A BLOCK THEORETIC ANALOGUE OF A

THEOREM OF GLAUBERMAN AND THOMPSON

Radha Kessar, Markus Linckelmann

May 2001

Abstract. If p is an odd prime, G a finite group and P a Sylow-p-subgroup of G, a theorem of Glauberman and Thompson states that G is p-nilpotent if and only if N_{G}(Z(J(P))) is p-nilpotent, where J(P) is the Thompson subgroup of P generated by all abelian subgroups of P of maximal order. Following a suggestion of G. R. Robinson, we prove a block-theoretic analogue of this theorem.

Theorem. Let p be an odd prime and let k be an algebraically closed field of characteristic p. Let G be a finite group, b a block of kG, and P a defect group of b. Set N = N_{G}(Z(J(P))) and let c be the unique block of kN such that Br_{P} (c) = Br_{P} (b); that is, c is the Brauer correspondent of b. Then kGb is nilpotent if and only if kNc is nilpotent.

We refer to [5] and [7] for accounts on the terminology from group theory and block theory, respectively, involved in the theorem above and its proof. Nilpotent blocks, introduced by Broué and Puig in [3], are the block theoretic analogue of the notion of p-nilpotent groups; the principal block of kG is nilpotent if and only if G is p-nilpotent. Thus, in this case, our theorem is equivalent to the theorem of Glauberman and Thompson. The proof proceeds in two steps. We reduce to the case where G is the normaliser of a b-centric Brauer pair (following the lines of the proof of [8, Ch. 8, Theorem 3.1]), and then we apply results of Külshammer and Puig in [6] to transport the problem back to the analogous group theoretic statement.

Proof. We 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. 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 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. [7, §48]). Note that O_{p}(G) ⊆ Q whenever the pair (Q, e_{Q}) is essential.

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By Brauer’s third main theorem (cf. [7, (40.17)]), if b is the principal block of kG, 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 kG, 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 unique isomorphism of categories. Note that we allways have F_{P} ⊆ F_{G,b}. Following [3], the block b is called nilpotent if F_{P} = F_{G,b}.

If H is any subgroup of G containing P C_{G}(P), the block e_{P} determines a unique block d of kH by Br_{P} (d)e_{P} = e_{P}. Then (P, e_{P}) is also a maximal d-Brauer pair, and this gives rise to the Brauer category F_{H,d} of kHd, defined as above for H and d instead of G and b.

We are going to use frequently the following fact:

1. If Q is a normal subgroup of P and H a subgroup of G such that P C_{G}(Q) ⊆ H ⊆ N_{G}(Q), then

F_{H,d} ⊆ F_{G,b} ,

where d is the unique block of kH such that Br_{P} (d)e_{P} = e_{P}. In particular, if kGb is nilpotent, then kHd is nilpotent.

Proof. If (R, f_{R}) is an essential d-Brauer pair contained in (P, e_{P}), then R contains Q as Q is normal in H. But then C_{G}(R) = C_{H}(R), and hence f_{R} = e_{R}. Thus N_{H}(R, f_{R})/C_{H}(R) is a subgroup of N_{G}(R, e_{R})/C_{G}(R). □

Statement 1 applies to N, c and Z(J(P)) instead of H, d, Q, respectively. Thus if kGb is nilpotent, so is kNc. In order to show the converse, we consider now a minimal counter example; that is, we assume that kGb is not nilpotent while kNc is nilpotent and that |G| is minimal with this property. Under this assumption, 1 implies the following statement:

2. If Q is a normal subgroup of P and H a subgroup of G such that P C_{G}(Q) ⊆ H ⊆ N_{G}(Q), then either H = G or kHd is nilpotent, where d is the unique block of kH such that Br_{P} (d)e_{P} = e_{P}.

