Broué conjecture holds for principal 3-blocks with elementary abelian defect group of order 9

Shigeo Koshitani*

Department of Mathematics, Faculty of Science Chiba University, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan e-mail koshitan@math.s.chiba-u.ac.jp

and

Naoko Kunugi

Department of Mathematics, Graduate School of Science and Technology Chiba University, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan e-mail mkunugi@g.math.s.chiba-u.ac.jp

April, 2000

Abstract. In representation theory of finite groups, there is a well-known and important conjecture due to M.Broué. He has conjectured that, for any prime p, if a finite group G has an abelian Sylow p-subgroup P, then the principal p-blocks of G and of the normalizer N_{G}(P) of P in G are derived equivalent. We prove in this paper that the Broué conjecture holds for the principal 3-block of an arbitrary finite group G with elementary abelian Sylow 3-subgroup P of order 9, by using initiated works for the case that G is simple, which are due to Puig, Okuyama, Waki, Miyachi and the authors. The result depends on the classification of finite simple groups.

0. INTRODUCTION AND NOTATION

In representation theory of finite groups, one of the most important problems now is to solve a conjecture given by M.Broué ([2], [4]). He conjectures the following:

(0.1) Conjecture (Broué [2, 6.2.Question], [4, 4.9.Conjecture]). Let G be a finite group with Sylow p-subgroup P, where p is a prime, and let H = N_{G}(P), the normalizer of P in G. Then, is it true that the principal p-blocks of G and H are derived equivalent provided P is abelian?

There are only several special cases where the Broué conjecture (0.1) has been checked, see [4, p.15] and [15, p.136] (see also [27], [28], [5], [24], [6], [20], [16], [17] and [18]).

The purpose of this paper is to present that the Broué conjecture (0.1) is true for an arbitrary finite group G with elementary abelian Sylow 3-subgroup P of order 9. As a matter of fact, it recently has been checked that the Broué conjecture (0.1) holds if the above G is moreover simple by works of Puig [27], Okuyama-Waki [26], Okuyama [24], Kunugi [20], Koshitani-Kunugi [16] and Koshitani-Miyachi ([17], [18]). Then, it looks quite natural to ask whether the Broué conjecture holds for an arbitrary finite group G with the same Sylow 3-subgroup P, by making use of the classification of finite simple groups. On the other hand, there is a work by A.Marcus [22] where he proves that the Broué conjecture is liftable from a finite group G to a finite group G̃ if G is a normal subgroup of G̃ such that G̃/G is a p′-group under some hypotheses. However, in general it is not trivial to know that the hypotheses in [22] are satisfied. Nevertheless, in the case that P is elementary abelian of order 9, we can show that the Broué conjecture holds by preparing tools (lemmas) which are a sort of generalizations or applications of theorems of E.C.Dade ([10], [11]) and A.Marcus [22].

*Partially supported by the JSPS (Japan Society for Promotion of Science) and the Ministry of Education, Science, Sports and Culture, Japan

The strategy of this paper is the following. Assume G ◁ G̃, G̃/G is a p′-group and P is an abelian Sylow p-subgroup of G. By a theorem of Alperin and Dade ([1] and [9]), the Broué conjecture (0.1) is liftable from G to a normal subgroup G·CG̃(P) of G̃. It is easy to know that the quotient group G̃/[G·CG̃(P)] is a subquotient group of NG̃(P)/CG̃(P) ↪ Aut(P). So, if P is small, then we might have a chance to lift the Broué conjecture from G·CG̃(P) to G̃. Actually, this is the case though it is not easy yet.

Our main theorem is the following:

(0.2) Theorem. Let k be an algebraically closed field of characteristic 3. Let G̃ be an arbitrary finite group with elementary abelian Sylow 3-subgroup P of order 9, and let H̃ = NG̃(P), the normalizer of P in G̃. Then, the principal block algebras B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent.

(0.3) Remark. First of all, the main result (0.2) depends on the classification of finite simple groups. Keep the notation as in (0.2), and let O be a complete discrete valuation ring of rank one whose residue field is k = O/J(O). In most cases of (0.2), B_{0}(kG̃) and B_{0}(kH̃) are even splendid (Rickard) equivalent. If we can show existences of splendid (Rickard) equivalences for a few cases which are not finished yet, then (0.2) can be stated that B_{0}(kG̃) and B_{0}(kH̃) are splendid (Rickard) equivalnent, and hence so are B_{0}(OG̃) and B_{0}(OH̃) by a result of J.Rickard [28, Theorem 5.2].

Throughout this paper we use the following notation and terminology. Let k be an algebraically closed field of characteristic p > 0. In this paper a module means always a finitely generated right module, unless stated otherwise. Let G and H be finite groups. Even in the case that M is a (kG, kH)-bimodule, we say that M is a (right) k[G×H]-module via an action m·(g, h) = g^{−1}mh for m ∈ M, g ∈ G and h ∈ H. For the same M, let M^{∨} = Hom_{k}(M, k) be the k-dual of M, so that M^{∨} is a (kH, kG)-bimodule. We say that G and H have the same p-local structure in the sense of Broué (see [4, 6.5.Theorem], [28, p.342] and [22, p.374]). We write δ(G) for {(g, g) ∈ G × G | g ∈ G}. Let G̃ ▷ G, and let N be a right kG-module. Then, we write Ng̃ or N g̃ for the g̃-conjugate of N where g̃ ∈ G̃, namely, Ng̃ = N as k-vector spaces and the action of G on Ng̃ is defined by n∗g = n·g̃^{−1}gg̃ for n ∈ N and g ∈ G. For an abelian Sylow p-subgroup P of G, let E(G) = N_{G}(P)/C_{G}(P), so that this is the inertial quotient of the principal p-block of G (note that E(G) is different from the largest semi-simple normal subgroup of G). Let n be a positive integer. Then, C_{n}, Σ_{n} and A_{n}, respectively, denote the cyclic group of order n, the symmetric group on n letters and the alternating group on n letters. We write n_{p} = p^{m} for a non-negative integer m if p^{m}|n and p^{m+1} |̸n. We denote by D_{n}, Q_{n} and SD_{n}, respectively, the dihedral, the quaternion and the semi-dihedral groups of order n. For two groups G and H we denote by G ⋊ H for a semi-direct product of G by H, namely G ◁ G ⋊ H. We denote by k_{G} for the trivial kG-module of k-dimension one. We write B_{0}(kG) for the principal block algebra of the group algebra kG. For a ring R, we denote by 1_{R}, R^{×}, Z(R) and J(R), respectively, the unit element, the set of all units, the center and the Jacobson radical of R. We say that two block algebras A and B of kG and kH, respectively, are Puig equivalent following Broué [4, p.22] (see also [22, 1.6.Theorem]).

Let P be a Sylow p-subgroup of G, and let H be a subgroup of G with H ⊇ N_{G}(P). Then, we can define the Green correspondences f↓_{G×H}^{G×G} : G × G → G × H and g↑_{H×H}^{H×G} : H × H → H × G with respect to a p-subgroup δ(P) of G × G. Then, let M(1) and N(1) be M(1) = M(1)_{(G,H)} = f↓_{G×H}^{G×G}(B_{0}(kG)) and N(1) = N(1)_{(H,G)} = g↑_{H×H}^{H×G}(B_{0}(kH)). Note that M(1) and N(1) are ppermutation (trivial source) k[G × H]- and k[H × G]-modules with vertex δ(P), respectively. We use the notation f↓_{H×H}^{G×H} and g↑_{H×G}^{G×G} in similar ways. For any subgroup H of G we denote by Scott_{G}(H) the Scott module of H in G (see [31, §3, p.1549] and [23, Chap.4 Theorem 8.4 and p.297]), and we denote by P_{H}(M) the (relatively) H-projective cover of a right kG-module M (see [31, §1, p.1538]). For kG-modules X and Y we write [X, Y]^{G} for dim_{k}[Hom_{kG}(X, Y)]. Let A be a finite dimensional k-algebra and M be an A-module. Then, we denote by P (M) the projective cover of M, by Soc(M) the socle of M, by Ω(M) the kernel of the projective cover P (M) ↠ M of M, namely Ω is the Heller functor (operator), by IBr(A) the set of all non-isomorphic simple A-modules, by mod-A the category of all finitely generated right A-modules, and by \underline{mod}-A the stable module category of finitely generated right A-modules (see [15, p.193, Definitions 9.3.1]).

We denote by c_{A}(S, T) or just by c(S, T) the Cartan invariant with respect to simple A-modules S and T. We write Y |X when Y is a direct summand of X. For other notation and terminology see the book of Nagao and Tsushima [23].

1. ELEMENTARY GROUP THEORY

In this section we give elementary facts on group theory. The proofs are easy. But, they are helpful to get our main result.

(1.1) Notation. We fix a finite group G and assume that G has an abelian Sylow p-subgroup P.

(1.2) Lemma. (i) The group E(G) is a p′-subgroup of Aut(P). If particular, if P = C_{3} × C_{3}, then E(G) ↪ SD_{16}.

In the following assume that L is a subgroup of a finite group G̃ such that p |̸|G̃ : L|, P ∈ Syl_{p}(L), G ◁ G̃, and that G̃/G is a p′-group (so that P ∈ Syl_{p}(G̃)), and let G_{0} = G·GG̃(P).

(ii) A correspondence φ_{L} : E(L) → E(G̃) given by ℓ·C_{L}(P) ↦ ℓ·CG̃(P) for ℓ ∈ N_{L}(P) is a groupmonomorphism, and hence we can consider E(L) ⩽ E(G̃) via φ_{L}. Moreover, φ_{G}(E(G)) ◁ E(G̃), so that we may consider E(G) ◁ E(G̃) via φ_{G}.

(iii) G_{0} ◁ G̃ and E(G) ≅ E(G_{0}).

(iv) G̃/G ≅ NG̃(P)/N_{G}(P).

(v) If, moreover, CG̃(P) ⊆ G, then we have that CG̃(P) = C_{G}(P) and G̃/G ≅ NG̃(P)/N_{G}(P) ≅ E(G̃)/E(G).

(vi) If CG̃(P) ⊆ G and |E(G̃)| = |E(G)|, then we have G̃ = G.

(vii) E(G) = E(G_{0}) and E(G̃)/E(G_{0}) ≅ G̃/G_{0}. In particular, if E(G̃) ≅ E(G_{0}), then G̃ = G_{0}.

(viii) If G̃/G ≅ E(G̃)/E(G) ≅ C_{q} for a prime q with q ≠ p, then CG̃(P) ⊆ G, namely, G_{0} = G.

(ix) If G̃/G ≅ C_{q} for a prime q with q ≠ p and E(G̃) ≅ E(G), then G̃ = G_{0}.

Proof. (i) Clear since Aut(C_{3} × C_{3}) ≅ GL_{2}(3).

(ii) Clearly, φ_{L} is well-defined and a group-homomorphism. If φ_{L}(ℓ·C_{L}(P)) = 1 for ℓ ∈ N_{L}(P), then ℓ ∈ CG̃(P), so that ℓ ∈ C_{L}(P). Hence, φ_{L} is a monomorphism.

Take any x ∈ E(G) and x̃ ∈ E(G̃). Write x = g·C_{G}(P) and x̃ = g̃·CG̃(P) for some g ∈ N_{G}(P) and g̃ ∈ NG̃(P). Then, x̃^{−1}·φ_{G}(x)·x̃ = g̃^{−1}gg̃·C_{G}(P) ∈ [N_{G}(P)·CG̃(P)]/CG̃(P) = Im(φ_{G}) since G ◁ G̃. Hence, φ_{G}(E(G)) ◁ E(G̃).

(iii) By Frattini argument, G̃ = G·NG̃(P). Then, G_{0} is a subgroup of G̃ since G ◁ G̃. Take any g̃ ∈ G̃ and g_{0} ∈ G_{0}. We can write g̃ = gn and g_{0} = g′c for some g, g′ ∈ G, n ∈ NG̃(P) and c ∈ CG̃(P). Now g̃g_{0}g̃^{−1} = g̃(g′c)g̃^{−1} = g̃g′g̃^{−1}·g̃cg̃^{−1}. Moreover, g̃cg̃^{−1} = gncn^{−1}g^{−1} ∈ G·CG̃(P)·G = G·CG̃(P) = G_{0} since CG̃(P) ◁ NG̃(P) and G ◁ G̃. Hence, g̃g_{0}g̃^{−1} ∈ G·G·CG̃(P) = G_{0}, which shows G_{0} ◁ G̃.

