A COLLECTION OF SUBGROUPS FOR THE GENERALIZED BURNSIDE RING

FUMIHITO ODA AND MASATO SAWABE

Abstract. In this paper, we will introduce a new collection of subgroups; which induces a generalized Burnside ring. This collection arises from the normalizers of certain pradical subgroups.

1. Introduction

Let G be a finite group, and D be a collection of subgroups of G; which is the union of G-conjugate classes of subgroups of G. Then, for a commutative ring R, the ordinary Burnside algebra R⊗_{ℤ}Ω(G) contains an R-submodule R⊗_{ℤ}Ω(G, D). In his paper [Yd90], Tomoyuki Yoshida introduced into the R-module R⊗_{ℤ}Ω(G, D) the notion of a generalized Burnside ring over R with respect to D (see Definition 1). However it seems to be a little hard to verify that whether or not R⊗_{ℤ}Ω(G, D) can be realized as a generalized Burnside ring. So Yoshida in particular focused on the localization ℤ_{(p)} of ℤ at a prime p as a coefficient ring, and gave a sufficient condition (C)_{p} on D for making ℤ_{(p)} ⊗_{ℤ} Ω(G, D) a generalized Burnside ring (see Sect. 2.2). But nevertheless, there are so far only a few interesting examples of D satisfying (C)_{p} such as p-centric subgroups in [Od08], Young subgroups of the symmetric group in [Yd90] and [OY01], line stabilizers of the general linear group in [Ta06]. From this reason, the purpose of this paper is to introduce a new collection N_{G}(𝔛) of subgroups of G, and to prove that under natural hypotheses, N_{G}(𝔛) satisfies (C)_{r} for any prime r. It follows that a ℤ-module Ω(G, N_{G}(𝔛)) itself is realized as a generalized Burnside ring over ℤ.

The paper is organized as follows: In Section 2, we will recall some of the theory of generalized Burnside rings (GBRs for short). In Section 3, standard collections of subgroups will be discussed. In particular, we will see that any collection of self-normalizing subgroups satisfies (C)_{r} for any prime r, and it follows that Ω(G, D) becomes a GBR over ℤ. In Section 4, we will define a collection N_{G}(𝔛) consisting of the normalizers of certain p-radical subgroups. Then any member of N_{G}(𝔛) is self-normalizing under some hypothesis (P). So from this fact, we have a GBR Ω(G, N_{G}(𝔛)) under (P) as mentioned above. On the other hand, we will establish another condition (Z) on D leading to the fact that any member of D is self-normalizing. Then we will show that, under hypotheses (P) and (W), our collection N_{G}(𝔛) satisfies the condition (Z) stronger than self-normalizing. Some examples of N_{G}(𝔛) will be given by using simple groups G such as the Mathieu group M_{24}, the Conway group Co_{1}, and the Monster M.

Throughout the paper, we use the following notation. Let p be a prime divisor of the order of a finite group G. For a subgroup H of G, denote by H_{p} and W_{G}(H) (or just W H for short), respectively, a Sylow p-subgroup of H and the quotient group N_{G}(H)/H called the Weyl group of H in G. Let (H) be the set of all G-conjugate subgroups of H; that is, (H) := {^{g}H | g ∈ G} where ^{g}H := gHg^{−1} for g ∈ G. If X is a finite G-set, denote by [X] the isomorphism class of finite G-sets containing X. Denote by |X| the cardinality of a finite set X. Let D be a collection of subgroups of G; which means that D is a family of subgroups of G closed under G-conjugation. The notation D(≤ H) (resp. D(≥ H)), for a subgroup H of G, stands for the set of all elements D of D such that D ≤ H (resp. D ≥ H). Denote by C(D) (resp. D/_{∼}) the set (resp. a set of representatives) of G-conjugacy classes of D.

Date: May 7, 2009.

Key words and phrases. Burnside ring, p-radical subgroups.

The first author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 18540059.

The second author was partially supported by MEXT Grant-in-Aid for Young Scientists (B) 19740005.