Proof. Let e be the unique block of N ∩ H such that Br_{P} (e)e_{P} = e_{P}. We have P C_{N}(Q) ⊆ N ∩ H ⊆ N_{N}(Q), and thus statement 1.1 implies that F_{N∩H,e} ⊆ F_{N,c}. But then k(N ∩ H)e is nilpotent, as kNc is so. Therefore, if H is a proper subgroup of G, then the induction hypothesis implies that the block kHd is nilpotent. □

3. We have O_{p}(G) ≠ {1}.

Proof. Since the block b of kG is not nilpotent, there exists a b-Brauer pair (Q, e_{Q}) with Q ≠ 1 such that kN_{G}(Q, e_{Q})e_{Q} is not nilpotent. This is because, for some non-trivial Brauer pair (Q, e_{Q}), N_{G}(Q, e_{Q})/QC_{G}(Q) is not a p-group. Amongst all such b-Brauer pairs, choose (Q, e_{Q}) such that a defect group R of kN_{G}(Q, e_{Q})e_{Q} has maximal order. After replacing, if necessary, (Q, e_{Q}) by a suitable G-conjugate, we may assume that R = N_{P} (Q). We are going to show that R = P, or equivalently that P ⊆ N_{G}(Q, e_{Q}). We assume that R is a proper subgroup of P, and derive a contradiction. Set H = N_{G}(Q, e_{Q}). Clearly R ⊆ H. Since Q ⊂ R, we have C_{G}(R) ⊂ C_{G}(Q) ⊂ H. Now (Q, e_{Q}) ⊆ (R, e_{R}) , and Q is normal in R, hence e_{Q} is the unique block of kC_{G}(Q) which is R-stable and for which Br_{R}(e_{Q})e_{R} = e_{R} (cf. [1]).

Set M = N_{G}(Z(J(R))). Since C_{G}(R) centralises Q and centralises Z(J(R)), we have C_{G}(R) ⊂ M ∩ H. Let d be the unique block of k(M ∩ H) (having R as defect group) such that Br_{R}(d)e_{R} = e_{R}. Let f be the unique block of kM (having R as defect group) such that Br_{R}(f)e_{R} = e_{R}. Since Z(J(R)) is a normal p-subgroup of M, f is a central idempotent of kC_{G}(Z(J(R)) (cf. [1]). Thus there exists a block f_{0} of C_{G}(Z(J(R))) such that ff_{0} = f_{0} and (Z(J(R)), f_{0}) ⊆ (R, e_{R}) in M, and hence in G. Since (R, e_{R}) ⊆ (P, e_{P}), by the uniqueness of inclusion of Brauer pairs, we must have f_{0} = e_{Z(J(R))}. Let M′ be the stabiliser of e_{Z(J(R))} in M. Then N_{P} (Z(J(R))), and hence N_{P} (R) is contained in a defect group of kM′e_{Z(J(R))}. In particular, the defect groups of kM′e_{Z(J(R))} have order strictly greater than |R|. By the maximality of |R|, we have that kM′e_{Z(J(R))} is nilpotent. Since kMf is the induced algebra Ind_{M′}^{M}(kM′e_{Z(J(R))}), it follows that kMf is nilpotent. Now RC_{G}(Q) ⊆ M ∩ H ⊆ N_{M}(Q), and by statement 1 again, it follows that k(M ∩ H)d is nilpotent. By the minimality of |G|, and the fact that kHe_{Q} is not nilpotent, it follows that H = G and hence R = P, contradicting the assumption R ≠ P. If R = P, then H satisfies the hypothesis of 2 with d = e_{Q}, and kHe_{Q} is not nilpotent, thus G = H. In particular, Q ⊆ O_{p}(G) ≠ 1. □

From now on set Q = O_{p}(G).

4. We have G = N_{G}(Q, e_{Q}) and b = e_{Q}.

Proof. Since G = N_{G}(Q), the block b is contained in kC_{G}(Q) (cf. [1]) and hence b = Tr_{N_{G}(Q,e_{Q})}^{G}(e_{Q}). Thus kGb ≅ Ind_{N_{G}((Q,e_{Q})}^{G}(kN_{G}((Q, e_{Q})e_{Q}), so that in particular, kN_{G}(Q, e_{Q})e_{Q} is not nilpotent. Since P is contained in N_{G}(Q, e_{Q}), it follows from 2 that G = N_{G}(Q, e_{Q}) and hence b = e_{Q}. □

Note that b is a block of any subgroup of G containing C_{G}(Q). We want to show, that actually the pair (Q, b) is b-centric (or self-centralising in the terminology of Puig, cf. [7, §41]); that is, the block kC_{G}(Q)b is nilpotent with Z(Q) as defect group. This notion goes back to Brauer [2]. We need the following technical statement.