Now, let ψ : E(G) → E(G_{0}) as in (ii). Then, by (ii), ψ(E(G)) ◁ E(G_{0}). Take any x_{0} ∈ E(G_{0}). Then, we can write x_{0} = g_{0}·C_{G_{0}}(P) for g_{0} ∈ N_{G_{0}}(P). We can write also that g_{0} = gc for g ∈ G and c ∈ CG̃(P), so that c = g^{−1}g_{0} ∈ G_{0}. Hence, c ∈ C_{G_{0}}(P). Thus, x_{0} = gc·C_{G_{0}}(P) = g·C_{G_{0}}(P). Furthermore, g = g_{0}c^{−1} ∈ N_{G_{0}}(P)·C_{G_{0}}(P) ⊆ N_{G_{0}}(P), so that g ∈ N_{G_{0}}(P) ∩ G = N_{G}(P). Therefore, x_{0} = g·C_{G_{0}}(P) = ψ(g·C_{G}(P)). Hence, ψ is onto. Then, by (ii), ψ is an isomorphism.

(iv) By Frattini argument, G̃/G = G·NG̃(P)/G ≅ NG̃(P)/[G ∩ NG̃(P)] = NG̃(P)/N_{G}(P).

(v) By the assumption, CG̃(P) = C_{G}(P). Hence, by (iv),

G̃/G ≅ NG̃(P)/N_{G}(P) = [NG̃(P)/CG̃(P)]/[N_{G}(P)/CG̃(P)] = E(G̃)/E(G).

(vi) Easy by (v).

(vii) This follows from (iii), (v) and (vi).

(viii) Assume CG̃(P)⊈G. Then, G ⪇ G·CG̃(P) = G_{0}, so that G_{0} = G̃ since |G̃ : G| is a prime. Hence, E(G̃) = E(G_{0}) = E(G) by (iii), a contradiction.

(ix) If G_{0} = G, then CG̃(P) ⊆ G, so that G̃/G ≅ E(G̃)/E(G) from (v), a contradiction. Hence G ⊊ G_{0}, which implies G_{0} = G̃ since |G̃ : G_{0}| is a prime. □

(1.3) Lemma (Gaschütz). Assume that G has a Sylow p-subgroup P such that P ≅ C_{p} × C_{p} and that C_{p} ↪ Z(G). Then, there is a normal subgroup L of G such that G = C_{p} × L.

Proof. By Sylow’s theorem we can assume P = P_{1} ×P_{2} such that P_{1} ≅ P_{2} ≅ C_{p} and P_{1} ⊆ Z(G). Hence P_{1} ◁ G and P_{1} is abelian. Namely, P splits on P ∩ P_{1}.

Take any prime q ≠ p, and any Q ∈ Syl_{q}(G). Then, Q ∩ P_{1} = 1, which means that Q splits on Q ∩ P_{1}. Thus, it follows from a theorem W.Gaschütz [30, Chap.2 Theorem 8.6] that G splits on P_{1}. That is, G = P_{1} ⋊ L for a subgroup L of G. Since P_{1} ⊆ Z(G), P_{1} ⋊ L = P_{1} × L. □

(1.4) Lemma. Assume that G ◁ G̃, G̃/G is a p′-group, G is a non-abelian simple group and O_{p′}(G̃) = 1. Then, G ◁ G̃ ↪ Aut(G) = G.Out(G).

Proof. Since G is non-abelian simple, GG̃(G) ∩ G = Z(G) = 1. Hence, there is a direct product CG̃(G) × G ◁ G̃. Since G̃/G is a p′-group, p |̸|CG̃(G)|. Hence, CG̃(G) = 1 since O_{p′}(G̃) = 1. Therefore, G̃ ≅ NG̃(G)/CG̃(G) ↪ Aut(G). □

(1.5) Lemma. Let G, H̃ and H be subgroups of G̃ such that G ∩ H̃ ⊇ H, and let M be a right kH-module.

\begin{matrix}G̃ & ⩾ & H̃ \\ | & & | \\ G & ⩾ & H \end{matrix}

Let h̃ ∈ H̃. Then, a map

M ⊗_{kH} kG·h̃ ⟶ (M ⊗_{kH} kH·h̃) ⊗_{k[h̃^{−1}Hh̃]} k[h̃^{1}Gh̃]

given by

m ⊗_{kH} gh̃ ↦ (m ⊗_{kH} h̃) ⊗_{k[h̃^{−1}Hh̃]} h̃^{−1}gh̃ for m ∈ M, g ∈ G

is an isomorphism of right k[h̃^{−1}Gh̃]-modules.

(1.6) Lemma. Let P ∈ Syl_{p}(G).

(i) B_{0}(kG)_{G×G} = P_{δ(P)}(k_{G×G}) = Scott_{G×G}(δ(P))

(ii) Let H be a subgroup of G with H ⊇ N_{G}(P). Then, it holds that M(1) = P_{δ(P)}(k_{G×H}) = Scott_{G×H}(δ(P)), and that there is a primitive idempotent e_{1} of (kG)^{H} such that M(1) = kGe_{1}, where (kG)^{H} = C_{kG}(H).

Proof. (i) Let G = ∪_{i=1}^{n}Hg_{i} be a coset decomposition of H in G, where n = |G : H|, g_{i} ∈ G and g_{1} = 1_{G}. Then, clearly, a map φ : k_{H}↑^{G} = ⊕_{i=1}^{n}(k_{H} ⊗_{kH} g_{i}) → k_{G} given by ∑_{i=1}^{n} α_{i} ⊗_{kH} g_{i} ↦ ∑_{i=1}^{n} α_{i}, α_{i} ∈ k, is an H-split epimorphism of kG-modules since a map ϕ : k_{H} → k_{H}↑^{G} given by α ↦ α⊗_{kH} 1 is a kH-homomorphism which satisfies φ∘ψ = id_{k}. On the other hand, it follows from a theorem of J.A.Green [23, Chap.5 Lemma 10.8] that B_{0}(kG) is a trivial source k[G × G]-module with vertex δ(P). Hence B_{0}(kG) = P_{δ(P)}(k_{G×G}) since k_{G} is in B_{0}(kG). The second equiality is given by [31, Proposition 3.1].

(ii) By the definition, M(1) is a uique indecomposable direct summand of B_{0}(kG)↓_{G×H}^{G×G} as a right k[G×H]-modules with vertex δ(P). On the other hand, by the definition of Scott modules, we know that Scott_{G×H}(δ(P)) | [Scott_{G×G}(δ(P))]↓_{G×H}^{G×G}. Therefore, by (i), we get M(1) = Scott_{G×H}(δ(P)) by looking at the vertices. The rest is easy. □

2. FINITE GROUPS WITH ELEMENTARY ABELIAN SYLOW 3-SUBGROUP OF ORDER 9

In this section we give a list of finite groups G such that G has an elementary abelian Sylow 3-subgroup of order 9, O_{3′}(G) = 1 and O^{3′}(G) = G, by making use of the classification of finite simple groups. Namely, we need to use the classification of finite simple groups to prove (2.1), (2.2) and (2.4). The following two propositions (2.1) – (2.2) were informed by S.Yoshiara.

(2.1) Proposition (Yoshiara). Let G be a finite group with elementary abelian Sylow 3subgroup of order 9 such that O_{3′}(G) = 1 and O^{3′}(G) = G. Then, G is one of the following (i)–(ii).

(i) G = X × Y for finite simple groups X and Y such that both of them have cyclic Sylow 3-subgroups of order 3.

(ii) G is a non-abelian finite simple group with elementary abelian Sylow 3-subgroup of order 9.

Proof. See [18, (1.2) Proposition]. □

(2.2) Proposition. If G is a non-abelian finite simple group with elementary abelian Sylow 3-subgroup of order 9, then G is one of the following nine types:

(i) A_{6}, A_{7}, A_{8}, M_{11}, M_{22}, M_{23}, HS. (ii) PSL_{3}(q) for a power q of a prime with q ≡ 4 or 7 (mod 9). (iii) PSU_{3}(q^{2}) for a power q of a prime with 2 < q ≡ 2 or 5 (mod 9). (iv) PSp_{4}(q) for a power q of a prime with q ≡ 4 or 7 (mod 9).

(v) PSp_{4}(q) for a power q of a prime with 2 < q ≡ 2 or 5 (mod 9). (vi) PSL_{4}(q) for a power q of a prime with 2 < q ≡ 2 or 5 (mod 9). (vii) PSU_{4}(q^{2}) for a power q of a prime with q ≡ 4 or 7 (mod 9). (viii) PSL_{5}(q) for a power q of a prime with q ≡ 2 or 5 (mod 9). (ix) PSU_{5}(q^{2}) for a pwer q of a prime with q ≡ 4 or 7 (mod 9).

(2.3) Remark. Note that in (iii) of (2.2) PSU_{3}(2^{2}) ≅ (C_{3} × C_{3}) ⋊ Q_{8}, in (v) of (2.2) PSp_{4}(2) = Sp_{4}(2) ≅ Σ_{6}, and that in (vi) of (2.2) PSL_{4}(2) = SL_{4}(2) ≅ A_{8} from [13, II 10.14 Satz], [13, II 9.21 Satz, II 9.22 Hauptsatz] and [13, II 6.14 Satz (5)], respectively.

(2.4) Proposition. Assume that G is a finite group with elementary abelian Sylow 3-subgroup P of order 9 such that O_{3′}(G) = 1 and O^{3′}(G) = G. Then we get the following :

(i) If G = C_{3} × C_{3}, then E(G) = 1. (ii) If G = C_{3} × X for a non-abelian simple group X with |X|_{3} = 3, then E(G) = C_{2}. (iii) If G = X×Y for non-abelian simple groups X and Y with |X|_{3} = |Y |_{3} = 3, then E(G) = C_{2}×C_{2}. (iv) If G = A_{6} or A_{7}, then E(G) = C_{4}.

(v) If G = A_{8}, then E(G) = D_{8}. (vi) If G = M_{11}, M_{23} or HS, then E(G) = SD_{16}. (vii) If G = M_{21}(≅ PSL_{3}(4)) or M_{22}, then E(G) = Q_{8}. (viii) If G = PSL_{3}(q) such that q ≡ 4 or 7 (mod 9), then E(G) = Q_{8}. (ix) If G = PSU_{3}(q^{2}) such that 2 < q ≡ 2 or 5 (mod 9), then E(G) = Q_{8}.

(x) If G = PSp_{4}(q) such that q ≡ 4 or 7 (mod 9), then E(G) = D_{8}. (xi) If G = PSp_{4}(q) such that 2 < q ≡ 2 or 5 (mod 9), then E(G) = D_{8}. (xii) If G = PSL_{4}(q) such that q ≡ 2 or 5 (mod 9), then E(G) = D_{8}. (xiii) If G = PSL_{5}(q) such that q ≡ 2 or 5 (mod 9), then E(G) = D_{8}. (xiv) If G = PSU_{4}(q) or PSU_{5}(q) such that q ≡ 4 or 7 (mod 9), then E(G) = D_{8}.

3. LEMMAS

In this section we give several lemmas which are useful to get our main result. Throughout this section we assume that k is an algebraically closed field of characteristic p > 0, and that G, G̃, G′, G̃′, X, X̃, H, H′, H̃ and H̃′ are all finite groups.

(3.1) Lemma (Alperin-Dade). Suppose that G ◁ G̃, G̃/G is a p′-group, P ∈ Syl_{p}(G̃) ∩ Syl_{p}(G) and G̃ = G·CG̃(P). Let ẽ and e be the block idempotents of B_{0}(kG̃) and B_{0}(kG), respectively. Then, B_{0}(kG̃) and B_{0}(kG) are category-isomorphic, namely, a map B_{0}(kG) → B_{0}(kG̃) defined by a ↦ aẽ for a ∈ B_{0}(kG) is a k-algebra-isomorphism, so that eẽ = ẽe = ẽ. Moreover, a right k[G̃ × G]-module B_{0}(kG̃) = ẽkG̃ = ẽkG̃e gives a Puig equivalence between B_{0}(kG̃) and B_{0}(kG).

Proof. This is proved in [1, Theorems 1 and 2] and [9, Theorem]. Note that (kG̃ẽ)↓_{G×G}^{G̃×G̃} ≅ kGe as right k[G × G]-modules as in [12, §4] (see also [3, Theorems 0.1 and 0.2] and [22, (5.4)]). □

(3.2) Lemma. Let A and B be finite dimensional k-algebras such that A and B are indecomposable as k-algebras. Assume that an (A, B)-bimodule M and a (B, A)-bimodule N realize a Morita equivalence between A and B. Then, M is indecomposable as an (A, B)-bimodule, and N is indecomposable as a (B, A)-bimodule.

