THE GENERALIZED BURNSIDE RING WITH RESPECT TO p-CENTRIC SUBGROUPS

FUMIHITO ODA

Abstract. Let 𝔛 be the set of all p-centric subgroups of a finite group G and a prime p. This paper shows that the certain submodule Ω(G, 𝔛)_{(p)} of the Burnside ring Ω(G)_{(p)} of G over the localization ℤ_{(p)} of ℤ at p has a unique ring structure such that the mark homomorphism φ_{(p)} relative to 𝔛 is an injective homomorphism. A key lemma of this paper is that 𝔛 satisfies the condition (C)_{p} that is discussed by [Yo90]. Díaz and Libman showed that certain ring A^{p-cent}(G)_{(p)} is isomorphic to the Burnside ring of the fusion system associated to G and a Sylow p-subgroup in [DL07]. This paper shows that A^{p-cent}(G)_{(p)} is isomorphic to Ω(G, 𝔛)_{(p)}.

1. Introduction

Let G be a finite group and let p be a prime. Let 𝔛 be a family of subgroups of G such that it is closed under taking G-conjugation. In [Yo90], some conditions such that the Rsubmodule R ⊗_{ℤ} Ω(G, 𝔛) of ordinary Burnside algebra R ⊗_{ℤ} Ω(G) over a commutative ring R has ring structure are considered. If 𝔛 satisfies the condition (C)_{p} (see 3.6 of [Yo90]), then Ω(G, 𝔛)_{(p)} has a canonical ring structure (see 3.11 of [Yo90]). The ring Ω(G, 𝔛)_{(p)} was called a generalized Burnside ring with respect to 𝔛 in [Yo90], [Od96] and [OY01]. Recently, there are some generalized version of Burnside ring for some mathematical objects. For instance the ring Ω(G, 𝔛) is called a partial Burnside ring relative to 𝔛 in [Ta06], if Ω(G, 𝔛) is a subring of the ordinary Burnside ring Ω(G).

The purpose of this paper is to show that if 𝔛 is the set of all p-centric subgroups of G then Ω(G, 𝔛)_{(p)} is a generalized Burnside ring with respect to 𝔛. A key lemma (Lemma 3.2) of the main results (Theorem 3.10 and Corollary 3.11) of this paper is used to show that the family 𝔛 satisfies the condition (C)_{p} that is discussed in [Yo90]. The lemma is used by Sawabe’s work on the reduced Lefschetz module (see Proposition 6 of [Sa06]). The family 𝔛 of all p-centric subgroups of G is the first example such that it satisfies the condition (C)_{p} but not closed under taking subgroups.

Díaz and Libman showed that the ring Ω(G, 𝔛)_{(p)} is isomorphic to the Burnside ring A(F)_{(p)} of the fusion system associated to G and a Sylow p-subgroup in [DL07]. They showed some properties of A(F)_{(p)} for any saturated fusion system F (Theorem 1.4 and Theorem 1.5 of [DL07]). Assuming Theorem 1.6 of [DL07], our Theorem 3.13 and 3.14 give alternative proofs in a particular case of Theorem 1.4 and 1.5 there. However, those are special case of their original results.

The paper is organized as follows: Section 2 recalls some definitions and results from the theory of generalized Burnside ring of [Yo90]. Section 3 is an application to the all p-centric subgroups of G, showing main results, relationship between our results and those of [DL07].

2000 Mathematics Subject Classification. Primary 20C15, Secondary 20C20.

Key words and phrases. Burnside rings, generalized Burnside rings, p-centric subgroups.

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

2. The generalized Burnside ring

2.1. Notation. If G is a finite group and p a prime, denote by G_{p} a Sylow p-subgroup of G, by |G|_{p} the order of G_{p} and by |G|_{p′} is the quotient |G|/|G|_{p}. Let 𝔛 be a family of subgroups of a finite group G such that it is closed under G-conjugation. If X is a finite G-set, denote by [X] the isomorphism class of finite G-sets containing X. Denote by W H the quotient group N_{G}(H)/H. If H is a subgroup of G, denote by (H) the G-conjugacy class {^{g}H|g ∈ G}, where gH = gHg^{−1}. If g ∈ G is an element of G, denote by ⟨g⟩ the cyclic group generated by g.

