ON BLOCKS OF STRONGLY P -SOLVABLE GROUPS

Radha Kessar, Markus Linckelmann

Abstract. We prove that a block of a finite strongly p-solvable group G with defect group P is Morita equivalent to its corresponding block of N_{G}(Z(J(P))) via a bimodule with endo-permutation source.

Let p be a prime. Following [8, Ch. 6, §5] a finite group G is called strongly p-solvable if G is p-solvable and either p ≥ 5 or p = 3 and Sl_{2}(3) is not involved in G. If G is a strongly p-solvable finite group and P a Sylow-p-subgroup of G, then G is p-constrained and p-stable (cf. [8, Ch. 8, §1]), and hence G = N_{G}(Z(J(P)))O_{p′}(G) by a theorem of Glauberman (cf. [8, Ch. 8, Theorem 2.11]). The purpose of this paper is to show that this theorem, which is used in the proof of the p-nilpotency theorem of Glauberman and Thompson, has a generalisation to blocks of strongly p-solvable groups. We denote by O a complete discrete valuation ring having an algebraically closed residue field k = O/J(O) of characteristic p. We state our Theorem and explain the notation below.

Theorem. Let G be a strongly p-solvable finite group, let b a block of OG and let P a defect group of b. Set N = N_{G}(Z(J(P))) and denote by c the Brauer correspondent of b in ON. Then the Brauer categories F_{G,b} and F_{N,c} are equal and there is an indecomposable OGb-ONc-bimodule M with the following properties.

(i) As O(G × N)-module, M has vertex ΔP = {(u, u)}_{u∈P} and an endo-permutation OΔP -module W as source.

(ii) The bimodule M and its O-dual M^{∗} induce a Morita equivalence between the block algebras OGb and ONc.

With the notation of the Theorem, the block c of ON is the unique block satisfying Br_{P} (c) = Br_{P} (b); this makes sense as C_{G}(P) ⊆ N. Fix a block e_{P} of kC_{G}(P) such that Br_{P} (b)e_{P} = e_{P}; that is, (P, e_{P}) is a maximal b-Brauer pair and a maximal c-Brauer pair.

By [1], for any subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e_{P}). Denote by F_{G,b} the category whose objects are the subgroups of P and whose set of morphisms from a subgroup Q of P to another subgroup R of P is the set of group homomorphisms φ : Q → R for which there exists an element x ∈ G

Typeset by AMS-TEX

satisfying φ(u) = xux^{−1} for all u ∈ Q and ^{x}(Q, e_{Q}) ⊆ (R, e_{R}). Thus the automorphism group of a subgroup Q of P as object of the category F_{G,b} is canonically isomorphic to N_{G}(Q, e_{Q})/C_{G}(Q). By Alperin’s fusion theorem, the category F_{G,b} is completely determined by the structure of P and the groups N_{G}(Q, e_{Q})/C_{G}(Q) where either Q = P or (Q, e_{Q}) is an essential b-Brauer pair (cf. [18, §48]). Note that O_{p}(G) ⊆ Q whenever the pair (Q, e_{Q}) is essential. The category F_{N,c} is defined similarly. Since O_{p}(N) contains Z(J(P)) and C_{N}(Q) = C_{G}(Q) for any subgroup Q of P containing Z(J(P)) we have an inclusion of categories F_{N,c} ⊆ F_{G,b}.

If b is the principal block of OG, then e_{Q} is the principal block of kC_{G}(Q), for any subgroup Q of P. Thus the above condition ^{x}(Q, e_{Q}) ⊆ (R, e_{R}) is equivalent to ^{x}Q ⊆ R. Therefore, if b is the principal block of OG, we write F_{G} instead of F_{G,b}.

In general, the definition of F_{G,b} depends on the choice of a maximal b-Brauer pair, but since all maximal b-Brauer pairs are G-conjugate, it is easy to see that F_{G,b} is unique up to isomorphism of categories. Note that we always have F_{P} ⊆ F_{G,b}.