Proof. Let M = M_{1} ⊕⋯⊕M_{n} for an integer n ⩾ 1 where each M_{i} is an indecomposable (A, B)- bimodule. Then,_{B}B_{B} ≅ N ⊗_{A}M ≅ ⊕_{i=1}^{n}(N ⊗_{A}M_{i}). Since_{B}B_{B} is indecomposable, we may assume that N ⊗_{A} M_{i} = 0 for any n ⩾ 2. Then, for i ⩾ 2, 0 = M ⊗_{B} (N ⊗_{A} M_{i}) ≅ _{A}A ⊗_{A} M_{i} ≅ M_{i}, so that M = M_{1}. □

(3.3) Lemma. Suppose that G ◁ G̃, G̃/G is a p′-group and P ∈ Syl_{p}(G̃) ∩ Syl_{p}(G) such that G̃ = G·CG̃(P). Let H̃ be a subgroup of G̃ with H̃ ⊇ NG̃(P), and let H = G ∩ H̃. Assume that H̃ = H·CH̃(P). Assume, moreover, that a Puig equivalence between B_{0}(kG) and B_{0}(kH) is given by a right k[G × H]-module M(1). Then, a right k[G̃ × H̃]-module M̃(1) gives a Puig equivalence between B_{0}(kG̃) and B_{0}(kH̃), where M̃(1) = M(1)_{(G̃,H̃)}.

Proof. Clearly, H ◁ H̃ and G̃ = G·H̃.

G̃ = G·CG̃(P) = G·H̃ −−→ H̃ = H·CH̃(P) ⊇ NG̃(P)
▽  ▽
G  ⟶ H = G ∩ H̃ ⊇ N_{G}(P)

Let Ã = B_{0}(kG̃), A = B_{0}(kG), B̃ = B_{0}(kH̃) and B = B_{0}(kH), and let ẽ, e, f̃ and f be the block idempotents of Ã, A, B̃ and B, respectively. Then, AÃ ⊆ Ã and ÃA ⊆ Ã since eẽ = ẽe = ẽ ∈ Z(kG̃) by [12, §4]. Hence, we can naturally consider _{A}ÃÃ and ÃÃ_{A}. Since Ã_{A} ≅ A_{A} by [12, §4], we get

Ã ⊗_{A} Ã ≅ Ã as (Ã, Ã)-bimodules. Again, by [12, §4], _{B}B̃_{B} = B̃˜↓_{H×H}^{H̃×H̃} ≅ _{B}B_{B}.

Let M̃ = Ã ⊗_{A} M(1) ⊗_{B} B̃. Then, since M(1) ⊗_{B} M(1)^{∨} ≅ _{A}A_{A}, we have

M̃ ⊗B̃ M̃^{∨} = (Ã ⊗_{A} M(1) ⊗_{B} B̃) ⊗B̃ (B̃ ⊗_{B} M(1)^{∨} ⊗_{A} Ã)
≅ Ã ⊗_{A} M(1) ⊗_{B} B̃ ⊗_{B} M(1)^{∨} ⊗_{A} Ã
≅ Ã ⊗_{A} M(1) ⊗_{B} B ⊗_{B} M(1)^{∨} ⊗_{A} Ã
≅ Ã ⊗_{A} M(1) ⊗_{B} M(1)^{∨} ⊗_{A} Ã
≅ Ã ⊗_{A} A ⊗_{A} Ã
≅ Ã ⊗_{A} Ã ≅ Ã

as (Ã, Ã)-bimodules. Similarly, we have M̃^{∨} ⊗Ã M̃ ≅ B̃ as (B̃, B̃)-bimodules. Thus, M̃ is indecomposable as a right k[G × H]-module by (3.2). Since M̃ = ẽ(M(1)↑^{G̃×H̃})f̃, (1.6)(ii)

implies that [M(1), k_{G×H}]^{G×H} = 1, so that [M̃, k_{G̃×H̃}]^{G̃×H̃} ≠ 0. Since [k_{δ(P)} ↑^{G̃×H̃}, k_{G̃×H̃}]^{G̃×H̃} = [Scott_{G̃×H̃}(δ(P)), k_{G̃×H̃}]^{G̃×H̃} = 1, it follows from a theorem of Krull-Schmidt that M̃ ≅ Scott_{G̃×H̃}(δ(P)). (Note that we get this also by [29, Lemma 2]). Hence we get the assertion from (1.6)(ii). □

(3.4) Lemma (Alperin-Dade-Marcus). Assume as in (3.1). Moreover, suppose that H ◁ H̃, P ∈ Syl_{p}(G̃) ∩ Syl_{p}(H̃), H̃ = H·CH̃(P) and G̃/G ≅ H̃/H. If B_{0}(kG) and B_{0}(kH) are Morita (resp. Puig, derived and splendid) equivalent, then B_{0}(kG̃) and B_{0}(kH̃) are also Morita (resp. Puig, derived and splendid) equivalent.

Proof. See [22, (5.4)]. □

(3.5) Lemma (Dade). Suppose that X ◁ X̃ and X̃/X is a cyclic p′-group. Let M be an indecomposable right kX-module such that M is X̃-invariant. Then, M is extendible to a kX̃- module.

Proof. Let G = X̃/X, E_{1} = End_{kX}(M) and E = End_{kX̃}(M↑X̃). Then, we know that Z(E_{1}) is a local commutative k-algebra (see [23, Chap.1 Theorem 3.10, Chap.4 Lemma 4.1(ii), and Chap.1 Theorem 5.7]). As is well-known, kX̃ is a crossed product of G over kX, and E is a crossed product of G over E_{1} (see [7]). Then, it follows from a result of Dade [10, Theorem 2.8, p.379] that the following two statements are equivalent:

(a) M is extendible to a kX̃-module.

(b) E is a skew group algebra of G over E_{1}. It is well-known that a condition H^{2}(G, Z(E_{1})^{×}) = 1 implies the condition (b). It is also well-known that R^{×} = k^{×} × (1_{R} + J(R)) (as multiplicative groups) for a local commutative finite-dimensional k-algebra R. Moreover, by [19, p.163, L.Proposition], H^{2}(G_{0}, 1 + J(Z(A_{1}))) = 1 for a crossed product A of a finite p′-group G_{0} over A_{1}. It holds H^{2}(G, k^{×}) = 1 since G is cyclic and k is a field. Therefore, H^{2}(G, Z(E_{1})^{×}) ≅ H^{2}(G, k^{×}) × H^{2}(G, 1 + J(Z(E_{1}))) = 1. □

(3.6) Lemma (Dade). Let G ◁ G̃ such that G̃/G is a p′-group, and let P ∈ Syl_{p}(G̃) ∩ Syl_{p}(G). Suppose, furthermore, that G̃ has a subgroup H̃ such that H̃ ⊇ NG̃(P), and let H = G ∩ H̃, so that G̃ = GH̃ and G̃/G ≅ H̃/H.

G̃ = GH̃ → H̃ ⊇ NG̃(P)
▽  ▽
G  → H = G ∩ H̃ ⊇ N_{G}(P)

(i) Let M(1) = M(1)_{(G,H)}. Then, M(1) is δ(H̃)-invariant, namely, M(1)·(h̃, h̃) ≅ M(1) as right k[G × H]-modules for any h̃ ∈ H̃.

(ii) Let f = f↓_{H×H}^{G×H} : G×H → H ×H be the Green correspondence with respect to δ(P). Then, f(M(1)) ≅ B_{0}(kH) as right k[H × H]-modules.

G̃ × H̃  →  H̃ × H̃
∪  ∪
(G × H)·δ(H̃) → (H × H)·δ(H̃)
▽  ▽
G × H  →  H × H

(iii) The following two statements are equivalent.

(a) The right k[G × H]-module M(1) is extendible to a right k[(G × H)·δ(H̃)]-module.

(b) The right k[H × H]-module B_{0}(kH) is extendible to a right k[(H × H)·δ(H̃)]-module. (iv) If, morever, G̃/G is cyclic, then the right k[G × H]-module M(1) is extendible to a

k[(G × H)·δ(H̃)]-module.

Proof. (i) Note, first of all, that B_{0}(kG)·(h̃, h̃) ≅ B_{0}(kG) as right k[G × G]-modules for any h̃ ∈ H̃. Then, by the definition of the Green correspondences and a theorem of Krull-Schmidt, we get the assertion.

(ii) Easy.

(iii) This follows from a result of E.C.Dade [11, (6.5) Corollary, p.94] and (ii).

(iv) Clearly, [(G × H)·δ(H̃)]/[G × H] ≅ δ(H̃)/δ(H) ≅ H̃/H ≅ [(H × H)·δ(H̃)]/[H × H], and these are cyclic p′-groups. Since B_{0}(kH) is δ(H̃)-invariant as a right k[H × H]-module as noted in the proof of (i), we know that B_{0}(kH) is extendible to a right k[(H × H)·δ(H̃)]-module by (3.5). Hence, M(1) is extendible to a right k[(G × H)·δ(H̃)]-module by (iii). □

(3.7) Lemma. Let G̃, G, H̃, H,P and M(1) be the same as in (3.6). Assume, moreover, that G̃ and H̃ have the same p-local structure, G̃/G is cyclic, CG̃(P) ⊆ G and the right k[G × H]-module M(1) gives a Puig equivalence between B_{0}(kG) and B_{0}(kH). Then, B_{0}(kG̃) and B_{0}(kH̃) are Puig equivalent.

Proof. Let ẽ e f̃ and f the block idempotents of B_{0}(kG̃), B_{0}(kG), B_{0}(kH̃) and B_{0}(kH), respectively. Since CG̃(P) ⊆ G, it follows from a result of Fong [23, Chap.5 Theorem 5.16(ii), Lemma 5.14 and Theorem 5.13(ii)] and the Brauer’s 3rd main theorem [23, Chap.5 Theorem 6.1] that e = ẽ and f = f̃. Now, by (3.6)(iv), the right k[G × H]-module M(1) is extendible to a right k[(G × H)·δ(H̃)]-module. Hence, we get the assertion from [22, 3.9.Corollary(a)]. □

(3.8) Lemma. Let G ◁ G̃, G′ ◁ G̃′, P ∈ Syl_{p}(G) ∩ Syl_{p}(G′), H = N_{G}(P), H̃ = NG̃(P), H′ = N_{G′}(P) and H̃′ = NG̃′(P). Assume that H̃/O_{p′}(H̃) ≅ H̃′/O_{p′}(H̃′), H/O_{p′}(H) ≅ H′/O_{p′}(H′), (so that we can identify B_{0}(kH̃) and B_{0}(kH̃′), and also B_{0}(kH) and B_{0}(kH′)), G̃/G ≅ H̃/H ≅ H̃′/H′ ≅ G̃′/G′ and these are p′-groups. We assume moreover that G̃ = G·CG̃(P), G̃′ = G′·CG̃′(P) and that a right k[G×G′]-module M(1)⊗_{B_{0}(kH)} N′(1) has a unique non-projective indecomposable direct summand 𝔐 such that 𝔐 realizes a Puig equivalence between B_{0}(kG) and B_{0}(kG′), where M(1) = M(1)_{(G,H)} and N′(1) = N(1)_{(H′,G′)}. Let 𝔐̃ = B_{0}(kG̃) ⊗_{B_{0}(kG)} 𝔐 ⊗_{B_{0}(kG′)} B_{0}(kG̃′).

G̃ −−→ H̃ ↠ H̃/O_{p′}(H̃) ≅ H̃′/O_{p′}(H̃′) ↞ H̃′ ←−− G̃′
▽ ▽ ▽
G ⟶ H ↠ H/O_{p′}(H) ≅ H′/O_{p′}(H′) ↞ H′ ⟵ G′

Then, a right k[G̃ × G̃′]-module 𝔐̃ is a unique non-projective indecomposable direct summand of M̃(1)⊗_{B_{0}(kH̃)} Ñ′(1) as a right k˜[G×G̃′]-module, and 𝔐̃ realizes a Puig equivalence between B_{0}(kG̃) and B_{0}(kG̃′), where M̃(1) = M(1)_{(G̃,H̃)} and Ñ′(1) = N(1)_{(H̃,G̃′)}.

