THE CROSSED BURNSIDE RING, THE DRINFEL’D DOUBLE, AND THE DADE GROUP OF A p-GROUP

FUMIHITO ODA

Abstract. Let p be a prime number. This paper solves the question of the difference between the rank of the crossed Burnside ring B^{c}(P) of a finite p-group P and of the rational representation ring R_{ℚ}(𝒟(P)) of the Drinfel’d double 𝒟(P) of the group algebra ℚP. The difference is represented by using the Dade groups of certain subgroups of P.

1. Introduction

This paper describes the difference between the rank of the crossed Burnside ring B^{c}(P) ([OY01], [Bo03a], [Bo03b], [OY04]) of a finite p-group P and of the rational representation ring R_{ℚ}(𝒟(P)) of the Drinfel’d double 𝒟(P) ([Ka94],[Ma95]) of the group algebra ℚP. The difference is represented by using the torsion-free ranks of the Dade groups, that was defined by E.C. Dade in 1978 ([Da78a], [Da78b]), in order to classify endo-permutation kP -modules, where k is a filed of characteristic p > 0. The question of the structure of the group D(P) of endo-permutation modules over an arbitrary finite p-group P was completely solved by S. Bouc [Bo06].

In his book [Ma95], S. Majid pointed out that a large class of modules for Drinfel’d double of a group algebra F G over a field F is provided by the crossed G-sets of J.H.C. Whitehead. This is a classic construction in algebraic topology.

Let G be a finite group. The Grothendieck ring of the category of finite crossed G-sets is called crossed Burnside ring [OY01]. A natural question then arises:

Is there a relationship between the structure of the crossed Burnside ring B^{c}(G) of G and of the representation ring R_{F} (𝒟(G)) of the Drinfel’d double 𝒟(G) of the group algebra F G ?

2000 Mathematics Subject Classification. Primary 20C20, Secondary 20D15, 18B99.

Key words and phrases. Crossed Burnside ring, Dade group, Drinfel’d double, Bouc functor.

Partially supported by JSPS, Grant-in-Aid for Scientific Research (C) 18540059.

For instance, the author proved an induction theorem for ℂ⊗R_{ℂ}(𝒟(G)) for a finite group by using a formula [OY01] for primitive idempotents of B^{c}(G) [Od07]. In the present paper, we compare the two ranks of the rings in case of G is a p-group and F = ℚ. One of the ingredients of the present paper is Theorem (10.4) of [BT00] that was proved by S. Bouc and J. Thévenaz by using a theory of Bouc functor [Th06] for finite p-groups, introduced in [BT00] and it was called a biset functor. In [BT00], Bouc and Thévenaz showed that there is an exact sequence of functors

[figure: 0; ℚD; α; ℚB; ε; ℚR_{ℚ}; 0,]

where ε(P) : ℚB(P) → ℚR_{ℚ}(P) is the morphism mapping a P -set to the corresponding permutation module over ℚ.

Our main result is as follows: Corollary 4.4 Let P be a p-group. Then there is an exact sequence of ℚ-vector spaces

[figure: 0; ℚD(P^{c}); α; ℚB^{c}(P); ε; ℚR_{ℚ}(D(P)); 0.]

In particular, we have

rankB^{c}(P) = rankR_{ℚ}(D(P)) + dim_{ℚ} ℚD(P^{c}).

The paper is organized as follows: Section 2 is devoted to definitions and results on Bouc functors. Section 3 recalls some definitions and results of Dade group, its functorial properties. Finally Section 4 describes some results as corollaries of theorem of Dade and Theorem (10.4) of [BT00].

2. Bouc functors

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

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

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

for k ∈ K, g ∈ G, u ∈ U, and v ∈ V.

Denote by 𝒞_{p} the following category:

• The objects of 𝒞_{p} are the finite p-groups.

THE CROSSED BURNSIDE RING, DRINFELD DOUBLE, DADE GROUP 3

• If P and Q are finite p-groups, then Hom_{𝒞_{p}}(P, Q) = B(Q×P^{op}) is the Burnside group of finite (Q, P)-bisets. An element of this group is called a virtual (Q, P)-biset.