Let P be a finite p-group. An O-free OP -module U is called an endo-permutation module if End_{O}(U) has a P -stable O-basis with respect to the action of P by conjugation. This concept is due to Dade [5, 6]. It is shown in [5], that an endo-permutation OP - module U has at most one isomorphism class of indecomposable direct summands of vertex P. Since the restriction to any subgroup Q of P of an endo-permutation OP - module U is an endo-permutation OQ-module, it follows that if U has an indecomposable direct summand with vertex P, then Res_{Q}^{P}(U) has exactly one isomorphism class of indecomposable direct summands with vertex Q for any subgroup Q of P. We will say that U is G-stable if the indecomposable summands with vertex Q of Res_{Q}^{P}(U) are isomorphic to those of Res_{φ}(U) for any subgroup Q of P and any group homomorphism φ : Q → P for which there is an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ Q. Here Res_{φ}(U) is the endo-permutation OQ-module obtained from restricting U via φ; that is, with u ∈ Q acting as φ(u) on U. In the terminology of [17] this is the same as saying that the interior P -algebra End_{O}(U) contains G-fusion. We state without proof two obvious Lemmas needed in the proof of the Theorem.

Lemma 1. Let G, H be finite groups and let P be a p-subgroup of both G and H. Let U, V be endo-permutation OP -modules having direct summands of vertex P.

(i) If U is G-stable, than Res_{Q}^{P}(U) is G-stable for any subgroup Q of P.

(ii) If U is G-stable and V is H-stable, then the O(P × P) -module U ⊗_{O} V is G × H-

stable.

Lemma 2. Let G be a finite group and let P be a p-subgroup of G. Let U be an endopermutation OP -module having an indecomposable direct summand V of vertex P. If U is G-stable then V is G-stable.

We quote the following well-known result on the local structure of certain blocks of p-constrained groups; recall that a finite group G is p-constrained if C_{G}(Q) ⊆ O_{p′,p}(G), where Q is a Sylow-p-subgroup of O_{p′,p}(G).

Proposition 3. ([9, 3.1, 3.4]) Let G be a finite p-constrained group and let b be a Gstable block of OO_{p′}(G). Then b is still a block of OG, any Sylow-p-subgroup P of G is a defect group of b as block of OG, and for any p-subgroup Q of G, the central idempotent Br_{Q}(b) is a block of kC_{G}(Q).

The last statement in Proposition 3 means that b is of principal type, which in turn implies that F_{G,b} = F_{G}. We will need Puig’s algebra theoretic formulation of Fong’s reduction [7], as outlined in [14].

Proposition 4. (cf. [9, 3.1(i)]) Let G be a finite p-solvable group and let b be a block of OG. There is a subgroup H of G containing O_{p′}(G) and an H-stable block e of OO_{p′}(H) such that

OGb ≅ Ind_{H}^{G}(OHe)

as interior G-algebras. Moreover, every Sylow-p-subgroup of H is then a defect group of e and b as blocks of OH and OG, respectively.

Proposition 5. (cf. [10, 4.4]) Let G be a finite p-solvable group, let b be a G-stable block of OO_{p′}(G) and set S = OO_{p′}(G)b. Denote by Ĝ the O^{×}-group opposite to that defined by the action of G on S, and set L̂ = Ĝ/N, where any element y of N is identified to its canonical image (y, yb) in Ĝ. There is a unique algebra isomorphism

OGb ≅ S ⊗_{O} O_{∗}L̂

mapping xb to s ⊗ \overline{(x, s)}, where x ∈ G and s ∈ S^{×} such that (x, s) ∈ Ĝ and where \overline{(x, s)} is the canonical image of (x, s) in L̂.

We show next, that this reduction is compatible with taking normalisers of Z(J(P)).

Proposition 6. Let G be a finite p-solvable group, let b be a block of OG, let H be a subgroup of G containing O_{p′}(G) and let e be an H-stable block of OO_{p′}(H) such that OGb ≅ Ind_{H}^{G}(OHe). Let P be a Sylow-p-subgroup of H and let Z be a subgroup of P such that N_{G}(P) is contained in N_{G}(Z). Denote by c and f the blocks of ON_{G}(Z) and ON_{H}(Z) which are the Brauer correspondents of b and e, respectively. Then f is an N_{H}(Z)-stable block of OO_{p′}(N_{H}(Z)) and we have an isomorphism of interior N_{G}(Z)- algebras

ON_{G}(Z)c ≅ Ind_{N_{H}(Z)}^{N_{G}(Z)}(ON_{H}(Z)f) .