Proof. As noted in the above, we can consider B_{0}(kH̃) = B_{0}(kH̃′) and also B_{0}(kH) = B_{0}(kH′). As in the proof of (3.3), B_{0}(kG̃)↓_{G̃×H̃}^{G̃×G̃} = 𝔐̃ ⊕ (⊕_{i} U_{i}), where each U_{i} is an indecomposable right k[G̃ × H̃]-module with vertex ⪇_{G̃×G̃}δ(P). Since M(1) = f↓_{G×H}^{G×G}(B_{0}(kG)) = g↑_{H×H}^{G×H}(B_{0}(kH)), we know M(1)·1_{B_{0}(kH)} = M(1), where 1_{B_{0}(kH)} is the block idempotent of B_{0}(kH). Thus, we have

that 𝔐|M(1) ⊗_{B_{0}(kH)} N′(1)|kG ⊗_{B_{0}(kH)} kG′ ≅ kG ⊗_{kH} B_{0}(kH) ⊗_{kH′} kG′ = B_{0}(kH)↑_{H×H′}^{G×G′}. Then,

by noting that B_{0}(kH̃)↓_{H×H}^{H̃×H̃} ≅ B_{0}(kH) by (3.1) and [12, §4], and that 𝔐 ⊗_{B_{0}(kG′)} 𝔐^{∨} ≅ B_{0}(kG) as right k[G × G]-modules, we get that

𝔐̃ ⊗_{B_{0}(kG̃′)} 𝔐̃^{∨} ≅ B_{0}(kG̃)

as right k[G̃ × G̃]-modules and that

𝔐̃^{∨} ⊗_{B_{0}(kG̃)} 𝔐̃ ≅ B_{0}(kG̃′)

as right k[G̃′ × G̃′]-modules. Thus, 𝔐̃ is indecomposable as a right k[G̃ × G̃′]-module by (3.2). Therefore, 𝔐̃ is a unique non-projective indecomposable direct summand of M̃(1) ⊗_{B_{0}(kH̃)} Ñ′(1) as a right k[G̃ × G̃′]-module. □

(3.9) Lemma. Let G ◁ G̃, G′ ◁ G̃′, P ∈ Syl_{p}(G) ∩ Syl_{p}(G′), H = N_{G}(P), H̃ = NG̃(P), H′ = N_{G′}(P) and H̃′ = NG̃′(P). Assume that H̃/O_{p′}(H̃) ≅ H̃′/O_{p′}(H̃′), H/O_{p′}(H) ≅ H′/O_{p′}(H′), (so that we can identify B_{0}(kH̃) and B_{0}(kH̃′), and also B_{0}(kH) and B_{0}(kH′)), G̃/G ≅ H̃/H ≅ H̃′/H′ ≅ G̃′/G′ and these are p′-groups. Let M(1) = M(1)_{(G,H)} and N′(1) = N(1)_{(H′,G′)} = f↓_{H′×G′}^{G′×G′}(B_{0}(kG′)) ≅ g↑_{H′×H′}^{H′×G′}(B_{0}(kH)). Moreover, let L(1) = g↑_{H×H′}^{G×G′}(B_{0}(kH)), where g↑_{H×H′}^{G×G′} : H × H′ → G × G′ is the Green correspondence with respect to δ(P).

G̃ −−→ H̃ ↠ H̃/O_{p′}(H̃) ≅ H̃′/O_{p′}(H̃′) ↞ H̃′ ←−− G̃′
▽ ▽ ▽
G ⟶ H ↠ H/O_{p′}(H) ≅ H′/O_{p′}(H′) ↞ H′ ⟵ G′

L(1)
B_{0}(kG) G × G  G × G′  G′ × G′ B_{0}(kG′)
↓ f  ↑ g  ↑ g
M(1) G × H ⟶^{f} H × H′ ⟶^{g} H′ × G′ N′(1)
B_{0}(kH)

Then, we get the following.

(i) L(1) is δ(H̃)-invariant as a right k[G × G′]-module, namely, L(1)·(h̃, h̃) ≅ L(1) as right k[G × G′]-modules for any h̃ ∈ H̃.

(ii) The following two statements (a) and (b) are equivalent.

(a) The right k[G × G′]-module L(1) is extendible to a k[(G × G′)·δ(H̃)]-module.

(b) The right k[H × H]-module B_{0}(kH) is extendible to a k[(H × H)·δ(H̃)]-module. (iii) If we assume, furthermore, that G̃/G is cyclic, then it holds that the right k[G × G′]-module

L(1) is extendible to a right k[(G × G′)·δ(H̃)]-module.

(iv) If we assume, moreover, that G̃/G is cyclic, the four groups G̃, H̃, G̃′ and H̃′ have the same p-local structure, CG̃(P) ⊆ G, CG̃′(P) ⊆ G′ and that a right k[G×G′]-module M(1)⊗_{B_{0}(kH)} N′(1) has a unique non-projective indecomposable direct summand 𝔐 as a right k[G × G′]-module such that 𝔐 realizes a Puig equivalence between B_{0}(kG) and B_{0}(kG′), then it holds that L(1) ≅ 𝔐 as right k[G × G′]-modules and that B_{0}(kG̃) and B_{0}(kG̃′) are Puig equivalent.

Proof. As in the proof of (3.8), we can consider B_{0}(kH̃) = B_{0}(kH̃′) and also B_{0}(kH) = B_{0}(kH′).

(i) This follows from (1.5).

(ii) We get the assertion by [11, (6.5) Corollary] and (3.6)(iii).

(iii) First, note that H̃/H ≅ δ(H̃)/δ(H) ↠ δ(H̃)/δ(G ∩ G′) ≅ [(G × G′)·δ(H̃)]/[G × G′], so that [(G × G′)·δ(H̃)]/[G × G′] is a cyclic p′-group. Hence, as in (3.6)(iv), we get the assertion by (i), (3.5) and (ii).

(iv) As in the proof of (3.8), 𝔐|M(1) ⊗_{B_{0}(kH)} N′(1)|B_{0}(kH)↑_{H×H′}^{G×G′}. Now, since 𝔐 realizes a Morita equivalence, 𝔐 is indecomposable as a right k[G × G′]-module by (3.2). Hence, by the definition of L(1) and the Green correspondences, we know that 𝔐 ≅ L(1) as right k[G × G′]- modules. By (iii), L(1) is extendible to a k[(G × G′)·δ(H̃)]-module. Therefore, [22, 3.9.Corollary (a)] implies that B_{0}(kG̃) and B_{0}(kG̃′) are Puig equivalent. □

(3.10) Lemma. Let G be an arbitrary finite group with cyclic Sylow 3-subgroup P of order 3, and let H = N_{G}(P). Then, the Green correspondence with respect to P gives a Puig equivalence between B_{0}(kG) and B_{0}(kH), that is, M(1) = M(1)_{(G,H)} gives a Puig equivalence.

Proof. Easy (see [24, Remark 1.6] and [17, (5.7) Lemma]). □

(3.11) Lemma. Let k be an algebraically closed field of characteristic 3, and assume that G has a Sylow 3-subgroup P such that P ≅ C_{3} × C_{3}. Let H = N_{G}(P). Then, B_{0}(kG) and B_{0}(kH) are stably equivalent of Morita type via M(1)_{(G,H)}.

Proof. Let Q be any subgroup of P with Q ≠ 1. If Q = P, then C_{G}(Q) = C_{H}(Q).

Assume Q ⪇ P, so that Q ≅ C_{3}. Thus, by (1.4), C_{G}(Q) = Q × L for a subgroup L of C_{G}(Q). Hence, C_{H}(Q) = Q × (L ∩ H). Clearly, B_{0}(k[C_{G}(Q)]) = kQ ⊗_{k} B_{0}(kL) and B_{0}(k[C_{H}(Q)]) = kQ ⊗_{k} B_{0}(k[L ∩ H]). Therefore, since C_{3} ∈ Syl_{3}(L), we get from (3.10) that M(1)_{(C_{G}(Q),C_{H}(Q))} gives a Puig equivalence between B_{0}(k[C_{G}(Q)]) and B_{0}(k[C_{H}(Q)]). Thus, by a result of M.Broué [4, 6.3.Theorem] (see also [28, Theorem 4.1]), a (B_{0}(kG), B_{0}(kH))-bimodule M(1)_{(G,H)} realizes a stable equivalence of Morita type between B_{0}(kG) and B_{0}(kH). □

4. BOURÉ CONJECTURE FOR NON-SIMPLE GROUPS

In this section we give several propositions where the Broué conjecture is checked for non-simple finite groups which have the elementary abelian Sylow 3-subgroup of order 9.

(4.1) Notation and assumption. Throughout this section we use the following notation and terminology. Let k be an algebraically closed field of characteristic 3, let G̃ be a finite group with elementary abelian Sylow 3-subgroup P of order 9, and let G ◁ G̃ such that 3 |̸|G̃/G|, so that P ∈ Syl_{3}(G̃) ∩ Syl_{3}(G). Furthermore, let H̃ be a subgroup of G̃ such that H̃ ⊇ NG̃(P), and let H be a subgroup of H̃ such that H ⊇ N_{G}(P). We denote by M_{10}, M_{11}, M_{21}, M_{22} and M_{23} for the Mathieu groups. For groups G, G̃, G′, G̃′, H, H̃ and H′, we denote by A, Ã, A′, Ã′, B, B̃ and B′, respectively, B_{0}(kG), B_{0}(kG̃), B_{0}(kG′), B_{0}(kG̃′), B_{0}(kH), B_{0}(kH̃) and B_{0}(kH′). We use the notation A_{6}.2_{i} for i = 1, 2, 3 and A_{6}.2^{2} as in [8, p.4], the notation L_{3}(4).2_{i} for i = 1, 2, 3 as in [8, p.23] and the notation M_{22}.2 as in [8, p.39].

(4.2) Proposition. B_{0}(kM_{22}) and B_{0}(kM_{10}) are Puig equivalent via a right k[M_{22} × M_{10}]- module M(1)_{(M_{22},M_{10})}.

Proof. Let G′ = M_{22}, G = M_{10} = A_{6}.2_{3} and P ∈ Syl_{3}(G′) ∩ Syl_{3}(G). Let H = N_{G}(P), so that H = N_{G′}(P) and H ≅ P ⋊ Q_{8}. Let M′(1) = M(1)_{(G′,H)} and N(1) = N(1)_{(H,G)}. Then, we can consider the Green correspondences

G′ ⇄^{f′}_{g′} H ⇄^{g}_{f} G.

First, we note that the right k[G′ × H]-module M′(1) induces a stable equivalence of Morita type between A′ and B, and that the right k[H ×G]-module N(1) induces a stable equivalence of Morita type between B and A by (3.11) (and actually P is a T.I. set in M_{10} by [8, p.4]).

Now, we can write IBr(A′) = {k_{G′}, 49, 49^{∨}, 55, 231}, IBr(A) = {k_{G}, 1_{G}, 4, 4^{∨}, 6}, and IBr(B) = {k_{H}, 1_{1}, 1_{2}, 1_{3}, 2} where the numbers of simple modules mean the k-dimensions. By [24, Example 4.5], we know that the Loewy structure of the Green correspondents f′(S′) for simple A′-modules S′ is

f′(55) = 1_{1}, f′(49) = 1_{2} 2 1_{3} , f′(49^{∨}) = 1_{3} 2 1_{2} , f′(231) = \begin{matrix}2 \\ k_{H} 1_{1} \\ 2 \end{matrix} .

On the other hand, we also know that

(^{∗})  f(1_{G}) ≅ f′(55), f(4) ≅ f′(49), f(4^{∨}) ≅ f′(49^{∨}), f(6) ≅ f′(231).

If follows from [21, Theorem 2.1(i)] that there is a unique non-projective indecomposable right k[G′ × G]-module 𝔐 such that 𝔐|M′(1) ⊗_{B} N(1) as right k[G′ × G]-modules. Then, (*) shows that, for any simple A′-module S′, S′⊗_{A′} 𝔐 is a simple right A-module. Therefore, by [21, Theorem 2.1(iii)] and [22, 1.6.Theorem], we know that 𝔐 realizes a Puig equivalence between A′ and A. □

(4.3) Lemma. Assume that G̃ = M_{10} = A_{6}.2_{3} and H̃ = NG̃(P). Then, B_{0}(kG̃) and B_{0}(kH̃) are derived (even splendid) equivalent.

Proof. Let G̃′ = M_{22}, so that H̃ = NG̃′(P) ≅ P ⋊ Q_{8} by the proof of (4.2). It follows from results of Okuyama [24, Example 4.5] and [25] that Ã′ and B̃ are splendid equivalent. Since Ã′ and Ã are Puig equivalent by (4.2), we get the assertion. □

(4.4) Lemma. B_{0}(k[M_{22}.2]) and B_{0}(k[M_{10}.2]) are Puig equivalent (note M_{10}.2 ≅ A_{6}.2^{2} by [8, p.4]).

Proof. Let G′ = M_{22}, G̃′ = M_{22}.2, G = M_{10} and G̃ = M_{10}.2. Note that N_{G′}(P) = N_{G}(P) = P ⋊ Q_{8} and NG̃′(P) = NG̃(P) = P ⋊ SD_{16} (see [8, p.4 and p.39]).

Ã′  Ã
G̃′ = M_{22}.2 ⊇ G̃ = M_{10}.2 ⊇ NG̃′(P) = NG̃(P) = P ⋊ SD_{16}
▽ ▽ ▽
G′ = M_{22} ⊇ G = M_{10} ⊇ N_{G′}(P) = N_{G}(P) = P ⋊ Q_{8}
A′  A

By (4.2), A′ and A are Puig equivalent via a right k[G′ ×G]-module M(1)_{(G′,G)}. Since E(G′) ≅ Q_{8} and E(G̃′) ≅ SD_{16}, we have CG̃′(P) ⊆ G′ by (1.2)(viii). Therefore, (3.7) implies that Ã′ and Ã are Puig equivalent. □.

