∈ [At_{p}] such that L ⊑ P^{@}, are non zero orthogonal central idempotents of RB(P, P), and their sum is equal to the identity of RB(P, P).

Proof : (1) The idempotent b_{L}^{P} is non zero if and only if there exists a minimal section (T, S) of P such that (T/S)^{@} ≅ L. Then L ⊑ P^{@}, by Proposition 6.7. Conversely, if L ⊑ P^{@}, then L ⊑ P, and there exists a minimal section (T, S) of P such that T/S ≅ L. Then (T/S)^{@} ≅ L^{@} ≅ L, so ϵ_{T,S}^{P} appears in the sum defining b_{L}^{p}, thus b_{L}^{P} ≠ 0.

(2) If b_{M}^{Q}RB(Q, P)b_{L}^{P} ≠ {0}, then there exist a minimal section (V, U) of Q with (V/U)^{@} ≅ M and a minimal section (T, S) of P with (T/S)^{@} ≅ L such that ϵ_{V,U}^{Q}RB(Q, P)ϵ_{T,S}^{P} ≠ 0. Then (V/U)^{@} ≅ (T/S)^{@} by Theorem 7.2, that is M ≅ L.

(3) The identity element of RB(P, P) is equal to the sum of the idempotents ϵ_{T,S}^{P}, for (T, S) ∈ [M(P)]. Grouping those idempotents ϵ_{T,S}^{P} for which (T/S)^{@} is isomorphic to a given L ∈ [At_{p}] shows that the identity element of RB(P, P) is equal to the sum of the idempotents b_{L}^{P}, for L ∈ [At_{p}] (and there are finitely many non zero b_{L}^{P}, by (1)). It follows that

b_{M}^{Q} a = b_{M}^{Q} a ∑_{L∈[At_{p}]} b_{L}^{P} = ∑_{L∈[At_{p}]} b_{M}^{Q} a b_{L}^{P}
= b_{M}^{Q} a b_{M}^{P} [by (2)]
= ∑_{L∈[At_{p}]} b_{L}^{Q} a b_{M}^{P} [by (2)]
= a b_{M}^{P} ,

since ∑_{L∈[At_{p}]} b_{L}^{Q} is the identity element of RB(Q, Q).

It follows that the family b_{L}^{P}, where P is a finite p-group, is an element b_{L} of the center of RC_{p}. Clearly b_{L}^{2} = b_{L}, and if L and M are non isomorphic atoric p-groups, then b_{L}b_{M} = 0, by (2). Moreover the infinite sum ∑_{L∈[At_{p}]} b_{L}

is actually locally finite, i.e. for each finite p-group P, the sum ∑_{L∈[At_{p}]} b_{L}^{P} has

only finitely many non zero terms. The sum ∑_{L∈[At_{p}]} b_{L} is clearly equal to the

identity endomorphism of the identity functor of RC_{p}.

(4) This is a straightforward consequence of (1) and (3).

7.5. Corollary: [ p ∈ R^{×}]

1. Let L be an atoric p-group. For a p-biset functor F, the family of maps F (b_{L}^{P}) : F (P) → F (P), for finite p-groups P, is an endomorphism of F, denoted by F (b_{L}).

2. If θ : F → G is a natural transformation of p-biset functors, the dia-

[figure: gram; F; θ; F (b_{L}); F; θ; G; G(b_{L}); G] is commutative. Hence the family of endomorphisms F (b_{L}), for p-biset functors F, is an idempotent of the center of the category F_{p,R}, denoted

by b̂_{L}.

3. The idempotents b̂_{L}, for L ∈ [At_{p}], are orthogonal idempotents of the center of F_{p,R}, and their sum is the identity.

4. If F is a p-biset functor over R, let b̂_{L}F denote the image of the endomorphism F (b_{L}) of F. Then F = ⊕_{L∈[At_{p}]} b̂_{L}F.

5. Let b̂_{L}F_{p,R} denote the full subcategory of F_{p,R} consisting of functors F such that F = b̂_{L}F. Then b̂_{L}F_{p,R} is an abelian subcategory of F_{p,R}. Moreover the functor

F ∈ F_{p,R} ↦ (b̂_{L}F)_{L∈[At_{p}]} ∈ ∏_{L∈[At_{p}]} b̂_{L}F_{p,R}

is an equivalence of categories.

Proof : All assertions are straightforward consequences of Theorem 7.4.

7.6. Notation: For an atoric p-group L, let RC_{p}^{L} denote the full subcategory of RC_{p} consisting of the class Y_{L} of finite p-groups P such that P^{@} ⊑ L. When p ∈ R^{×}, Let moreover

b_{L}^{+} = ∑_{{H∈[At_{p}] \atop H⊑L}} b_{H}

be the sum of the idempotents b_{H} corresponding to atoric subquotients of L, up to isomorphism.

The class Y_{L} is closed under taking subquotients, by Proposition 6.7. It follows that we can apply the results of Section 6 (Appendix) of [12]: if F is a p-biset functor over R, we can restrict F to an R-linear functor from RC_{p}^{L} to R-Mod. This yields a forgetful functor O_{Y_{L}} : F_{p,R} → Fun_{R}(RC_{p}^{L}, R-Mod). The right adjoint R_{Y_{L}} of this functor is described in full detail in Section 6 of [12], as follows: if G is an R-linear functor from RC_{p}^{L} to R-Mod, and P is a finite p-group, set

(7.7)  R_{Y_{L}}(G)(P) = lim ; ⟵_{(X,M)∈Σ_{L}(P)} G(X/M)

the inverse limit of modules G(X/M) on the set Σ_{L}(P) of sections (X, M) of P such that (X/M)^{@} ⊑ L, i.e. the set of sequences (l_{X,M})_{(X,M)∈Σ_{L}(P)} with the following properties:

1. if (X, M) ∈ Σ_{L}(P), then l_{X,M} ∈ G(X/M).

2. if (X, M), (Y, N) ∈ Σ_{L}(P) and M ≤ N ≤ Y ≤ X, then

Defres_{Y/N}^{X/M}l_{X,M} = l_{Y,N} .

3. if x ∈ P and (X, M) ∈ Σ_{L}(P), then ^{x}l_{X,M} = l_{^{x}X,^{x}M}.

Recall now that for finite groups P and Q, and for a finite (Q, P)-biset U, for a subgroup T of Q and an element u of U, the subgroup T^{u} of P is defined by T^{u} = {x ∈ P | ∃t ∈ T tu = ux}. By Lemma 6.4 of [12], if (T, S) is a section of Q, then (T^{u}, S^{u}) is a section of P, and T^{u}/S^{u} is a subquotient of T/S.