2.2. The mark homomorphism. Let Ω(G, 𝔛) be the submodule of the ordinary Burnside ring Ω(G) of a finite group G generated by elements [G/H] for H ∈ 𝔛. Then Ω(G, 𝔛) is a free ℤ-module with basis {[G/H]|(H) ∈ C(𝔛)}, where C(𝔛) is the set of the G-conjugacy classes of a family 𝔛 of subgroups of G. The direct product ∏_{(S)∈C(𝔛)} ℤ of copies of the ring ℤ will be denoted by Ω̃(G, 𝔛). Let φ_{S} denote the additive map from Ω(G, 𝔛) to ℤ defined by [G/H] ↦ |(G/H)^{S}|, where |(G/H)^{S}| is the cardinality of the S-fixed points of the G-set G/H for a subgroup S of G. Thus we have an additive homomorphism relative to 𝔛

φ := (φ_{S})_{(S)} : Ω(G, 𝔛) → Ω̃(G, 𝔛) : x ↦ (φ_{S}(x))

and φ is called a mark homomorphism. For any element x ∈ Ω(G, 𝔛) and any subgroup S in 𝔛, we often write x(S) = φ_{S}(x). For a prime p, let ℤ_{(p)} be the localization of ℤ at p :

ℤ_{(p)} := {a/b|a ∈ ℤ, b ∈ ℤ\pℤ} ⊆ ℚ.

For a ℤ-module M, we set M_{(p)} := ℤ_{(p)} ⊗_{ℤ} M, and denote by φ_{(p)} : Ω(G, 𝔛)_{(p)} → Ω̃(G, 𝔛)_{(p)} the map induced by the mark homomorphism φ.

2.3. Condition (C)_{p}. Let p be a prime. We use the notation for a subgroup H of G that \overline{H} := ∩\{S ∈ 𝔛|H ≤ S\}. We put \overline{H} = G if there is no element U ∈ 𝔛 containing H. In [Yo90] Yoshida introduced the following condition:

(C)_{p} gS ∈ (W S)_{p}, S ∈ 𝔛 ⇒ \overline{⟨g⟩S} ∈ 𝔛, where (W S)_{p} is a Sylow p-subgroup of W S.

2.4. The Cauchy-Frobenius homomorphism. The direct product ∏_{(S)∈C(𝔛)}(ℤ/|W S|ℤ) of the quotient groups will be denoted by Obs(G, 𝔛). The Cauchy-Frobenius homomorphism ψ_{(p)} from Ω̃(G, 𝔛)_{(p)} to Obs(G, 𝔛)_{(p)} is defined by

(y(S))_{(S)} ⟼ ( ∑_{gS∈(W S)_{p}} y(\overline{⟨g⟩S}) mod |W S|_{p})_{(S)}_{.}

Theorem 2.5. (Yoshida [Yo90] 3.10, 3.11). Let p be a prime. Then under the condition (C)_{p}, the following sequence of ℤ_{(p)}-modules is exact:

(2.1)  0 ⟶ Ω(G, 𝔛)_{(p)} ⟶^{φ_{(p)}} Ω̃(G, 𝔛)_{(p)} ⟶^{ψ_{(p)}} Obs(G, 𝔛)_{(p)} ⟶ 0.

Moreover, Ω(G, 𝔛)_{(p)} has a unique ring structure such that φ_{(p)} is a ring homomorphism.

Remark 2.6. If x and y are elements in Ω(G, 𝔛)_{(p)}, denote by x • y the product of x and y over Ω(G, 𝔛)_{(p)} defined by

x • y := (φ_{(p)})^{−1} (φ_{(p)}(x)φ_{(p)}(y)) .

Definition 2.7. Let R be a commutative ring. The R-module R⊗_{ℤ}Ω(G, 𝔛) is called a generalized Burnside ring with respect to 𝔛 provided it has ring structure with identity element such that the homomorphism

1 ⊗ φ : R ⊗_{ℤ} Ω(G, 𝔛) → R⊗_{ℤ}Ω̃(G, 𝔛)

is an injective ring homomorphism.

Remark 2.8. If 𝔛 satisfies the condition (C)_{p}, and if the commutative ring R is p-torsion free and |G|_{p′} is invertible in R, then by Theorem 2.5, R ⊗_{ℤ} Ω(G, 𝔛), particularly Ω(G, 𝔛)_{(p)} is a generalized Burnside ring with respect to 𝔛.

