A functorial presentation of units of Burnside rings

Serge Bouc

Abstract: Let B^{×} be the biset functor over 𝔽_{2} sending a finite group G to the group B^{×}(G) of units of its Burnside ring B(G), and let B̂^{×} be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from B^{×} in the dual Burnside functor 𝔽̂_{2}B, or equivalently, an explicit set of generators G_{S} of the kernel L of the natural surjection 𝔽_{2}B → B̂^{×}. This yields a two terms projective resolution of B̂^{×}, leading to some information on the extension functors Ext^{1}(−, B^{×}). For a finite group G, this also allows for a description of B^{×}(G) as a limit of groups B^{×}(T/S) over sections (T, S) of G such that T/S is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor B^{×} is not finitely generated, and that its dual B̂^{×} is finitely generated, but not finitely presented. The last result of the paper shows in addition that G_{S} is a minimal set of generators of L, and it follows that the lattice of subfunctors of L is uncountable.

MSC2020: 19A22, 16U60, 20J15.

Keywords: Burnside ring, units, biset functor.

1. Introduction

The Burnside ring B(G) is a fundamental invariant attached to a finite group G. It is a commutative (unital) ring, and most of its structural properties have been described several decades ago, e.g. the prime spectrum by Dress ([10]), or the primitive idempotents of the algebra ℚB(G) by Gluck ([12]) and Yoshida ([19]).

An important missing item in this list is the group of multiplicative units B(G)^{×}. More precisely, it follows from Burnside’s theorem that B(G)^{×} is a finite elementary abelian 2-group, but the rank of this group is known only under additional assumptions on G, by the work of many different people ([13], [14], [20], [18], [6], [1]). A very efficient algorithm for computing this rank has also been obtained in [2]. However, no general formula for the rank of B(G)^{×} is known so far. Let us recall that, according to an observation of tom Dieck ([17], Proposition 1.5.1) based on a theorem of Dress ([10]), Feit-Thompson’s theorem ([11]) is equivalent to the assertion that B^{×}(G) has order 2 if |G| is odd.

It was observed in [6] that the assignment G ↦ B(G)^{×} is a biset functor B^{×} with values in 𝔽_{2}-vector spaces, and that B^{×} embeds in the 𝔽_{2}- dual functor Hom_{ℤ}(B, 𝔽_{2}) of the Burnside functor. In the present paper, we give (Theorem 3.4) a characterization of the image of this embedding ı : B^{×} → Hom_{ℤ}(B, 𝔽_{2}), or equivalently, we describe generators for the kernel L of the natural transposed surjection ȷ : 𝔽_{2}B → Hom_{ℤ}(B^{×}, 𝔽_{2}). This yields a functorial presentation

\xymatrix{0 \ar[r] & L \ar[r] & 𝔽_{2}B \ar[r]^{ȷ} & Hom_{ℤ}(B^{×}, 𝔽_{2}) \ar[r] & 0}

of the (dual) functor Hom_{ℤ}(B^{×}, 𝔽_{2}). Unfortunately, the generators we obtain for L are not linearly independent, so at least at this stage, the previous exact sequence doesn’t provide any obvious formula for the 𝔽_{2}-dimension of the evaluations of the functor B^{×}.

It still admits some interesting consequences, that we develop in the three last sections. Section 4 is concerned with the extension groups Ext^{1}(M, B^{×}) for some biset functors M over 𝔽_{2}. In Section 5, we give a sectional characterization of B^{×}(G) as a limit of groups B^{×}(T/S), where (T, S) runs through sections (T, S) of G such that T/S is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Finally, in Section 6, it is shown that the biset functor B^{×} is not finitely generated, and that its dual is finitely generated (by a single element), but not finitely presented. We also show that the generating set we obtain for the above functor L is minimal, and that the lattice of subfunctors of L is uncountable. These last results may be seen as another sign of the difficulty of determining B^{×}(G) for an arbitrary finite group G, at least with our present methods.

2. Review of Burnside rings and biset functors

2.1. Burnside rings. We quickly recall first some basic definitions on Burnside rings. Missing details can be found in [3].

Let G be a finite group. The Burnside ring B(G) of G is the Grothendieck ring of the category of finite (left) G-sets, for relations given by disjoint union decompositions. In other words B(G) is the quotient of the free abelian group on the set of isomorphism classes of finite G-sets by the subgroup generated by all the elements of the form [X ⊔ Y] − [X] − [Y], where [X] denote the isomorphism class of the G-set X.

The multiplication in B(G) is induced by the cartesian product of G-sets. In other words [X][Y] = [X × Y] for any two finite G-set X and Y. The identity element is the (isomorphism class of a) G-set of cardinality 1.

The additive group B(G) is free with basis the set of isomorphism classes of transitive G-sets. Each transitive G-set is isomorphic to a G-set of the form G/H, where H is a subgroup of G. The transitive G sets G/H and G/K are isomorphic if and only if the subgroups H and K of G are conjugate. For sake of simplicity, we denote by G/H instead of [G/H] the image in B(G) of the transitive G-set G/H. With this abuse of notation, the abelian group B(G) has a basis consisting of the elements G/H, where H runs through a set of representatives of conjugacy classes of subgroups of G.

The commutative ℚ-algebra ℚB(G) = ℚ⊗_{ℤ} B(G) is split semisimple. Its primitive idempotents (see [12] or [19]) are indexed by the subgroup of G, up to conjugation. The idempotent e_{H}^{G} indexed by H ≤ G is equal to

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

where μ is the Möbius function of the poset of subgroups of G. Its defining property is that

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

Here the map α ∈ ℚB(G) ↦ |α^{K}| ∈ ℚ is the linear form sending (the class of a finite) G-set X to the cardinality |X^{K}| of the set X^{K}| of K-fixed points in X. 2.2. Biset functors. (see [7] for details) For finite groups G and H, an (H, G)-biset is a H × G^{op}-set, i.e. a set U endowed with a left H-action and a right G-action which commute. The Burnside group B(H × G^{op}) of (finite) (H, G)-bisets is denoted by B(H, G).

Let C denote the following category:

• The objects of C are all the finite groups.

• For finite groups G and H, the set of morphisms Hom_{C}(G, H) is equal to B(H, G).

• The composition of morphisms in C is defined by bilinearity from the standard “tensor product” (also called composition) of bisets: for finite groups G, H and K, for a (K, H)-biset V and an (H, G)-biset U, set

V ×_{H} U = (V × U)/H ,

where H acts on the right on (V × U) by (v, u)h = (vh, h^{−1}u). Then V ×_{H} U is a (K, G)-biset in the obvious way.

• The identity element of 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 all ℤ-modules. Biset functors, together with natural transformations between them, form an abelian category F. More generally, for a commutative (unital) ring k, one can consider the k-linearization kC of C, i.e. the category with the same objects, and such that

Hom_{kC}(G, H) = k ⊗_{ℤ} Hom_{C}(G, H) = kB(H, G) ,

the composition in kC being defined as the k-bilinear extension of the composition in C. A biset functor over k is a k-linear functor from kC to the category k-Mod of all k-modules. These functors, together with natural transformations between them, form an abelian category F_{k}.

There is a one to one parametrization (H, V) ↦ S_{H,V} of simple bisets functors over k (up to isomorphism of functors) by pairs (H, V) of a finite group H and a simple kOut(H)-module V (up to isomorphism of such pairs). If G is a finite group, and if S_{H,V} (G) ≠ {0}, then H is isomorphic to a subquotient of G.

Let F be a biset functor, and let G, H be finite groups. We define the opposite biset U^{op} as the (G, H)-biset U, where the action of (g, h) ∈ G × H on u ∈ U is defined by

g · u · h = h^{−1}ug^{−1} .

This definition extends uniquely to a k-linear map α ↦ α^{op} from kB(H, G) to kB(G, H).

For α ∈ kB(H, G) and m ∈ F (G), we denote by α(m) or simply α m the image of m by the map F (α). In particular, when U is a finite (H, G)-biset, we set U m = F ([U])(m).

When F is a biset functor over k, and M is a k-module, the M-dual of F is the functor Hom_{k}(F, M) defined by

\{ \begin{matrix}∀G, Hom_{k}(F, M)(G) = Hom_{k}(F (G), M) \\ ∀G, H, ∀α ∈ kB(H, G), Hom_{k}(F, M)(α) = ^{t}F (α^{op}) . \end{matrix}

When p is a prime number, let kC_{p} denote the full subcategory of kC consisting of finite p-groups. A k-linear functor from kC_{p} to k-Mod is called a p-biset functor (over k). The abelian category of p-biset functors over k is denoted by F_{p,k}.