• The composition of morphisms is ℤ-bilinear, and if P, Q, R are finite p-groups, if U is a finite (Q, P)-biset, and V is a finite (R, Q)-biset, then the composition of (the isomorphism classes of) V and U is the (isomorphism class) of V ×_{Q} U. The identity morphism Id_{P} of the p-group P is the class of the set P, with left and right action given by multiplication.

Let ℱ_{p} denote the category of additive functors from 𝒞_{p} to the category ℤ-Mod of abelian groups. An object of ℱ_{p} is called a Bouc functor (defined over p-groups, with values in ℤ-Mod)(see [Th06], [Bo06]).

2.2. Notation. The Bouc functor of Burnside group will be denoted by B. The Bouc functor of rational representations will be denoted by R_{ℚ}. The G-set of the family of all subgroups of a finite group G will be denoted by 𝒮(G). If X is a G-set, denote by G\X a family of G-orbits, and by [G\X] a set of representatives of G\X.

2.3. Examples.

Recall that a formalism of bisets gives a single framework for the usual operations of induction, restriction, inflation, deflation, and transport by isomorphism via the following correspondences:

• If Q is a subgroup of P, then let Ind_{Q}^{P} ∈ Hom_{𝒞_{p}}(Q, P) denote the set P, with left action of P and right action of Q given by multiplication.

• If Q is a subgroup of P, then let Res_{Q}^{P} ∈ Hom_{𝒞_{p}}(P, Q) denote the set P, with left action of Q and right action of P given by multiplication.

• If N ⊴ P, and Q = P/N, then let Inf_{Q}^{P} ∈ Hom_{𝒞_{p}}(Q, P) denote the set Q, with left action of P given by projection and multiplication, and right action of Q given by multiplication.

• If N ⊴ P, and Q = P/N, then let Def_{Q}^{P} ∈ Hom_{𝒞_{p}}(P, Q) denote the set Q, with left action of Q given by multiplication, and right action of P given by projection and multiplication.

• If φ : P → Q is a group isomorphism, then let Iso_{P}^{Q} = Iso_{P}^{Q}(φ) ∈ Hom_{𝒞_{p}}(P, Q) denote the set Q, with left action of Q given by multiplication, and right action of P given by taking image by φ, and then multiplying in Q.

2.4. Section. A section (T, S) of the group P is a pair of subgroups of P such that S ⊴ T ≤ P.

2.5. Remark: If P and Q are p-groups, then any element φ ∈ Hom_{𝒞_{p}}(P, Q) is a ℤ-linear combination of transitive (Q, P)-bisets, and by Lemma 7.4 of [BT00], every transitive (Q, P)-biset can be factored in the category 𝒞_{p} as a composition

Ind_{V}^{Q} ∘ Inf_{V/U}^{V} ∘ Iso_{T/S}^{V/U} ∘ Def_{T/S}^{T} ∘ Res_{T}^{P}

where (T, S) is a section of P, and (V, U) is a section of Q, such that there exists a group isomorphism φ : T/S → V/U.

3. The Dade group

3.1. Let k be a field of characteristic p. If P is a p-group, the Dade group D(P) = D_{k}(P) can be defined as the group of equivalence classes of capped endo-permutation kP -modules, or as the group of equivalence classes of Dade P -algebras over k, for a suitable equivalence relation in each case. It is an abelian group, and Puig ([Pu90]) has shown that it is finitely generated.

3.2. Some known Dade groups: The structure of D(P) is known for any 2-group P of normal 2-rank 1: when P is generalized quaternion, the result is due to Dade, and the other cases have been solved by Carlson and Thévenaz:

Theorem 3.3. (Dade [Da78a], Carlson-Thévenaz [CT00])

(1) D(C_{2^{n}}) ≅ (ℤ/2ℤ)^{n−1}, and D(C_{p^{n}}) ≅ (ℤ/2ℤ)^{n}, if p ≥ 3.

(2) D(D_{2^{n}}) ≅ ℤ^{2n−3}.

