A new proof of Glauberman’s Z^{∗}-theorem

Martin Hertweck

Universität Stuttgart, Fachbereich Mathematik, IGT, 70550 Stuttgart, Germany

Abstract

It is shown that for a finite group G of order divisible by a prime p, an algebraically closed field k of characteristic p and H ≤ G such that the summand of the induced module k_{H} ↑^{G} which belongs to the principal block of kG is just the trivial kGmodule k_{G}, the indecomposable projective kG-module which covers k_{G} is also a projective cover of k_{H}; in particular, [G, Z(H)] ≤ O_{p′}(G). When combined with a recent result of John Murray this yields another proof of Glauberman’s Z^{∗}-theorem.

Key words: Z^{∗}-theorem, weakly embedded subgroup, Burry–Carlson–Puig theorem, Broué’s theorem

1 The result

The aim of this short note is to present the following lemma which we found while thinking about the question whether a direct “representation-theoretic” proof of the Z_{p}^{∗}-theorem (stated below) may exist for p odd—important work of Robinson [13,14] makes this plausible. Here we point out that the lemma, when combined with a recent result of Murray [9], yields another proof of Glauberman’s Z^{∗}-theorem (the case p = 2).

Let G be a finite group of order divisible by a prime p, and let k denote an algebraically closed field of characteristic p. By e_{0} we denote the principal block idempotent of kG, and k_{U}, for U ≤ G, denotes k viewed as trivial right kU-module.

Lemma 1 Let H be a subgroup of G and suppose that (k_{H}↑^{G})e_{0} ≅ k_{G}. Let j be a primitive idempotent of kH such that j(kH) is a projective cover of k_{H}. Then e_{0}j(kG) = e_{0}j(kH) and e_{0}j(kG) is a projective cover of both k_{G} and k_{H}. In particular, [G, Z(H)] ≤ O_{p′}(G).

  Email address: hertweck@mathematik.uni-stuttgart.de (Martin Hertweck).

Preprint submitted to Elsevier Science May 23, 2006

This will be applied using the next lemma which is implied by a much more general theorem due to Broué (see [3, 6.4]; a proof thereof can be found in [1, Theorem 8.3]). The necessary argument in our case, which rests upon the Burry–Carlson–Puig theorem, is repeated below.

A proper subgroup H of G of order divisible by p is said to be weakly pembedded in G provided N_{G}(Q) = N_{H}(Q)O_{p′}(N_{G}(Q)) whenever Q is a nontrivial p-subgroup of H. We remark that then H contains a Sylow p-subgroup of G, as one can see by choosing for Q a Sylow p-subgroup of H.

Lemma 2 Suppose that G contains a weakly p-embedded subgroup H. Then, as kG-modules, (k_{H}↑^{G})e_{0} ≅ k_{G} ⊕ (projective).

We quote the mentioned result of Murray in a weak form which suffices for our purpose (see [9] for more details). We remark that the proof is likewise based on the Burry–Carlson–Puig theorem.

Theorem 3 Assume that p = 2 and let t be an involution of G. Then any indecomposable projective direct summand of k_{C_{G}(t)}↑^{G} is irreducible, whence belongs to a 2-block of G that has defect zero. □

The immediate consequence of the above is:

Theorem 4 Suppose that p = 2 and that the centralizer C_{G}(t) of some involution t in G is a weakly 2-embedded subgroup of G. Then (k_{C_{G}(t)}↑^{G})e_{0} ≅ k_{G} and G = C_{G}(t)O_{2′}(G).

To complete the picture we finally state the Z_{p}^{∗}-theorem (in the form given in [3, 6.5]). For p = 2 it is due to Glauberman [6] while for p odd it is known to be a consequence of the classification of the finite simple groups.

We say that a subgroup H of G strongly controls p-fusion in G if H contains a Sylow p-subgroup of G and whenever Q is a p-subgroup of H and g is an element of G such that Q^{g} ⊆ H, then g ∈ C_{G}(Q)H.

Z_{p}^{∗}-Theorem Suppose that H is a subgroup of G which strongly controls pfusion in G, and that H is the centralizer in G of a nontrivial p-subgroup of G. Then G = HO_{p′}(G).

We remark that a weakly p-embedded subgroup H of G strongly controls pfusion in G, by Alperin’s fusion theorem (see [7, Theorem 7.2.6]). Furthermore, for a suspected minimal counterexample G to the Z_{p}^{∗}-theorem we have (cf. [2]; the straightforward proof is nevertheless included below):

Remark 5 Suppose that for any group of smaller order than G the Z_{p}^{∗}-theorem holds true and that H is a subgroup of G which satisfies the suppositions of the Z_{p}^{∗}-theorem, H = C_{G}(S) for some nontrivial p-subgroup S of G. Then for 1 ≠ T ≤ S, either C_{G}(T) = G or C_{G}(T) is weakly p-embedded in G.

Thus the Z^{∗}-theorem follows immediately from Theorem 4.

2 Proofs

The following remark is used in the proofs of Lemma 1 and Lemma 2.