With this notation, when P and Q are finite p-groups, when U is a finite (Q, P)-biset, and l = (l_{X,M})_{(X,M)∈Σ_{L}(P)} is an element of R_{Y_{L}}(G)(P), we denote by Ul the sequence indexed by Σ_{L}(Q) defined by

(Ul)_{Y,N} = ∑_{u∈[Y \U/P]_{L}}(N\Y u)(l_{Y^{u},N^{u}})

where [Y \U/P] is a set of representatives of (Y × P)-orbits on U, and N\Y u is viewed as a (Y/N, Y^{u}/N^{u})-biset. It shown in Section 6 of [12] that Ul ∈ R_{Y_{L}}(G)(Q), and that R_{Y_{L}}(G) becomes a p-biset functor in this way. Moreover^{1}:

7.8. Theorem: [[12] Theorem 6.15] The assignment G ↦ R_{Y_{L}}(G) is an Rlinear functor R_{Y_{L}} from Fun_{R}(RC_{p}^{L}, R-Mod) to F_{p,R}, which is right adjoint to the forgetful functor O_{Y_{L}}. Moreover the composition O_{Y_{L}} ∘R_{Y_{L}} is isomorphic to the identity functor of Fun_{R}(RC_{p}^{L}, R-Mod).

7.9. Theorem: [ p ∈ R^{×}] For an atoric p-group L, let b̂_{L}^{+}F_{p,R} be the full subcategory of F_{p,R} consisting of functors F such that b̂_{L}^{+}F = F. Then the forgetful functor O_{Y_{L}} and its right adjoint R_{Y_{L}} restrict to quasi-inverse equivalences of categories

\xymatrix{b̂_{L}^{+}F_{p,R} \ar[r]^{O_{Y_{L}}} & Fun_{R}(RC_{p}^{L}, R-Mod) . \ar[l]^{R_{Y_{L}}}}

1In Theorem 6.15 of [12], only the case R = ℤ is considered, but the proofs extend trivially to the case of an arbitrary commutative ring R

Proof : \underline{First step:} The first thing to check is that the image of the functor R_{Y_{L}} is contained in b̂_{L}^{+}F_{p,R}. We first prove that if H is an atoric p-group, if F ∈ F_{p,R}, and if O_{Y_{L}}(b̂_{H}F) ≠ 0, then H ⊑ L: indeed in that case, there exists P ∈ Y_{L} such that b_{H}^{P}F (P) ≠ 0. In particular b_{H}^{P} ≠ 0 by Theorem 7.4, hence H ⊑ P^{@}. Since P^{@} ⊑ L as P ∈ Y_{L}, it follows that H ⊑ L, as claimed.

In particular

O_{Y_{L}}(F) = O_{Y_{L}}( ∑_{{H∈[At_{p}] \atop H⊑L}} b̂_{H}F ) = O_{Y_{L}}(b̂_{L}^{+}F ) .

Set G_{p}^{L} = Fun_{R}(RC_{p}^{L}, R-Mod), and let G ∈ G_{p}^{L}. Let H be an atoric p-group such that H ⋢ L. If F ∈ F_{p,R}, then

Hom_{F_{p,R}}(F, b̂_{H}R_{Y_{L}}(G)) = Hom_{F_{p,R}}(b̂_{H}F, b̂_{H}R_{Y_{L}}(G))
= Hom_{F_{p,R}}(b̂_{H}F, R_{Y_{L}}(G))
≅ Hom_{G_{p}^{L}}(O_{C_{Y_{L}}}(b̂_{H}F ), G) = {0} .

So the functor F ↦ Hom_{F_{p,R}}(F, b̂_{H}R_{Y_{L}}(G)) is the zero functor, and it follows from Yoneda’s lemma that b̂_{H}R_{Y_{L}}(G) = 0 if H ⋢ L. In other words R_{Y_{L}}(G) = b̂_{L}^{+}R_{Y_{L}}(G), as was to be shown.

\underline{Second step:} The first step shows that we have adjoint functors

\xymatrix{b̂_{L}^{+}F_{p,R} \ar[r]^{O_{Y_{L}}} & Fun_{R}(RC_{p}^{L}, R-Mod) = G_{p}^{L} . \ar[l]^{R_{Y_{L}}}}

Moreover, the composition O_{Y_{L}} ∘ R_{Y_{L}} is isomorphic to the identity functor, by Theorem 7.8. All we have to show is that the unit of the adjunction is also an isomorphism, in other words, that for any F ∈ b̂_{L}^{+}F_{p,R} and any finite p-group P, the natural map

(7.10)  η_{P} : F (P) → R_{Y_{L}}O_{Y_{L}}(F)(P) = lim ; ⟵_{(X,M)∈Σ_{L}(P)} F (X/M)

sending u ∈ F (P) to the sequence (Defres_{X/M}^{P}u)_{(X,M)∈Σ_{L}(P)}, is an isomorphism.

The map η_{P} is injective: indeed, if u ∈ F (P), then u = ∑_{{H∈[At_{p}] \atop H⊆L}} b_{H}^{P}u, as

F = b̂_{L}^{+}F. If Defres_{X/M}^{P}u = 0 for any section (X, M) of P with (X/M)^{@} ⊑ L, then F (ϵ_{T,S}^{P})(u) = 0 for any section (T, S) of P such that (T/S)^{@} ⊑ L, by Proposition 4.6 and Proposition 6.7. In particular b_{H}^{P}u = 0 for any atoric subquotient H of L, hence u = 0.

To prove that η_{P} is also surjective, we generalize the construction of Theorem A.2 of [11] (which is the case L = 1), and we define, for an element v = (v_{X,M})_{(X,M)∈Σ_{L}(P)} in R_{Y_{L}}O_{Y_{L}}(F)(P), an element u = ι_{P} (v) of F (P) by

u = \frac{1}{|P |} ∑_{{(T,S)∈M(P) \atop (T/S)^{@}⊑L}} ∑_{{X≤T, M ⊴ T \atop S≤M≤Φ(T)≤X≤T}} |X|μ(X, T)μ_{⊴ T} (S, M)Indinf_{X/M}^{P}v_{X,M} .

This yields an R-linear map ι_{P} : R_{Y_{L}}O_{Y_{L}}(F)(P) → F (P).

For (Y, N) ∈ Σ_{L}(P), set u_{Y,N} = Defres_{Y/N}^{P}u. Then:

u_{Y,N} = ∑_{{(T,S)∈M(P) \atop (T/S)^{@}⊑L}} ∑_{{X≤T, M ⊴ T \atop S≤M≤Φ(T)≤X≤T}} \frac{|X|}{|P |}μ(X, T)μ_{⊴ T} (S, M)Defres_{Y/N}^{P}Indinf_{X/M}^{P}v_{X,M}.

Moreover

Defres_{Y/N}^{P}Indinf_{X/M}^{P}v_{X,M} = ∑_{g∈[Y \P/X]} Indinf_{J_{g}/J′_{g}}^{Y/N}Iso(ϕ_{g})Defres_{I_{g}/I′_{g}}^{^{g}X/^{g}M}^{g}v_{X,N} ,

where J_{g} = N(Y ∩^{g}X), J′_{g} = N(Y ∩^{g}M), I_{g} = ^{g}M(Y ∩^{g}X), I′_{g} = ^{g}M(N∩^{g}X), and ϕ_{g} is the isomorphism I_{g}/I′_{g} → J_{g}/J′_{g} sending xI′_{g} to xJ′_{g}, for x ∈ Y ∩^{g}X. Hence

Defres_{Y/N}^{P}Indinf_{X/M}^{P}v_{X,M} = ∑_{g∈[Y \P/X]} Indinf_{J_{g}/J′_{g}}^{Y/N}Iso(ϕ_{g})v_{I_{g},I′_{g}}

= \frac{|Y ∩^{g}X|}{|Y ||X|} ∑_{g∈P} Indinf_{J_{g}/J′_{g}}^{Y/N}Iso(ϕ_{g})v_{I_{g},I′_{g}} .