More generally, if T is a class of finite groups closed under taking subquotients, one can consider the full subcategory kT of kC consisting of groups in T . The abelian category of k-linear functors from kT to k-Mod is denoted by F_{T ,k}.

Let F be an object of F_{T,k}. An element of F is a pair (R, v), where R ∈ T and v ∈ F (R). For a set G of elements of F, the subfunctor ⟨G⟩ of F generated by G is defined as the intersection of all subfunctors L of F such that v ∈ L(R) for all (R, v) ∈ G. Its evaluation at a group H ∈ T is given by

⟨G⟩(H) = ∑_{\begin{matrix}(R,v)∈G \\ α∈kB(H,R) \end{matrix}} F (α)(v) .

The functor F is finitely generated if there exists a finite set G of elements of F such that ⟨G⟩ = F. The functor F is finitely presented if there exists an exact sequence Q → P → F → 0 in F_{T ,k}, where P and Q are finitely generated projective functors.

Let F be an object of F_{T,k}, and H ∈ T . The residue (or Brauer quotient) \overline{F} (H) of F at H is the k-module defined by

\overline{F} (H) = F (H)/ ∑_{\begin{matrix}T ∈T \\ |T |<|H| \\ α∈kB(H,T) \end{matrix}} F (α)(F (T)) .

2.3. Elementary bisets. Bisets of one of the following forms are called elementary bisets:

• Let H be a subgroup of a finite group G. The set G, viewed as a (G, H)-biset (by multiplication), is called induction (from H to G), and denoted by Ind_{H}^{G}. The set G, viewed as an (H, G)-biset (by multiplication), is called restriction (from G to H), and denoted by Res_{H}^{G}.

• Let N be a normal subgroup of a finite group G. The set G/N, viewed as a (G, G/N)-biset (by multiplication on the right, and projection followed by multiplication on the left), is called inflation (from G/N to G) and denoted by Inf_{G/N}^{G}. The set G/N, viewed as a (G/N, G)-biset (by multiplication on the left, and projection followed by multiplication on the right), is called deflation (from G to G/N), and denoted by Def_{G/N}^{G}.

• Let φ : G → G′ be an isomorphism of groups. The set G′, viewed as a (G′, G)-biset (by multiplication on the left, and multiplication by the image under φ on the right), is called transport by isomorphism (by φ), and denoted by Iso(φ) or Iso_{G}^{G′} if φ is clear from the context.

In addition to these elementary bisets, it is convenient to distinguish the following ones, when (T, S) is a section of a finite group G, i.e. a pair of subgroups of G with S ⊴ T:

• The set G/S, viewed as a (G, T/S)-biset, is called induction-inflation (from T/S to G), and denoted by Indinf_{T/S}^{G}. It is isomorphic to the composition Ind_{T}^{G} ×_{T} Inf_{T/S}^{T}.

• The set S\G, viewed as a (T/S, G)-biset, is called deflation-restriction (from G to T/S), and denoted by Defres_{T/S}^{G}. It is isomorphic to the composition Def_{T/S}^{T} ×_{T} Res_{T}^{G}.

For finite groups G and H, any transitive (H, G)-biset is isomorphic to a biset of the form (H × G)/X, where X is a subgroup of H × G, and the biset structure is given by h · (a, b)X · g = (ha, g^{−1}b)X for h, a in H and g, b in G. We set

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

With this notation, we have k_{1}(X) ⊴ p_{1}(X) and k_{2}(X) ⊴ p_{2}(X), and there is a canonical group isomorphism f : p_{2}(X)/k_{2}(X) → p_{1}(X)/k_{1}(X) sending gk_{2}(X) to hk_{1}(X) if (h, g) ∈ X. Moreover (see [7], Lemma 2.3.26), there is an isomorphism of (H, G)-bisets

(2.4)  (H × G)/X ≅ Indinf_{p_{1}(X)/k_{1}(X)}^{H} Iso(f) Defres_{p_{2}(X)/k_{2}(X)}^{G}

where the concatenation on the right hand side denotes the composition of bisets. In other words, any transitive biset is isomorphic to a composition of elementary bisets.

It follows that if T is a class of finite groups closed under taking subquotients, if F ∈ F_{T ,k} and H ∈ T , then

\overline{F} (H) = F (H)/ ∑_{\begin{matrix}A ⊴ B≤H \\ (B,A)≠(H,1) \end{matrix}} Indinf_{B/A}^{H}F (B/A)

= F (H)/( ∑_{\begin{matrix}B<H \\ B maximal \end{matrix}} Ind_{B}^{H}F (B) + ∑_{\begin{matrix}1<A ⊴ H \\ A minimal \end{matrix}} Inf_{H/A}^{H}F (H/A)) .

2.5. Faithful elements. (see [7], Section 6.3) Let F be a biset functor over a commutative ring k. For a finite group G, let

∂F (G) = {u ∈ F (G) | ∀1 ≠ N ⊴ G, Def_{G/N}^{G}u = 0} .

The k-submodule ∂F (G) is called the submodule of faithful elements of F (G). If is always a direct summand of F (G). More precisely, the element

f_{1}^{G} = ∑_{N ⊴ G} μ_{⊴ G}(1, N) [(G × G)/L]

of kB(G, G) is an idempotent endomorphism of G in the category kC, and one can show that ∂F (G) = F (f_{1}^{G})(F (G)). 2.6. Genetic bases of p-groups. ([7] Definition 6.4.3, Lemma 9.5.2 and Theorem 9.6.1) Let p be a prime number, and P be a finite p-group. For a subgroup Q of P, let Z_{P} (Q) ≥ Q denote the subgroup of N_{P} (Q) defined by

Z_{P} (Q)/Q = Z(N_{P} (Q)/Q) .

The subgroup Q is a genetic subgroup of P if the two following properties hold:

• The group N_{P} (Q)/Q is a Roquette p-group, i.e. it has normal p-rank 1. Recall that the Roquette p-groups of order p^{n} are the cyclic p-groups C_{p^{n}} (n ≥ 0), and in addition when p = 2, the generalized quaternion groups Q_{2^{n}} (n ≥ 3), the dihedral groups D_{2^{n}} (n ≥ 4) and the semidihedral groups SD_{2^{n}} (n ≥ 4).

• For any x ∈ G, the intersection ^{x}Q ∩ Z_{P} (Q) is contained in Q if and only if ^{x}Q = Q.

For two genetic subgroups Q and R of P, write

Q _{P} R ⇔ ∃x ∈ P such that ^{x}Q ∩ Z_{P} (R) ≤ R and R^{x} ∩ Z_{P} (Q) ≤ Q .

One can show that this defines an equivalence relation on the set of genetic subgroups of P. A genetic basis of P is a set of representatives of equivalence classes of genetic subgroups of P for the relation _{P} . 2.7. Rational p-biset functors. ([7] Theorem 10.1.1, Definition 10.1.3 and Theorem 10.1.5) Let p be a prime number, and F be a p-biset functor over a commutative ring k. If P is a finite p-group and B is a genetic basis of P, the map

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

is split injective, with left inverse

D_{B} : ⨁_{Q∈B} f_{1}^{N_{P} (Q)/Q} ∘ Defres_{N_{P} (Q)/Q}^{P} : F (P) → ⨁_{Q∈B} ∂F (N_{P} (Q)/Q) .

The functor F is called rational if for any finite p-group P, the map I_{B} is an isomorphism for some - equivalently for any - genetic basis B of P. Rational p-biset functors form a Serre subcategory of F_{p,k}. Moreover, for any k-module M, the M-dual Hom_{k}(F, M) of a rational p-biset functor F is rational.

3. The main theorem

3.1. Notation: Let R denote the class of finite groups which are isomorphic to one of the following groups:

• A cyclic group C_{p} of odd prime order p.

• A cyclic group C_{4} of order 4.

• An elementary abelian group (C_{2})^{2} of order 4.

• A dihedral 2-group D_{2^{n}} of order 2^{n} ≥ 8.

• A semidihedral 2-group SD_{2^{n}} of order 2^{n} ≥ 16.

For each group R in R, let ε_{R} denote the element of B(R) defined by:

• ε_{R} = R/1 − R/R if R ≅ C_{p}.

• ε_{R} = R/1 − R/S if R ≅ C_{4}, where S is the unique subgroup of order 2

of R.

• ε_{R} = R/1 − (R/A + R/B + R/C) + 2R/R if R ≅ (C_{2})^{2}, where A, B, C

are the the three subgroups of order 2 of R.

• ε_{R} = (R/I − R/IZ) − (R/J − R/JZ), if R ≅ D_{2^{n}}, where I and J are non conjugate non central subgroups of order 2 of R, and Z is the center of R.