Remark 6 Let V be a kG-module (of finite dimension) and Q a p-subgroup of G. The Burry–Carlson–Puig theorem (see [4], [12, Corollary 1.4] or [10, Theorem 4.4.6]) gives a bijection between indecomposable summands of V with vertex Q and indecomposable summands of V ↓_{N_{G}(Q)} with vertex Q, with corresponding summands being Green correspondents. Since Green correspondents lie in blocks which are Brauer correspondents (see [5, (59.9)]), it follows from Brauer’s Third Main Theorem (see [5, (61.16)]) that this bijection gives a bijection between the indecomposable summands with vertex Q lying in the principal blocks.

Proof of Lemma 1 The trivial module k_{G} has a Sylow p-subgroup P of G as vertex. Since k_{G} is a summand of k_{H}↑^{G} we can therefore assume that P ≤ H.

We first show that N_{G}(P) = N_{H}(P)O_{p′}(N_{G}(P)). Set L = N_{G}(P), N = N_{H}(P) and K = O_{p′}(L). By Remark 6 and the assumption, k_{H}↑^{G}↓_{L} has k_{L} as the only indecomposable summand with vertex P lying in the principal block. Mackey decomposition gives k_{N} ↑^{L} as a summand of k_{H} ↑^{G}↓_{L}. Any indecomposable summand of k_{N}↑^{L} has vertex P (here we use that P ≤ H). For a subgroup X of G we write X̂ for the sum of its elements in kG. Set i = |K|^{−1}K̂. This is the principal block idempotent of kL since L is p-constrained (see [8, VII, 13.6]). The module k_{N}↑^{L} is the permutation representation of L on the right cosets of N, so k_{N}↑^{L} = N̂(kL), and (k_{N}↑^{L})i = N̂(kL)i = N̂K̂(kL) ≠ 0 (since K is a p′-group). As noted at the beginning, this implies N̂K̂(kL) ≅ k, so N̂K̂ is a multiple of L̂, i.e., L = NK as desired.

In particular, the principal blocks of kN_{G}(P) and kN_{H}(P) coincide. We note that this implies that as k(H × H)-modules, the principal block of kH is a summand of the principal block of kG. Indeed, by Brauer’s Third Main Theorem, the principal block of kL is a summand of (kG)e_{0} as k(L × L)- module, so the principal block of kN is a summand of (kG)e_{0} as k(N × N)- module. Moreover, the principal block of kN as k(N × N)-module is a Green correspondent of the principal block of kH as k(H × H)-module, again by Brauer’s Third Main Theorem—note that N × N contains the normalizer in H ×H of the diagonally embedded P which is a vertex of both modules. Thus the desired conclusion follows from the Burry–Carlson–Puig theorem.

Let f_{0} be the principal block idempotent of kH. We have seen that there is m in (kG)e_{0} and a k(H ×H)-submodule V of (kG)e_{0} such that (kH)f_{0} ≅ (kH)f_{0}m as k(H × H)-modules, an isomorphism being given by mapping f_{0} to m, and

(kG)e_{0} = (kH)f_{0}m ⊕ V.  (1)

Note that m commutes with all elements of H. We consider (1) as a decomposition of (kG)e_{0} as left kH-module. By assumption

k ≅ k ⊗_{kH} (kG)e_{0} = (k ⊗_{kH} (kH)f_{0}m) ⊕ (k ⊗_{kH} V ) = k ⊕ (k ⊗_{kH} V ).

Thus k ⊗_{kH} V = 0. Consequently, no submodule of the left kH-module V is a nonzero homomorphic image of kHj, the projective cover of the trivial module k. Indeed, let U be a nonzero homomorphic image of kHj. Then we have a surjection (which in fact is an isomorphism)

k ⊗_{kH} U → k ⊗_{kH} U/rad(U) ≅ k ⊗_{kH} k ≅ k,

showing that k ⊗_{kH} U ≠ 0, whence U cannot lie in V.

Let g ∈ G and set M = (kH)jge_{0} ≤ (kG)e_{0}. Note that (kH)f_{0} ≅ (kH)f_{0}e_{0}, so M is a projective cover of the trivial left kH-module k. Thus the projection of M to V with respect to the decomposition (1) is zero, that is, M ≤ (kH)f_{0}me_{0}. Hence

(kH)j(kG)e_{0} ≤ (kH)f_{0}me_{0},
j(kG)e_{0} ≤ f_{0}m(kH)e_{0},
j(kG)e_{0} ≤ jm(kH)e_{0} (since jf_{0} = j).

Thus e_{0}j(kG) = e_{0}mj(kH). In particular e_{0}j(kH) ≤ e_{0}mj(kH) ≅ e_{0}j(kH), so e_{0}mj(kH) = e_{0}j(kH) and e_{0}j(kG) = e_{0}j(kH). Since e_{0}j(kH) is a projective cover of k_{H}, the restriction of the kG-module e_{0}j(kG) to H is indecomposable and it follows that e_{0}j(kG) is a projective cover of k_{G}.