(3) D(SD_{2^{n}}) ≅ ℤ^{2n−4} ⊕ ℤ/2ℤ.

(4) D(Q_{2^{n}}) ≅ ℤ^{2n−5} ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ, for n ≥ 4.

(5) D(Q_{8}) ≅ ℤ ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ, if the ground field contains all cubic roots of unity, and D(Q_{8}) ≅ ℤ ⊕ ℤ/4ℤ otherwise.

3.4. Functorial properties. The Dade group has some important functorial properties: if Q is a subgroup of P, and if N is a normal subgroup of P, or if φ : P → P′ is a group isomorphism, then there are maps of restriction, induction, deflation, inflation, and transport by isomorphism

(3.0.1)

[figure: D(P); Res_{Q}^{P}; D(Q); Ten_{Q}^{P}; D(P)]

[figure: D(P); Def_{P/N}^{P}; D(P/N); Inf_{P/N}^{P}; D(P)]

D(P) \overline{^{Iso_{P}^{P′}}} D(P′)

coming respectively from the restriction, tensor induction, Brauer quotient, inflation, and transport by group isomorphism of Dade algebras (note the change of notation for induction).

These five operations can be unified in a single formalism using bisets: if P and Q are p-groups, if U is a finite (Q, P)-biset, then in Corollary 2.3 of [BT00], they introduced a map D(U) : D(P) → D(Q), such that five types of morphism in (3.0.1) are equal to D(U), where U is the corresponding biset Res_{Q}^{P}, Ind_{Q}^{P}, Def_{P/N}^{P}, Inf_{P/N}^{P}, and Iso_{Q}^{P} defined in 2.3. The definition of this map D(U) associated to an arbitrary finite (Q, P)-biset U followed from the existence of a corresponding functor T_{U} : \underline{Perm}_{k}(P) → \underline{Perm}_{k}(Q), which is a sort of generalized tensor induction, where \underline{Perm}_{k}(P) is a category equivalent to the category of finitely generated permutation kP -modules (see Sect.2 of [BT00], in particular Lemma (2.2) and Corollary (2.13), for details).

Moreover the correspondence U → D(U) is additive (by Proposition 2.10 of [BT00]), so it can be extended to a correspondence

φ ∈ Hom_{𝒞_{p}}(P, Q) ↦ D(φ) ∈ Hom_{𝒞_{p}}(D(P), D(Q)).

However, these constructions do not define a Bouc functor structure: the reason is that in general, with obvious notation

D(ψ) ∘ D(φ) ≠ D(ψ ∘ φ),

as can be read from Proposition 3.5 and Proposition 3.10 of [BT00].

3.5. A category 𝒞_{ℚ}. We consider the category 𝒞_{ℚ} obtained by tensoring everything with ℚ : the objects of 𝒞_{ℚ} are the same as the objects of 𝒞_{p}, but

Hom_{𝒞_{ℚ}}(P, Q) = ℚ ⊗_{ℤ} B(Q, P).

The composition of morphisms is defined by ℚ-linearity. We denote by ℱ_{ℚ} the category whose objects are the ℚ-linear functors from 𝒞_{ℚ} to the category of ℚ-modules. The morphisms in ℱ_{ℚ} are the natural transformations of functors. For simplicity, we write ℚD(P) = ℚ ⊗_{ℤ} D(P). The following proposition is (7.6) of [BT00].

Proposition 3.6. ℚD is a ℚ-linear functor from 𝒞_{ℚ} to the category of ℚ-vector spaces. In other words, ℚD is a Bouc functor defined over p-groups, with values in ℚ-Mod.

The main ingredient of the present paper is

Theorem 3.7. (Bouc-Thévenaz [BT00] Theorem 10.4) There is an exact sequence of functors

[figure: 0; ℚD; α; ℚB; ε; ℚR_{ℚ}; 0]

where ε(P) : ℚB(P) → ℚR_{ℚ}(P) is the morphism mapping a P -set to the corresponding permutation module over ℚ.