(4.5) Lemma. Let G̃ = M_{10}.2 = A_{6}.2^{2} and H̃ = NG̃(P). Then, B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent.

Proof. Let G = M_{10} and H = N_{G}(P). Then, we can write IBr(Ã) = {kG̃, 1_{a}, 1_{b}, 1_{c}, 6_{a}, 6_{b}, 8} and IBr(A) = {k_{G}, 1_{G}, 6, 4, 4^{∨}} such that k_{G}↑G̃ = kG̃ ⊕ 1_{a}, 1_{G}↑G̃ = 1_{b} ⊕ 1_{c}, 6↑G̃ = 6_{a} ⊕ 6_{b} and 4↑G̃ = 4^{∨}↑G̃ = 8. Similarly, we can write IBr(B̃) = {kH̃, 1_{0}, 1_{1a}, 1_{1b}, 2′, 2_{a}, 2_{b}} and IBr(B) = {k_{H}, 1_{1}, 1_{2}, 1_{3}, 2} such that k_{H}↑H̃ = kH̃ ⊕ 1_{0}, 1_{1}↑H̃ = 1_{1a} ⊕ 1_{1b}, 1_{2}↑H̃ = 1_{3}↑H̃ = 2′, 2↑H̃ = 2_{a} ⊕ 2_{b} (note that, in [24, §4 Case 4], the simple B̃-modules here are represented in different notatin, say, 0, 3, 1, 2, 6, 4, 5 for kH̃, 1_{0}, 1_{1a}, 1_{1b}, 2′, 2_{a}, 2_{b}, respectively). Then, we can consider the Green correspondences

G̃ ⇄^{f̃}_{g̃} H̃, G ⇄^{f}_{g} H

with respect to P. Let M̃(1) = M(1)_{(G̃,H̃)}. Then, by (3.11), M̃(1) gives a stable equivalence of Morita type between Ã and B̃. It follows from the proof of (4.2) that the Loewy structure of f(S) for simple A-modules S is

f(k_{G}) = k_{H}, f(1_{0}) = 1_{1}, f(6) = \begin{matrix}2 \\ k_{H} 1_{1} \\ 2 \end{matrix} , f(4) = 1_{2} 2 1_{3} , f(4^{∨}) = 1_{3} 2 1_{2} .

Next, we claim that the Loewy structure of f̃(S̃) for simple Ã-modules S̃ is

(*)  \begin{matrix}f̃(kG̃) = kH̃, f̃(1_{a}) = 1_{0}, f̃(1_{b}) = 1_{1b}, f̃(1_{c}) = 1_{1a}, \\ f̃(6_{a}) = \begin{matrix}2_{a} \\ 1_{1a} 1_{0} \\ 2_{b} \end{matrix} , f̃(6_{b}) = \begin{matrix}2_{b} \\ kH̃ 1_{1b} \\ 2_{a} \end{matrix} , f̃(8) = \begin{matrix}2′ \\ 2_{b} 2_{a} \\ 2′ \end{matrix} . \end{matrix}

First, consider f̃(6_{a}). Since f̃(6_{a})↓_{H} = 6_{a}↓_{H̃}↓_{H} = 6_{a}↓_{G}↓_{H} = 6↓_{H} = f(6). Hence, we may assume that f̃(6_{a})/f̃(6_{a})·J(kH̃) ≅ 2_{a}. On the other hand, it is easy to know the Loewy structure of projective indecomposable kH̃-modules. Hence, by looking at the quiver with relations for kH̃ = B̃ in [24, §4 Case 4], we get the structure of f̃(6_{a}) as desired. Similar for the rest.

Let G′ = HS (the Higman-Sims simple group) and H′ = N_{G′}(P′) where P′ ∈ Syl_{3}(G′). Then, we may assume P′ = P and H′ = H̃. We use the notation k_{G′}, 22, 748, 154_{0}, 1176, 321, 1253 which are simple A′-modules as in [24, Example 4.8]. Let f′ : G′ → H′ = H̃ be the Green correspondence with respect to P. Then we know that, except the structure of f̃(1_{b}) = 1_{1b} ≅ f′(748), f̃(S̃) = f′(S′) for each pair (S̃, S′) ∈ {(kG̃, k_{G′}), (1_{a}, 22), (1_{c}, 154_{0}), (6_{a}, 1176), (6_{b}, 321), (8, 1253)}. Namely, we get the following deformations of Ã and A′ by M̃(1) and M′(1), respectively, where M′(1) = M(1)_{(G′,H̃)}. That is:

S̃  kG̃ 1_{a}  1_{b}  1_{c}  6_{a}  6_{b}  8
⇓f̃  ↓  ↓  ↓  ↓  ↓  ↓  ↓

(S̃ ⊗Ã M̃(1))B̃ (modulo proj.)

k 1_{0}  1_{1b}  1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix} \begin{matrix}2′ \\ ⧸ & ⧹ \\ 2_{a} & 2_{b} \\ ⧹ & ⧸ \\ 2′ \end{matrix}

S′  k_{G′} 22  748  154_{0}  1176  321  1253
⇓f′  ↓  ↓  ↓  ↓  ↓  ↓  ↓

(S′ ⊗_{A′} M′(1))B̃ (modulo proj.)

k 1_{0}

\begin{matrix}1_{1b} & 2′ \\ | & ⧸ & | \\ 2_{a} & 2_{b} \\ | & ⧸ & | \\ 2′ & 1_{1b} \end{matrix}

1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix} \begin{matrix}2′ \\ ⧸ & ⧹ \\ 2_{a} & 2_{b} \\ ⧹ & ⧸ \\ 2′ \end{matrix}

Note that in the above table the modules at the rows are represented modulo projectives. Thus, as in [24, Example 4.8 (1)–(4)], take I_{0}^{(1)} = {2′}, which is so-called a nice index set in [24, §2, (III)]. Then, just as done in [24, Example 4.8 (3), (4)], there are a k-algebra B̃_{1} and a (B̃, B̃_{1})-bimodule M̃_{1} such that by an Okuyama’s surgery with respect to the index set I_{0}^{(1)}, the k-algebra B̃ is deformed into a new k-algebra B̃_{1}, which satisfies the following:

(S̃ ⊗Ã M̃(1))B̃ (modulo proj.)

k 1_{0}  1_{1b}  1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix} \begin{matrix}2′ \\ ⧸ & ⧹ \\ 2_{a} & 2_{b} \\ ⧹ & ⧸ \\ 2′ \end{matrix}

⇓I_{0}^{(1)} = {2′}  ↓ ↓  ↓  ↓  ↓  ↓  ↓

(S̃ ⊗Ã M̃(1) ⊗B̃ M̃_{1})B̃_{1}

(modulo proj.)

k 1_{0}  1_{1b}  1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix}

2′

(S′ ⊗_{A′} M′(1))B̃ (modulo proj.)

k 1_{0}

\begin{matrix}1_{1b} & 2′ \\ | & ⧸ & | \\ 2_{a} & 2_{b} \\ | & ⧸ & | \\ 2′ & 1_{1b} \end{matrix}

1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix} \begin{matrix}2′ \\ ⧸ & ⧹ \\ 2_{a} & 2_{b} \\ ⧹ & ⧸ \\ 2′ \end{matrix}

⇓I_{0}^{(1)} = {2′}  ↓ ↓  ↓  ↓  ↓  ↓  ↓

(S′ ⊗_{A′} M′(1) ⊗B̃ M̃_{1})B̃_{1}

(modulo proj.)

k 1_{0}

\begin{matrix}1_{1b} \\ 2′ \\ 1_{1b} \end{matrix}

1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix}

2′

Note that, since 1_{1b} ∉ {2′} = I_{0}^{(1)}, the Okuyama’s surgery does not make any change for 1_{1b} in B̃.

Next, as in [24, Example 4.8 (5)–(6)], take I_{0}^{(2)} = {2_{a}, 2_{b}}, which is a nice index set again. Then, just as in [24, Example 4.8 (5)–(6)], there are a k-algebra B̃_{2} and a (B̃_{1}, B̃_{2})-bimodule M̃_{2} such that by an Okuyama’s surgery with respect to I_{0}^{(2)}, the k-algebra B̃_{1} is deformed into another k-algebra B̃_{2}, which satisfies the following:

(S̃ ⊗Ã M̃(1) ⊗B̃ M̃_{1})B̃_{1}

(modulo proj.)

k 1_{0} 1_{1b} 1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix}

2′

⇓I_{0}^{(2)} = {2_{a}, 2_{b}}  ↓ ↓  ↓  ↓  ↓  ↓  ↓
(S̃ ⊗Ã M̃(1) ⊗B̃ M̃_{1} ⊗B̃_{1} M̃_{2})B̃_{2}

(modulo proj.)

k 1_{0} 1_{1b} 1_{1a}  2_{a}  2_{b}  2′

(S′ ⊗_{A′} M′(1) ⊗B̃ M̃_{1})B̃_{1}

(modulo proj.)

k 1_{0}

\begin{matrix}1_{1b} \\ 2′ \\ 1_{1b} \end{matrix}

1_{1a}

\begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix}

2′

⇓I_{0}^{(2)} = {2_{a}, 2_{b}}  ↓ ↓  ↓  ↓  ↓  ↓  ↓

(S′ ⊗_{A′} M′(1) ⊗B̃ M̃_{1} ⊗B̃_{1} M̃_{2})B̃_{2}

(modulo proj.)

k 1_{0}

\begin{matrix}1_{1b} \\ 2′ \\ 1_{1b} \end{matrix}

1_{1a}  2_{a}  2_{b}  2′

Note that 1_{1a} ∉ {2_{a}, 2_{b}} = I_{0}^{(2)}, so that an Okuyama’s surgery with respect to the index set I_{0}^{(2)} does not make any change for 1_{1b} in B̃_{1}. Therefore, we finally get that

(S̃ ⊗Ã M̃(1) ⊗B̃ M̃_{1} ⊗B̃_{1} M̃_{2})B̃_{2} = (simple) ⊕ (projective)

for any simple Ã-module S̃ as it is seen in the above table. Hence, as in [24, §2 (IV)], Ã and B̃_{2} are Morita (even Puig) equivalent. Consequently, Ã and B̃ are derived equivalent. □

(4.6) Lemma. Let G̃ = M_{22}.2 and H̃ = NG̃(P). Then, B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent.

Proof. Let G′ = M_{10}.2 ≅ A_{6}.2^{2}. We know that H̃ = N_{G′}(P) ≅ P ⋊ SD_{16} from the proof of (4.4). Then, by (4.5), A′ and B̃ are derived equivalent. Thus, we get the assertion since Ã and A′ are Puig equivalent by (4.4). □.

(4.7) Lemma. Let G̃ = A_{6}.2_{2} = PGL_{2}(9) and H̃ = NG̃(P). Then, B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent.

Proof. Let G = A_{6} and H = N_{G}(P). By [8, p.2], H = P ⋊ C_{4} and H̃ = P ⋊ C_{8}.

G̃ = A_{6}.2_{2} −−→ H̃ = P ⋊ C_{8}
▽  ▽
G = A_{6} −−→ H = P ⋊ C_{4}

We can write IBr(Ã) = {kG̃, 1G̃, 3_{a1}, 3_{a2}, 3_{b1}, 3_{b2}, 4_{1}, 4_{2}} and IBr(A) = {k_{G}, 3_{a}, 3_{b}, 4} such that k_{G}↑G̃ = kG̃ ⊕ 1G̃, 3_{a}↑G̃ = 3_{a1} ⊕ 3_{a2}, 3_{b}↑G̃ = 3_{b1} ⊕ 3_{b2} and 4↑G̃ = 4_{1} ⊕ 4_{2}. Similarly, we can write IBr(B̃) = {k, 1H̃, 1_{2}, 1_{2}^{∨}, 1_{3}, 1_{3}^{∨}, 1_{4}, 1_{4}^{∨}} and IBr(B) = {k_{H}, 1_{H}, 1′_{H}, 1′_{H}^{∨}} such that k_{H}↑H̃ = kH̃ ⊕ 1H̃, 1_{H}↑H̃ = 1_{2} ⊕ 1_{2}^{∨}, 1′_{H}↑H̃ = 1_{3} ⊕ 1_{4}, (1′_{H}^{∨})↑H̃ = 1_{3}^{∨} ⊕ 1_{4}^{∨}. Then, by making the proof of [24, Example 4.2] duplicated, we get the assertion. □

(4.8) Lemma. Let G̃ = L_{3}(4).2_{1} and H̃ = NG̃(P). Then, B_{0}(kG̃) and B_{0}(kH̃) are derived (even splendid) equivalent.