• ε_{R} = R/I − R/IZ if R ≅ SD_{2^{n}}, where I is a non central subgroup of order 2 of R, and Z is the center of R.

Let moreover \overline{ε}_{R} denote the image of ε_{R} in 𝔽_{2}B(G).

3.2. Remark: When R is cyclic, elementary abelian of order 4, or dihedral, the elements ε_{R} ∈ B(R) already appear in [9] (Notation 3.6) and [5] (Corollary 6.5 and Notation 6.9, where ε_{D_{2^{n}}} is denoted δ_{n}). Also observe that ε = f_{1}^{R} R/1 ∈ B(R) if R is neither dihedral nor semidihedral. Moreover ε_{R} = f_{1}^{R} (R/I − R/J) if R ≅ D_{2^{n}}, and ε_{R} = f_{1}^{R} R/I when R ≅ SD_{2^{n}}. Hence f_{1}^{R} ε_{R} = ε_{R} for any R ∈ R.

3.3. Notation: Let 𝔽_{2,+} denote the additive group of 𝔽_{2}, and

s ∈ {±1} ↦ s_{+} ∈ 𝔽_{2,+}

be the group isomorphism is defined by (−1)_{+} = 1_{𝔽_{2}} and (+1)_{+} = 0_{𝔽_{2}}. Let t ∈ 𝔽_{2,+} ↦ t_{×} ∈ {±1} denote the inverse group isomorphism.

Recall that for a finite group G, any unit element u ∈ B^{×}(G) in the Burnside ring of G defines a linear form ı_{G}(u) on B(G), with values in 𝔽_{2}, by

ı_{G}(u)(G/H) = |u^{H}|_{+} .

When G runs through finite groups, these maps ı_{G} form an injective morphism of biset functors ı : B^{×} → Hom_{ℤ}(B, 𝔽_{2}) ([6], Proposition 7.2).

Moreover, by a theorem of Yoshida ([20], Proposition 6.5), a linear form φ : B(G) → 𝔽_{2} lies in the image of ı_{G} if and only if for any subgroup H of G, the map

φ̃_{H} : x ∈ N_{G}(H)/H ↦ φ(G/H⟨x⟩) − φ(G/H)

is a group homomorphism from N_{G}(H)/H to 𝔽_{2,+}.

3.4. Theorem: Let G be a finite group, and φ ∈ Hom_{ℤ}(B(G), 𝔽_{2}). The following assertions are equivalent:

1. φ lies in the image of the natural map ı_{G} : B^{×}(G) → Hom_{ℤ}(B(G), 𝔽_{2}).

2. (Defres_{T/S}^{G}φ)(ε_{T/S}) = 0_{𝔽_{2}} for every section (T, S) of G such that

T/S ∈ R.

Proof: 1 ⇒ 2 We check that if φ ∈ Im ı_{G}, then (Defres_{T/S}^{G}φ)(ε_{T/S}) = 0_{𝔽_{2}} for each section (T, S) of G such that T/S ∈ R. Since Defres_{T/S}^{G}φ ∈ Im ı_{T/S} as ı is a morphism of biset functors, we can assume that G ∈ R, and check that φ(ε_{G}) = 0_{𝔽_{2}} when φ = ı_{G}(u) for some u ∈ B^{×}(G). In this case φ(G/H) = |u^{H}|_{+} for any subgroup H of G.

But since f_{1}^{G} ε_{G} = ε_{G} by Remark 3.2, and since f_{1}^{G} = (f_{1}^{G})^{op}, we have

φ(ε_{G}) = i_{G}(u)(f_{1}^{G} ε_{G})
= (f_{1}^{G}i_{G}(u))(ε_{G})
= i_{G}(f_{1}^{G} u)(ε_{G}) .

Moreover f_{1}^{G} u ∈ ∂B^{×}(G) by definition. But ∂B^{×}(G) = {0} if |Z(G)| > 2 by Lemma 6.8 of [6], and ∂B^{×}(G) also vanishes if G is generalized quaternion or semidihedral, by Corollary 6.10 there. It follows that ∂B^{×}(G) = {0} if G ∈ R, unless perhaps if G ≅ D_{2^{n}} for some n ≥ 3. In particular φ(ε_{G}) = 0_{𝔽_{2}} if G ∈ R, unless G is a dihedral 2-group of order at least 8.

Now by Corollary 6.12 of [6] (and its proof), if G ≅ D_{2^{n}} for n ≥ 3, the group ∂B^{×}(G) has order 2, generated by the element

υ_{G} = 1 − 2(e_{I}^{G} + e_{J}^{G}) = G/G + G/1 − (G/I + G/J)

of B(G). Moreover ε_{G} = f_{1}^{G} (G/I − G/J) by Remark 3.2. Thus

φ(ε_{G}) = i_{G}(u)(f_{1}^{G} (G/I − G/J))
= (f_{1}^{G}i_{G}(u))(G/I − G/J)
= i_{G}(f_{1}^{G} u)(G/I − G/J)
= |(f_{1}^{G} u)^{I}|_{+} − |(f_{1}^{G} u)^{J}|_{+} .

We can moreover assume that f_{1}^{G} u is the generator υ_{G} of ∂B^{×}(G), i.e. that f_{1}^{G} u = 1 − 2(e_{I}^{G} + e_{J}^{G}). Then clearly |(f_{1}^{G} u)^{I}| = |(f_{1}^{G} u)^{J}|, so φ(ε_{G}) = 0_{𝔽_{2}} also in this case, as was to be shown.

2 ⇒ 1 We prove the converse by induction on the order of G: we consider φ ∈ Hom_{ℤ}(B(G), 𝔽_{2}) such that (Defres_{T/S}^{G}φ)(ε_{T/S}) = 0_{𝔽_{2}} for every section (T, S) of G such that T/S ∈ R. Assuming the result holds for groups of order smaller than |G|, by transitivity of deflation-restrictions, we can assume that Defres_{T/S}^{G}φ lies in the image of B^{×}(T/S) for any section (T, S) such that |T/S| < |G|, in other words any section (T, S) different from (G, 1). By Yoshida’ theorem above, it follows that the map φ̃_{H} is a group homomorphism for any H ≠ 1. So proving that φ lies in the image of ı_{G} amounts to proving that the map φ̃_{1} : G → 𝔽_{2} is a group homomorphism.

Equivalently, all we have to show is that the map

ψ : x ∈ G ↦ φ̃_{1}(x)_{×} ∈ {±1}

is a group homomorphism. Now ψ is a central function on G, with values in {±1}, and moreover ψ(1) = 1 since φ̃_{1}(1) = 0_{𝔽_{2}}. Hence all we have to show is that ψ is a generalized character of G. Indeed in this case, there exist integers n_{χ} ∈ ℤ, indexed by the irreducible complex characters of G such that ψ = ∑_{χ∈Irr(G)} n_{χ}χ. Moreover

∑_{χ∈Irr(G)} n_{χ}^{2} = \frac{1}{|G|} ∑_{x∈G} |ψ(x)|^{2} = 1 ,

so n_{χ} = 0, except for a single irreducible character χ for which n_{χ} = ±1. Since ψ(1) = 1, it follows that ψ = χ, so ψ is a character of G, of degree 1, hence a group homomorphism from G to {±1}.

Now by Brauer’s induction theorem, the map ψ is a generalized character of G if and only if its restriction to any Brauer elementary subgroup H of G is a generalized character of H. But for x ∈ H

(Res_{H}^{G}ψ)(x) = φ̃_{1}(x)_{×} = φ(G/⟨x⟩)_{×}/φ(G/1)_{×}
= φ(Ind_{H}^{G}(H/⟨x⟩ − H/1))_{×}
= ((Res_{H}^{G}φ)(H/⟨x⟩ − H/1))_{×} .

It follows that if H is a proper subgroup of G, since Res_{H}^{G}φ ∈ Im ı_{H}, the map Res_{H}^{G}ψ is a generalized character of H. In other words we can assume that G itself is Brauer elementary.

Then there exists a prime number p such that G = C ×P, where P is a pgroup, and C is a p′-cyclic group. In particular G is nilpotent, so G = Q×R, where Q is a 2-group and R is nilpotent of odd order. Then any subgroup T of G is equal to A × B, for some subgroup A of Q and some subgroup B of R. If B is non trivial, there exists a normal subgroup C of B of (odd) prime index l. Set S = A × C. Then (T, S) is a section of G, and T/S ≅ C_{l}. Thus

(Defres_{T/S}^{G}φ)(ε_{T/S}) = 0_{𝔽_{2}} = φ(Indinf_{T/S}^{G}(T/S−T/T)) = φ(G/S)−φ(G/T) .