Thus

u_{Y,N} = ∑_{\begin{matrix}(T,S)∈M(P) \\ (T/S)^{@}⊑L \\ X≤T, M ⊴ T \\ S≤M≤Φ(T)≤X≤T \\ g∈P \end{matrix}} \frac{|Y ∩^{g}X|}{|P ||Y |}μ(X, T)μ_{⊴ T} (S, M)Indinf_{J_{g}/J′_{g}}^{Y/N}Iso(ϕ_{g})v_{I_{g},I′_{g}}.

Now μ(X, T) = μ(^{g}X,^{g}T) and μ_{⊴ T} (S, M) = μ_{⊴}_{^{g}T} (^{g}S,^{g}M), so summing over (^{g}T ,^{g}S,^{g}X,^{g}M) instead of (T, S, X, M) we get

u_{Y,N} = ∑_{\begin{matrix}(T,S)∈M(P) \\ (T/S)^{@}⊑L \\ X≤T, M ⊴ T \\ S≤M≤Φ(T)≤X≤T \end{matrix}} \frac{|Y ∩ X|}{|Y |}μ(X, T)μ_{⊴ T} (S, M)Indinf_{J_{1}/J′_{1}}^{Y/N}Iso(ϕ_{1})v_{I_{1},I′_{1}}.

Setting W = Y ∩ X, we have J_{1} = NW, J′_{1} = N(W ∩ M), I_{1} = MW, I′_{1} = M(N ∩ W), and these four groups only depend on W, once M and N are given. Hence, for given T, S and M, we can group together the terms of the above summation for which Y ∩ X is a given subgroup W of Y ∩ T. This gives

u_{Y,N} = ∑_{\begin{matrix}(T,S)∈M(P) \\ (T/S)^{@}⊑L \\ M ⊴ T \\ S≤M≤Φ(T) \\ W ≤Y ∩T \end{matrix}} ( ∑_{\begin{matrix}Φ(T)≤X≤T \\ X∩Y =W \end{matrix}} μ(X, T))\frac{|W |}{|Y |}μ_{⊴ T} (S, M)Indinf_{J_{1}/J′_{1}}^{Y/N}Iso(ϕ_{1})v_{I_{1},I′_{1}}.

Moreover ∑_{\begin{matrix}Φ(T)≤X≤T \\ X∩Y =W \end{matrix}} μ(X, T) = ∑_{\begin{matrix}X≤T \\ X∩(Y ∩T)=W \end{matrix}} μ(X, T), since μ(X, T) = 0 unless

X ≥ Φ(T), and the latter summation vanishes unless Y ∩T = T, by classical combinatorial lemmas ([15] Corollary 3.9.3). This gives:

u_{Y,N} = ∑_{\begin{matrix}(T,S)∈M(P) \\ (T/S)^{@}⊑L \\ M ⊴ T \\ S≤M≤Φ(T)≤W ≤T ≤Y \end{matrix}} \frac{|W |}{|Y |}μ(W, T)μ_{⊴ T} (S, M)Indinf_{J_{1}/J′_{1}}^{Y/N}Iso(ϕ_{1})v_{I_{1},I′_{1}}.

Moreover in this summation J_{1} = NW, J′_{1} = N(W ∩ M) = NM, I_{1} = MW = W, I′_{1} = M(N ∩W) = MN ∩W. All these groups remain unchanged if we replace M by M(N ∩ Φ(T)), so for given T, S and W, we can group together those terms for which M(N ∩ Φ(T)) is a given normal subgroup U of T with U ≤ Φ(T). The sum ∑_{\begin{matrix}S≤M ⊴ T \\ M(N∩Φ(T))=U \end{matrix}} μ_{⊴ T} (S, M) is equal to 0 (by the

same above-mentioned classical combinatorial lemmas) unless N ∩Φ(T) ≤ S. Hence

u_{Y,N} = ∑_{\begin{matrix}(T,S)∈M(P) \\ (T/S)^{@}⊑L \\ U ⊴ T \\ N∩Φ(T)≤S≤U≤Φ(T)≤W ≤T ≤Y \end{matrix}}\frac{|W |}{|Y |}μ(W, T)μ_{⊴ T} (S, U)Indinf_{J_{1}/J′_{1}}^{Y/N}Iso(ϕ_{1})v_{I_{1},I′_{1}},

where J_{1} = NW, J′_{1} = NU, I_{1} = W, I′_{1} = UN ∩ W.

Now if N ∩ Φ(T) ≤ S ≤ Φ(T) ≤ T ≤ Y, then (T N/N)^{@} ⊑ (Y/N)^{@}. Moreover the normal subgroup (N ∩ T)/(N ∩ Φ(T)) of T/(N ∩ Φ(T)) intersects trivially the Frattini subgroup

Φ(T/(N ∩ Φ(T))) = Φ(T)(N ∩ Φ(T))/(N ∩ Φ(T)) ,

so (T/(N ∩ Φ(T)))^{@} ≅ (T/(T ∩ N)^{@} ≅ (T N/N)^{@} by Proposition 6.6. Then

(T/S)^{@} ⊑ (T/(N ∩ Φ(T)))^{@} ⊑ (T N/N)^{@} ⊑ (Y/N)^{@}. As (Y/N)^{@} ⊑ L by assumption, it follows that

u_{Y,N} = ∑_{\begin{matrix}S ⊴ T ≤Y \\ U ⊴ T \\ N∩Φ(T)≤S≤U≤Φ(T)≤W ≤T ≤Y \end{matrix}}\frac{|W |}{|Y |}μ(W, T)μ_{⊴ T} (S, U)Indinf_{J_{1}/J′_{1}}^{Y/N}Iso(ϕ_{1})v_{I_{1},I′_{1}}.

Now the sum ∑_{\begin{matrix}S ⊴ T \\ N∩Φ(T)≤S≤U \end{matrix}} μ_{⊴ T} (S, U) is equal to zero unless U = N ∩ Φ(T).

Hence

u_{Y,N} = ∑_{Φ(T)≤W ≤T ≤Y} \frac{|W |}{|Y |}μ(W, T)Indinf_{J_{1}/J′_{1}}^{Y/N}Iso(ϕ_{1})v_{I_{1},I′_{1}}.