2. The generalized Burnside rings

In this section, we will recall, from [Yd90], some of the basis of the theory of generalized Burnside rings. Let D be a collection of subgroups of G. A finite G-set X is called a (G, D)-set if the stabilizer in G of any element x ∈ X lies in D. Let Ω(G, D) be the Grothendieck group of the category of (G, D)-sets, for relation given by disjoint union. Since a finite G-set is decomposed into the disjoint union of uniquely determined transitive G-sets, we have that Ω(G, D) is the free abelian group with basis {[G/H] | (H) ∈ C(D)}.

The ordinary Burnside ring Ω(G) is defined by Ω(G, Sgp(G)) for the totality Sgp(G) of all subgroups of G (namely, it is defined on all G-sets), with the multiplication coming from the Cartesian product. Sometimes Ω(G, D) for a proper sub-collection D of Sgp(G) will have a multiplication (which is, roughly speaking, a ”generalized Burnside ring”), and then this multiplication may, or may not, be inherited from the ordinary Burnside ring. In the following, we will explain more clearly.

2.1. The mark homomorphism. The direct product ∏_{(S)∈C(D)} ℤ of |C(D)|-copies of the ring ℤ will be denoted by Ω̃(G, D). This free ℤ-module also has a natural multiplication, arising from that in ℤ. Below, we define a map φ of ℤ-modules into it, and then we can ask when the multiplication can be pulled back to a multiplication also in the domain. Now let φ_{S}, for (S) ∈ C(D), denote the additive map from Ω(G, D) to ℤ defined by φ_{S}([X]) := |X^{S}| where X^{S} is the S-fixed points of the (G, D)-set X. Thus we have an additive homomorphism relative to D

φ := (φ_{S})_{(S)} : Ω(G, D) → Ω̃(G, D)

defined by

φ([X]) := (φ_{S}([X]))_{(S)∈C(D)}

for any [X] ∈ Ω(G, D), and φ is called a mark homomorphism. Note that the definition of φ is not restricted to any particular chosen collection D ⊆ Sgp(G). In fact, take D to be the whole Sgp(G) (namely, consider ”all G-sets”), but really this is the general case, not a special case.

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, D)_{(p)} → Ω̃(G, D)_{(p)} the map induced by the mark homomorphism φ. It is convenient to extend the notation above to M_{(∞)} := M and φ_{(∞)} := φ for p = ∞.

2.2. Condition (C)_{p} and (C′)_{p}. For a subgroup H of G, define a subgroup \overline{H} associated to both H and D as follows:

\overline{H} := \{ \begin{matrix}⋂_{S∈D(≥H)} S & if & D(≥ H) ≠ ∅, \\ G & if & D(≥ H) = ∅. \end{matrix}

It is not a priori clear whether \overline{H} should also be in D. Then the following conditions on D have been introduced by Yoshida[Yd90]. Let p be a prime. Recall that, for S ∈ D, (W S)_{p} is a Sylow p-subgroup of W S := N_{G}(S)/S.

(C)_{p} S ∈ D, gS ∈ (W S)_{p} ⟹ \overline{⟨g⟩S} ∈ D. (C′)_{p} S ∈ D, gS ∈ (W S)_{p} ⟹ ⟨g⟩S ∈ D.

Moreover, for p = ∞, Yoshida interpreted (C)_{p} as follows: (C)_{∞} S ∈ D, gS ∈ W S ⟹ ⟨g⟩S ∈ D.

Yoshida showed the relationship between (C)_{p} and (C)_{∞} in [Yd90].

Lemma 1. [Yd90, Lemma 3.7 (c)] A collection D of subgroups of G satisfies (C)_{∞} if and only if it satisfies (C)_{p} for all prime p.

Lemma 2. Suppose that a collection D of subgroups of G satisfies (C′)_{p} for a prime p. Then, for S ∈ D and gS ∈ (W S)_{p}, we have that ⟨g⟩S = \overline{⟨g⟩S}. In particular, the condition (C′)_{p} on D implies the condition (C)_{p} on D.