5. Let H be a subgroup of G containing P and let d be a block of kH having P as defect group. Put H̄ = H/Q and for any element a of kH let ā denote the image of a under the canonical surjection of kH onto kH̄. Then \overline{Br_{P} (d)} = BrP̄(d̄).

Proof. Since Q is normal in H, the block idempotent d is a k-linear combination over the set C_{H}(Q)_{p′} of p′-elements in C_{H}(Q). Write d = ∑_{g∈C_{H}(Q)_{p′}} α_{g}g with coefficients α_{g} ∈ k. So d̄= ∑_{g∈C_{H}(Q)_{p′}} α_{g}ḡ and BrP̄(d̄) = ∑_{g∈C_{H}(Q)_{p′} ∩C_{H}(P̄)} α_{g}ḡ, where C_{H}(P̄) denotes the inverse image in H of CH̄(P̄).

We claim that C_{H}(Q)_{p′} ∩ C_{H}(P̄) = C_{H}(P)_{p′}. To see this, consider the action of an element g ∈ C_{H}(Q)_{p′} ∩ C_{H}(P̄) on an element u of P. Since g normalises P and centralises P/Q, ^{g}(u) = uv for some v in Q. Let n be the order of g. Since g centralises Q, it follows that u = ^{g^{n}}u = uv^{n}. But p and n are relatively prime, hence v = 1, thereby proving the claim.

The statement is immediate from the above expression for d̄ □ 6. The blocks kP C_{G}(Q)b and kC_{G}(Q)b are nilpotent.

Proof. By a result of Cabanes [4], normal p-extensions of nilpotent blocks are nilpotent; thus kP C_{G}(Q)b is nilpotent if and only if kC_{G}(Q)b is nilpotent. If P C_{G}(Q) is a proper subgroup of G, then, by 2, b is nilpotent as block of P C_{G}(Q), and hence of C_{G}(Q). Thus we may assume that G = P C_{G}(Q). We have to show that kGb is a nilpotent block. Set Ḡ = G/Q and let b̄ denote the image of b under the canonical surjection of kG onto kḠ. Identify C_{G}(Q)/Z(Q) with its canonical image in Ḡ; this is a normal subgroup of Ḡ of index a p-power. Since b is a k-linear combination of p′-elements in C_{G}(Q) and Z(Q) = Q ∩ C_{G}(Q) is a central subgroup of C_{G}(Q), it is clear that b̄ is a block of kC_{G}(Q)/Z(Q) and hence of kḠ. Furthermore, P̄ is a defect group of kḠb̄. Let Z be the inverse image in G of Z(J(P̄)) and set H = N_{G}(Z). Then H is the inverse image in G of the group H̄ = kNḠ(Z(J(P̄))). Let f be the block of kH̄ which corresponds to the block b̄ of kḠ; that is, BrP̄(b̄) = BrP̄(f). Clearly, P and C_{G}(Z) are both subgroups of H. Since Z properly contains Q and Q = O_{p}(G), H is a proper subgroup of G. Thus by 2, the block kHd is nilpotent where d is the block of kH satisfying Br_{P} (d)e_{P} = e_{P}. Since N_{G}(P) is contained in H, we have in fact that Br_{P} (d) = Br_{P} (b).