For a given subgroup W of Y, the sum ∑_{Φ(T)≤W ≤T ≤Y} μ(W, T) is equal to

∑_{W ≤T ≤Y} μ(W, T) since μ(W, T) = 0 unless W ≥ Φ(T), and the latter is equal

to zero if W ≠ Y, and to 1 if W = Y. Thus

u_{Y,N} = \frac{|Y |}{|Y |}Indinf_{J_{1}/J′_{1}}^{Y/N}Iso(ϕ_{1})v_{I_{1},I′_{1}},

where J_{1} = NY = Y, J′_{1} = N(Y ∩ U) = N, I_{1} = Y =, I′_{1} = UN ∩ Y = N. Hence I_{1} = J_{1} = Y and I′_{1} = J′_{1} = N, so ϕ_{1} is equal to the identity. It follows that u_{Y,N} = v_{Y,N} for any (Y, N) ∈ Σ_{L}(P), so η_{P} (u) = v. This proves that the map η_{P} is surjective, hence an isomorphism, with inverse ι_{P}. This completes the proof of Theorem 7.9.

7.11. Definition: Let RC_{p}^{♯L} be the following category:

• The objects of RC_{p}^{♯L} are the finite p-groups P such that P^{@} ≅ L.

• If P and Q are finite p-groups such that P^{@} ≅ Q^{@} ≅ L, then

Hom_{RC_{p}^{♯L}}(P, Q) = RB(Q, P)/ ∑_{L⋢S} RB(Q, S)B(S, P)

is the quotient of RB(Q, P) by the R-submodule generated by all morphisms from P to Q in RC_{p} which factor through a p-group S which do not admit L as a subquotient.

• The composition of morphisms in RC_{p}^{♯L} is induced by the composition of morphisms in RC_{p}.

7.12. Remark: Morphisms in RC_{p} which factor through a p-group S such that L ⋢ S clearly generate a two-sided ideal, so the composition in RC_{p}^{♯L} is well defined. Moreover the category RC_{p}^{♯L} is R-linear. Let Fun_{R}(RC_{p}^{♯L}, R-Mod) denote the category of R-linear functors from RC_{p}^{♯L} to the category R-Mod of R-modules.

7.13. Lemma: Let p be a prime, and L be an atoric p-group. Let P and Q be finite p-groups.

1. If P^{@} ≅ L or Q^{@} ≅ L, and if M ≤ (Q × P), then q(M)^{@} ⊑ L. Moreover q(M)^{@} ≅ L if and only if L ⊑ q(M).

2. If P^{@} ≅ Q^{@} ≅ L, then

Hom_{RC_{p}^{♯L}}(P, Q) = RB(Q, P)/ ∑_{S^{@}⊏L} RB(Q, S)B(S, P)

is also the quotient of RB(Q, P) by the R-submodule generated by all morphisms from P to Q in RC_{p} which factor through a p-group S such that S^{@} is a proper subquotient of L.

3. If P^{@} ≅ Q^{@} ≅ L, then Hom_{RC_{p}^{♯L}}(P, Q) has an R-basis consisting of the (images of the) transitive (Q, P)-bisets (Q × P)/M, where M is a subgroup of (Q × P) such that q(M)^{@} ≅ L (up to conjugation).

Proof : (1) Indeed q(M) is a subquotient of P, and a subquotient of Q. Hence q(M)^{@} is a subquotient of P^{@} and a subquotient of Q^{@}, thus q(M) ⊑ L^{@} ≅ L. Now suppose that q(M)^{@} ≅ L. Then L is a quotient of q(M), so L ⊑ q(M). Conversely, if L ⊑ q(M), then L ≅ L^{@} is a subquotient of q(M)^{@}, which is a subquotient of L. So q(M)^{@} ≅ L.

(2) Let S be a finite p-group such that L ⋢ S, or equivalently L ⋢ S^{@}. Any element of RB(Q, S)B(S, P) is a linear combination of (Q, P)-bisets of the form (Q × P)/(M ∗ N), for M ≤ (Q × S) and N ≤ (S × P). This biset (Q × P)/(M ∗ N) also factors though T = q(M ∗ N), by 2.6. Moreover T is a subquotient of q(M) and q(N), hence a subquotient of Q, S, and P. Hence T^{@} ⊑ Q^{@} ≅ L, and T^{@} ≇ L, since L ⋢ S^{@}. Hence T ⊏ L.

(3) The (images of the) elements (Q × P)/M, where M is a subgroup of (Q × P) such that q(M)^{@} ≅ L (up to conjugation), clearly generate Hom_{RC_{p}^{♯L}}(P, Q). Moreover, the proof of (2) shows that they are linearly independent, since any transitive (Q, P)-biset (Q × P)/N appearing in an element of the sum ∑_{S^{@}⊏L} RB(Q, S)B(S, P) is such that q(N)^{@} ⊏ L. .

7.14. Remark: If G is an R-linear functor from RC_{p}^{♯L} to the category R-Mod of R-modules, we can extend G to an R-linear functor from RC_{p}^{L} to R-Mod by setting G(P) = {0} if P is a finite p-group such that P^{@} is a proper subquotient of L. Conversely, an R-linear functor from RC_{p}^{L} to R-Mod which vanishes on p-groups P such that P^{@} ≇ L can be viewed as an R-linear functor from RC_{p}^{♯L} to R-Mod. In the sequel, we will freely identify those two types of functors, and consider Fun_{R}(RC_{p}^{♯L}, R-Mod) as the full subcategory of Fun_{R}(RC_{p}^{L}, R-Mod) consisting of functors which vanish on p-groups P such that P^{@} ≇ L.

7.15. Theorem: [ p ∈ R^{×}] Let L be an atoric p-group.

1. If F is a p-biset functor over R such that F = b̂_{L}F, and P is a finite p-group such that L ⋢ P, then F (P) = {0}.

2. If G is an R-linear functor from RC_{p}^{♯L} to R-Mod, then b̂_{L}R_{Y_{L}}(G) =

R_{Y_{L}}(G).

3. The forgetful functor O_{Y_{L}} and its right adjoint R_{Y_{L}} restrict to quasiinverse equivalences of categories

\xymatrix{b̂_{L}F_{p,R} \ar[r]^{O_{Y_{L}}} & Fun_{R}(RC_{p}^{♯L}, R-Mod) . \ar[l]^{R_{Y_{L}}}}

Proof : (1) Since b̂_{L}F = F, then in particular F (b_{L}^{P})F (P) = F (P). If L ⋢ P, then there is no minimal section (T, S) of P with (T/S)^{@} ≅ L, thus b_{L}^{P} = 0, and F (P) = {0}.