3.8. Remark. We have no natural description of the map α in terms of arbitrary endo-permutation modules, but we can give an explicit description of the map on a basis

{Teninf_{Q/Q_{0}}^{P}(Ω_{Q/Q_{0}})|Q ∈ [P \𝒮(P)]}

of ℚD(P) (see 4.11 of [BT00]). By 10.5 of [BT00], we have

α_{P} (Teninf_{Q/Q_{0}}^{P}(Ω_{Q/Q_{0}})) = ∑_{{X≤Q \atop XQ_{0}=Q}} \frac{|N_{P} (X)|}{|Q|}e_{X}^{P},

where e_{X}^{P} is the primitive idempotent of ℚB(P).

We could determine the difference rankB^{c}(P) − rankR_{ℚ}(𝒟(P)) by using a result as follows:

Theorem 3.9. (Bouc-Thévenaz [BT00] Theorem A) The torsion-free rank of the Dade group D(P) is equal to the number of conjugacy classes of non-cyclic subgroups of P.

4. The crossed Burnside ring and the rational representations of Drinfel’d double

4.1. Definition. Let M be one of the Bouc functors ℚD, ℚB and ℚR_{ℚ}. We use a construction of Dress for Mackey functors for obtaining a module from M. Let P be a p-group. Now we set

M(X) = (⨁_{x∈X} M(P_{x}))^{P}

= {(m(x)) ∈ ⨁_{x∈X} M(P_{x})|^{g}(m(x)) = m(gx) ∀g ∈ P }_{,}

where P_{x} is the stabilizer of x in P.

Corollary 4.2. Let P be a p-group and X a P -set. Then there is an exact sequence of ℚ-vector spaces

[figure: 0; ℚD(X); α; ℚB(X); ε; ℚR_{ℚ}(X); 0.]

Proof. Let X = ⋃_{x∈[P \X]} P/P_{x} be a decomposition of X into transitive P -sets. Since P_{x} is a p-group, we obtain an exact sequence

[figure: 0; ℚD(P_{x}); α_{Px}; ℚB(P_{x}); ε_{Px}; ℚR_{ℚ}(P_{x}); 0]

of ℚ-vector spaces by Theorem 3.7. Hence we obtain an exact sequence

[figure: 0; ⨁_{x} ℚD(P_{x}); α; ⨁_{x} ℚB(P_{x}); ε; ⨁_{x} ℚR_{ℚ}(P_{x}); 0]

of ℚ-vector spaces, where α = (α_{P_{x}}), ε = (ε_{P_{x}}) and x ∈ [P \X]. It is easy to see that

⨁_{x∈[P \X]} M(P_{x}) ≅ (⨁_{x∈X} M(P_{x}))^{P} = M(X)

for M, this completes the proof of Corollary 4.2 □

4.3. Notation. Let P^{c} be the P -set P with P -action defined by conjugation. We recall the category of finite crossed P -sets over P^{c} : a crossed P -set (X, α) is a pair consisting of a finite P -set X, together with a map of P -sets α from X to P^{c}, and a morphism of crossed P - sets from (X, α) to (Y, β) is a morphism of P -sets φ from X to Y such that βφ = α. The product of two crossed G-sets (X, α) and (Y, β) is the crossed G-set (X × Y, α · β), where X × Y is the direct product of X and Y, with diagonal G-action, and α · β is the map from X × Y to G^{c} defined by (α · β)(x, y) = α(x) · β(y). We denote by B^{c}(P) the crossed Burnside ring of P, i.e. the Grothendieck ring of the category of finite crossed P -sets over P^{c}, for relations given by decomposition into disjoint union of crossed P -sets, the ring structure being induced by the product of crossed P -sets. Also we denote by R_{ℚ}(𝒟(P)) the rational representation ring of the Drinfel’d double 𝒟(P) = (ℚP)^{∗} ⊗ ℚP for the group algebra ℚP.

Corollary 4.4. Let P be a p-group. Then there is an exact sequence of ℚ-vector spaces

[figure: 0; ℚD(P^{c}); α; ℚB^{c}(P); ε; ℚR_{ℚ}(D(P)); 0.]