It follows by induction that φ(G/(A×B)) = φ(G/(A×1)) for any subgroup A of Q and any subgroup B of R. In particular, for x ∈ Q and y ∈ R, the subgroup ⟨(x, y)⟩ of G = Q × R is equal to ⟨x⟩ × ⟨y⟩, and

ψ(x, y) = φ(G/(⟨x⟩ × ⟨y⟩))_{×}/φ(G/1)_{×}
= φ(G/(⟨x⟩ × 1⟩))_{×}/φ(G/1)_{×}
= ψ(x, 1) .

In other words it is enough to prove that the restriction of ψ to Q × 1 is a generalized character. Equivalently, all we have to do is consider the case where G = Q × 1, i.e. we can assume that G is a 2-group.

We consider the linearization morphism B(G) → R_{ℚ}(G), which is surjective by the Ritter-Segal theorem ([15], [16], [4]), and hence fits in a short exact sequence

\xymatrix{0 \ar[r] & K(G) \ar[r] & B(G) \ar[r] & R_{ℚ}(G) \ar[r] & 0} .

By Corollary 6.16 of [5], the kernel K(G) of the linearization morphism is linearly generated by the elements of the form Indinf_{T/S}^{G}ε_{T/S}, where (T, S) is a section of G such that T/S is elementary abelian of order 4 or dihedral of order at least 8. Since

φ(Indinf_{T/S}^{G}ε_{T/S}) = (Defres_{T/S}^{G}φ)(ε_{T/S}) = 0_{𝔽_{2}} ,

the linear form φ vanishes on K(G), so we can consider φ as an element of Hom_{ℤ}(R_{ℚ}(G), 𝔽_{2}).

Now the functor R_{ℚ} is a rational 2-biset functor (see Definition 10.1.3 and Proposition 9.6.12 of [7]), so its 𝔽_{2}-dual Hom_{ℤ}(R_{ℚ}, 𝔽_{2}) is also rational. It follows that if G is a genetic basis of G, we have

(3.5)  φ = ∑_{S∈G} Indinf_{N_{G}(S)/S}^{G}f_{1}^{N_{G}(S)/S}Defres_{N_{G}(S)/S}^{G} φ .

For each S ∈ G, the group N_{G}(S)/S is a Roquette 2-group, i.e. it is cyclic, generalized quaternion of order at least 8, dihedral or semidihedral of order at least 16. Recall moreover that Im ı_{R} = Hom_{ℤ}(R_{ℚ}(R), 𝔽_{2}) if R is trivial, of order 2, or dihedral of order at least 16. If we show that f_{1}^{N_{G}(S)/S}Defres_{N_{G}(S)/S}^{G} φ = 0 when N_{G}(S)/S is not one of these Roquette 2-groups, then Equation 3.5 shows that φ ∈ Im ı_{G}, and we are done.

So all we have to do is to show that f_{1}^{N_{G}(S)/S}Defres_{N_{G}(S)/S}^{G} φ = 0 when R = N_{G}(S)/S is cyclic of order at least 4, generalized quaternion, or semidihedral. In each of these cases R has a unique central subgroup Z of order 2, and f_{1}^{R} = R/1 − R/Z ∈ B(R, R).

Let X be any subgroup of R. Then

(f_{1}^{N_{G}(S)/S}Defres_{N_{G}(S)/S}^{G} φ)(R/X) = (Defres_{N_{G}(S)/S}^{G} φ)(f_{1}^{R} ×_{R} (R/X))

= (Defres_{N_{G}(S)/S}^{G} φ)(R/X − R/ZX) .

This is zero if Z ≤ X, so we can assume Z ≰ X, that is Z ∩ X = 1. If R is cyclic or generalized quaternion, this implies X = 1. Then

(f_{1}^{R}Defres_{R}^{G} φ)(R/X) = (Defres_{R}^{G} φ)(R/1 − R/Z)
= (Defres_{R}^{G} φ)(Ind_{D}^{R}(D/1 − D/Z))
= (Res_{D}^{R}Defres_{R}^{G}φ)(D/1 − D/Z)
= (Defres_{D}^{G}φ)(ε_{D}) ,

where D is a cyclic subgroup of order 4 of R. By assumption, this is equal to 0_{𝔽_{2}}, hence f_{1}^{N_{G}(S)/S}Defres_{N_{G}(S)/S}^{G} φ = 0 in these cases, as was to be shown.

Now if R ≅ SD_{2^{n}} with n ≥ 4, the same argument shows that

(f_{1}^{R}Defres_{R}^{G} φ)(R/X) = 0_{𝔽_{2}}

if Z ≤ X or X = 1. Up to conjugation, the only remaining subgroup X of G is X = I, and then

(f_{1}^{R}Defres_{R}^{G} φ)(R/X) = (Defres_{R}^{G} φ)(R/I − R/IZ)

= (Defres_{R}^{G} φ)(ε_{R}) = 0_{𝔽_{2}} .

This completes the proof of Theorem 3.4.

3.6. Corollary: Let G be a finite group. Then the kernel of the natural map ȷ_{G} : 𝔽_{2}B(G) → Hom_{𝔽_{2}}(B^{×}(G), 𝔽_{2}) is the group generated by the elements Indinf_{T/S}^{G}\overline{ε}_{T/S}, for all sections (T, S) of G such that T/S ∈ R.

Proof: This is just a dual reformulation of Theorem 3.4, since the natural map ȷ_{G} : 𝔽_{2}B(G) → Hom_{𝔽_{2}}(B^{×}(G), 𝔽_{2}) is the transposed map of ı_{G}, up to the identification of 𝔽_{2}B(G) with the 𝔽_{2}-bidual of B(G).

For a finite group R, let B_{R} = Hom_{C}(R, −) = B(−, R) denote the representable biset functor defined by R, and let 𝔽_{2}B = 𝔽_{2}⊗_{ℤ}B_{R} ≅ Hom_{𝔽_{2}C}(R, −).

3.7. Corollary: Let S be the set of finite groups defined by

S = {C_{p} | p odd prime} ∪ {C_{4}} ∪ {SD_{2^{n}} | n ≥ 4} ,

(by which we mean that S contains exactly one group of order p for each odd prime p, one cyclic group of order 4, and one semidihedral group of each order 2^{n} ≥ 16), and let G_{S} = \{(R, \overline{ε}_{R}) | R ∈ S\} be the associated set of elements of 𝔽_{2}B.

1. Let L be the biset subfunctor of 𝔽_{2}B generated by G_{S}. Then there is an exact sequence of biset functors

\xymatrix{0 \ar[r] & L \ar[r] & 𝔽_{2}B \ar[r]^{ȷ} & Hom_{𝔽_{2}}(B^{×}, 𝔽_{2}) \ar[r] & 0 .}

2. When R ∈ R, let d_{R} : 𝔽_{2}B_{R} → 𝔽_{2}B be the morphism of biset functors induced by adjunction from \overline{ε}_{R} ∈ 𝔽_{2}B(R). Then the sequence

\xymatrix{⨁_{R∈S} 𝔽_{2}B_{R} \ar[r]^{d} & 𝔽_{2}B \ar[r]^{ȷ} & Hom_{𝔽_{2}}(B^{×}, 𝔽_{2}) \ar[r] & 0}

is exact, where d is the sum of all the maps d_{R}, for R ∈ S.

3. For R ∈ R, let 𝔽_{2}B_{R}f_{1}^{R} be the direct summand of 𝔽_{2}B_{R} equal to the image of the idempotent endomorphism f_{1}^{R} ∈ B(R, R) = End_{F}(B_{R}). Then the morphism d_{R} : 𝔽_{2}B_{R} → 𝔽_{2}B factors through a morphism d̂_{R} : 𝔽_{2}B_{R}f_{1}^{R} → 𝔽_{2}B, and this yields an exact sequence of functors

\xymatrix{⨁_{R∈S} 𝔽_{2}B_{R}f_{1}^{R} \ar[r]^{d̂} & 𝔽_{2}B \ar[r]^{ȷ} & Hom_{𝔽_{2}}(B^{×}, 𝔽_{2}) \ar[r] & 0 .}

where d̂ is the sum of the morphisms d̂_{R}, for R ∈ S.

Proof: For Assertion 1, all we need to show is that the element \overline{ε}_{R} belongs to L(R), when R is elementary abelian of order 4 or dihedral of order at least 8. But one checks easily that Res_{D_{2^{n−1}}}^{SD_{2^{n}}} \overline{ε}_{SD_{2^{n}}} = \overline{ε}_{D_{2^{n−1}}} for n ≥ 4, and that Res_{(C_{2})^{2}}^{D_{8}} \overline{ε}_{D_{8}} = \overline{ε}_{(C_{2})^{2}}. Now Assertion 2 follows from the fact that L is equal to the image of d, and Assertion 3 from the fact that f_{1}^{R}ε_{R} = ε_{R} by Remark 3.2.