2.9. Example. Let G be the symmetric group of 3-letters generated by the permutations s := (1, 2, 3) and t := (1, 2). Let C be the cyclic 2-subgroup generated by t and let 𝔛 be the family {^{g}C|g ∈ G}. Then 𝔛 satisfies the condition (C)_{2}, Ω(G, 𝔛)_{(2)} is a generalized Burnside ring with respect to 𝔛 by Theorem 2.5. Since we have C\G/C = CeC ∪ CsC,

[G/C][G/C] = [G/C × G/C] = [G/C ∩^{e}C] + [G/C ∩^{s}C] = [G/C] + [G/1]

over the ordinary Burnside ring Ω(G). Therefore the product [G/C][G/C] in Ω(G) is not contained in Ω(G, 𝔛)_{(2)}. On the other hand, the product [G/C] • [G/C] over Ω(G, 𝔛)_{(2)} is equal to [G/C], because

[G/C] • [G/C] = (φ_{(2)})^{−1} ((φ_{(2)}([G/C])) · (φ_{(2)}([G/C])))
= (φ_{(2)})^{−1} (|(G/C)^{C}| · |(G/C)^{C}|)
= (φ_{(2)})^{−1} (1 · 1)
= (φ_{(2)})^{−1} (1)
= [G/C].

Hence [G/C] is the identity element of Ω(G, 𝔛)_{(2)} and [G/C] • [G/C] in Ω(G, 𝔛)_{(2)} is not equal to [G/C][G/C] in Ω(G). Those ring structures are different.

Proposition 2.10. Let • be the multiplication of Ω(G, 𝔛)_{(p)} defined by Remark 2.6. Then x • y = xy in Ω(G)_{(p)} for x, y ∈ Ω(G, 𝔛)_{(p)} if and only if 𝔛 is closed by intersection.

Proof : If x • y = xy in Ω(G)_{(p)} for x, y ∈ Ω(G, 𝔛)_{(p)}, then we have that

[G/H] • [G/K] = [G/H][G/K] = ∑_{g∈[H\G/K]}[G/H ∩^{g}K]

for any H, K ∈ 𝔛 and Ω(G, 𝔛)_{(p)} must contain ∑_{g∈[H\G/K]}[G/H ∩ ^{g}K]. Hence the family 𝔛 must contain H ∩^{g}K for any g ∈ G.

On the other hand, if 𝔛 is closed by intersection, then Ω(G, 𝔛) contains ∑_{g∈H\G/K}[G/H∩^{g}K]. Since ∑_{g∈H\G/K}[G/H ∩^{g}K] = [G/H][G/K] in Ω(G), Ω(G, 𝔛)_{(p)} contains [G/H][G/K]. Then we have [G/H]•[G/K] = [G/H][G/K], because φ_{(p)}([G/H])·φ_{(p)}([G/K]) = φ_{(p)}([G/H][G/K]). Proposition 2.10 follows by Remark 2.6. □

2.11. Condition (C′)_{p}. In [Yo90] Yoshida introduced the following condition: (C′)_{p} gS ∈ (W S)_{p}, S ∈ 𝔛 ⇒ ⟨g⟩S ∈ 𝔛. If 𝔛 satisfies the condition (C′)_{p}, then the condition (C)_{p} holds trivially, and so Ω(G, 𝔛)_{(p)} is a generalized Burnside ring with respect to 𝔛 by Theorem 2.5.

2.12. The equivalence relation ∼_{p}. We assume the condition (C)_{p}, so that by Theorem 2.5, Ω(G, 𝔛)_{(p)} is a generalized Burnside ring with respect to 𝔛. Let ∼_{p} be the equivalence relation on C(𝔛) generated by the relation

(2.2)  (\overline{⟨g⟩S}) ∼_{p} (S) for S ∈ 𝔛, gS ∈ (W S)_{p}.

This relation can be lifted to 𝔛, that is, S ∼_{p} T if and only if (S) ∼_{p} (T).