Proof: Straightforward. □

2.3. The Cauchy-Frobenius homomorphism. The obstruction group Obs(G, D) is defined by the direct product ∏_{(S)∈C(D)}(ℤ/|W S|ℤ) of the quotient groups ℤ/|W S|ℤ for (S) ∈ C(D). Then, assuming the condition (C)_{p}, the Cauchy-Frobenius homomorphism

ψ_{(p)} : Ω̃(G, D)_{(p)} ⟶ Obs(G, D)_{(p)}

is defined by

ψ_{(p)}(x) := ( ∑_{gS∈(W S)_{p}} m\overline{⟨g⟩S} mod |W S|_{p})_{(S)∈C(D)}

for any x = (m_{S})_{(S)} ∈ Ω̃(G, D)_{(p)} where m_{S} is a p-local integer at (S) ∈ C(D). Note that, by (C)_{p}, the group \overline{⟨g⟩S} in the definition of ψ_{(p)} lies in D always. It is convenient, under (C)_{∞}, to extend the notation above to ψ := ψ_{(∞)} for p = ∞ as ψ : Ω̃(G, D) ⟶ Obs(G, D) defined by

ψ(x) := ( ∑_{gS∈W S} m\overline{⟨g⟩S} mod |W S|)_{(S)∈C(D)}

for any x = (m_{S})_{(S)} ∈ Ω̃(G, D). The following is the exact nature of the obstruction group Obs(G, D).

Lemma 3. [Yd90, Theorem 3.10] Let p be a prime or ∞. Then under the condition (C)_{p}, the following sequence of ℤ_{(p)}-modules is exact:

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

In particular, under the condition (C)_{∞}, the following sequence of abelian groups is exact:

0 ⟶ Ω(G, D) ⟶^{φ} Ω̃(G, D) ⟶^{ψ} Obs(G, D) ⟶ 0.

Note that this Lemma shows that Obs(G, D) is therefore the obstruction to φ being an isomorphism; and that this fact will be used later at Lemma 7

Definition 1. [Yd90, Definition 3.12] Let R be a commutative unital ring; in other words, R is a commutative ring with the identity element. Suppose that an R-module R ⊗_{ℤ} Ω(G, D) satisfies the following two conditions:

(1) The map φ_{R} := 1 ⊗ φ : R ⊗_{ℤ} Ω(G, D) → R ⊗_{ℤ} Ω̃(G, D) is injective.

(2) The image Im(φ_{R}) of R ⊗_{ℤ} Ω(G, D) by φ_{R} is a unital subring of the commutative

ring R ⊗_{ℤ} Ω̃(G, D). Then we can define a multiplication ”•” on R ⊗_{ℤ} Ω(G, D) obtained by pulling back from the natural multiplication in a product of copies of ℤ; namely

• : R ⊗_{ℤ} Ω(G, D) × R ⊗_{ℤ} Ω(G, D) → R ⊗_{ℤ} Ω(G, D)

defined by

x • y = (φ_{R})^{−1} (φ_{R}(x)φ_{R}(y))

for x, y ∈ R ⊗_{ℤ} Ω(G, D). This makes R ⊗_{ℤ} Ω(G, D) a commutative unital ring, and we call this ring a generalized Burnside ring with respect to D of G over R. In this case, it is clear that φ_{R} becomes a ring homomorphism.

Remark 1. We should mention that the original definition, say A, stated in [Yd90, Definition 3.12] is as follows: ”The R-module R ⊗ Ω(G, D) is called a generalized Burnside ring with respect to D provided it has a unital ring structure with a multiplication ⋆ such that φ_{R} is an injective ring homomorphism”. However the multiplication ⋆ is uniquely determined by Definition 1 above. Indeed, for x, y ∈ R ⊗ Ω(G, D), we have that x • y = (φ_{R})^{−1}(φ_{R}(x)φ_{R}(y)) = (φ_{R})^{−1}(φ_{R}(x ⋆ y)) = x ⋆ y. Therefore Definition 1 implies the original A. The coverse is now clear.

Remark 2. The R-module R ⊗_{ℤ} Ω(G, D) is called a partial Burnside ring relative to D if R ⊗_{ℤ} Ω(G, D) is a ”subring” of the Burnside algebra R ⊗_{ℤ} Ω(G) of G over R (see [Ta06]).

Lemma 4. [Yd90, Theorem 3.11] (1) Under the condition (C)_{p}, Ω(G, D)_{(p)} is realized as a generalized Burnside ring.

(2) In particular, under the condition (C)_{∞}, Ω(G, D) is realized as a generalized Burnside ring. Furthermore, for a prime p, the two ring structures on Ω(G, D)_{(p)} defined by (1) and (2) coincide.