Proof. Let G = L_{3}(4), H = N_{G}(P) and G_{0} = G·CG̃(P). By [8, p.23], H̃ = (P ⋊ Q_{8}) × C_{2}, so that E(G̃) ≅ Q_{8} ≅ E(G). If G_{0} = G, then CG̃(P) ⊆ G, so that (1.2)(v) implies that G̃/G ≅ E(G̃)/E(G) = 1, a contradiction. Hence, G ⪇ G_{0}, so that G̃ = G_{0}. Then, by (3.1), Ã and A are Puig equivalent. On the other hand, we get from [24, Example 4.6] and [25] that A and B are derived (even splendid) equivalent. Since B ≅ B̃ ≅ k[P ⋊ Q_{8}], we know that Ã and B̃ are derived (even splendid) equivalent. □

(4.9) Lemma. Let G̃ = L_{3}(4).2_{2} and H̃ = NG̃(P). Then, B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent.

Proof. Let G = L_{3}(4) and H = N_{G}(P). Then, H̃ ≅ P ⋊ SD_{16} and H = P ⋊ Q_{8} from [8, p.23]. Then, we can use the same notation for simple B-modules and B̃-modules as in the proof of (4.5). We can write that IBr(Ã) = {kG̃, 1G̃, 15_{1a}, 15_{1b}, 30, 19_{a}, 19_{b}} and IBr(A) = {k_{G}, 15_{1}, 15_{2}, 15_{3}, 19} such that k_{G}↑G̃ = kG̃ ⊕ 1G̃, 15_{1}↑G̃ = 15_{1a} ⊕ 15_{1b}, 15_{2}↑G̃ = 15_{3}↑G̃ = 30 and 19↑G̃ = 19_{a} ⊕ 19_{b}. Let

G̃ ⇄^{f̃}_{g̃} H̃, G ⇄^{f}_{g} H

be the Green correspondences with respect to P. Note that M̃(1) gives a stable equivalence of Morita type between Ã and B̃ by (3.11), where M̃(1) = M(1)_{(G̃,H̃)}. We know the structure of the Green correspondents f(S) for each simple A-module S, which was calculated by G.Schneider (see [20, Lemma 6.1]). Then, we can calculate the Loewy structure of f̃(S̃) for each simple Ã-module S̃, namely,

S̃  kG̃ 1G̃  19_{a}  19_{b}  15_{1a}  15_{1b}  30
⇓f̃  ↓  ↓  ↓  ↓  ↓  ↓  ↓

(S̃ ⊗Ã M̃(1))B̃ (modulo proj.)

k 1_{0}

\begin{matrix}1_{1b} & 2′ \\ | & ⧸ & | \\ 2_{a} & 2_{b} \\ | & ⧸ & | \\ 2′ & 1_{1b} \end{matrix} \begin{matrix}1_{1a} & 2′ \\ | & ⧸ & | \\ 2_{b} & 2_{a} \\ | & ⧸ & | \\ 2′ & 1_{1a} \end{matrix} \begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned} \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix} \begin{aligned}& 2_{a} & 2_{b} \\ ⧸ & & ⧹⧸ & ⧹ \\ 1_{0} & & 2′ & k \\ ⧹ & & ⧸⧹ & ⧸ \\ & 2_{b} & 2_{a} \end{aligned}

(1)

We may assume that B̃ is a basic k-algebra. Namely, all simple B̃-modules are of k-dimension one. Now, compare (1) with [24, Example 4.8 (2)]. So, as in the proof of (4.5) and [24, Example 4.8 (2)–(4)], take I_{0}^{(1)} = {2′}, which is a nice index set in [24, §2 (III)]. We denote by K^{−}(proj-R) the homotopy category of bounded above complexes of finitely generated projective right R-modules for a k-algebra R. Let B̃_{1} = End_{K^{−}(proj-B̃)}(P (I_{0}^{(1)})^{•}) and let M̃_{1} be a (B̃, B̃_{1})-bimodule which gives a stable equivalence of Morita type between B̃ and B̃_{1}, see [24, Theorem 1.2 and §2]. Then, we get a unique non-projective indecomposable direct summand of X̃ ⊗B̃ M̃_{1} as a right B̃_{1}-module for a B̃-module X̃ in several cases. That is, we get the following table:

(2)

X̃  k 1_{0} 1_{1b} 1_{1a} 2_{a} 2_{b}  \begin{matrix}2′ \\ ⧸ & ⧹ \\ 2_{a} & 2_{b} \\ ⧹ & ⧸ \\ 2′ \end{matrix}

⇓I_{0}^{(1)} = {2′}  ↓ ↓ ↓  ↓  ↓ ↓  ↓

(X̃ ⊗B̃ M̃_{1})B̃_{1} (modulo proj.)

k 1_{0} 1_{1b} 1_{1a} 2_{a} 2_{b} 2′

We claim the following table, too.

(3)

X̃  \begin{matrix}1_{1b} & 2′ \\ | & ⧸ & | \\ 2_{a} & 2_{b} \\ | & ⧸ & | \\ 2′ & 1_{1b} \end{matrix}  \begin{matrix}1_{1a} & 2′ \\ | & ⧸ & | \\ 2_{b} & 2_{a} \\ | & ⧸ & | \\ 2′ & 1_{1a} \end{matrix}  \begin{aligned}& 2_{a} & 2_{b} \\ ⧸ & & ⧹⧸ & ⧹ \\ 1_{0} & & 2′ & k \\ ⧹ & & ⧸⧹ & ⧸ \\ & 2_{b} & 2_{a} \end{aligned}

⇓I_{0}^{(1)} = {2′}  ↓  ↓  ↓

(X̃ ⊗B̃ M̃_{1})B̃_{1} (modulo proj.)

\begin{matrix}1_{1b} \\ 2′ \\ 1_{1b} \end{matrix} \begin{matrix}1_{1a} \\ 2′ \\ 1_{1a} \end{matrix}

\begin{matrix}2_{b} & 2′ & 2_{a} \\ ⧸⧹ & ⧸⧹ & ⧸⧹ \\ k 1_{1b} & k 1_{0} & 1_{1a} 1_{0} \\ ⧹⧸ & ⧹⧸ & ⧹⧸ \\ 2_{a} & 2′ & 2_{b} \end{matrix}

Since M̃_{1} gives a stable equivalence of Morita type between B̃ and B̃_{1}, we get the second and third columns in (3) by looking at the last column in (2). Now, we know the Loewy and socle structure of P (2′) in B̃ since B̃ = k[P ⋊ SD_{16}], see [24, Appendix Case 4] for instance. Hence we get the last column in (3) as well since B̃/(B̃e_{2′} B̃) ≅ B̃_{1}/(B̃_{1}e_{2′} B̃_{1}), where e_{2′} is a primitive idempotent of B̃ and B̃_{1} corresponding to the simples 2′.

Next, by using (2), as in [24, Example 4.8], we know the quiver and relations for B̃_{1} as follows. Namely,

(4)  the quiver of B̃_{1}  =  \begin{aligned}& k \\ & 1_{1b} \\ ↓ & ↑ \\ 2_{a} & 2′ & 2_{b} \\ ↓ & ↑ \\ & 1_{0} \\ & 1_{1a} \end{aligned}

Next, we claim that the Cartan matrix CB̃_{1} for B̃_{1} has the following form:

(5)  CB̃_{1} =  \begin{aligned}& k & 1_{1b} & 1_{1a} & 1_{0} & 2_{a} & 2_{b} & 2′ \\ k & 3 & 1 & 2 & 1 & 1 & 1 & 2 \\ 1_{1b} & 1 & 3 & 1 & 2 & 1 & 1 & 2 \\ 1_{1a} & 2 & 1 & 3 & 1 & 1 & 1 & 2 \\ 1_{0} & 1 & 2 & 1 & 3 & 1 & 1 & 2 \\ 2_{a} & 1 & 1 & 1 & 1 & 2 & 1 & 1 \\ 2_{b} & 1 & 1 & 1 & 1 & 1 & 2 & 1 \\ 2′ & 2 & 2 & 2 & 2 & 1 & 1 & 3 \end{aligned}

It is well-known that

∑_{n∈ℤ}(−1)^{n} dim_{k} [Hom_{K^{−}(proj-B̃)}(P (k)^{•}, P (1_{1b})^{•}[n])]

= ∑_{r,s∈ℤ}(−1)^{r−s} dim_{k} [HomB̃((P (k)^{•})^{r}, (P (1_{1b})^{•})^{s})].

Then, by using this and the definition of a tilting complex P (I_{0}^{(1)})^{•} in B̃, we get (5), see [24, §1].

Actually, we can know the relations for the quiver in (4) by making use of the relations for the quiver of B̃ stated in [24, Appendix Case 4] and the definition of B̃_{1} which is given as an endomorphism algebra of the tilting complex P (I_{0}^{(1)})^{•}.

Especially, we know the structure of P (2_{a}) which has the form

(6)  P (2_{a}) in B̃_{1}  =  \begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ | & × & | \\ 2′ & & 2_{b} \\ | & × & | \\ k & & 1_{1b} \\ ⧹ & & ⧸ \\ & 2_{a} \end{aligned}

Now, take I_{0}^{(2)} = {2_{a}, 2_{b}}, and let B̃_{2} = End_{K^{−}(proj-B̃_{1})}(P (I_{0}^{(2)})^{•}) and let M̃_{2} be a (B̃_{1}, B̃_{2})- bimodule giving a stable equivalence of Morita type between B̃_{1} and B̃_{2}. Then, just as in [24, Example 4.8 (4)–(6)], we get the following table by making use of (6):

(7)

X̃  k 1_{0} 1_{1b} 1_{1a}  \begin{aligned}& 2_{a} \\ ⧸ & & ⧹ \\ 1_{1a} & & 1_{0} \\ ⧹ & & ⧸ \\ & 2_{b} \end{aligned}  \begin{matrix}2_{b} \\ ⧸ & ⧹ \\ k & 1_{1b} \\ ⧹ & ⧸ \\ 2_{a} \end{matrix} 2′

⇓I_{0}^{(2)} = {2_{a}, 2_{b}} ↓ ↓ ↓  ↓  ↓  ↓  ↓

(X̃ ⊗B̃_{1} M̃_{2})B̃_{2} (modulo proj.)

k 1_{0} 1_{1b} 1_{1a}  2_{a}  2_{b}  2′

Then, especially, the Okuyama’s surgery given in the sixth and seventh columns in (9) imply the second column of the next table:

(8)

X̃  \begin{matrix}2_{b} & 2′ & 2_{a} \\ ⧸⧹ & ⧸⧹ & ⧸⧹ \\ k 1_{1b} & k 1_{0} & 1_{1a} 1_{0} \\ ⧹⧸ & ⧹⧸ & ⧹⧸ \\ 2_{a} & 2′ & 2_{b} \end{matrix}  \begin{matrix}1_{1b} \\ 2′ \\ 1_{1b} \end{matrix}  \begin{matrix}1_{1a} \\ 2′ \\ 1_{1a} \end{matrix}

⇓I_{0}^{(2)} = {2_{a}, 2_{b}}  ↓  ↓  ↓

(X̃ ⊗B̃_{1} M̃_{2})B̃_{2} (modulo proj.)

\begin{aligned}& 2′ \\ & ⧸⧹ \\ 2_{b} & k 1_{0} & 2_{a} \\ & ⧹⧸ \\ & 2′ \end{aligned}

\begin{matrix}1_{1b} \\ 2′ \\ 1_{1b} \end{matrix} \begin{matrix}1_{1a} \\ 2′ \\ 1_{1a} \end{matrix}

The third and fourth columns in (8) are easily obtained by (7).