In particular, we have

rankB^{c}(P) = rankR_{ℚ}(D(P)) + dim_{ℚ} ℚD(P^{c}).

Proof. By Corollary 4.2, there is an exact sequence of ℚ-vector spaces

[figure: 0; ℚD(P^{c}); α; ℚB(P^{c}); ε; ℚR_{ℚ}(P^{c}); 0.]

There is an isomorphism of P -sets

P^{c} ≅ ⋃_{g∈[P \P^{c}]} P/C_{P} (g).

By Definition 4.1, B(P^{c}) is isomorphic to ⊕_{g∈[P \P^{c}]}B(C_{P} (g)) as ℤmodules. The latter module is isomorphic to the crossed Burnside ring B^{c}(P) of P as a ℤ-module (see 2.3 of [Bo03a]). Similarly, R_{ℚ}(P^{c}) ≅ R_{ℚ}(D(P)) and D(P^{c}) ≅ ⊕_{g}D(C_{P} (g)) as ℤ-modules (see 3.5 of [Bo03b] or 5.5 of [OY04]), and Corollary 4.4 follows by the facts that the natural bases of B(C_{P} (g)) and R_{ℚ}(C_{P} (g)) are ℚ-bases of ℚB(C_{P} (g)) and ℚR_{ℚ}(C_{P} (g)), respectively. □

Corollary 4.5. Let P be a p-group. Then the following numbers are equal:

(1) rankB^{c}(P)

(2) rankR_{ℚ}(D(P)) + ∑_{g∈[P \P^{c}]} dim_{ℚ} ℚD(C_{P} (g))

(3) ∑_{Q∈[P \𝒮(P)]} \frac{|C_{P} (Q)|}{|N_{P} (Q)|} · |Q| ( ∑_{x∈Q/Q′} \frac{1}{|x|})

(4) ∑_{Q∈[P \𝒮(P)]} |N_{P} (Q)\C_{P} (Q))|

(5) ∑_{g∈[P \P^{c}]} |C_{P} (g)\𝒮(C_{P} (g))|

Proof. Since the natural basis ((3.1.c) of [OY01]) of B^{c}(P) is a ℚ-basis of ℚ⊗B^{c}(P) and the irreducible rational representations form a ℚ-basis of R_{ℚ}(P^{c}) ≅ R_{ℚ}(D(P)), Corollary 4.2 shows (1) = (2). The equality of (1) = (3) = (4) is easy to see from Proposition 3.2 of [OY01]. As an immediate consequence of 3.4 of [Bo03a] (or 2.1 of [Bo03b]), we have (1) = (5). □

Corollary 4.6. Let P be a p-group. Then

rankB^{c}(P) = rankR_{ℚ}(D(P)) + ∑_{g∈[P \P^{c}]} |C_{P} (g)\𝒮_{non}(C_{P} (g))|,

where 𝒮_{non}(C_{P} (g)) is the C_{P} (g)-set of non-cyclic subgroups of C_{P} (g) with C_{P} (g)-action defined by conjugation.

Proof. Theorem 3.9 of Bouc and Thévenaz shows that the torsion-free rank of the Dade group D(C_{P} (g)) is equal to the number of conjugacy classes of non-cyclic subgroups of C_{P} (g), and Corollary 4.6 follows by the equality of (1) and (2) of Corollary 4.5. □

Corollary 4.7. Let P be an abelian p-group. Then

rankB^{c}(P) − rankR_{ℚ}(D(P)) = |P | dim_{ℚ} ℚD(P).

Proof. The number of the elements of [P \P^{c}] is equal to |P |, because P is an abelian group. The equality of (1) and (2) of Corollary 4.5 completes Corollary 4.7. □

Corollary 4.8. Let P be a cyclic p-group. Then

rankB^{c}(P) = rankR_{ℚ}(D(P)).