(2) Let G be an R-linear functor from RC_{p}^{♯L} to R-Mod, in other words an Rlinear functor from FC_{p}^{L} to R-Mod which vanishes on p-groups P such that P^{@} is a proper subquotient of L. By Theorem 7.9, we have b̂_{L}^{+}R_{Y_{L}}(G) = R_{Y_{L}}(G). If H is an atoric p-group which is a proper subquotient of L, then G vanishes over any subquotient Q of H, since Q^{@} ⊑ H ⊏ L if Q ⊑ H. In particular b_{H}^{P} acts by 0 on R_{Y_{L}}(G)(P), for any finite p-group P: indeed b_{H}^{P} is a linear combination of terms of the form Indinf_{X/M}^{P}Defres_{X/M}^{P}, where (X, M) is a section of P such that S ≤ M ≤ Φ(T) ≤ X ≤ T, for some section (T, S) of P with (T/S)^{@} ≅ H. For such a section (X, M) of P, we have (X/M)^{@} ⊑ (T/S)^{@} ⊑ H, thus G vanishes on any subquotient of X/M, so R_{Y_{L}}(G)(X/M) = {0}, hence b_{H}^{P} = 0 on R_{Y_{L}}(G)(P), as claimed. It follows that b̂_{H}R_{Y_{L}}(G) = 0, hence b̂_{L}^{+}R_{Y_{L}}(G) = R_{Y_{L}}(G) = b̂_{L}R_{Y_{L}}(G).

(3) This is a straightforward consequence of (1) and (2), by Theorem 7.9. The following proposition gives some detail on the structure of the cate-

gory RC_{p}^{♯L}:

7.16. Proposition: Let p be a prime, and L be an atoric p-group.

1. Let P be a finite p-group. Then P^{@} ≅ L if and only if there exists an elementary abelian p-group E such that P ≅ E × L.

2. Let P = E × L and Q = F × L, where E and F are elementary abelian p-groups. If M ≤ (Q × P), then q(M)^{@} ≅ L if and only if

p_{1,2}(M) = p_{2,2}(M) = L and k_{1,2}(M) = k_{2,2}(M) = 1 ,

where p_{1,2} and p_{2,2} are the morphisms from ((H × L) × (G × L)) to L defined by p_{1,2}((h, x), (g, y)) = x and p_{2,2}((h, x), (g, y)) = y, and

k_{1,2}(M) = {x ∈ L | ((1, x), (1, 1)) ∈ M} ,
k_{2,2}(M) = {x ∈ L | ((1, 1), (1, x)) ∈ M} .

Proof : (1) This follows from Proposition 6.8.

(2) By Lemma 7.13, the R-module RB(Q, P) has a basis consisting of the isomorphism classes of (Q, P)-bisets of the form (Q × P)/M, where M is a subgroup of (Q×P), up to conjugation, and q(M)^{@} ≅ L. If M is such a subgroup, then L ≅ (p_{1}(M)/k_{1}(M))^{@} ⊑ (p_{1}(M))^{@} ⊑ Q^{@} ≅ L, so p_{1}(M)^{@} ≅ L, and similarly p_{2}(M)^{@} ≅ L. By Proposition 6.8 p_{1}(M)^{@} ≅ L if and only if Ep_{1}(M) = P, which in turn is equivalent to p_{1,2}(M) = L. Similarly p_{2}(M)^{@} ≅ L if and only if p_{2,2}(M) = L.

Then (p_{1}(M)/k_{1}(M))^{@} ≅ L if and only if k_{1}(M) ∩ Φ(p_{1}(M)) = 1, by Proposition 6.6. Moreover Φ(p_{1}(M)) = Φ(P), as there exists an elementary abelian subgroup E′ of P such that P = E′ × p_{1}(M), by Proposition 6.8 again. Since Φ(P) = 1 × Φ(L), it follows that k_{1}(M) ∩ (1 × Φ(L)) = 1. Now N = k_{1}(L)∩(1×L) is a normal subgroup of (1×L) (since p_{1,2}(M) = L), which intersect trivially (1 × Φ(L)). Since L is atoric, by Lemma 6.3, any central element of order p of (1×L) is contained in (1×Φ(L)), so N contains no non trivial central element of (1 × L), hence N = 1. Thus k_{1}(L) ∩ (1 × L) = 1, or equivalently k_{1,2}(M) = 1. Similarly k_{2,2}(M) = 1. Hence q(M)^{@} ≅ L if and only if p_{1,2}(M) = p_{2,2}(M) = L and k_{1,2}(M) = k_{2,2}(M) = 1.

8. L-enriched bisets

8.1. Notation: Let G and H be finite groups. If U is an (H, G)-biset, and u ∈ U, let (H, G)_{u} denote the stabilizer of u in (H × G), i.e.

(H, G)_{u} = {(h, g) ∈ (H × G) | hu = ug} .

Let H_{u} = k_{1}((H, G)_{u}) denote the stabilizer of u in H, and _{u}G = k_{2}((H, G)_{u}) denote the stabilizer of u in G. Set moreover

q(u) = q((H, G)_{u}) = (H, G)_{u}/(H_{u} ×_{u}G) .

8.2. Definition: Let L be a finite group. For two finite groups G and H, an L-enriched (H, G)-biset is a (H × L, G × L)-biset U such that L ⊑ q(u), for any u ∈ U. A morphism of L-enriched (H, G)-bisets is a morphism of

(H × L, G × L)-bisets.

The disjoint union of two L-enriched (H, G)-bisets is again an Lenriched (H, G)-biset. Let B[L](H, G) denote the Grothendieck group of finite L-enriched (H, G)-bisets for relations given by disjoint union decompositions. The group B[L](H, G) is called the Burnside group of L-enriched

(H, G)-bisets.

8.3. Lemma: Let G, H, L be finite groups, and U be an (H × L, G × L)- biset. Let U^{♯L} denote the set of elements u ∈ U such that L ⊑ q(u). Then U^{♯L} is the largest sub-L-enriched (H, G)-biset of U.

Proof : It suffices to show that U^{♯L} is a sub-(H × L, G × L)-biset of U, for then it is clearly the largest sub-L-enriched (H, G)-biset of U. And this is straightforward, since for any (u, g, h, x, y) ∈ (U × G × H × L × L), if v = (h, y)u(g, x)^{−1}, then

(H × L, G × L)_{v} = ^{((h,y),(g,x))}(H × L, G × L)_{u} ,

and this conjugation induces a group isomorphism q(v) ≅ q(u).

8.4. Lemma: Let G, H, L be finite groups.

1. Let U be an L-enriched (H, G)-biset. If V is a sub-(H ×L, G×L)-biset of U, then V is an L-enriched (H, G)-biset.

2. The group B[L](H, G) has a ℤ-basis consisting of the transitive bisets ((H ×L)×(G×L))/M, where M is a subgroup of ((H ×L)×(G×L)) (up to conjugation) such that L ⊑ q(M).

Proof : (1) This is straightforward.

(2) It follows from (1) that B[L](H, G) has a basis consisting of the isomorphism classes of L-enriched (H, G)-bisets which are transitive (H ×L, G×L)- bisets. These are of the form U = ((H ×L)×(G×L))/M, for some subgroup M of ((H × L) × (G × L)). Now if u is the element ((1, 1), (1, 1))M of U, the group (H × L, G × L)_{u} is equal to M, hence q(u) ≅ q(M).

8.5. Lemma: Let G, H, K, L be finite groups.

1. For an (H, G)-biset U, endow U × L with the (H × L, G × L)-biset structure defined by

∀h ∈ H, ∀g ∈ G, ∀x, y, z ∈ L, ∀u ∈ U, (h, x)(u, y)(g, z) = (hug, xyz) .