Now, as before, by using (7) we get the quiver and relations for B̃_{2}. In particular, we know the structures of P (1_{1b}), P (1_{1a}) and P (2′) for B̃_{2} which have the following forms:

(9)  P (1_{1b}) in B̃_{2} = \begin{matrix}1_{1b} \\ 2′ \\ k 1_{1b} 1_{0} 2_{a} \\ 2′ \\ 1_{1b} \end{matrix} ,  P (1_{1a}) in B̃_{2} = \begin{matrix}1_{1a} \\ 2′ \\ k 1_{1a} 1_{0} 2_{b} \\ 2′ \\ 1_{1a} \end{matrix} ,

(10)  P (2′) in B̃_{2} = \begin{matrix}2′ \\ 1_{1a} 1_{1b} k 1_{0} 2_{a} 2_{b} \\ 2′ 2′ 2′ \\ 1_{1a} 1_{1b} k 1_{0} 2_{a} 2_{b} \\ 2′ \end{matrix}

Take another nice index set, namely, let I_{0}^{(3)} = {1_{1b}, 1_{1a}}. Let B̃_{3} = End_{K^{−}(proj-B̃_{2})}(P (I_{0}^{(3)})^{•}) and let M̃_{3} be a (B̃_{2}, B̃_{3})-bimodule which gives a stable equivalence of Morita type between B̃_{2} and B̃_{3}. Then, as in [24, Example 4.8 (6)–(8)], we obtain the following table by using (9):

(11)

X̃  k 1_{0}  \begin{matrix}1_{1b} \\ 2′ \\ 1_{1b} \end{matrix}  \begin{matrix}1_{1a} \\ 2′ \\ 1_{1a} \end{matrix} 2_{a} 2_{b}  \begin{aligned}& 2′ \\ & ⧸⧹ \\ 2_{b} & k 1_{0} & 2_{a} \\ & ⧹⧸ \\ & 2′ \end{aligned}

⇓I_{0}^{(3)} = {1_{1b}, 1_{1a}} ↓ ↓  ↓  ↓  ↓ ↓  ↓

(X̃ ⊗B̃_{2} M̃_{3})B̃_{3} (modulo proj.)

k 1_{0} 1_{1b} 1_{1a} 2_{a} 2_{b}

\begin{aligned}& 2′ \\ & ⧸⧹ \\ 2_{b} & k 1_{0} & 2_{a} \\ & ⧹⧸ \\ & 2′ \end{aligned}

Next, we claim that

(12)  cB̃_{3}(2′, 2′) = 3.

For the index set I_{0}^{(3)} above, it follows from [24, §1] and the structure of P (2′) for B̃_{2} in (10) that a direct summand P (2′)^{•} of the tilting complex P (I_{0}^{(3)})^{•} in B̃_{2} has the form

(13)  P (2′)^{•} : ⋯ → 0 → P (1_{1a}) ⊕ P (1_{1b}) → P (2′) → 0 → ⋯

where the term P (2′) in (13) is of degree zero. We denote by (P (2′)^{•})^{r} the term of degree r in a complex P (2′)^{•} for r ∈ ℤ. Let c(S, T) = cB̃_{2}(S, T) for simple B̃_{2}-modules S and T. Then, we get by (13), (9) and (10) that

∑_{r,s∈ℤ}(−1)^{r−s} dim_{k} [HomB̃_{2}((P (2′)^{•})^{r}, (P (2′)^{•})^{s})]

= c(2′, 2′) − (c(2′, 1_{1a}) + c(2′, 1_{1b})) − (c(1_{1a}, 2′) + c(1_{1b}, 2′))

+ (c(1_{1a}, 1_{1a}) + c(1_{1a}, 1_{1b}) + c(1_{1b}, 1_{1a}) + c(1_{1b}, 1_{1b}))

= 3.

On the other hand,

cB̃_{3}(2′, 2′) = dim_{k} [HomB̃_{3}(P (2′), P (2′))] = dim_{k} [Hom_{K^{−}(proj-B̃_{2})}(P (2′)^{•}, P (2′)^{•})].

Hence, as for the proof (5), we get (12).

Next, take I_{0}^{(4)} = {2′}, and let B̃_{4} = End_{K^{−}(proj-B̃_{3})}(P (I_{0}^{(4)})^{•}) and let M̃_{4} be a (B̃_{3}, B̃_{4})-bimodule giving a stable equivalence of Morita type between B̃_{3} and B̃_{4}. Let X̃ be a B̃_{3}-module appearing at the last column in (11). Let e be a primitive idempotent of B̃_{3} corresponding to a simple B̃_{3}- module 2′. Then, we know (Soc(X̃))e = Soc(X̃) ≅ 2′. Moreover, it follows from (12) that ΩX̃/2′ has no composition factors isomorphic to 2′. That is, (ΩX̃)e ≅ 2′. Hence, as we have done many times already, we get by [24, Lemma 2.1 (2)] that

(14)

X̃  \begin{aligned}& 2′ \\ & ⧸⧹ \\ 2_{b} & k 1_{0} & 2_{a} \\ & ⧹⧸ \\ & 2′ \end{aligned}

⇓I_{0}^{(4)} = {2′}  ↓

(X̃ ⊗B̃_{3} M̃_{4})B̃_{4} (modulo proj.)

2′

Then, we finally know from (1), (2), (3), (7), (8), (11) and (14) that

(S̃ ⊗Ã M̃(1) ⊗B̃ M̃_{1} ⊗B̃_{1} M̃_{2} ⊗B̃_{2} M̃_{3} ⊗B̃_{3} M̃_{4})B̃_{4} = (simple) ⊕ (projective)

for any simple Ã-module S̃. Therefore, again by [21, Theorem 2.1(iii)], we get the assertion. □

(4.10) Lemma. Let G̃ = L_{3}(4).2_{3} or L_{3}(4).2^{2}, and let H̃ = NG̃(P). Then, B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent.

Proof. Case 1: G̃ = L_{3}(4).2^{2}. Let G = L_{3}(4).2_{2}. Then, we know E(G̃) ≅ E(G) ≅ SD_{16} since E(G) ◁ E(G̃) ↪ SD_{16} by (1.2)(i)–(ii) and since E(G) ≅ SD_{16} by the proof of (4.9). Hence, by (1.2)(ix), we get G̃ = G·CG̃(P). Thus, H̃ = H·CH̃(P). Therefore, we get from (4.9) and (3.4) that Ã and B̃ are derived equivalent.

Case 2: G̃ = L_{3}(4).2_{3}. Then, H̃ ≅ P ⋊ SD_{16} from [8, p.23]. Hence, just as in (4.9), B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent. □

(4.11) Remark. It follows from (3.1) and the proofs of (4.9) and (4.10) that the three k-algebras B_{0}(k[L_{3}(4).2_{2}]), B_{0}(k[L_{3}(4).2_{3}]) and B_{0}(k[L_{3}(4).2^{2}]) are all Puig equivalent.

5. PROOF OF THE MAIN RESULT

In this section we give a proof of the main result (0.2), which is a case-by-case analysis and which is divided into several parts.

(5.1) Notation. Throughout this section except the proof of (0.2), let G̃ be an arbitrary finite group such that O_{3′}(G̃) = 1 and that G̃ has a Sylow 3-subgroup P with P ≅ C_{3} × C_{3}, and let G = O^{3′}(G̃), so that P ∈ Syl_{3}(G̃) ∩ Syl_{3}(G). Let H̃ = NG̃(P), H = N_{G}(P), G_{0} = G·CG̃(P) and H_{0} = N_{G_{0}}(P). Then, H_{0} = H·CG̃(P) = H·CH̃(P). By Frattini argument, G̃ = G·H̃. We assume that k is an algebraically closed field of characteristic 3.

We say that (⋆) holds for G̃ if there is a derived equivalence between B_{0}(kG̃) and B_{0}(kH̃)

and (⋆) holds for G̃ if there is a Puig equivalence between B_{0}(kG̃) and B_{0}(kH̃).

Of course, (⋆) implies (⋆).

(5.2) Lemma. If E(G) = 1, then (⋆) holds for G̃.

Proof. Since G is 3-nilpotent, G̃ is 3-solvable of 3-length one. So, it is well-known that (⋆) holds for G̃ since B_{0}(kG̃) and B_{0}(kH̃) are isomorphic via restriction just as in (3.1) by a result of Isaacs-Smith [14] (see [12, §4], [2, p.71, ℓ. − 9]). □

(5.3) Lemma. If E(G) = 2, then (⋆) holds for G̃.

Proof. By (2.4), G = Q × L′ where Q ≅ C_{3} and L′ is a non-abelian simple group such that C_{3} ∈ Syl_{3}(L′). Then, Q = O_{3}(G) since G is not 3-solvable. Then, Q char G ◁ G̃, so that Q = O_{3}(G̃). Let C̃ = CG̃(Q). Then, G̃/C̃ = NG̃(Q)/CG̃(Q) ↪ Aut(Q) ≅ C_{2}, and hence |G̃/C̃| = 1 or 2.

Case 1. G̃ = C̃. Since C_{3} ≅ Q ⊆ Z(G̃), we get by (1.3) that G̃ = Q × L for some L ◁ G̃. Let R ∈ Syl_{3}(L) and N = N_{L}(R). Then, by (3.10), a right k[L × N]-module M′(1) = M(1)_{(L,N)} gives a Puig equivalence between B_{0}(kL) and B_{0}(kN). Hence, M̃(1) = M(1)_{(G̃,H̃)} gives a Puig equivalence between B_{0}(kG̃) and B_{0}(kH̃). Namely (⋆) holds for G̃.

Case 2. |G̃/C̃| = 2. Again, by (1.3), C̃ = Q × L for some L ◁ C̃. Let Ñ = NC̃(P). Then, just as in Case 1, M(1) = M(1)_{(C̃,Ñ)} gives a Puig equivalence between B_{0}(kC̃) and B_{0}(kÑ). Clearly, CG̃(P) ⊆ C̃. Therefore, (3.7) implies that (⋆) holds for G̃. □

(5.4) Lemma. If |E(G)| = 16, then (⋆) holds for G̃ (even a splendid equivalence exists).

Proof. By (2.4), G ∈ {M_{11}, M_{23}, HS}. By [24, Examples 4.7, 4.8 and 4.9] and [25], there is a splendid equivalence between B_{0}(kG) and B_{0}(kH). Then, by (1.2)(i), (ii) and (iii), we know E(G) = E(G_{0}) = E(G̃), so that G̃ = G_{0} by (1.2)(iv). Now, by (3.4), the splendid equivalence between B_{0}(kG) and B_{0}(kH) is liftable to that between B_{0}(kG̃) and B_{0}(kH̃). □

(5.5) Lemma. If E(G) ≅ C_{2} × C_{2}, then (⋆) holds for G̃.

Proof. Since C_{2} × C_{2} ◁̸ SD_{16}, we get by (1.2)(i)-(ii) that E(G̃) ∈ {C_{2} × C_{2}, D_{8}}. By (2.4), G = X × Y for non-abelian simple groups X and Y such that C_{3} ≅ Q ∈ Syl_{p}(X) and C_{3} ≅ R ∈ Syl_{3}(Y). Let H_{X} = N_{X}(Q), H_{Y} = N_{Y} (R). Then, by (3.10), there are Puig equivalences between B_{0}(kX) and B_{0}(kH_{X}), and also between B_{0}(kY) and B_{0}(kH_{Y}), both of them are given by the Green correspondences with respect to Q and R, respectively. Hence, M(1) = M(1)_{(G,H)} gives a Puig equivalence between B_{0}(kG) and B_{0}(kH). Let M_{0}(1) = M(1)_{(G_{0},H_{0})}. Then, by (3.3), M_{0}(1) gives a Puig equivalence between B_{0}(kG_{0}) and B_{0}(kH_{0}). Hence, by (i), (ii) and (v) of (1.2), we have G̃/G_{0} ≅ E(G̃)/E(G_{0}) ≅ 1 or C_{2}. Hence, by (3.7), (⋆) holds for G̃. □

(5.6) Lemma. Assume that G is one of (ix), (x), (xiii) or (xiv) in the list of (2.4). Then, (⋆) holds for G̃.

Proof. First, assume that G is of type (ix) of (2.4). Consider the following diagram

G̃ = G·H̃  −−→  H̃ = NG̃(P)
▽  ▽
G_{0} = G·CG̃(P) −−→  H_{0} = N_{G_{0}}(P)
▽  ▽
G = PSU_{3}(q^{2}) → H = N_{G}(P) = P ⋊ Q_{8}

Clearly, H_{0} = H·C_{H_{0}}(P). Let M(1) = M(1)_{(G,H)} and M_{0}(1) = M(1)_{(G_{0},H_{0})}. It follows from [16] that M(1) gives a Puig equivalence between B_{0}(kG) and B_{0}(kH). Hence, by (3.3), M_{0}(1) gives a Puig equivalence between B_{0}(kG_{0}) and B_{0}(kH_{0}). Since E(G) = Q_{8} by (2.4), we get from (i), (ii) and (v) of (1.2) that G̃/G_{0} ≅ E(G̃)/E(G_{0}) = 1 or C_{2}. Hence, as in (5.5), (3.7) implies the assertion.

For the cases (x), (xiii) and (xiv) of (2.4), we similarly get the assertion by using [27, Corollaire 3.6], [17] and [18] instead of [16]. □

(5.7) Lemma. If G ∈ {A_{6}, A_{7}, A_{8}, M_{21} ≅ PSL_{3}(4), M_{22}}, then (⋆) holds for G̃

Proof. Case 1. G = A_{6}. By (1.4) and [8, p.4], G̃ ∈ {A_{6}, Σ_{6}, PGL_{2}(9), M_{10}, M_{10}.2 = Aut(Σ_{6})}. If G̃ = A_{6} or Σ_{6}, then (⋆) holds for G̃ by [24] and, [24] or [5], respectively.

If G̃ = PGL_{2}(9), then (⋆) holds for G̃ by (4.7).

If G̃ = M_{10}, then (⋆) holds for G̃ by (4.3).