3. Some collections with (C)_{p} and the p-radical collection

In this section, we will recall some standard collections D of subgroups of G satisfying the condition (C)_{p}. Then, by Lemma 4, such collection D gives us a generalized Burnside ring Ω(G, D)_{(p)}. Furthermore, we will make some comments on the collection of p-radicals.

Let S_{p}(G) be the set of all non-trivial p-subgroups of G. For a member U of S_{p}(G), we say that U is p-radical (resp. p-centric) if O_{p}(N_{G}(U)) = U (resp. if any p-element in C_{G}(U) lies in U). Denote by B_{p}(G) (resp. C_{p}(G)) the set of all p-radicals of G but excluding the unique minimal element O_{p}(G) (resp. the set of all p-centrics of G). And we set B_{p}^{cen}(G) := B_{p}(G) ∩ C_{p}(G). Then it is known that there are homotopy equivalences between Δ(S_{p}(G)) andΔ(B_{p}(G)), and also Δ(C_{p}(G)) and Δ(B_{p}^{cen}(G)) (see e.g. [Be98, Theorem 6.6.6] and [Gr02, pp. 420, 421]). Recall that Δ(D), for a collection D, is a simplicial complex defined by strict inclusion-chains in the poset (D, ≤) as simplices.

Example 1. (1) Let D be a collection of subgroups of G closed under intersection, and suppose that the whole group G is contained in D. Then D satisfies (C)_{p} for any prime p. This is because that a subgroup \overline{⟨g⟩S} appearing in (C)_{p} is defined by the intersection of certain subgroups in D, or by the whole group G.

(2) Let D be a collection of self-normalizing subgroups S of G; that is, N_{G}(S) = S. Then since W S is trivial for any S ∈ D, it is clear that S ∈ D and gS ∈ (W S)_{p} imply ⟨g⟩S = S ∈ D for any prime p. Thus D satisfies (C′)_{p} for any prime p. It follows that (C)_{p} holds for D by Lemma 2.

(3) The collection S_{p}(G) satisfies (C′)_{p}. Indeed, for S ∈ S_{p}(G) and gS ∈ (W S)_{p}, the group ⟨g⟩S is nothing else but a non-trivial p-subgroup of G.

(4) The collection C_{p}(G) satisfies (C′)_{p}. It follows from the fact that C_{p}(G) is closed under super-p-groups; that is, the condition Q ≤ P for Q ∈ C_{p}(G) and P ∈ S_{p}(G) implies P ∈ C_{p}(G) (e.g. see Proposition 6 in [Sa06]). The generalized Burnside ring Ω(G, C_{p}(G))_{(p)} is investigated in [Od08]

Then it is natural to ask the next question that whether or not the collection B_{p}(G) (or B_{p}^{cen}(G)) satisfies (C)_{p}. However we can find the answer is ”No” from the case of G ≅ GL_{4}(2) ≅ A_{8} and p = 2. The details are described in the following.

3.1. Failure of (C)_{p} for B_{p}(G) and B_{p}^{cen}(G). Let V be a 4-dimensional vector space over the field of two elements, and set G = GL(V) ≅ GL_{4}(2). Then, by the Borel-Tits Theorem (c.f. [Be98, Theorem 6.8.4]), the collection D := B_{2}^{cen}(G) = B_{2}(G) gives us the set of all non-trivial unipotent radicals of parabolics of G. Let ({0} < V_{p} < V_{ℓ} < V_{π} < V) be a maximal flag of V, and I := {p, ℓ, π}. Consider the stabilizer of each V_{i} (i ∈ I); that is a maximal parabolic subgroup of G.

G_{p} := Stab_{G}(V_{p}) ≅ 2^{3} : L_{3}(2)
G_{ℓ} := Stab_{G}(V_{ℓ}) ≅ 2^{4} : (L_{2}(2) × L_{2}(2))
G_{π} := Stab_{G}(V_{π}) ≅ 2^{3} : L_{3}(2)

Furthermore, for i ∈ I and ∅ ≠ F ⊆ I, we set U_{i} := O_{2}(G_{i}) and U_{F} := ⟨U_{i} | i ∈ F ⟩. Then P := U_{pℓπ} is a Sylow 2-subgroup of G of order 2^{6}, and we have that D(≤ P) = {U_{F} | ∅ ≠ F ⊆ I}. In particular, {U_{p} ≅ 2^{3}, U_{ℓ} ≅ 2^{4}, U_{π} ≅ 2^{3}} forms the set of all minimal members of D contained in P. The following is more precise description of U_{F} ’s.