Then U × L is an L-enriched (H, G)-biset.

2. In particular, for any finite group G, the identity biset of G × L is an L-enriched (G, G)-biset.

3. If U is an (H, G)-biset and V is a (K, H)-biset, then there is an isomorphism

(V × L) ×_{(H×L)} (U × L) ≅ (V ×_{H} U) × L

of L-enriched (H, G)-bisets.

Proof : (1) For u ∈ U and l ∈ L,

(H × L, G × L)_{(u,l)} = {((h,^{l}x), (g, x)) | hug = u, l ∈ L} ≅ (H, G)_{u} × L .

In particular (H × L)_{(u,l)} = H_{u} × 1 and _{(u,l)}(G × L) = _{u}G × 1, and q((u, l)) ≅ q(u) × L has a (sub)quotient isomorphic to L.

(2) In particular, if H = G and U is the identity (G, G)-biset, then U × L is the identity biset of (G × L).

(3) It is straightforward to check that the maps

\xymatrix{[(v, x), (u, y)] ∈ (V × L) ×_{(H×L)} (U × L) \ar@{|->}[r] & ([v, u], xy) ∈ (V ×_{H} U) × L \\ [(v, 1), (u, l)] ∈ (V × L) ×_{(H×L)} (U × L) & ([v, u], l) ∈ (V ×_{H} U) × L \ar@{|->}[l]}

are well defined isomorphisms of (K × L, G × L)-bisets, inverse to one another.

8.6. Notation: Let G, H, K, L be finite groups. If U is an L-enriched (H, G)-biset and V is an L-enriched (K, H)-biset, let V ×^{L}_{H}U denote the Lenriched (K, G)-biset defined by

V ×^{L}_{H}U = (V ×_{(H×L)} U)^{♯L} .

8.7. Lemma: Let G, H, J, K, L be finite groups.

1. If V is a (K × L, H × L)-biset and U is an (H × L, G × L)-biset, then

(V ×_{(H×L)} U)^{♯L} = V^{♯L}×^{L}_{H}U^{♯L} .

In particular, if V and U are L-enriched bisets, so is V ×^{L}_{H}U.

2. If U and U′ are L-enriched (H, G)-bisets, if V, V′ are L-enriched (K, H)-bisets, then there are isomorphisms

V ×^{L}_{H}(U ⊔ U′) ≅ (V ×^{L}_{H}U) ⊔ (V ×^{L}_{H}U′)
(V ⊔ V′)×^{L}_{H}U ≅ (V ×^{L}_{H}U) ⊔ (V′×^{L}_{H}U)

of L-enriched (K, G)-bisets.

3. If moreover W is an L-enriched (J, K)-biset, then there is a canonical

isomorphism

(W ×^{L}_{K}V)×^{L}_{H}U ≅ W ×^{L}_{K}(V ×^{L}_{H}U)

of L-enriched (J, G)-bisets.

Proof : (1) Denote by [v, u] the image in V ×_{(H×L)}U of a pair (v, u) ∈ (V ×U). By Lemma 2.3.20 of [7],

(K × L, G × L)_{[v,u]} = (K × L, H × L)_{v} ∗ (H × L, G × L)_{u} ,

so by Lemma 2.3.22 of [7], the group q([v, u]) is a subquotient of q(v) and q(u). So if [v, u] ∈ (V ×_{(H×L)} U)^{♯L}, then L is a subquotient of q([v, u]), hence it is a subquotient of q(v) and q(u), that is v ∈ V^{♯L} and u ∈ U^{♯L}. Hence

(V ×_{(H×L)} U)^{♯L} ⊆ (V^{♯L} ×_{(H×L)} U^{♯L})^{♯L} = V^{♯L}×^{L}_{H}U^{♯L} ,

and the reverse inclusion (V^{♯L} ×_{(H×L)} U^{♯L})^{♯L} ⊆ (V ×_{(H×L)} U)^{♯L} is obvious. Hence (V ×_{(H×L)} U)^{♯L} = V^{♯L}×^{L}_{H}U^{♯L}. If V and U are L-enriched bisets, i.e.

if V = V^{♯L} and U = U^{♯L}, this gives (V ×_{(H×L)} U)^{♯L} = V ×^{L}_{H}U, so V ×^{L}_{H}U is

an L-enriched biset.

(2) This is straightforward.

(3) With the above notation, there is a canonical isomorphism

α : (W ×_{(K×L)} V) ×_{(H×L)} U → W ×_{(K×L)} (V ×_{(H×L)} U)

sending [[w, v], u] to [w, [v, u]]. Hence

(W ×^{L}_{K}V)×^{L}_{H}U = ((W ×^{L}_{K}V) ×_{(H×L)} U)^{♯L}
= ((W ×_{(K×L)} V)^{♯L} ×_{(H×L)} U)^{♯L}
= ((W ×_{(K×L)} V) ×_{(H×L)} U)^{♯L} [by (1)]

Similarly

W ×^{L}_{K}(V ×^{L}_{H}U) = (W ×_{(K×L)} (V ×^{L}_{H}U))^{♯L}
= (W ×_{(K×L)} (V ×_{(H×L)} U)^{♯L})^{♯L}
= (W ×_{(K×L)} (V ×_{(H×L)} U))^{♯L} [by (1)] .

Hence α induces an isomorphism (W ×^{L}_{K}V)×^{L}_{H}U ≅ W ×^{L}_{K}(V ×^{L}_{H}U).

8.8. Definition: Let L be a finite group, and R be a commutative ring. The L-enriched biset category RC[L] of finite groups over R is defined as follows:

• The objects of RC[L] are the finite groups.

• For finite groups G and H,

Hom_{RC[L]}(G, H) = R ⊗_{ℤ} B[L](H, G) = RB[L](H, G)

is the R-linear extension of the Burnside group of L-enriched (H, G)- bisets.

• The composition in RC[L] is the R-linear extension of the product (V, U) ↦ V ×^{L}_{H}U defined in 8.6.

• The identity morphism of the group G is (image in RB[L](G, G) of) the identity biset of G × L, viewed as an L-enriched (G, G)-biset.

The category RC[L] is R-linear. An L-enriched biset functor over R is an R-linear functor from RC[L] to R-Mod. The category of L-enriched biset functors over R is denoted by F_{R}[L]. It is an abelian R-linear category.

8.9. Theorem: Let p be a prime number, and R be a commutative ring.

1. If L is an atoric p-group, the category RC_{p}^{♯L} of Definition 7.11 is equivalent to the full subcategory REl_{p}[L] of RC[L] consisting of elementary abelian p-groups.

2. If p ∈ F^{×}, the category F_{p,R} of p-biset functors over R is equivalent to the direct product of the categories Fun_{R}(REl_{p}[L], R-Mod) of R-linear functors from REl_{p}[L] to R-Mod, for L ∈ [At_{p}].