4. Extensions

4.1. Notation: Let T denote the class of finite groups which are subquotient of some group in S, that is the class of groups isomorphic to a group in the following set

\{ \begin{matrix}\{C_{p} | p odd prime\} ∪ \{C_{2^{n}} | n ≥ 0\} ∪ \{Q_{2^{n}} | n ≥ 3\} \\ ∪ \{(C_{2})^{2}\} ∪ \{D_{2^{n}} | n ≥ 3\} ∪ \{SD_{2^{n}} | n ≥ 4\} \end{matrix}

In other words T is the class of groups which are cyclic of odd prime order, elementary abelian of order 4, dihedral of order 8, or a Roquette 2-group.

4.2. Proposition:

1. Set B̂^{×} = Hom_{𝔽_{2}}(B^{×}, 𝔽_{2}). If S be a biset functor over 𝔽_{2}, such that S(1) = {0}, then Ext_{F_{𝔽_{2}}}^{1}(B̂^{×}, S) is isomorphic to a subspace of ∏_{R∈S} ∂S(R), where S is the set defined in Corollary 3.7.

2. Let H be a finite group. If there exists a simple 𝔽_{2}Out(H)-module V such that Ext_{F_{𝔽_{2}}}^{1}(S_{H,V} , B^{×}) ≠ {0}, then H ∈ T .

Proof: The functor L of Corollary 3.7 is equal to Im d = Imd̂= Ker ȷ, so we have an exact sequence

\xymatrix{0 \ar[r] & L \ar[r] & 𝔽_{2}B \ar[r]^{ȷ} & B̂^{×} \ar[r] & 0 ,}

and a surjective morphism \xymatrix{⨁_{R∈S} 𝔽_{2}B_{R}f_{1}^{R} \ar@{->>}[r]^{d̂} & L} . Let M be any biset func-

tor over 𝔽_{2}. Applying first the functor Hom_{F_{𝔽_{2}}}(−, M) to the short exact sequence, we get the beginning of a long exact sequence

\xymatrix{0 \ar[r] & Hom(B̂^{×}, M) \ar[r] & Hom(𝔽_{2}B, M) \ar[r] & Hom(L, M) \ar[r] & Ext^{1}(B̂^{×}, M) \ar[r] & 0 ,}

where Hom and Ext^{1} are taken in the category F_{𝔽_{2}}. Indeed Ext^{1}(𝔽_{2}B, M) is equal to zero as 𝔽_{2}B = Hom_{F_{𝔽_{2}}}(1, −) is representable, hence projective in F_{𝔽_{2}}. Moreover Hom(𝔽_{2}B, M) ≅ M(1), so we get an exact sequence

(4.3) \xymatrix{0 \ar[r] & Hom(B̂^{×}, M) \ar[r] & M(1) \ar[r] & Hom(L, M) \ar[r] & Ext^{1}(B̂^{×}, M) \ar[r] & 0 .}

In the case M = S, since S(1) = {0}, this gives an isomorphism Hom(L, S) ≅ Ext^{1}(B̂^{×}, S). On the other hand, since L is a quotient of ⨁_{R∈S} 𝔽_{2}B_{R}f_{1}^{R}, we get

an inclusion

\xymatrix{Hom(L, S) \ar@{^{(}->}[r] & Hom( ⨁_{R∈S} 𝔽_{2}B_{R}f_{1}^{R}, S) ≅ ∏_{R∈S} Hom(𝔽_{2}B_{R}f_{1}^{R}, S) .}

Moreover Hom(𝔽_{2}B_{R}f_{1}^{R}, S) ≅ f_{1}^{R}S(R) = ∂S(R). This proves Assertion 1.

For Assertion 2, we take M = S_{H,W} in the exact sequence 4.3, where W = V^{∗} is the dual module. If H ≠ 1, then S_{H,W} (1) = {0}. And if H = 1, then W = 𝔽_{2}, and by duality

Hom(B̂^{×}, S_{1,𝔽_{2}}) ≅ Hom(S_{1,𝔽_{2}}, B^{×}) ≅ 𝔽_{2} ≅ S_{1,𝔽_{2}}(1) .

In both cases, we get an isomorphism

Hom(L, S_{H,W}) ≅ Ext^{1}(B̂^{×}, S_{H,W}) .

Now by duality, this is isomorphic to Ext^{1}(S_{H,V} , B^{×}). As before, it embeds into ∏_{R∈S} ∂S_{H,W} (R). Hence if Ext^{1}(S_{H,V} , B^{×}) ≠ {0}, then there exists some

R in S such that ∂S_{H,W} (R) ≠ {0}. In particular S_{H,W} (R) ≠ {0}, so H is a subquotient of R. This completes the proof.

5. Sectional characterization

Let O_{T} denote the forgetful functor F_{k} → F_{T ,k}. It was shown in Section 5 of [8] that the functor O_{T} has a right adjoint R_{T} defined as follows^{1}. For a finite group G, let T (G) denote the set of sections (T, S) of G such that T/S ∈ T . If F ∈ F_{T ,k}, then

1The construction in [8] actually dealt only with p-biset functors, for a fixed prime number p, and their restriction to a class of finite p-groups closed under taking subquotients. But it extends verbatim to the categories F_{k} and F_{T ,k}, for any class T of finite groups closed under taking subquotients.

R_{T} (F)(G) = lim ; ⟵_{(T,S)∈T (G)} F (T/S) ,

that is the set of sequences of elements l_{T,S} ∈ F (T/S), for (T, S) ∈ T (G), subject to the following conditions:

1. If (T, S) and (T′, S′) are elements of T (G) such that S ≤ S′ ≤ T′ ≤ T,

then

Defres_{T′/S′}^{T/S}l_{T,S} = l_{T′,S′} ,

2. If (T, S) ∈ T (G) and x ∈ G, then

xl_{T,S} = l_{^{x}T,^{x}S} .

The biset functor structure on R_{T} (F) is obtained as follows. For a finite group H and a finite (H, G)-biset U, the image v = U l of l ∈ R_{T} (F)(G) by U is the sequence m_{T,S}, for (T, S) ∈ T (H), defined by

m_{T,S} = ∑_{u∈[T \U/G]}(S\T u) l_{T^{u},S^{u}} ,

where for a subgroup X of H, we set X^{u} = {g ∈ G | ∃x ∈ X, xu = ug}. This makes sense as one can show that (T^{u}, S^{u}) is a section of G, and that T^{u}/S^{u} is isomorphic to a subquotient of T/S, hence in T if T/S ∈ T .

The unit η : Id → R_{T} ∘ O_{T} of the adjunction between O_{T} and R_{T}, evaluated for N ∈ F_{k} at a finite group G, is the map η_{N,G} from N(G) to R_{T} O_{T} (N)(G) sending n ∈ N(G) to the sequence of elements l_{T,S} ∈ O_{T} (N)(T/S) = N(T/S), for (T, S) ∈ T (G), defined by

l_{T,S} = Defres_{T/S}^{G}n .

5.1. Theorem: The unit of the adjunction between O_{T} : F_{𝔽_{2}} → F_{T ,𝔽_{2}} and R_{T} : F_{T ,𝔽_{2}} → F_{𝔽_{2}} induces an isomorphism of biset functors over 𝔽_{2}

η_{B^{×}} : B^{×} → R_{T} O_{T} (B^{×}) .

In other words, for any finite group G, the map

η_{B^{×},G} : B^{×}(G) → lim ; ⟵_{(T,S)∈T (G)} B^{×}(T/S)

is an isomorphism.

Proof: Set B̂ = Hom_{ℤ}(B, 𝔽_{2}), and let C denote the cokernel of the morphism ı : B^{×} → B̂. The functor O_{T} is exact, so its right adjoint R_{T} is left exact. Hence we get a commutative diagram

(5.2)  \xymatrix{0 \ar[r] & B^{×} \ar[r]^{ı} \ar[d]^{η_{B^{×}}} & B̂ \ar[r]^{π} \ar[d]^{ηB̂} & C \ar[r] \ar[d]^{η_{C}} & 0 \\ 0 \ar[r] & R_{T} O_{T} (B^{×}) \ar[r]^{f} & R_{T} O_{T} (B̂) \ar[r]^{g} & R_{T} O_{T} (C)}

with exact rows, where π is the projection morphism, f = R_{T} O_{T} (ı), and g = R_{T} O_{T} (π).