2.13. Primitive idempotents. If Q ∈ 𝔛, denoted by e_{Q}^{p} the sum ∑ e_{H} of idempotents e_{H} ∈ ℚ ⊗_{ℤ} Ω(G, 𝔛), where (H) ∈ C(𝔛) with H ∼_{p} Q (see Theorem 4.2 of [Yo90]). Let μ_{𝔛} : 𝔛 × 𝔛 → ℤ be the Möbius function on the poset 𝔛 with the order relation by inclusion. Yoshida computed the idempotent e_{Q}^{p} of Ω(G, 𝔛)_{(p)} for Q ∈ 𝔛 as follows:

e_{Q}^{p} = ∑_{(D)∈C(𝔛)} \frac{1}{|W D|} ( ∑_{H∼_{p}Q} μ_{𝔛}(D, H)) [G/D].

Theorem 2.14. (Yoshida [Yo90] 4.12). Under the condition (C)_{p}, the element e_{Q}^{p} is a primitive idempotent of Ω(G, 𝔛)_{(p)}, and conversely any primitive idempotent of Ω(G, 𝔛)_{(p)} has this form. Thus the set of primitive idempotents of Ω(G, 𝔛)_{(p)} is bijectively corresponding to the equivalence classes of the equivalence relation ∼_{p} in C(𝔛).

Yoshida determined the prime ideals of the generalized Burnside ring Ω(G, 𝔛)_{(p)} by the method of Dress.

Theorem 2.15. (Yoshida [Yo90] 5.12 and 5.13). Assume that the condition (C)_{p} holds for 𝔛. Then there is a bijective correspondence between the connected components of prime ideals of Ω(G, 𝔛)_{(p)} and ∼_{p}-equivalence classes. Moreover, the set {(D) ∈ C(𝔛)|W D is a p′-group } is a complete set of representatives of ∼_{p}-equivalence classes.

3. p-centric subgroups

Definition 3.1. A p-subgroup P of G is said to be p-centric if the centralizer C_{G}(P) is the product of the center of P and a group of order prime to p. This is equivalent to the condition that the center Z(P) of P be a Sylow p-subgroup of C_{G}(P). The subgroup P is p-centric if and only if Z(P) = C_{S}(P) for any Sylow subgroup S of G containing P. Denote by C_{p}(G) the set of all p-centric subgroups of G.

Lemma 3.2. (e.g. see Proposition 6 of [Sa06]).Let 𝔛 be C_{p}(G). If Q ≤ P for Q ∈ 𝔛 and P is a p-subgroup of G, then P ∈ 𝔛. In particular, any Sylow p-subgroup is p-centric.

Proof : Suppose that Q ≤ P for Q ∈ 𝔛 and P ≤ G is a p-subgroup. Then C_{G}(P) ≤ C_{G}(Q), and thus any p-element g in C_{G}(P) is contained in C_{G}(Q). Since Q is p-centric, we have x ∈ Q ≤ P, which implies that P is p-centric. □

Proposition 3.3. Let 𝔛 be C_{p}(G). Then 𝔛 satisfies the condition (C)_{p}.

Proof : Let P ∈ 𝔛 and gP ∈ (W P)_{p}. The groups P and (W P)_{p} are p-groups, so is ⟨g⟩. Since P ≤ ⟨g⟩P and ⟨g⟩P is a p-group, ⟨g⟩P ∈ 𝔛 by Proposition 3.2. Hence 𝔛 satisfies the condition (C′)_{p}, hence also the condition (C)_{p}. □

Proposition 3.3 and Proposition 2.5 show the following corollary.

Corollary 3.4. Let 𝔛 be C_{p}(G). Then Ω(G, 𝔛)_{(p)} is a generalized Burnside ring with respect to 𝔛. In particular, the ring Ω(G, 𝔛)_{(p)} has an identity element.

If 𝔛 = C_{p}(G), then we can determine the complete set of representatives of ∼_{p}-equivalence classes.

Proposition 3.5. Let 𝔛 be C_{p}(G). Then the Sylow p-subgroups are the only p-centric subgroups of G such that the order |W Q| is not divisible by p for Q ∈ 𝔛.

Proof :Let S be a Sylow p-subgroup of G. Then |W S| is not divisible by p. Let Q be a proper subgroup of S. Then |W Q| is divisible by p because Q is a non maximal p-subgroup of G. □

Theorem 2.15, Proposition 3.5 and 2.13 show the following corollary.