Proof. By [10, 4.2], we have O_{p′}(N_{H}(Z)) = O_{p′}(C_{H}(Z)) = O_{p′}(H) ∩ C_{H}(Z). By Proposition 3, Br_{Z}(e) is a block of kC_{H}(Z). Since e is H-stable and every block of kN_{H}(Z) is contained in kC_{H}(Z), it follows that Br_{Z}(e) is a block of kN_{H}(Z). As e ∈ OO_{p′}(H), it follows that Br_{Z}(e) ∈ kO_{p′}(N_{H}(Z)). Thus Br_{Z}(e) lifts to a unique N_{H}(Z)-stable block f of OO_{p′}(N_{H}(Z)), which is the Brauer correspondent of e. If x ∈ N_{G}(Z) − N_{H}(Z), then ^{x}Br_{Z}(e)Br_{Z}(e) = Br_{Z}(^{x}e · e) = 0, because ^{x}e · e = 0 as OGb ≅ Ind_{H}^{G}(OHe). Thus xf · f = 0. Therefore, Ind_{N_{H}(Z)}^{N_{G}(Z)}(ON_{H}(Z)f) ≅ ON_{G}(Z)Tr_{N_{H}(Z)}^{N_{G}(Z)}(f); in particular, the

algebras ON_{H}(Z)f and ON_{G}(Z)Tr_{N_{H}(Z)}^{N_{G}(Z)}(f) have isomorphic centers. As f is primi-

tive in Z(ON_{H}(Z)), it follows that Tr_{N_{H}(Z)}^{N_{G}(Z)}(f) is primitive in Z(ON_{G}(Z)). But then

c = Tr_{N_{H}(Z)}^{N_{G}(Z)}(f), which implies the result. □

Proof of the Theorem. By Proposition 4, there is a subgroup H of G containing O_{p′}(G) and an H-stable block e of OO_{p′}(H) such that OGb ≅ Ind_{H}^{G}(OHe). By general properties of algebra induction (cf. [15]), the blocks OGb and OHe have isomorphic source algebras. In view of Proposition 6, we may therefore assume that G = H and b = e, and then c is an N-stable block of OO_{p′}(N). Set S = OO_{p′}(G)b. Then S is a matrix algebra of rank n^{2} prime to p, on which G acts. We consider the subgroup

Ĝ = {(x, s)|x ∈ G, s ∈ S^{×} such that ^{x}t = sts^{−1} for all t ∈ S}

of G × S^{×}, endowed with the group homomorphism O^{×} → Ĝ mapping λ ∈ O^{×} to (1_{G}, λ^{−1}1_{S}); that is, Ĝ is the O^{×} -group opposite to that defined by the action of G on S (cf. [10] for more details on this terminology). By the Skolem-Noether theorem, every algebra automorphism of S is inner, and thus, for any x ∈ G, there is s_{x} ∈ S^{×} such that (x, s_{x}) ∈ Ĝ. Equivalently, the canonical projection Ĝ → G onto the first component is surjective. We may choose the s_{x} in such a way that det(s_{x}) = 1 for all x ∈ G, and then s_{x}s_{y} = λ_{x,y}s_{xy} for any x, y ∈ G and some n^{th} roots of unity λ_{x,y} in O^{×} (all this is well-known; see e. g. [10]). Note that this means that the 2-cocycle λ representing the central extension Ĝ of G by O^{×} has actually values in the canonical image of k^{×} in O^{×} (this will be relevant when we apply [16, (e)] below, which is formulated for k^{×}- groups). There is a unique group homomorphism σ : P → S^{×} such that det(σ(y)) = 1 for all y ∈ P and such that (y, σ(y)) ∈ Ĝ; in other words, the action of P on S “lifts canonically”. Since S is a matrix algebra, we may write S = End_{O}(U) for some free O-module U of rank n. Then U becomes an OP -module via the group homomorphism σ. In fact, U becomes an endo-permutation OP -module, because S is a direct summand of the group algebra OO_{p′}(G), which has O_{p′}(G) as P -stable O-basis, and thus S has a P -stable O-basis.