Now, it follows from 5 that BrP̄(d̄) = \overline{Br_{P} (d)} = \overline{Br_{P} (b)} = BrP̄(b̄) = BrP̄(f). In particular d̄f ≠ 0. Since kHd is nilpotent, this means that f = d̄ and hence that kH̄f is nilpotent. As G is a minimal counter example to the Theorem, it follows that kḠb̄ is nilpotent, which implies that kGb is nilpotent. □

7. The group Q is a defect group of kQC_{G}(Q)b.

Proof. Let R be a defect group of kQC_{G}(Q)b. We may assume that R = QC_{P} (Q). The pair (R, e_{R}) is a maximal Brauer pair for the block kQC_{G}(Q)b, and hence, by the Frattini argument, G = N_{G}(R, e_{R})QC_{G}(Q) = N_{G}(R, e_{R})C_{G}(Q). Suppose, if possible, that Q is a proper subgroup of R. Then, N_{G}(R, e_{R}) is a proper subgroup of G because Q = O_{p}(G). On the other hand N_{G}(R, e_{R}) satisfies the hypothesis of 2 with R instead of Q, since P normalises R, and consequently (R, e_{R}). So kN_{G}(R, e_{R})e_{R} is nilpotent. In particular, N_{G}(R, e_{R})/C_{G}(R) is a p-group, and hence so is G/C_{G}(Q). In other words, G = P C_{G}(Q), and hence kGb is nilpotent by 6, a contradiction. □

We are now in the situation where kGb is an extension of the nilpotent block kQC_{G}(Q)b, and this is where the results of Külshammer and Puig in [6] come in.

8. There exists a short exact sequence of groups

1 ⟶ Q ⟶ L ⟶ G/QC_{G}(Q) ⟶ 1

such that P is a Sylow p-subgroup of L and such that we have F_{G,b} = F_{L}.

Proof. The existence of a canonical short exact sequence of finite groups as stated such that P is a Sylow-p-subgroup of L is a particular case of [6, 1.8]. The equality F_{G,b} = F_{L} is a translation of the statement [6, 1.8.2], which requires a brief explanation. Since Q is normal in L and in G, it suffices to show that the images in Aut(R) of N_{G}(R, e_{R})/C_{G}(R) and N_{L}(R)/C_{L}(R) are equal, where R is a subgroup of P containing Q. As (Q, e_{Q}) is b-centric and Q is p-centric in L, it follows from a result of Puig (cf. [7, (41.1), (41.4)]) that (R, e_{R}) is b-centric and R is p-centric in L (that is, Z(R) is a Sylow-p-subgroup of C_{L}(R)). In particular, kC_{G}(R)e_{R} has a unique conjugacy class of primitive idempotents. Setting H = QC_{G}(Q), we have C_{G}(R) = C_{H}(R), hence there is a unique point γ_{R} of R on kH such that Br_{R}(i)e_{R} = i for some (and hence any) element i of γ_{R}. In this way, we get an inclusion preseving bijection, R_{γ_{R}} → (R, e_{R}) between local pointed groups R_{γ_{R}} on kHb for which Q_{γ_{Q}} ⊆ R_{γ_{R}} ⊆ P_{γ_{P}} and kGb-Brauer pairs, (R, e_{R}) with (Q, e_{Q}) ⊆ (R, e_{R}) ⊆ (P, e_{P}). Further, it is clear that N_{G}(R, e_{R}) = N_{G}(R_{γ_{R}}), and thus, with the notation in [6], E_{G}(R, e_{R}) = E_{L}(R) for any subgroup R such that Q ≤ R ≤ P, which implies the equality F_{G,b} = F_{L}. □

We conclude the proof of the Theorem as follows. Let L be as in 8 above. The equality F_{G,b} = F_{L} and the fact that F_{N,c} ⊆ F_{G,b} implies immediately that F_{N,c} = F_{N_{L}(Z(J(P)))}. Since kGb is not nilpotent, L is not a p-nilpotent group. However, kNc is nilpotent and hence N_{L}(Z(J(P))) is p-nilpotent. This contradicts the normal p-complement theorem [5, Ch. 8, Theorem 3.1] of Glauberman and Thompson. □

References

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

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

Radha Kessar University College High Street Oxford OX14BH

U.K.

Markus Linckelmann CNRS, Université Paris 7 UFR Mathématiques 2, place Jussieu 75251 Paris Cedex 05

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