Proof : (1) Let E be an elementary abelian p-group. Then (E × L)^{@} ≅ L, so E × L is an object of RC_{p}^{♯L}. Set I(E) = E × L. If E and F are elementary abelian p-groups, and if U is a finite L-enriched (F, E)-biset, then U is in particular an (F × L, E × L)-biset, an we can consider its image I(U) in the quotient Hom_{RC_{p}^{♯L}}(E × L, F × L) of RB(F × L, E × L). This yields a unique R-linear map RB[L](F, E) → Hom_{RC_{p}^{♯L}}(E × L, F × L), still denoted by I.

We claim that these assignments define a functor I from REl_{p}[L] to RC_{p}^{♯L}: indeed, the identity (E × L, E × L)-biset is clearly mapped to the identity morphism of I(E). Moreover, if G is an elementary abelian p-group, if V is an L-enriched (G, F)-biset and U is an L-enriched (F, E)-biset, it is clear that

I(V ×^{L}_{F} U) = I(V) ∘ I(U) ,

where the right hand side composition is in the category RC_{p}^{♯L}: indeed, the transitive bisets (Q × P)/M with q(M)^{@} ⊏ L appearing in the product V ×_{(F ×L)} U are exactly those vanishing in Hom_{RC_{p}^{♯L}}(I(E), I(F)), by Lemma 7.13. Hence I is an isomorphism

I : RB[L](F, E) → Hom_{RC_{p}^{♯L}}(I(E), I(F)) .

In other words I is a fully faithful functor from REl_{p}[L] to RC_{p}^{♯L}. Moreover, by Proposition 6.8, if P is a finite p-group with P^{@} ≅ L, there exists an elementary abelian p-group E such that P is isomorphic to E × L, hence P is isomorphic to E × L in the category RC_{p}^{♯L}.

It follows that the functor I is fully faithful and essentially surjective, so it is an equivalence of categories.

(2) This is a straightforward consequence of (1), Assertion 5 of Corollary 7.5, and Assertion 3 of Theorem 7.15.

9. The category b̂_{L}F_{p,R}, for an atoric p-group L (p ∈ R^{×})

Let L be a fixed atoric p-group. In this section, we give some detail on the structure of the category b̂_{L}F_{p,R} of p-biset functors invariant by the idempotent b̂_{L}.

We start by straightforward consequences of Theorem 7.15. For a finite p-group P, we denote by Σ_{♯L}(P) the subset of Σ_{L}(P) consisting of sections (X, M) of P such that (X/M)^{@} ≅ L. When G is an R-linear functor from RC_{p}^{♯L} to R-Mod, we can compute R_{Y_{L}}(G) at P by restricting the inverse limit of 7.7 to the subset Σ_{♯L}(P), i.e. by

R_{Y_{L}}(G)(P) = lim ; ⟵_{(X,M)∈Σ_{♯L}(P)} G(X/M) .

9.1. Proposition: [ p ∈ R^{×}] Let L be an atoric p-group. If F is a p-biset functor in b̂_{L}F_{p,R}, and P is a finite p-group, then

F (P) ≅ lim ; ⟵_{(X,M)∈Σ_{♯L}(P)} F (X/M) ,
≅ ⊕_{\begin{matrix}(T,S)∈[M(P)] \\ (T/S)^{@}≅L \end{matrix}} δ_{Φ}F (T/S)^{N_{P} (T,S)/T} .

Proof : The isomorphism F (P) ≅ lim ; ⟵_{(X,M)∈Σ_{♯L}(P)} F (X/M) is Assertion 3 of

Theorem 7.15. The second isomorphism follows from Theorem 5.4, which implies that for (T, S) ∈ M(P)

δ_{Φ}F (T/S)^{N_{P} (T,S)/T} ≅ F (ϵ_{T,S}^{P})(F (P)) .

Moreover F (b_{L}^{P})F (P) = F (P) since F ∈ b̂_{L}F_{p,R}, and

F (ϵ_{T,S}^{P})F (b_{L}^{P}) = F (ϵ_{T,S}^{P}b_{L}^{P}) = 0

unless (T/S)^{@} ≅ L. Thus δ_{Φ}F (T/S)^{N_{P} (T,S)/T} = {0} unless (T/S)^{@} ≅ L, which completes the proof.

The decomposition of the category F_{p,R} of p-biset functors stated in Corollary 7.5 leads to the following natural definition:

9.2. Definition: [ p ∈ R^{×}] Let F be an indecomposable p-biset functor over R. There exists a unique atoric p-group L (up to isomorphism) such that F = b̂_{L}F. The group L is called the vertex of F.

9.3. Remark: It follows in particular from this definition that if F and F′ are indecomposable p-biset functors over R with non-isomorphic vertices, then Ext_{F_{p,R}}^{∗}(F, F′) = {0}.

9.4. Theorem: [ p ∈ R^{×}] Let F be an indecomposable p-biset functor over R and let L be a vertex of F. If Q is a finite p-group such that F (Q) ≠ {0}, but F vanishes on any proper subquotient of Q, then L ≅ Q^{@}.

Proof : Let Q be a finite p-group such that F (Q) ≠ {0} and F (Q′) = {0} for any proper subquotient Q′ of Q. By Proposition 4.6, if (T, S) is a minimal section of Q, then

ϵ_{T,S}^{Q} = \frac{1}{|N_{Q}(T, S)|} ∑_{{X≤T,M ⊴ T \atop S≤M≤Φ(T)≤X≤T}} |X|μ(X, T)μ_{⊴ T} (S, M) Indinf_{X/M}^{Q}∘Defres_{X/M}^{Q} .

Now if X/M is a proper subquotient of Q, i.e. if X ≠ Q or M ≠ 1, then F (X/M) = {0}, and F (Indinf_{X/M}^{Q} ∘ Defres_{X/M}^{Q}) = 0. Hence F (ϵ_{T,S}^{Q}) = 0 unless T = Q and S = 1, and moreover

F (ϵ_{Q,1}^{Q}) = \frac{1}{|Q|}|Q|μ(Q, Q)μ_{⊴ Q}(1, Q)F (Indinf_{Q/1}^{Q}Defres_{Q/1}^{Q}) = Id_{F (Q)} .

If b̂_{L}F = F, then in particular F (b_{L}^{Q}) is equal to the identity map of F (Q). This can only occur if the idempotent ϵ_{Q,1}^{Q} appears in the sum defining b_{L}^{Q}, in other words if (Q/1)^{@} ≅ L, i.e. Q^{@} ≅ L. Conversely, if Q^{@} ≅ L, then F (b_{L}^{Q}) = F (ϵ_{Q,1}^{Q}) = Id_{F (Q)} ≠ 0. It follows that b̂_{L}F ≠ 0, hence b̂_{L}F = F, since F is indecomposable. Hence Q^{@} is (isomorphic to) the vertex of F, as was to be shown.

We assume from now on that R = k is a field. Recall ([7] Chapter 4) that the simple p-biset functors over k are indexed by pairs (Q, V) consisting of a p-group Q and a simple kOut(Q)-module V.