We observe next that U is G-stable. If Q is a subgroup of P and x ∈ G such that xQ ⊆ P, denote by φ : Q ≅ ^{x}Q the isomorphism sending y ∈ Q to ^{x}y. By the uniqueness of σ we have σ(^{x}y) = ^{x}σ(y) for all y ∈ Q; indeed, both sides have determinant 1 and act by conjugation as ^{x}y on S. Choose s ∈ S^{×} such that (x, s) ∈ Ĝ, or equivalently, such that ^{x}t = sts^{−1} for all t ∈ S. Then the map sending u ∈ U to s(u) is an isomorphism Res_{φ}(U) ≅ Res_{Q}^{P}(U); in particular, U is G-stable.

We identify O_{p′}(G) to its canonical image {(y, yb) | y ∈ O_{p′}(G)} in Ĝ, and set L̂ = Ĝ/O_{p′}(G). Thus L̂ is a central O^{×}-extension of the group L = G/O_{p′}(G).

Analogously, set T = OO_{p′}(N)c. This is a matrix algebra over O of rank m^{2} prime to p on which N acts. Set

Ñ = {(y, t) ∈ N × T^{×}|^{y}(t′) = tt′t^{−1} for all t′ ∈ T }

endowed with the group homomorphism O^{×} → Ñ sending λ ∈ O^{×} to (1_{N}, λ^{−1}1_{T}) For any y ∈ N, denote by t_{y} an element in T^{×} such that (y, t_{y}) ∈ Ñ and such that det(t_{y}) = 1. Denote by τ : P → T^{×} the unique group homomorphism such that (u, τ(u)) ∈ Ñ and such that det(τ(u)) = 1 for all u ∈ P. Write T = End_{O}(V) for some free O-module V of rank m. As before, V becomes an endopermutation OP -module via τ. and V is H-stable by the uniqueness of τ.

Similarly, identify O_{p′}(N) to its canonical image in Ñ and set L̃ = Ñ/O_{p′}(N).

Let W be an indecomposable direct summand of U^{∗} ⊗_{O} V, viewed as OΔP -module, such

that W has vertex ΔP. It follows from Lemma 1 and Lemma 2 that W is G × H-stable.

By Glauberman’s theorem [8, Ch. 8, Theorem 2.11], we have G = NO_{p′}(G), Since, by Proposition 4, both b and c are of principal type, we have the equality of the Brauer categories F_{G,b} = F_{N,c}. By [10, 4.2], we have O_{p′}(G) ∩ N = O_{p′}(N). Thus the inclusion N ⊆ G induces a group isomorphism N/O_{p′}(N) ≅ G/O_{p′}(G) = L. Identify N/O_{p′}(N) to L through this isomorphism. The crucial step is to show that there is an isomorphism of O^{×}-groups

L̂ ≅ L̃

which induces the identity on the canonical quotients L of L̂, L̃, and which preserves the canonical images of P in L̂ and L̃ elementwise. Before we prove the existence of such a group isomorphism, let us show how this concludes the proof of the Theorem.

Identifying O_{∗}L̃ and O_{∗}L̂ through the algebra isomorphism induced by this group isomorphism, together with Proposition 5, shows that we have isomorphisms of interior P -algebras

OGb ≅ S ⊗_{O} O_{∗}L̂ and ONc ≅ T ⊗_{O} O_{∗}L̂ .

If we consider U ⊗_{k} V^{∗} as S-T -bimodule, then through the above algebra isomorphisms,

the bimodule M = U ⊗_{O} O_{∗}L̂ ⊗_{O} V^{∗} has vertex ΔP and W as source. Clearly M and its

dual induce a Morita equivalence between OGb and ONc.

It remains to prove the existence of a group isomorphism L̂ ≅ L̃ as claimed. By the above remarks the central extensions L̂, L̃ of L by O^{×} can be represented by 2-cocycles with values in the canonical image of k^{×} in O^{×}, and hence we may assume that O = k.