If G̃ = M_{10}.2 = A_{6}.2^{2} = Aut(Σ_{6}), then (⋆) holds for G̃ by (4.5).

Case 2. G = A_{7} or A_{8}. By (1.4) and [8, p.10 and p.22], G̃ = A_{n} or Σ_{n}, where n = 7 or 8. Hence, (⋆) holds for G̃ by [24] and [5].

Case 3. G = M_{21} ≅ PSL_{3}(4) = L_{3}(4). By (1.4) and [8, p.23], G̃ ∈ {L_{3}(4), L_{3}(4).2_{1}, L_{3}(4).2_{2}, L_{3}(4).2_{3}, L_{3}(4).2^{2}} (the notation here addapts to that in [8, p.23]). Case 3-(a): G̃ = L_{3}(4). Then, (⋆) holds for G̃ by [24] (even a splendid equivalence exists in this case by [25]). Case 3-(b): G̃ = L_{3}(4).2_{1}. Then, (⋆) holds for G̃ by (4.8). Case 3-(c): G̃ = L_{3}(4).2_{2}. Then, (⋆) holds for G̃ by (4.9). Case 3-(d): G̃ = L_{3}(4).2_{3}. Then, (⋆) holds for G̃ by (4.10). Case 3-(e): G̃ = L_{3}(4).2^{2}. Then, (⋆) holds for G̃ by (4.10).

Case 4. G = M_{22}. Then, by (1.4) and [8, p.39], G̃ = M_{22} or M_{22}.2. For G̃ = M_{22}, (⋆) holds by [24, Example 4.5] and [25].

Assume, next, that G̃ = M_{22}.2. In this case the assertion is obtained by (4.6). □

(5.8) Lemma. Assume that G is one of (viii), (xi) or (xii) in the list of (2.4). Then, (⋆) holds for G̃.

Proof. Case 1. Assume that G is of type (viii) of (2.4). By (2.4), E(G) ≅ Q_{8}, so that E(G̃) ≅ Q_{8} or SD_{16} by (1.2)(i)-(ii). Case 1-(a): E(G̃) ≅ Q_{8}. By (1.2)(ii)-(iii), E(G̃) = E(G_{0}) = E(G). Hence, by (1.2)(vi) (or (vii)), G̃ = G_{0}, so that H̃ = H_{0}. Now, it follows from [20], [24] and [25] that there exists a splendid equivalence between B_{0}(kG) and B_{0}(kH). Hence, by (3.4), the splendid equivalence is lifted to that between B_{0}(kG_{0}) and B_{0}(kH_{0}). This means that (⋆) holds for G̃ (even splendid equivalence exists). Case 1-(b): E(G̃) ≅ SD_{16}. Let G = L_{3}(q) and G′ = L_{3}(4). We can consider P ∈ Syl_{3}(G) ∩ Syl_{3}(G′). Hence, let H′ = N_{G′}(P). We then may consider H = H′ (see [20]). Let A = B_{0}(kG), A′ = B_{0}(kG′) and B = B_{0}(kH) = B_{0}(kH′). Let M(1) = M(1)_{(G,H)} and N′(1) = N(1)_{(H,G′)}. It follows from [20] that A and A′ are Puig equivalent via 𝔐, where 𝔐 is a unique non-projective indecomposable direct summand of M(1) ⊗_{B} N′(1). Let G_{0} = G·CG̃(P), A_{0} = B_{0}(kG_{0}), B_{0} = B_{0}(k[N_{G_{0}}(P)]) and 𝔐_{0} = A_{0} ⊗_{A} 𝔐. Then, (3.1) implies that 𝔐_{0} realizes a Puig equivalence between A_{0} and A′. Now, let G̃′ = L_{3}(4).2_{2} as in §4 (see [8, p.23]). We know from (1.2)(v) and (1.2)(vii) that G̃/G_{0} ≅ C_{2} ≅ G̃′/G′. Clearly, B ≅ B_{0} ≅ k[P ⋊Q_{8}]. Therefore, we get from (3.9)(iv) that Ã and Ã′ are Puig equivalent, where Ã = B_{0}(kG̃) and Ã′ = B_{0}(kG̃′). We have known already that Ã′ and B̃′ are derived equivalent, where B̃′ = B_{0}(k[NG̃′(P)]). Clearly, B̃′ ≅ B̃ ≅ k[P ⋊SD_{16}], where B̃ = B_{0}(k[NG̃(P)]). Thus, we finally know that Ã and B̃ are derived equivalent.

Case 2. Assume that G is of type (xi) of (2.4). By (2.4), E(G) ≅ D_{8}, so that E(G̃) ≅ D_{8} or SD_{16} by (1.2)(i)-(ii). Case 2-(a): E(G̃) ≅ D_{8}. As in the proof of Case 1-(a), G̃ = G_{0}. Then, we get the assertion just as in Case 1-(a) by using [26], [24] and [25]. Case 2-(b): E(G̃) ≅ SD_{16}. We get the assertion as in Case 1-(b) by using (4.5) instead of (4.9).

Case 3. Assuem that G is of type (xii) of (2.4). By (2.4), E(G) ≅ D_{8}, so that E(G̃) ≅ D_{8} or SD_{16} by (1.2)(i)-(ii). Case 3-(a): E(G̃) ≅ D_{8}. As in the proof of Case 1-(a), we get the assertion by [17], [24] and [25]. Case 3-(b): E(G̃) ≅ SD_{16}. We obtain the assertion as in Case 1-(b) by using the fact that (0.1) is checked for G = Σ_{8} ≅ A_{8}.2 ≅ PSL_{4}(2).2 by Chuang in [5] instead of (4.9). □

Proof of Main Theorem (0.2). Let G̃ be an arbitrary finite group such that C_{3} × C_{3} ≅ P ∈ Syl_{3}(G̃). Since we consider the principal 3-blocks, we may assume O_{3′}(G̃) = 1. Then, (2.4) and (5.2)–(5.8) imply that B_{0}(kG̃) and B_{0}(kH̃) are derived equivalent. □

ACKNOWLEDGEMENTS

The authors would like to express their gratitude to S.Yoshiara, who taught the first author how to use the classification of finite simple groups. The authors would like to thank also T.Okuyama for his useful advice and comments on the subject of this paper. A part of this paper was done while the first author was staying at Aachen Technical University, Essen University and Jena University in Germany during March – April, 1998, and also at Jena University, Magdeburg University, Essen University during March–August, 1999. He would like to thank C.Bessenrodt, G.Hiss, B.Külshammer and G.O.Michler for their kind hospitality. The author would like to thank also the JSPS (Japan Society for Promotion of Science). His stay of 1998 was financially supported by the JSPS: the Joint Research Project ”Representation Theory of Finite Groups and Algebraic Groups” 1997 – 1999 under the Japanese-German Cooperative Science Promotion Program supported by JSPS and DFG, and his stay of 1999 in Germany was financially supported by the Ministry of Education, Science, Sports and Culture, Japan.

This work was also supported by the RIMS (Research Institute for Mathematical Sciences) at Kyoto University, 1998. The first author would like to express his great gratitude to E.Bannai as well.

Finally, the first author would like to thank A.Marcus for answering to some questions on his paper [22], and J.Chuang for stimulating conversations in Bielefeld and Oxford in 1999.

REFERNCES

[1] J.L.Alperin, Isomorphic blocks, J. Algebra 43 (1976), 694–698.

[2] M.Broué, Isométries parfaites, Types de blocs, Catégories dérivées, Astérisque 181–182 (1990), 61–92.

[3] M.Broué, Isométries de caractères et équivalences de Morita ou Dérivées, Publ.Math. IHES 71 (1990), 45–63.

[4] M.Broué, Equivalences of blocks of group algebras, in :Finite Dimensional Algebras and Related Topics (edited by V.Dlab and L.L.Scott), Kluwer Acad. Pub., Dordrecht, 1994, pp.1–26.

[5] J.Chuang, The derived categories of some blocks of symmetric groups and a conjecture of Broué, J. Algebra 217 (1999), 114–155.

[6] J.Chuang, On SL_{2}(p^{2}), preprint, 1999.

[7] S.B.Conlon, Twisted group algebras and their representations, J. Austral. Math. Soc. 4 (1964), 152–173.

[8] J.H.Conway, R.T.Curtis, S.P.Norton, R.A.Parker and R.A.Wilson, ”Atlas of Finite Groups”, Clarendon Press, Oxford, 1985.

[9] E.C.Dade, Remarks on isomorphic blocks, J. Algebra 45 (1977), 254–258.

[10] E.C.Dade, Extending irreducible modules, J. Algebra 72 (1981), 374–403.

[11] E.C.Dade, Extending group modules in a relative prime case, Math. Z. 186 (1984), 81–98.

[12] A.Hida and S.Koshitani, Morita equivalent blocks in non-normal subgroups and p-radical blocks in finite groups, J. London Math. Soc. 59 (1999), 541–556.

[13] B.Huppert, ”Endliche Gruppen I”, Springer-Verlag, Berlin, 1967.

[14] I.M.Isaacs and S.D.Smith, A note on groups of p-length 1, J. Algebra 38 (1976), 531–535.

[15] S.König and A.Zimmermann, Derived Equivalences for Group Rings, Springer Lecture Notes in Mathematics, Vol.1685, Springer-Verlag, Berlin, 1998.

[16] S.Koshitani and N.Kunugi, The principal 3-blocks of the 3-dimensional projective special linear groups in non-defining characteristic, preprint (December, 1997).

[17] S.Koshitani and H.Miyachi, The principal 3-blocks of four- and five-dimensional projective special linear groups in non-defining characteristic, to appear in J. Algebra.

[18] S.Koshitani and H.Miyachi, Donovan conjecture and Loewy length for principal 3-blocks of finite groups with elementary abelian Sylow 3-subgroup of order 9, preprint (December, 1999).

[19] B.Külshammer, Crossed products and blocks with normal defect groups, Commun. Algebra 13 (1985), 147–168.

[20] N.Kunugi, Morita equivalent 3-blocks of the 3-dimensional projective special linear groups, Proc. London Math. Soc. (3) 80 (2000), 575–589.

[21] M.Linckelmann, Stable equivalences of Morita type for self-injective algebras and p-groups, Math.Z. 223 (1996), 87–100.

[22] A.Marcus, On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra 184 (1996), 372–396.

[23] H.Nagao and Y.Tsushima, ”Representations of Finite Groups”, Academic Press, New York, 1990.

[24] T.Okuyama, Some examples of derived equivalent blocks of finite groups, preprint (1998).

[25] T.Okuyama, Remarks on splendid tilting complexes, in :Representation Theory of Finite Groups and Related Topics (edited by S.Koshitani), RIMS Kokyuroku (Proceedings of Research Institute for Mathematical Sciences) Vol.1149 (Kyoto University, 2000), pp.53–59.

[26] T.Okuyama and K.Waki, Decomposition numbers of Sp(4,q), J. Algebra 199 (1998), 544–555.

[27] L.Puig, Algèbres de source de certains blocs des groupes de Chevalley, Astérisque 181–182 (1990), 221–236.

[28] J.Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. (3) 72 (1996), 331–358.

[29] R.Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, in :Groups 1993 Galway /St Andrews Vol.2 (edited by C.M.Campbell et al), London Mathematical Society Lecture Note Series 212, Cambridge University Press, Cambridge 1995, pp.512–523.

[30] M.Suzuki, ”Group Theory I and II”, Springer-Verlag, Berlin, 1982 and 1986.

[31] J.Thévenaz, Relative projective covers and almost split sequences, Commun. Algebra 13 (1985), 1535–1554.

Appendix. In order to make the diagrams given in (3), (4), (8), (11) and (14) in the proof of (4.9)Lemma completed, you may need the following data: (3): At the south east corner, two lines between 1_{1b} and 2′; two lines between 2′ and 1_{1a} should be added. (4): The following 16 arrows should be here:

\begin{aligned}1_{1b} & → & 2′, & 2′ & → & 1_{1b}, & 1_{0} & → & 2′, & 2′ & → & 1_{0}, \\ k & → & 2_{a}, & 1_{1b} & → & 2_{a}, & 2_{a} & → & 1_{1a}, & 2_{a} & → & 1_{0}, \\ 1_{1a} & → & 2_{b}, & 1_{0} & → & 2_{b}, & 2_{b} & → & 1_{1b}, & 2_{b} & → & k, \\ k & → & 2′, & 2′ & → & k, & 1_{1a} & → & 2′, & 2′ & → & 1_{1a} \end{aligned}

(8): The lines stated in (3) above should be added, and besides at the bottom of the second column, two lines between 2_{b} and 2′; and two lines between 2′ and 2_{a} should be added. (11): At the last column, the same remark given in the latter part of (8). (14): At the upper part of the second column, the same remark given in the latter part of (8).