U_{pℓ} ≅ U_{p} : K_{1} ≅ U_{ℓ} : H_{1}, U_{pπ} ≅ U_{p} : K_{2} ≅ U_{π} : L_{1}, U_{ℓπ} ≅ U_{ℓ} : H_{2} ≅ U_{π} : L_{2}

where

B_{2}(W U_{p})/_{∼} = B_{2}(L_{3}(2))/_{∼} = {K_{1} ≅ 2^{2}, K_{2} ≅ 2^{2}, K_{12} ≅ D_{8}}
B_{2}(W U_{ℓ})/_{∼} = B_{2}(L_{2}(2) × L_{2}(2))/_{∼} = {H_{1} ≅ 2, H_{2} ≅ 2, H_{12} ≅ 2^{2}}
B_{2}(W U_{π})/_{∼} = B_{2}(L_{3}(2))/_{∼} = {L_{1} ≅ 2^{2}, L_{2} ≅ 2^{2}, L_{12} ≅ D_{8}}.

Lemma 5. Take an involution z ∈ K_{1} ∩ K_{2}. Then, for U_{p} ∈ D and zU_{p} ∈ (W U_{p})_{2}, we have that ⟨z⟩U_{p} = \overline{⟨z⟩U_{p}} ∉ D. In particular, neither the condition (C)_{2} nor (C′)_{2} holds for the collection D.

Proof: Looking at the orders, we have that |⟨z⟩U_{p}| = 2×|U_{p}| = 2^{4}, and that |U_{pℓ}∩U_{pπ}| = 2^{4}. On the other hand, ⟨z⟩U_{p} is contained in both U_{p}K_{1} = U_{pℓ} and U_{p}K_{2} = U_{pπ}; which implies that ⟨z⟩U_{p} ≤ \overline{⟨z⟩U_{p}} ≤ U_{pl} ∩ U_{pπ}. Then comparing the orders, H := ⟨z⟩U_{p} = \overline{⟨z⟩U_{p}} = U_{pl} ∩ U_{pπ}.

Suppose next that H lies in D. Then H must be G-conjugate to a unique representative U_{ℓ} of order 2^{4}. But since H properly contains U_{p}, this contradicts that U_{ℓ} is a minimal member of D. Therefore H ∉ D. This completes the proof. □

As we saw above, the collections B_{p}(G) and B_{p}^{cen}(G) themselves do not satisfy (C)_{p} in general. However, a collection of the ”normalizers” of some p-radicals works well; which is the main subject of this paper. In the next section, we will explain more clearly.

4. A collection N_{G}(𝔛) of subgroups

In this section, we will introduce, for a sub-collection 𝔛 of B_{p}(G), a collection N_{G}(𝔛) by taking the normalizers of certain p-radical subgroups. Then we will see that, under a natural hypothesis (P), N_{G}(𝔛) is a self-normalizing collection. This implies that Ω(G, N_{G}(𝔛)) is a generalized Burnside ring (see Example 1(2)). On the other hand, we will establish another hypothesis (Z) on a collection D; which makes D self-normalizing. Then we will show that, under suitable hypotheses (P) and (W), N_{G}(𝔛) satisfies the condition (Z) stronger than self-normalizing.

4.1. Standing notations and hypotheses. Let 𝔛 be a sub-collection of B_{p}(G). Denote by 𝔛_{min} the set of all minimal elements in 𝔛 with respect to inclusion-relation. For a fixed Sylow p-subgroup P of G, we set

𝔛_{min}(≤ P) = {U_{1}, U_{2}, …, U_{l}}.

Let I = {1, …, l} be an index set. For ∅ ≠ F ⊆ I, define U_{F} := ⟨U_{i} | i ∈ F ⟩. Then we introduce a collection N_{G}(𝔛) associated to both G and 𝔛 ⊆ B_{p}(G) by taking the normalizers of U_{F} ’s as follows:

N_{G}(𝔛) := {^{g}N_{G}(U_{F}) | g ∈ G, ∅ ≠ F ⊆ I}.