We first claim that the vertical morphism ηB̂ is an isomorphism. Indeed, let G be a finite group. The map η_{B̂,G} : B̂(G) → lim ; ⟵_{(T,S)∈T (G)} B̂(T/S) is the

map sending the linear form φ : B(G) → 𝔽_{2} to the sequence of linear forms d_{T,S} = Defres_{T/S}^{G}φ : B(T/S) → 𝔽_{2}, for (T, S) ∈ T (G). Since

d_{T,S}((T/S)/(T/S)) = φ(Indinf_{T/S}^{G}(T/S)/(T/S)) = φ(G/T) ,

and since (T, T) ∈ T for any subgroup T of G, we see that η_{B̂,G} is injective. Now let l = (ψ_{T,S})_{(T,S)∈T (G)} be an element of lim ; ⟵_{(T,S)∈T (G)} B̂(T/S). So for

each (T, S) ∈ T (G), we get a linear map ψ_{T,S} : B(T/S) → 𝔽_{2}. For a subgroup U of G, we set φ(G/U) = ψ_{U,U}((U/U)/(U/U)). Then for any x ∈ G, we have φ(G/^{x}U) = φ(G/U), since

φ(G/^{x}U) = ψ_{^{x}U,^{x}U}((^{x}U/^{x}U)/(^{x}U/^{x}U))
= (^{x}ψ_{U,U})((^{x}U/^{x}U)/(^{x}U/^{x}U))
= ψ_{U,U}((U/U)/(U/U))

because ^{x}ψ_{T,S} = ψ_{^{x}T,^{x}S} for any (T, S) ∈ T (G). It follows that φ extends linearly to an element of B̂(G). Moreover, for any (T, S) ∈ T (G) and any subgroup U/S of T/S, we have ψ_{U,U} = Defres_{U/U}^{T/S}ψ_{T,S}, so

Defres_{T/S}^{G}φ((T/S)/(U/S)) = φ(Indinf_{T/S}^{G}((T/S)/(U/S)))

= φ(G/U) = ψ_{U,U}((U/U)/(U/U))

= (Defres_{U/U}^{T/S}ψ_{T,S})((U/U)/(U/U))

= ψ_{T,S}(Indinf_{U/U}^{T/S}((U/U)/(U/U))

= ψ_{T,S}((T/S)/(U/S)) .

Hence Defres_{T/S}^{G}φ = ψ_{T,S}, for any (T, S) ∈ T (G), so the map η_{B̂,G} is surjective. Hence it is an isomorphism, for any finite group G, and ηB̂ is an isomorphism, as claimed.

Now the Snake’s lemma, applied to Diagram 5.2, shows that the morphism η_{B^{×}} is injective, and that its cokernel is isomorphic to the kernel of η_{C}. For a finite group G, an element \overline{φ} of Ker η_{C,G} is represented by a linear form φ : B(G) → 𝔽_{2} such that Defres_{T/S}^{G}φ belongs to ı(B^{×}(T/S)), for any (T, S) ∈ T (G), hence in particular for any section (T, S) of G such that T/S ∈ R. By Theorem 3.4, it follows that (Defres_{T/S}^{G}φ)(ϵ_{T,S}) = 0. As this holds for any section (T, S) of G with T/S ∈ R, the form φ lies in the image of ı_{G}, by Theorem 3.4 again. In other words the element \overline{φ} is equal to zero. Hence Ker η_{C,G} = {0} for any G, so η_{C} is injective. It follows that η_{B^{×}} is surjective. Hence it is an isomorphism. This completes the proof of the theorem.

6. Finite generation and presentation

In this section, we show that the biset functor B^{×} is not finitely generated, and that its dual B̂^{×} is finitely generated (by a single element!), but not finitely presented. Recall that k is a commutative ring, and that kB_{R} = k ⊗_{ℤ} B(−, R) is the representable functor Hom_{kC}(R, −).

6.1. Lemma: Let R and H be finite groups. If \overline{kB_{R}}(H) ≠ \{0\}, then H is a subquotient of R.

Proof: Recall that kB_{R}(H) = kB(H, R) has a k-basis consisting of the transitive bisets (H × R)/X, for X in a set of representatives of conjugacy classes of subgroups of H ×R. Let X be one of these groups, and Y = p_{1}(X) be its first projection. Then (H × R)/X = kB_{R}(Ind_{Y}^{H})((Y × R)/X), where we abuse notation in the right hand side by viewing X as a subgroup of Y × R, and (Y × R)/X as an element of kB_{R}(Y). Hence if the image of (H × R)/X in \overline{kB_{R}}(H) is non zero, then Y = H.

In this case let N = k_{1}(X). Then N is a normal subgroup of H = p_{1}(X), and (H × R)/X = kB_{R}(Inf_{H/N}^{H})(((H/N) × R)/\overline{X}), where

\overline{X} = \{(hN, g) | (h, g) ∈ X\} .

Hence if the image of (H ×R)/X in \overline{kB_{R}}(H) is non zero, we also have N = 1. Then H ≅ Y/N ≅ T/S, where T = p_{2}(X) and S = k_{2}(X). In particular H is a subquotient of R. This completes the proof.

6.2. Proposition: Let k be a commutative ring and T be a class of finite groups closed under taking subquotients. Let F ∈ F_{T ,k}. The following conditions are equivalent:

1. The functor F is finitely generated.

2. There exists a finite family E of groups in T and an epimorphism

⊕_{R∈E} kB_{R} → F.

3. For any H ∈ T , the k-module \overline{F} (H) is finitely generated, and there exists an integer n ∈ ℕ such that \overline{F} (H) = \{0\} whenever |H| > n.

Proof: The equivalence of 1 and 2 is classical. If 1 holds, then there is a finite set G of elements of F such that ⟨G⟩ = F. For each (R, v) ∈ G, we get a morphism ṽ : kB_{R} → F associated to v ∈ F (R) by Yoneda’s lemma. The sum of these morphisms

⊕_{(R,v)∈G} ṽ: ⊕_{(R,v)∈G} kB_{R} → F

is surjective if ⟨G⟩ = F, so 2 holds. Conversely, each representable functor kB_{R} is generated by the single element (R, Id_{R}), where Id_{R} ∈ kB(R, R) is the identity endomorphism of R. Hence if 2 holds, then F is a quotient of a finite sum of finitely generated functors, so F is finitely generated, and 1 holds.

Now if 2 holds, then for each H ∈ T , the k-module F (H) is a quotient of ⊕_{R∈E} kB(H, R), and each kB(H, R) is a finitely generated k-module.

Hence F (H) is a finitely generated k-module. Moreover \overline{F} (H) is a quotient of ⊕_{R∈E} \overline{kB_{R}}(H), and \overline{kB_{R}}(H) = \{0\} unless H is a subquotient of R, by

Lemma 6.1. In particular \overline{kB_{R}}(H) = \{0\} if |H| > |R|, so \overline{F} (H) = \{0\} if |H| > n, where n = max{|R| | R ∈ E}. Hence 3 holds.

Now if 3 holds, there is a finite set of isomorphism classes of finite groups R such that \overline{F} (R) ≠ \{0\}. Let U be a set of representatives of this set. For each R ∈ U, we can lift to F (R) a finite generating set of the k-module \overline{F} (R). We get a finite subset V_{R} of F (R), and this gives a finite set

G = {(R, v) | R ∈ U, v ∈ V_{R}}

of elements of F, which in turns gives a morphism

π : P = ⊕_{(R,v)∈G} kB_{R} → F .

Our choice of U and V_{R}, for R ∈ U, shows that the induced morphism \overline{π_{H}} : \overline{P} (H) → \overline{F} (H) is surjective for any H ∈ T : if H is not isomorphic to a group in U, then this is trivially true because \overline{F} (H) = \{0\}. And otherwise, we can assume H ∈ U, and then \overline{π_{H}} is surjective because \overline{π_{H}}(V_{H}) generates \overline{F} (H) by construction.

We deduce by induction on n = |H| that π_{H} : P (H) → F (H) is surjective for any H ∈ T . For n = 1, this is clear, since

π_{1} = \overline{π_{1}} : \overline{P} (1) = P (1) → F (1) = \overline{F} (1)

is surjective. Now assume π_{K} is surjective for any K ∈ T with |K| < n = |H|, and let v ∈ F (H). Since \overline{π_{H}} is surjective, there is an element w ∈ P (H), a set Σ of proper sections of H (i.e. sections different from (H, 1)), and elements v_{T,S} ∈ F (T/S), for (T, S) ∈ Σ, such that

v = π_{H}(w) + ∑_{(T,S)∈Σ} Indinf_{T/S}^{H}v_{T,S} .