Corollary 3.6. Let 𝔛 be C_{p}(G). Then the generalized Burnside ring Ω(G, 𝔛)_{(p)} with respect to 𝔛 is a local ring. The identity element of Ω(G, 𝔛)_{(p)} is

e_{G_{p}}^{p} = ∑_{(D)∈C(𝔛)} \frac{1}{|W D|} ( ∑_{H∼_{p}G_{p}} μ_{𝔛}(D, H)) [G/D].

Remark 3.7. Let S_{p}(G) be the family of all p-subgroups of G and let C_{p}^{n}(G) be the family S_{p}(G)\C_{p}(G) of non-p-centric p-subgroups of G. Since S_{p}(G) satisfies the condition (C′)_{p}, it satisfies the condition (C)_{p}. By Theorem 2.5, Ω(G, S_{p}(G))_{(p)} is a generalized Burnside ring. The module Ω(G, C_{p}^{n}(G))_{(p)} is an ideal of Ω(G, S_{p}(G))_{(p)}. Díaz and Libman introduced a quotient ring A^{p-cent}(G)_{(p)} whose free basis is C(C_{p}(G)) as ℤ_{p}-module (see 5.1 of [DL07]). Denote by H^{g} the subgroup g^{−1}Hg of G for g ∈ G. The product of basis elements (P) and (Q) is ∑_{g}(Q^{g} ∩ P) where the sum ranges through the double cosets QgP such that Q^{g} ∩ P is p-centric. We can regard the quotient Ω(G, S_{p}(G))_{(p)}/Ω(G, C_{p}^{n}(G))_{(p)} as a ℤ_{(p)}-module A^{p-cent}(G)_{(p)} by identifying a representative [G/P] of a basis of the quotient ring with an element (P), where P ∈ C_{p}(G). Moreover, the map is a ring isomorphism (see Corollary 3.11).

Lemma 3.8. Let Q ∈ C_{p}^{n}(G). Then φ_{P} ([G/Q]) = 0 for any P ∈ C_{p}(G).

Proof: For a pair H and K of subgroups of G φ_{K}([G/H]) = ♯Map_{G}(G/K, G/H), where Map_{G}(G/K, G/H) is the set of G-maps from G/K to G/H. Hence φ_{K}([G/H]) ≠ 0 if and only if K ≤_{G} H. If Q ∈ C_{p}^{n}(G) then P is not G-conjugate subgroup of Q for any P ∈ C_{p}(G). So φ_{P} ([G/Q]) = 0. □

Lemma 3.9. Let Q ∈ S_{p}(G). If φ_{P} ([G/Q]) = 0 for any P ∈ C_{p}(G), then Q ∈ C_{p}^{n}(G).

Proof: Let Q ∈ C_{p}(G). Then φ_{Q}([G/Q]) = |W Q| ≠ 0. □

Theorem 3.10. The generalized Burnside ring Ω(G, C_{p}(G))_{(p)} is isomorphic to the quotient ring Ω(G, S_{p}(G))_{(p)}/Ω(G, C_{p}^{n}(G))_{(p)}.

Proof: We give a linear map ρ from Ω(G, S_{p}(G))_{(p)} to Ω(G, C_{p}(G))_{(p)}. Let x be an element of Ω(G, S_{p}(G))_{(p)}. We note that φ_{Q}([G/P]) = 0 for Q ∈ C_{p}(G) and P ∈ C_{p}^{n}(G) by Lemma 3.8. Let χ be the element (x(Q))_{(Q)∈C(C_{p}(G))} in Ω̃(G, C_{p}(G))_{p}, where x(Q) is the image of x by

Ω(G, C_{p}(G))_{(p)} ↪ Ω(G, S_{p}(G))_{(p)} →^{φ_{Q}} ℤ_{(p)}. Since C_{p}(G) satisfies the condition (C’)_{p} by the proof of Proposition 3.3, \overline{⟨g⟩Q} = ⟨g⟩Q for any S ∈ C_{p}(G). We have that

ψ_{(p)}(χ) = ∑_{gQ∈(W Q)_{p}} χ(\overline{⟨g⟩Q}) = ∑_{gQ∈(W Q)_{p}} χ(⟨g⟩Q) ≡ 0 ( mod |W Q|_{p})

for any Q ∈ C_{p}(G) by the lemma of Cauchy-Frobenius. By Theorem 2.5, we have that χ is contained in Ω(G, C_{p}(G))_{(p)}. The linear map ρ is obtained by ρ(x) = χ. Lemma 3.8 and Lemma 3.9 show that Ker(ρ) = Ω(G, C_{p}^{n}(G))_{(p)}. The map ρ makes the following diagram of ℤ_{(p)}-modules commutative:

[figure: Ω(G, S_{p}(G))_{(p)}; ρ; Ω(G, C_{p}(G))_{(p)}; Ω̃(G, S_{p}(G))_{p}; φ′_{(p)}; π; Ω̃(G, C_{p}(G))_{(p)},; φ_{(p)}]

where π is the projection.