Proof. Since C_{P} (g) for any element g ∈ P is a subgroup of the cyclic p-group P, it is immediate from Corollary 4.6. □

4.9. Abelian p-groups. If P is a finite abelian p-group, it is a direct sum of cyclic subgroups, of orders p^{λ_{1}}, p^{λ_{2}}, …, p^{λ_{r}}, say, where we may suppose that λ_{1} ≥ λ_{2} ≥ … ≥ λ_{r}. A partition is any sequence

λ = (λ_{1}, λ_{2}, …, λ_{r}, …)

of non-negative integers in decreasing order:

λ_{1} ≥ λ_{2} ≥ … ≥ λ_{r} ≥ …

and containing only finitely many non-zero terms. The sequence of exponents λ = (λ_{1}, … λ_{r}, 0, …) of the orders of cyclic subgroups of P is a partition, called the type of P, which describes P up to isomorphism. We write l(λ) = r for the length of the partition λ. The sum

λ_{1} + λ_{2} + …

of the terms of λ is denoted by |λ|. If λ and μ are partitions, we write λ ⊃ μ to mean that λ_{i} ≥ μ_{i} for all i ≥ 1. A combinatorial method of the enumeration of the number of subgroups of P is well-known (see, for instance, [Bu87]).

Theorem 4.10. Let P be an abelian p-group of type λ = (λ_{1}, … λ_{r}). Then the number of subgroups of type μ = (μ_{1}, … μ_{s}) is

∏_{i≥1} p^{μ′_{i+1}(λ′_{i}−μ′_{i})}\left[{λ′_{i} − μ′_{i+1} \atop μ′_{i} − μ′_{i+1}}\right]_{p}_{,}

where λ′ and μ′ are the conjugate partitions of λ and μ, respectively, and \left[{n \atop k}\right]_{p} is the p-binomial coefficient.

Corollary 4.11. Let P be an abelian p-group of type λ. Then

dim_{ℚ} ℚD(P) = ∑_{{λ⊃μ \atop l(μ)>1}} ∏_{i≥1} p^{μ′_{i+1}(λ′_{i}−μ′_{i})}\left[{λ′_{i} − μ′_{i+1} \atop μ′_{i} − μ′_{i+1}}\right]_{p}

Proof. An abelian p-group Q of type μ is isomorphic to a subgroup of P if and only if λ ⊃ μ. Since a subgroup of type μ of P is cyclic if and only if l(μ) = 1 or l(μ) = 0, Theorem 3.9 completes Corollary 4.11. □

Corollary 4.12. Let P be an abelian p-group of type λ. Then

rankB^{c}(P) − rankR_{ℚ}(D(P)) = p^{|λ|} ∑_{{λ⊃μ \atop l(μ)>1}} ∏_{i≥1} p^{μ′_{i+1}(λ′_{i}−μ′_{i})}\left[{λ′_{i} − μ′_{i+1} \atop μ′_{i} − μ′_{i+1}}\right]_{p}

Proof. Corollary 4.7 and Corollary 4.11 show Corollary 4.12. □

4.13. Some small 2-groups. We summarize basic facts on the structure of the centralizers of the representative of a conjugacy class of dihedral, semi-dihedral and generalized quaternion 2-groups (see, for instance, III.17 of [Er90]). In the rest of the paper, we always denote by z the central elements of order 2 of the group considered. Suppose that

D_{2^{n}} = ⟨x, y|x^{2^{n−1}} = y^{2} = 1, y^{−1}xy = x^{−1}⟩

is a dihedral group of order 2^{n} (n ≥ 2). Then the centralizers of 1 and z are D_{2^{n}}. The centralizers of y and xy are Klein four groups. The centralizers of the representative of the other 2^{n−2}−1 conjugacy classes are cyclic subgroups (n ≥ 3). Suppose that

SD_{2^{n}} = ⟨x, y|x^{2^{n−1}} = y^{2} = 1, y^{−1}xy = x^{−1+2^{n−2}}⟩

is a semi-dihedral group of order 2^{n} (n ≥ 4). Then the centralizers of 1 and z are SD_{2^{n}}. The centralizer of y is a Klein four group. The centralizers of the representative of the other 2^{n−2} conjugacy classes are cyclic subgroups. Suppose that

Q_{2^{n}} = ⟨x, y|x^{2^{n−2}} = y^{2}, y^{4} = 1, y^{−1}xy = x^{−1+2^{n−2}}⟩

is a generalized quaternion group of order 2^{n} (n ≥ 3). Then the centralizers of 1 and z are Q_{2^{n}}. The centralizers of the representative of the other 2^{n−2} + 1 conjugacy classes are cyclic subgroups.