Since |T/S| < n for any (T, S) ∈ Σ, the map π_{T/S} : P (T/S) → F (T/S) is surjective, and there is an element w_{T,S} ∈ P (T/S) such that v_{T/S} = π_{T/S}(w_{T,S}). It follows that

v = π_{H}(w) + ∑_{(T,S)∈Σ} Indinf_{T/S}^{H}π_{T/S}(w_{T,S})

= π_{H}(w) + ∑_{(T,S)∈Σ} π_{H}(Indinf_{T/S}^{H}w_{T,S}) = π_{H}(w + ∑_{(T,S)∈Σ} Indinf_{T/S}^{H}w_{T,S}) .

Hence π_{H} is surjective, and this completes the inductive step.

It follows that π is an epimorphism, so 3 implies 2, completing the proof of Proposition 6.2.

6.3. Corollary: The biset functor B^{×} is not finitely generated.

Proof: It has been show by Barsotti ([1], Proposition 6.8) that if p is a prime number congruent to 1 mod 4, then \overline{B^{×}}(D_{2p}) ≠ \{0\}, where D_{2p} is a dihedral group of order 2p (in Barsotti’s terminology, the group D_{2p} is residual). It follows that there are arbitrary large finite groups H such that \overline{B^{×}}(H) ≠ \{0\}. By Proposition 6.2, the functor B^{×} is not finitely generated.

Recall from Corollary 3.7 that there is an exact sequence of biset functors

(6.4)  \xymatrix{0 \ar[r] & L \ar[r] & 𝔽_{2}B \ar[r]^{ȷ} & B̂^{×} \ar[r] & 0 ,}

where

S = {C_{p} | p odd prime} ∪ {C_{4}} ∪ {SD_{2^{n}} | n ≥ 4} ,

and L is the biset subfunctor of 𝔽_{2}B generated G_{S} = \{(R, \overline{ε}_{R}) | R ∈ S\}.

6.5. Proposition:

1. Let G = {(1, u)}, where u is the non zero element of B̂^{×}(1) ≅ 𝔽_{2}.

Then ⟨G⟩ = B̂^{×}.

2. The functor L is not finitely generated.

3. The functor B̂^{×} is not finitely presented.

Proof: (1) This follows from the fact that B̂^{×} is a quotient of 𝔽_{2}B, and that 𝔽_{2}B is generated by (1, e), where e ∈ 𝔽_{2}B(1) is the class of a set of cardinality one, endowed with the trivial action of the trivial group. Indeed if K ≤ H are finite groups, then H/K = Indinf_{K/K}^{H}Iso(f_{K})(e), where f_{K} : 1 → K/K is the unique group isomorphism.

(2) The exact sequence (6.4) shows that if p is an odd prime number, then L(C_{p}) ≅ 𝔽_{2} and L(1) = \{0\}. Hence \overline{L}(C_{p}) ≅ L(C_{p}) ≅ 𝔽_{2}, so there exist arbitrary large finite groups H such that \overline{L}(H) ≠ \{0\}. By Proposition 6.2, the functor L is not finitely generated.

(3) Suppose that there exists an exact sequence in F_{𝔽_{2}}

(6.6)  \xymatrix{N \ar[r] & M \ar[r] & B̂^{×} \ar[r] & 0 ,}

where M is projective and N is finitely generated. This gives an exact sequence

\xymatrix{0 \ar[r] & K \ar[r] & M \ar[r] & B̂^{×} \ar[r] & 0 ,}

where K is the image of N in M. In particular K is finitely generated. Then, since M and 𝔽_{2}B are projective in F_{𝔽_{2}}, Shanuel’s lemma gives an isomorphism of functors

L ⊕ M ≅ K ⊕ 𝔽_{2}B .

Then L is a quotient of K ⊕ 𝔽_{2}B, which is finitely generated. Hence L is finitely generated, contradicting 2. So no short exact sequence like (6.6) can exists, hence B̂^{×} is not finitely presented.

Recall that in Corollary 3.7, the set

S = {C_{p} | p odd prime} ∪ {C_{4}} ∪ {SD_{2^{n}} | n ≥ 4}

was introduced. We finally prove that G_{S} = \{(R, \overline{ε}_{R}) | R ∈ S\} is a minimal set of generators of L.

6.7. Theorem: Let S′ be a proper subset of the set S introduced in Corollary 3.7, and G_{S′} = \{(R, \overline{ε}_{R}) | r ∈ S′\}. Then ⟨G_{S′}⟩ is a proper subfunctor of

⟨G_{S}⟩ = L.

Proof: It suffices to show that for any R ∈ S, if we set G_{S}^{R} = G_{S} − \{(R, \overline{ε}_{R})\} and L^{R} = ⟨G_{S}^{R}⟩, then \overline{ε}_{R} ∉ L^{R}(R). So we assume that \overline{ε}_{R} ∈ L^{R}(R), for contradiction, i.e. that there exists a finite set E of pairs (H, X), where H ∈ S − {R} and X is a subgroup of R × H, such that

(6.8)  \overline{ε}_{R} = ∑_{(H,X)∈E} ((R × H)/X)\overline{ε}_{H} .

Since f_{1}^{R}\overline{ε}_{R} = \overline{ε}_{R} for any R, by Remark 3.2, this also reads

\overline{ε}_{R} = ∑_{(H,X)∈E} f_{1}^{R}((R × H)/X)f_{1}^{H} \overline{ε}_{H} .

We now observe that each element H of S has a unique minimal normal (hence central of prime order) subgroup Z_{H}. And if k_{1}(X) ≥ Z_{R}, then

f_{1}^{R}((R × H)/X) = f_{1}^{R}Inf_{R/Z_{R}}^{R}Def_{R/Z_{R}}^{R}((R × H)/X) = 0

by Lemma 6.3.2 of [7]. Similarly ((R × H)/X)f_{1}^{H} = 0 if k_{2}(X) ≥ Z_{H}. So we can assume that k_{1}(X) ∩ Z_{R} = 1 and k_{2}(X) ∩ Z_{H} = 1 for any (H, X) ∈ E.

This forces k_{1}(X) = k_{2}(X) = 1, unless R is semidihedral and k_{1}(X) is a non central subgroup of order 2 of R, or H is semidihedral and k_{2}(X) is a non central subgroup of order 2 of H (both cases may occur simultaneously). In the first case p_{1}(X) ≤ N_{R}(k_{1}(X)) = k_{1}(X)Z_{R}, and p_{1}(X)/k_{1}(X) has order 1 or 2. Similarly, in the second case p_{2}(X)/k_{2}(X) has order 1 or 2. In any of these two cases, the morphism f_{1}^{R}((R × H)/X)f_{1}^{H} of C_{𝔽_{2}} factors through a group of order 1 or 2 (see 2.4). Since L(1) = L(C_{2}) = {0}, it follows that f_{1}^{R}((R × H)/X)f_{1}^{H} \overline{ε}_{H} = 0.

So we can assume that k_{1}(X) = k_{2}(X) = 1 for any (H, X) ∈ E. In this case X is a twisted diagonal subgroup of R × H, that is, there is a subgroup K of H, a subgroup S of R, and a group isomorphism f : K → S, such that

(6.9)  (R × H)/X ≅ Ind_{S}^{R} Iso(f) Res_{K}^{H} .

We can also assume that Res_{K}^{H} \overline{ε}_{H} ≠ 0, and in particular that L(K) ≠ \{0\}.

Suppose first that R = C_{p} for an odd prime number p. Then S = 1 or S = R. Since L(1) = {0}, we have S = R. Then K ≅ S = C_{p} is a subgroup of H ∈ S. The only element H of S admitting a subgroup of odd prime order p is C_{p} itself. But H ∈ S − {C_{p}} by assumption, so we get a contradiction.

Suppose now that R = C_{4}. Again, since L(1) = L(C_{2}) = {0}, we have S = R, and H ≥ K ≅ C_{4}. Since H ∈ S − {C_{4}}, it follows that H = SD_{2^{n}} for some n ≥ 4. Then K is contained in the unique generalized quaternion subgroup Q of index 2 of H. One checks easily that Res_{Q}^{H} \overline{ε}_{H} = Q/1 − Q/Z_{H}, and it follows that Res_{K}^{H} \overline{ε}_{H} = 0. We get a contradiction also in this case.

We are left with the case R = SD_{2^{n}}, for some n ≥ 4. Then if (H, X) ∈ E, the first projection S of the twisted diagonal subgroup X of R×H is a proper subgroup of R: otherwise indeed, its second projection K is a semidihedral subgroup of H ∈ S, which can only occur if H itself is semidihedral and isomorphic, hence equal, to R. This is a contradiction, since H ∈ S − {R}.