In the following, we establish two hypotheses on 𝔛 (in particular on U_{F} ’s); which were used in [Sa03]. They will play a crucial role in our present investigation too.

Hypothesis (W). Each U_{i} (1 ≤ i ≤ l) is weakly closed in P with respect to G. In other words, if ^{g}U_{i} ≤ P for some g ∈ G then ^{g}U_{i} = U_{i}.

Hypothesis (P). For any ∅ ≠ F ⊆ I, U_{F} ∈ 𝔛.

Note that if we assume (W) then 𝔛_{min}(≤ P) gives a complete set of G-conjugate representatives of 𝔛_{min}. The following lemma shows that, under (P), N_{G}(𝔛) is a selfnormalizing collection.

Lemma 6. Keeping the above notation, assume (P). Then N_{G}(U_{F}) is a self-normalizing subgroup of G. Consequently, Ω(G, N_{G}(𝔛)) is a generalized Burnside ring (see Example 1(2)).

Proof: Any element g ∈ N_{G}(N_{G}(U_{F})) acts on O_{p}(N_{G}(U_{F})). Since U_{F} lies in 𝔛 by (P) and since 𝔛 was chosen in B_{p}(G), we have that U_{F} is p-radical. Thus g normalizes U_{F}, and so g ∈ N_{G}(U_{F}). □

We will see in the next sub-section that N_{G}(𝔛) satisfies another condition stronger than self-normalizing.

4.2. A more general setting. Let D be a collection of subgroups of G. In this subsection, we will establish another hypothesis (Z) on the mark homomorphism on Ω(G, D). Then we will show that, D under (Z) is a self-normalizing collection. Furthermore the structure of the generalized Burnside ring Ω(G, D) will be also investigated. Then we will prove that N_{G}(𝔛) satisfies (Z) under natural (P) and (W). Recall that the mark homomorphism φ_{S}, for (S) ∈ C(D), is an additive map from Ω(G, D) to ℤ (see Section 2.1).

Hypothesis (Z). For (S), (T) ∈ C(D),

φ_{T} ([G/S]) = \{ \begin{matrix}1 & if T ≤ ^{g}S for some g ∈ G, \\ 0 & otherwise. \end{matrix}

Note that {[G/S] | (S) ∈ C(D)} is a basis of the free abelian group Ω(G, D).

Remark 3. The Hypothesis (Z) has the following interpretations.

(1) Recall that the table of marks M_{G}(D) of G with respect to D is a matrix indexed by C(D), and an ((S), (T))-entry for (S), (T) ∈ C(D) is the value φ_{T} ([G/S]). If Ω(G, D) is realized as a generalized Burnside ring then M_{G}(D) gives us the information of the product on the ring Ω(G, D). Then (Z) implies that M_{G}(D) coincides with the transposed^{t}E_{C(D)} of the incidence matrix E_{C(D)} of C(D); which is a matrix indexed by C(D), and an ((S), (T))-entry 1 if S ≤ ^{g}T for some g ∈ G, or 0 otherwise. So in this case, M_{G}(D) can be easily obtained by just looking at the inclusions among C(D).

(2) By the definition, φ_{T} ([G/S]) is the number of the T -fixed points of the G-set G/S, and hence (Z) implies that there is at most one such fixed point. This relates to a more group-theoretic condition like weakly closed property (W) (see Theorem 1).

The following are fundamental results obtained by (Z).

Lemma 7. Assume (Z) for a collection D of subgroups of G.

(1) Any element S in D is a self-normalizing subgroup of G.

(2) D satisfies (C′)_{r} for any prime r, and hence (C)_{∞} holds for D. Consequently, Ω(G, D)_{(r)} for any prime r (or Ω(G, D)) is a generalized Burnside ring.

(3) The following sequence of abelian groups is exact:

0 ⟶ Ω(G, D) ⟶^{φ} Ω̃(G, D) ⟶^{ψ} Obs(G, D) ⟶ 0.