We will show, that there is an isomorphism of k^{×}-groups Ñ ≅ N̂ preserving the canonical images of P and of O_{p′}(N) and inducing the identity on the canonical quotients N of N̂, Ñ. Since O_{p′}(N) and Z = Z(J(P)) are normal subgroups of coprime order in N, they commute, and thus the Brauer construction with respect to Z applied to S yields S(Z) = T; in particular, Br_{Z}(b) = c. By Puig’s version [16, (e)] of Dade’s splitting theorem on fusion for endo-permutation modules (applied to Z instead of P), there is a group homomorphism f : N_{S^{×}}(σ(Z)) → T^{×} which extends the group homomorphism (S^{Z})^{×} → T^{×} induced by Br_{Z} and which satisfies f(s)Br_{Z}(s′)f(s^{−1}) = Br_{Z}(ss′s^{−1}) for all s ∈ N_{S^{×}}(σ(Z)) and all s′ ∈ S^{Z}. The latter condition implies that if (x, s) ∈ N̂ then (x, f(s)) ∈ Ñ, and the map sending (x, s) to (x, f(s)) is in fact a k^{×}-group isomorphism inducing the identity on N. Now τ and f ∘σ are two group homomorphisms from P to T^{×} lifting the action of P on T, hence they are equal, and therefore the above isomorphism N̂ ≅ Ñ preserves the canonical images of P elementwise. Finally, since f extends Br_{Z} we get for any x ∈ O_{p′}(N) that f(xb) = Br_{Z}(xb) = xc, and so the isomorphism N̂ ≅ Ñ preserves the canonical images of O_{p′}(N). Taking the quotients by these images yields a k^{×}-isomorphism L̂ ≅ L̃ with the required properties. □

Remarks. (1) If G is a strongly p-solvable finite group then SL_{2}(p) is not isomorphic to a subquotient of G, and hence every block of OG is SL_{2}(p)-free in the sense of [12, 1.2]. By [12, Theorem 1.4] the equality F_{G,b} = F_{N,c} holds more generally for any block b of a finite group whenever the block b is SL_{2}(p)-free.

(2) If b is the principal block of a strongly p-solvable finite group G, then c is the principal block of N and the inclusion N ⊆ G induces an algebra ismomorphism ONc ≅ OGb. This in turn is equivalent to Glauberman’s Theorem [8, Ch. 8, Theorem 2.11] because the kernel of the principal block b of OG is well-known to be O_{p′}(G).

(3) Glauberman’s Theorem [8, Ch. 8, Theorem 2.11] is used in the proof of the pnilpotency Theorem of Glauberman and Thompson, stating that for an odd prime p a finite group G is p-nilpotent if and only if N = N_{G}(Z(J(P)) is p-nilpotent, where here P is a Sylow-p-subgroup of G. The latter has a block theoretic version as well; cf. [11].

References

1. J. L. Alperin, M. Broué, Local methods in block theory, Ann. Math. 110 (1979), 143–157.

2. R. Brauer, On the structure of blocks of characters of finite groups, Lecture Notes in Mathematics 372 (1974), 103–130.

3. M. Broué, L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.

4. M. Cabanes, Extensions of p-groups and construction of characters, Comm. Alg. 15 (1987), 1297– 1311.

5. E. C. Dade, Endo-permutation modules over p-groups I, Ann. Math. 107 (1978), 459–494.

6. E. C. Dade, Endo-permutation modules over p-groups II, Ann. Math. 108 (1978), 317–346.

7. P. Fong, On the characters of p-solvable groups, Trans. Amer. Math. Soc. 98 (1961), 263–284.

8. G. Gorenstein, Finite Groups, Chelsea Publishing Company, New York, 1980.

9. M. E. Harris, M. Linckelmann, Splendid derived equivalences for blocks of finite p-solvable groups, J. London Math. Soc. 62 (2000), 85–96.

10. M. E. Harris, M. Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite p-solvable groups, Trans. Amer. Math. Soc. 354 (2002), 3435–3453.

11. R. Kessar, M. Linckelmann, A block theoretic analogue of a theorem of Glauberman and Thompson, Proc. Amer. Math. Soc. 131 (2003), 35–40.

12. R. Kessar, M. Linckelmann, G. R. Robinson, Local control in fusion systems of p-blocks of finite groups, J. Algebra 257 (2002), 393–413.

13. B. Külshammer, L. Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), 17–71.

14. L. Puig, Local block theory in p-solvable groups, Proceedings of Symp. Pure Math. 37 (1980), 385–388.

15. L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265–292.

16. L. Puig, Local extensions in endo-permutation modules split: a proof of Dade’s theorem, Séminaire sur les groupes finis, Publ. Math. Univ. Paris VII (1986), 199–205.

17. L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77–116.

18. J. Thévenaz, G-Algebras and Modular Representation Theory, Oxford Science Publications, Clarendon Press, Oxford, 1995.

Radha Kessar, Markus Linckelmann Math Tower 231 West 18th Avenue Columbus, Ohio 43210

U.S.A.