It follows from (6.8) and (6.9) that \overline{ε}_{R} is a sum of elements of the form Ind_{S}^{R}u_{S}, for proper subgroups S of R and elements u_{S} of L(S). Let D, C, and Q be the subgroups of index 2 of R, where D is dihedral, C is cyclic, and Q is generalized quaternion. We can write

\overline{ε}_{R} = Ind_{D}^{R}v_{D} + Ind_{C}^{R}v_{C} + Ind_{Q}^{R}v_{Q} ,

for some v_{D} ∈ L(D), v_{C} ∈ L(C), and v_{Q} ∈ L(Q). We set Z = Z_{R} for simplicity. If M ∈ {D, C, Q}, then M ≥ Z, and one checks easily that f_{1}^{R}Ind_{M}^{R} = Ind_{M}^{R}f_{1}^{M}. It follows that

\overline{ε}_{R} = Ind_{D}^{R}f_{1}^{D}v_{D} + Ind_{C}^{R}f_{1}^{C}v_{C} + Ind_{Q}^{R}f_{1}^{Q}v_{Q} .

Moreover f_{1}^{M}M/N = 0 if N ∩ Z(M) ≠ 1. It follows that f_{1}^{C}v_{C} is a multiple of C/1 − C/Z, and that f_{1}^{Q}v_{Q} is a multiple of Q/1 − Q/Z. Hence

\overline{ε}_{R} = Ind_{D}^{R}w_{D} + λ(R/1 − R/Z) ,

for some λ ∈ 𝔽_{2} and some w_{D}(= f_{1}^{D}v_{D}) ∈ ∂L(D).

Now cutting the exact sequence

\xymatrix{0 \ar[r] & L(D) \ar[r] & 𝔽_{2}B(D) \ar[r] & B̂^{×}(D) \ar[r] & 0}

by the idempotent f_{1}^{D} gives the exact sequence

\xymatrix{0 \ar[r] & ∂L(D) \ar[r] & ∂𝔽_{2}B(D) \ar[r] & ∂B̂^{×}(D) \ar[r] & 0 .}

The vector space ∂𝔽_{2}B(D) has a basis consisting of the elements D/1−D/Z, D/I − D/IZ, and D/J − D/JZ, where I and J are two non conjugate non central subgroups of order 2 of D. The vector space ∂B̂^{×}(D) is isomorphic to the dual of ∂B^{×}(D), which is one dimensional (see e.g. Corollary 6.12 of [6], or the proof of Theorem 3.4). It follows that ∂L(D) has dimension 2. Now ∂L(D) contains the two elements \overline{ε}_{D} = (D/I − D/IZ) − (D/J − D/JZ) and D/1 − D/Z = Ind_{C_{4}}^{D}\overline{ε}_{C_{4}}, which are obviously linearly independent. Hence these two elements form a basis of ∂L(D). It follows that

\overline{ε}_{R} = αInd_{D}^{R}\overline{ε}_{D} + βInd_{C_{4}}^{R}\overline{ε}_{C_{4}} + λ(R/1 − R/Z) ,

for some α, β in 𝔽_{2}. But one checks easily that Ind_{D}^{R}\overline{ε}_{D} = 0, for I and J are conjugate in R. Moreover Ind_{C_{4}}^{R}\overline{ε}_{C_{4}} = R/1 − R/Z. Thus \overline{ε}_{R} = R/I − R/IZ is a scalar multiple of R/1 − R/Z, which is obviously wrong. This final contradiction completes the proof of Theorem 6.7.

6.10. Corollary: Let 2^{S} be the set of subsets of S, ordered by inclusion of subsets, and [0, L] be the poset of subfunctors of L, ordered by inclusion of subfunctors. Let g : 2^{S} → [0, L] be the map sending A ⊆ S to

g(A) = ⟨\{(H, \overline{ε}_{H}) | H ∈ A\}⟩ ⊆ L ,

and f : [0, L] → 2^{S} be the map sending the subfunctor M of L to

f(M) = \{H ∈ S | \overline{ε}_{H} ∈ M(H)\} ⊆ S .

Then:

1. Let A, A′ be subsets of S, and M, M′ be subfunctors of L. Then

g(A ∪ A′) = g(A) + g(A′) and f(M ∩ M′) = f(M) ∩ f(M′) .

In particular f and g are maps of posets.

2. f ∘ g = Id_{2^{S}}.

3. The poset [0, L] is uncountable.

Proof: Assertion 1 is straightforward. For Assertion 2, let A ⊆ S and A′ = f ∘ g(A). Then clearly A ⊆ A′. If this inclusion is strict, let S ∈ A′ − A. Then

(S, \overline{ε}_{S}) ∈ g(A) = ⟨\{(H, \overline{ε}_{H}) | H ∈ A\}⟩ ⊆ g(S′) ,

where S′ = S − {S}. Then g(S′) = g(S) = L, and by Theorem 6.7, it follows that S′ = S, a contradiction. Hence A = A′, and f ∘ g = Id_{2^{S}}. In particular g is injective, and Assertion 3 follows, since the set of subsets of the (infinite) countable set S is uncountable.

6.11. Remark: The map g is not surjective, and not a map of lattices (that is, the image by g of an intersection of subsets need not be the intersection of the images of the subsets). Indeed, if S is any semidihedral group in S, and C its cyclic subgroup of index 2, we have

Res_{C}^{S} \overline{ε}_{S} = C/1 − C/Z = Ind_{C′}^{C}\overline{ε}_{C′} ,

where Z is the center of S and C′ the subgroup of order 4 of C. It follows that u = C/1 − C/Z is a non-zero element of M(C), where M is the intersection of the subfunctors of L generated by \{(S, \overline{ε}_{S})\} and \{(C_{4}, \overline{ε}_{C_{4}})\}. In other words if A = {S} and A′ = {C_{4}}, we have 0 ≠ u ∈ (g(A) ∩ g(A′))(C), so g(A) ∩ g(A′) ≠ {0}. Now if g(A) ∩ g(A′) belongs to the image of g, there is a subset A′′ of S such that g(A) ∩ g(A′) = g(A′′), and then

A′′ = fg(A′′) = f(g(A) ∩ g(A′)) = fg(A) ∩ fg(A′) = A ∩ A′ = ∅ .

This is a contradiction since g(A′′) ≠ {0} but g(∅) = {0}. It follows that g is not surjective, and not a map of lattices, since g(A) ∩ g(A′) ≠ g(A ∩ A′).

References

[1] J. Barsotti. On the unit group of the Burnside ring as a biset functor for some solvable groups. Journal of Algebra, 508, 05 2018.

[2] R. Boltje and G. Pfeiffer. An algorithm for the unit group of the Burnside ring of a finite group. In Groups St Andrews 2005, volume 339, pages 230–236. London Math. Soc. Lectures Notes Series, 2007.

[3] S. Bouc. Burnside rings. In Handbook of Algebra, volume 2, chapter 6E, pages 739–804. Elsevier, 2000.

[4] S. Bouc. A remark on a theorem of Ritter and Segal. J. of Group Theory, 4:11–18, 2001.

[5] S. Bouc. The Dade group of a p-group. Inv. Math., 164:189–231, 2006.

[6] S. Bouc. The functor of units of Burnside rings for p-groups. Comm. Math. Helv., 82:583–615, 2007.

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

[8] S. Bouc and J. Thévenaz. A sectional characterization of the Dade group. Journal of Group Theory, 11(2):155–298, 2008.

[9] S. Bouc and E. Yalçın. Borel-Smith functions and the Dade group. J. Algebra, 311:821–839, 2007.

[10] A. Dress. A characterization of solvable groups. Math. Zeit., 110:213– 217, 1969.

[11] W. Feit and J. G. Thompson. Solvability of groups of odd order. Pacific J. Math., 13:775–1029, 1963.

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

[13] T. Matsuda. On the unit group of Burnside rings. Japan. J. Math., 8(1):71–93, 1982.

[14] T. Matsuda and T. Miyata. On the unit groups of the Burnside rings of finite groups. J. Math. Soc. Japan, 35(1):345–354, 1983.

[15] J. Ritter. Ein Induktionssatz für rational Charaktere von nilpotenten Gruppen. J. f. reine u. angew. Math., 254:133–151, 1972.

[16] G. Segal. Permutation representations of finite p-groups. Quart. J. Math. Oxford, 23:375–381, 1972.

[17] T. tom Dieck. Transformation groups and representation theory, volume 766 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

[18] E. Yalçın. An induction theorem for the unit groups of Burnside rings of 2-groups. J. Algebra, 289:105–127, 2005.

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

[20] T. Yoshida. On the unit groups of Burnside rings. J. Math. Soc. Japan, 42(1):31–64, 1990.

Serge Bouc LAMFA-CNRS UMR 7352 Université de Picardie-Jules Verne 33, rue St Leu, 80039 Amiens Cedex 01 France

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