(4) The generalized Burnside ring Ω(G, D) is isomorphic to Ω̃(G, D) as rings. In particular, Obs(G, D) = 0.

Proof: (1) For any S ∈ D, we have that φ_{S}([G/S]) = 1 by (Z). But this is just equal to the index of S in N_{G}(S) by the definition of φ_{S}. Thus N_{G}(S) = S.

(2) See Example 1(2) and Lemmas 1, 2, 4.

(3) The exactness follows from Lemma 3.

(4) Since M_{G}(D) is an unitriangular matrix by (Z), we have that M_{G}(D) ∈ SL(n, ℤ) where n := |C(D)|. Thus the inverse M_{G}(D)^{−1} is also unitriangular with integer entries. It follows that the mark homomorphism φ : Ω(G, D) ⟶ Ω̃(G, D) is an isomorphism of abelian groups. Furthermore, we can see that φ induces a ring isomorphism by Lemma 4. The last claim Obs(G, D) = 0 is now clear from the above exact sequence. □

In the following, we summarize some implications discussed in Lemma 7.

(Z) for D

self-normalizing collection D

Lem 7(1)

(C′)_{r} for D for all prime r

Ex. 1(2)

(C)_{∞} for D Lem 1, 2

Ω(G, D) is a generalized Burnside ring

Lem 4

Theorem 1. Assume (W) and (P) for 𝔛 ⊆ B_{p}(G). Then the collection N_{G}(𝔛) satisfies (Z). Consequently, Ω(G, N_{G}(𝔛)) is realized as a generalized Burnside ring.

Proof: Take any elements

H := ^{h_{1}}N_{G}(U_{F}) and K := ^{h_{2}}N_{G}(U_{J}) ∈ N_{G}(𝔛)

for h_{1}, h_{2} ∈ G and ∅ ≠ F, J ⊆ I. Since φ_{H}([G/K]) = |{gK ∈ G/K | H ≤ ^{g}K}|, it suffices to show that if H ≤ ^{x}K and H ≤ ^{y}K for x, y ∈ G then x^{−1}y ∈ K.

Suppose now that H ≤ ^{x}K. Then

N_{G}(U_{F}) ≤ ^{h_{1}^{−1}xh_{2}}N_{G}(U_{J}) = N_{G}(^{h_{1}^{−1}xh_{2}}U_{J}) =: M.

Since U_{F} is p-radical by (P), O_{p}(M) ≤ U_{F} by [SY97, Lemma1.9]. Again ^{h_{1}^{−1}xh_{2}}U_{J} is also p-radical by (P), so

h_{1}^{−1}xh_{2}U_{J} = O_{p}(M) ≤ U_{F} ≤ P ≥ U_{J}.

Then by (W), ^{h_{1}^{−1}xh_{2}}U_{J} = U_{J}. It follows that h_{1}^{−1}xh_{2} ∈ N_{G}(U_{J}). By the same way, we have that h_{1}^{−1}yh_{2} ∈ N_{G}(U_{J}) from H ≤ ^{y}K. Then

h_{2}^{−1}x^{−1}yh_{2} = (h_{1}^{−1}xh_{2})^{−1}(h_{1}^{−1}yh_{2}) ∈ N_{G}(U_{J}),

and thus x^{−1}y ∈ ^{h_{2}}N_{G}(U_{J}) = K as desired. □

Remark 4. (1) As shown in Lemma 6, N_{G}(𝔛) under (P) is a collection of self-normalizing subgroups, and this implies that Ω(G, N_{G}(𝔛)) is a generalized Burnside ring. However, Theorem 1 tells us that, assuming (W) further, N_{G}(𝔛) satisfies the condition (Z) stronger than self-normalizing.

(2) Assume (W) and (P). Then it is easy to see that {N_{G}(U_{F}) | ∅ ≠ F ⊆ I} is a set of representatives of G-conjugate classes of N_{G}(𝔛). Thus the rank of Ω(G, N_{G}(𝔛)) as the free abelian group is just equal to 2^{|I|} − 1. In examples below, we will see that this number is relatively small.

(3) There were already interesting known examples of (W) and (P) (see Section 4.3 below), and now Theorem 1 shows that these have a generalized Burnside ring Ω(G, N_{G}(𝔛)). Here we mention that Ω(G, N_{G}(𝔛)) is not a ”subring” of the ordinary Burnside ring Ω(G). Because the collection N_{G}(𝔛) does not contain the whole group G, and also is not closed under intersection.