We have to show that ρ is a ring homomorphism. Let x, y ∈ Ω(G, S_{p}(G))_{(p)}. We show that ρ(x) • ρ(y) = ρ(xy) in Ω(G, C_{p}(G))_{(p)}. We have that

ρ(x) • ρ(y) = φ_{(p)}^{−1}(φ_{(p)}(ρ(x))φ_{(p)}(ρ(y))) = φ_{(p)}^{−1}(πφ′_{(p)}(x)πφ′_{(p)}(y)) = φ_{(p)}^{−1}πφ′_{(p)}(xy) = ρ(xy)

by Remark 2.6 and the commutativity of the diagram above. □

Corollary 3.11. The generalized Burnside ring Ω(G, C_{p}(G))_{(p)} is isomorphic to A^{p-cent}(G)_{(p)}.

Proof: There exists an isomorphisms of ℤ_{p}-module from Ω(G, C_{p}(G))_{(p)} to A^{p-cent}(G)_{(p)} defined by sending [G/P] to (P) for P ∈ C_{p}(G). Since Ω(G, C_{p}(G))_{(p)} is commutative and [G/Q ∩^{g}P] = [G/Q^{g} ∩ P] for any g ∈ G, so it suffices to show that [G/Q] • [G/P] = ∑_{g∈[Q\C_{p}(G)/P]}[G/Q ∩^{g}P], where [Q\C_{p}(G)/P] is the subset of [Q\G/P] consisting of elements g such that Q ∩ ^{g}P ∈ C_{p}(G), by the note on the product of A^{p-cent}(G)_{(p)} in Remark 3.7. Let ρ be the surjective ring homomorphism from Ω(G, S_{p}(G))_{(p)} to Ω(G, C_{p}(G))_{(p)} in Theorem 3.10. Then

[G/Q] • [G/P] = ρ([G/Q]) • ρ([G/P]) = ρ([G/Q][G/P]) = ρ ( ∑_{g∈[Q\G/P]}[G/Q ∩^{g}P])

= ∑_{g∈[Q\G/P]} ρ([G/Q ∩^{g}P]) = ∑_{g∈[Q\C_{p}(G)/P]} ρ([G/Q ∩^{g}P]) = ∑_{g∈[Q\C_{p}(G)/P]}[G/Q ∩^{g}P],

because Ker(ρ) = Ω(G, C_{p}^{n}(G)). □

In this paper, we use the following theorem.

Theorem 3.12. (Díaz and Libman [DL07] Theorem 5.11). Let F be the fusion system associated to a finite group G and a Sylow p-subgroup S. Then the rings A(F)_{(p)} and A^{p-cent}(G)_{(p)} are isomorphic.

Then we have the following theorems of Díaz and Libman those are restricted to the case of their assumption.

Theorem 3.13. Let F be the fusion system associated to a finite group G and a Sylow p-subgroup S. Then the ring A(F)_{(p)} has a unit.

Proof: Theorem 3.12, Corollary 3.11 and Corollary 3.4 show the theorem. □

Theorem 3.14. Let F be the fusion system associated to a finite group G and a Sylow p-subgroup S. Then the ring A(F)_{(p)} is local.

Proof: Theorem 3.12, Corollary 3.11 and Corollary 3.6 show the theorem. □

References

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[OY01] F. Oda and T. Yoshida, On the generalized Burnside ring with respect the Young subgroups of the symmetric group, J. Algebra 236 (2001), 349–354.

[Sa06] M. Sawabe, On the reduced Lefschetz module and the centric p-radical subgroups II., J. London Math. Soc.(2) 73 (2006), 126–140.

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[Yo90] T. Yoshida, The generalized Burnside ring of a finite group, Hokkaido Math. J. 19 (1990), 509–574.

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

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