Corollary 4.14. Let P be a dihedral group D_{2^{n}} of order 2^{n} (n ≥ 2). Then

rankB^{c}(P) − rankR_{ℚ}(D(P)) = 4n − 4.

Proof. By Corollary 4.5, we have

rankB^{c}(P) − rankR_{ℚ}(D(P)) = ∑_{g∈[P \P^{c}]} dim_{ℚ} ℚD(C_{P} (g)).

Since the Dade group of a cyclic group is a torsion group by (1) of Theorem 3.3, the structures of the centralizer of an element of P mentioned above show that the right hand side of the equation is equal to

dim_{ℚ} ℚD(C_{P} (1))+dim_{ℚ} ℚD(C_{P} (z))+dim_{ℚ} ℚD(C_{P} (y))+dim_{ℚ} ℚD(C_{P} (xy)).

Moreover, (2) of Theorem 3.3 shows that the number is equal to

2 dim_{ℚ} ℚD(P) + 2 dim_{ℚ} ℚD(D_{4}) = 2(2n − 3) + 2 · 1 = 4n − 4,

and Corollary 4.14 follows. □

Corollary 4.15. Let P be a semi-dihedral group SD_{2^{n}} of order 2^{n} (n ≥ 4). Then

rankB^{c}(P) − rankR_{ℚ}(D(P)) = 4n − 7.

Proof. The same argument as for D_{2^{n}} and (2) and (3) of Theorem 3.3 show that the difference rankB^{c}(P) − rankR_{ℚ}(D(P)) is equal to

dim_{ℚ} ℚD(C_{P} (1)) + dim_{ℚ} ℚD(C_{P} (z)) + dim_{ℚ} ℚD(C_{P} (y))
= 2 dim_{ℚ} ℚD(P) + dim_{ℚ} ℚD(D_{4})
= 2(2n − 4) + 1
= 4n − 7,

and Corollary 4.15 follows. □

Corollary 4.16. Let P be a generalized quaternion group Q_{2^{n}} of order 2^{n} (n ≥ 3). Then

rankB^{c}(P) − rankR_{ℚ}(D(P)) = 4n − 10.

Proof. The same argument as for D_{2^{n}} and (4) and (5) of Theorem 3.3 show that the difference rankB^{c}(P) − rankR_{ℚ}(D(P)) is equal to

dim_{ℚ} ℚD(C_{P} (1)) + dim_{ℚ} ℚD(C_{P} (z))
= 2 dim_{ℚ} ℚD(P)
= 2(2n − 5)
= 4n − 10,

and Corollary 4.16 follows. □

Acknowledgments

I am deeply indebted to Serge Bouc for his helpful suggestions and invaluable advice so generously given during the seminar at Université de Picardie-Jules Verne. I would like to express my gratitude to Hiroshi Horimoto for helpful discussions and comments on the seminar at Toyama National College of Techonology.

References

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[Od07] F. Oda, Crossed Burnside rings and Bouc’s construction of Green functors, J. Algebra 315 (2007), 18–30.

[OY01] F. Oda and T. Yoshida, Crossed Burnside rings I. The Fundamental Theorem, J. Algebra 236 (2001), 29–79.

[OY04] F. Oda and T. Yoshida, Crossed Burnside rings II. The Dress construction of a Green functor, J. Algebra 282 (2004), 58–82.

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[Th06] J. Thévenaz, Endo-permutation modules, a guide tour, Group representation theory, EPFL Press, Lausanne, 2007, 115-147.

(F. Oda) Department of Liberal Arts, Toyama National College of Technology, Toyama 939-8630, Japan

  E-mail address: oda@toyama-nct.ac.jp