4.3. Examples. In this sub-section, we will give some examples of 𝔛 ⊆ B_{p}(G) satisfying (W) and (P), and then we have a generalized Burnside ring Ω(G, N_{G}(𝔛)) (see Theorem 1).

(1) Lie type groups in characteristic p: Let G be a group of Lie type in characteristic p, and put 𝔛 := B_{p}^{cen}(G) = B_{p}(G) the set of all non-trivial unipotent radicals of parabolics of G. Set

𝔛_{min}(≤ P) = {U_{1}, U_{2}, …, U_{l}}

where P is a Sylow p-subgroup of G, and l is the Lie rank of G. Then it is well known that 𝔛 satisfies the hypotheses (W) and (P) (c.f. [Gr02, Lemma 4.2]). Therefore, we have a generalized Burnside ring Ω(G, N_{G}(𝔛)) of rank 2^{l} − 1. Note that our collection N_{G}(𝔛) = {^{g}N_{G}(U_{F}) | g ∈ G, ∅ ≠ F ⊆ I} is nothing else but just the set of all proper parabolic subgroups of G. Here it is worth mentioning that the poset of unipotent radicals is homotopy equivalent to the poset of parabolics, and they are often used more or less interchangeable. But for the purposes of this paper, the poset of unipotent radicals will not have the generalized Burnside condition; only the parabolics will.

(2) Mathieu group M_{24} and p = 2: Let G be the Mathieu simple group M_{24} of degree 24, and put 𝔛 := B_{2}^{cen}(G) = B_{2}(G) (see for example [Yr00]). Then, for a Sylow 2-subgroup P of G, we have that

𝔛_{min}(≤ P) = {U_{O} ≅ 2^{4}, U_{T} ≅ 2^{6}, U_{S} ≅ 2^{6}}

with normalizers N_{G}(U_{O}) ≅ U_{O} : GL_{4}(2), N_{G}(U_{T}) ≅ U_{T} : (L_{2}(2)×L_{3}(2)), N_{G}(U_{S}) ≅ U_{S} : 3Sp_{4}(2). Recall that they are respectively the stabilizers in M_{24} of an octad, a trio, and a sextet. Then it is shown in [Sa02, Section 6.2] that (W) and (P) hold for 𝔛. Therefore, we have a generalized Burnside ring Ω(M_{24}, N_{M_{24}}(𝔛)) of rank 2^{3} − 1 = 7.

(3) Conway group Co_{1}, Monster M, and p = 2: Let G be the largest Conway simple group Co_{1} or the Monster simple group M, and put 𝔛 := B_{2}^{cen}(G) ⊂ B_{2}(G) (see [Sa99, Yr05]). Then it is shown in [Sa02, Section 6.2] and [Sa06, Appendix] respectively that 𝔛 satisfies (W) and (P) for Co_{1} and M. Therefore, we have generalized Burnside rings Ω(Co_{1}, N_{Co_{1}}(𝔛)) of rank 2^{4} − 1 = 15, and Ω(M, N_{M}(𝔛)) of rank 2^{5} − 1 = 31.

Remark 5. Assume (W) and (P) for 𝔛 ⊆ B_{p}(G). Then it is easy to see that, for any ∅ ≠ K ⊆ F, a sub-collection 𝔛_{K} := {^{g}U_{J} | g ∈ G, ∅ ≠ J ⊆ K} of 𝔛 satisfies again those two. Thus, by Theorem 1, Ω(G, N_{G}(𝔛_{K})) is also a generalized Burnside ring.

Acknowledgments: We express our sincere thanks to the referee for his/her very careful reading, and also for giving a lot of valuable suggestions; which surely improved the original version of the paper.

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(F. Oda) Department of Mathematical Sciences, Yamagata University, Yamagata 990-8560, Japan

  E-mail address: odaf@sci.kj.yamagata-u.ac.jp

(M. Sawabe) Department of Mathematics, Faculty of Education, Chiba University, Chiba 263-8522, Japan

  E-mail address: sawabe@faculty.chiba-u.jp