9.5. Corollary: Let k be a field of characteristic different from p.

1. If Q is a finite p-group, and V is a simple kOut(Q)-module, then the vertex of the simple p-biset functor S_{Q,V} is isomorphic to Q^{@}.

2. Let Q (resp. Q′) be a finite p-group, and V (resp. V′) be a simple kOut(Q)-module (resp. a simple kOut(Q′)-module). If Q^{@} ≇ Q^{′@},

then Ext_{F_{p,k}}^{∗}(S_{Q,V} , S_{Q′,V′}) = {0}.

Proof : (1) Indeed Q is a minimal group for S_{Q,V}, so S_{Q,V} (Q) ≠ {0}, but S_{Q,V} vanishes on any proper subquotient of Q.

(2) Follows from (1) and Remark 9.3.

9.6. Definition: Let F be a p-biset functor. A functor S is a subquotient of F (notation S ⊑ F) if there exist subfunctors F_{2} < F_{1} ≤ F such that F_{1}/F_{2} ≅ S. A composition factor of F is a simple subquotient of F.

9.7. Lemma: Let k be a field, and F be a p-biset functor over k.

1. If F is a non zero, then F admits a composition factor.

2. If S is a family of simple p-biset functors over k, there exists a greatest subfunctor of F all composition factors of which belong to S.

Proof : (1) Let P be a finite p-group such that F (P) ≠ {0}. Then F (P) is a kB(P, P)-module. Choose m ∈ F (P) − {0}, and consider the kB(P, P)- submodule M of F (P) generated by m. Since kB(P, P) is finite dimensional over k, the module M is also finite dimensional over k, hence it contains a simple submodule V. By Proposition 3.1 of [8], there exists a simple pbiset functor S such that S(P) ≅ V as kB(P, P)-module. Then S(P) is a subquotient of F (P), so by Proposition 3.5 of [8], there exists a subquotient of F isomorphic to S.

(2) Observe first that if M, N are subfunctors of F, then any composition factor of M + N is a composition factor of M or a composition factor of N: indeed, if S is a composition factor of M + N, let F_{2} < F_{1} ≤ M + N with S ≅ F_{2}/F_{1}, and consider the images F′_{1} and F′_{2} of F_{1} and F_{2}, respectively, in the quotient (M + N)/N ≅ M/(M ∩ N). If F′_{1} ≠ F′_{2}, that is if F_{1} + N ≠ F_{2} + N, then F′_{1}/F′_{2} ≅ (F_{1} + N)/(F_{2} + N) ≅ F_{1}/F_{2} ≅ S is a subquotient of (M + N)/N ≅ M/(M ∩ N), hence S is a subquotient of M. Otherwise F_{1}+N = F_{2}+N, so F_{1} = F_{2}+(F_{1}∩N), hence S ≅ F_{1}/F_{2} ≅ (F_{1}∩N)/(F_{2}∩N) is a subquotient of N. It follows by induction that any subquotient S of a finite sum ∑_{M∈I} M of subfunctors of F is a subquotient of some M ∈ I.

The latter also holds when I is infinite: let Σ = ∑_{M∈I} M be an arbitrary

sum of subfunctors of F, and S be a composition factor of Σ. Let F_{2} < F_{1} be subfunctors of Σ such that S ≅ F_{1}/F_{2}. If P is a p-group such that S(P) ≅ F_{1}(P)/F_{2}(P) ≠ 0, let U be a finite subset of F_{1}(P) such that F_{1}(P)/F_{2}(P) is generated as a kB(P, P)-module by the images of the elements of U (such a set exists because S(P) is finite dimensional over k, for any P). If V is the kB(P, P)-submodule of F_{1}(P) generated by U, then V maps surjectively on the module F_{1}(P)/F_{2}(P), so there is a kB(P, P)-submodule W of V such that V/W ≅ S(P). Now since U is finite, there exists a finite subset J of I such that U ⊆ ∑_{M∈J} M(P). Setting Σ_{1} = ∑_{M∈J} M, it follows that V/W ≅ S(P) is a

subquotient of Σ_{1}(P), so by Proposition 3.5 of [8], there exists a subquotient of Σ_{1} isomorphic to S. By the observation above S is a subquotient of some M ∈ J ⊆ I.

Now let I the set of subfunctors M of F such that all the composition factors of M belong to S, and N = ∑_{M∈I} M. The above discussion shows that

N ∈ I, so N is the greatest element of I.

9.8. Theorem: Let k be a field of characteristic different from p, and L be an atoric p-group. Let F_{p,k}[L] the full subcategory of F_{p,k} consisting of functors whose composition factors all have vertex L, i.e. are all isomorphic to S_{P,V}, for some p-group P such that P^{@} ≅ L, and some simple kOut(P)-

module V.

1. If F is a p-biset functor, then b̂_{L}F_{p,k} is the greatest subfunctor of F which belongs to F_{p,k}[L].

2. In particular b̂_{L}F_{p,k} = F_{p,k}[L].

Proof : (1) Let F be a p-biset functor over k, and let F_{1} = b̂_{L}F. If S is a composition factor of F_{1}, then S = b̂_{L}S, as S is a subquotient of F_{1}. Hence S has vertex L, by Definition 9.2. It follows that F_{1} is contained in the greatest subfunctor F_{2} of F which belongs to F_{p,k}[L] (such a subfunctor exists by Lemma 9.7).

Conversely, we know that F_{2} = ⊕_{Q∈[At_{p}]} b̂_{Q}F_{2}. For Q ∈ [At_{p}], any composi-

tion factor S of b̂_{Q}F_{2} has vertex Q, by Definition 9.2. But S is also a direct summand of F_{2}, so Q ≅ L. It follows that if Q ≇ L, then b̂_{Q}F_{2} has no composition factor, so b̂_{Q}F_{2} = {0}, by Lemma 9.7. In other words F_{2} = b̂_{L}F_{2}, hence F_{2} ≤ F_{1}, and F_{2} = F_{1}, as was to be shown.

(2) Let F be a p-biset functor. Then F ∈ b̂_{L}F_{p,k} if and only if F = b̂_{L}F, i.e. by (1) if and only if all the composition factors of F have vertex L.

9.9. Example: the Burnside functor. Let k be a field of characteristic q ≠ p (q ≥ 0). It was shown in [10] Theorem 8.2 (see also [7] 5.6.9) that the Burnside functor kB is uniserial, hence indecomposable. As kB(1) ≠ 0, the vertex of kB is the trivial group, by Theorem 9.4, thus kB is an object of b̂_{1}F_{p,k} = F_{p,k}[1]. It means that all the composition factors of kB have to be of form S_{Q,V}, where Q^{@} = 1, i.e. Q is elementary abelian. And indeed by [10] Theorem 8.2, the composition factors of kB are all of the form S_{Q,k}, where Q runs through a specific set of elementary abelian p-groups which depends on the order of p modulo q (suitably interpreted when q = 0).

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Serge Bouc LAMFA-CNRS UMR7352 33 rue St Leu, 80039 - Amiens - Cedex 01 France

email:  serge.bouc@u-picardie.fr