Now let x ∈ Z(H) and g ∈ G, h ∈ H. Then x commutes with each element from e_{0}j(kG) (= e_{0}j(kH)), so (e_{0}jhg)x = x(e_{0}jhg) = e_{0}jhxg and (e_{0}jh)gxg^{−1}x^{−1} = e_{0}jh. Since e_{0}j(kG) = e_{0}j(kH) it follows that [G, Z(H)] lies in the kernel K_{0} of the projective cover of k_{G}. Finally it is easy to see that the normal subgroup K_{0} of G lies in O_{p′}(G): since the idempotent e_{0}j affords the projective cover of k_{G} its sum-of-coefficients is one, and the elements of an K_{0}-orbit under right multiplication on G all have the same coefficient in e_{0}j, which implies that K_{0} does not contain nontrivial p-elements since k has characteristic p (cf. [11, Proposition 1]).

Proof of Lemma 2 Suppose that Q is a nontrivial p-group contained in H which is a vertex of an indecomposable summand of (k_{H}↑^{G})e_{0}. We determine the indecomposable summands of k_{H} ↑ ^{G} ↓ _{N_{G}(Q)} with vertex Q lying in the principal block. Mackey decomposition gives

k_{H}↑^{G}↓_{N_{G}(Q)} ≅ ⨁_{H\g/N_{G}(Q)} k_{H^{g}∩N_{G}(Q)}↑^{N_{G}(Q)}.

If the summand belonging to the double coset H\g/N_{G}(Q) has an indecomposable summand with vertex Q, then Q ≤ H^{g} ∩ N_{G}(Q), i.e., Q^{g^{−1}} ≤ H and g ∈ HN_{G}(Q) by Alperin’s fusion theorem, as we remarked earlier. Thus the indecomposable summands of k_{H} ↑ ^{G} ↓ _{N_{G}(Q)} with vertex Q all occur in k_{N_{H}(Q)}↑^{N_{G}(Q)}. Set L = N_{G}(Q), N = N_{H}(Q) and K = O_{p′}(L). For a subgroup X of G we write X̂ for the sum of its elements in kG. Set i = |K|^{−1}K̂. The module k_{N}↑^{L} is the permutation representation of L on the right cosets of N, so its summand lying in the principal block appears in N̂(kL)i. But L = NK by assumption, so N̂(kL)i = N̂K̂(kL) = L̂(kL) ≅ k_{L}. By Remark 6 it follows that Q = P, that k_{G} is a summand of (k_{H} ↑ ^{G})e_{0} which appears with multiplicity 1, and that all other summands of (k_{H}↑^{G})e_{0} are projective.

Proof of Theorem 4 Since the principal block (kG)e_{0} has the Sylow psubgroups of G as defect groups, it follows from Lemma 2 and Theorem 3 that (k_{C_{G}(t)}↑^{G})e_{0} ≅ k_{G}. Now Lemma 1 gives [G, t] ≤ O_{2′}(G). Thus ⟨t⟩ acts by conjugation on each coset of O_{2′}(G), and since the action is coprime, we have G = C_{G}(t)O_{2′}(G).

Proof of Remark 5 In the situation to be discussed we have H ≤ C_{G}(T), so C_{G}(T) strongly controls p-fusion in G, by Sylow’s Theorem. Thus it suffices to assume H < G and to show that H is weakly p-embedded in G.

First, suppose that G has a nontrivial normal p-subgroup Q. Set Ḡ = G/Q. If g is an element in G with ḡ ∈ CḠ(S̄), then S^{g} ≤ SQ ≤ H and g ∈ C_{G}(S)H = H. Thus CḠ(S̄) = H̄. In particular S̄ ≠ 1 as H̄ < Ḡ. If R is a p-subgroup of H containing Q and R̄ḡ ≤ H̄ for some g ∈ G then R^{g} ≤ H and g ∈ C_{G}(R)H. Thus H̄ strongly controls p-fusion in Ḡ, and as the Z^{∗}-theorem holds true for Ḡ we have Ḡ = H̄O_{p′}(Ḡ). Let L ⊴ G with Q ≤ L and L̄ = O_{p′}(Ḡ). Then G = HL. Set G_{1} = C_{G}(Q) and H_{1} = C_{H}(Q). Then S ≤ H_{1} ≤ G_{1} and C_{G_{1}}(S) = G_{1} ∩ H = H_{1}. If R is a p-subgroup of H_{1} and R^{g} ≤ H_{1} for some g ∈ G_{1} then (RQ)^{g} ≤ H and g ∈ C_{G}(RQ)H, so g ∈ C_{G_{1}}(RQ)H_{1} ≤ C_{G_{1}}(R)H_{1}. If additionally R is a Sylow p-subgroup of H_{1} and R^{g} = R then C_{G_{1}}(R) ≤ G_{1} ∩ C_{G}(S) = H_{1}, so g ∈ H_{1}, which shows that R is a Sylow psubgroup of G_{1}. Thus H_{1} strongly controls p-fusion in G_{1}. If C_{G}(Q) = G then L = Q×K for a p′-group K by the Schur–Zassenhaus Theorem and K ⊴ G, so G = HL = HK = HO_{p′}(G). If C_{G}(Q) < G then C_{G}(Q) = C_{H}(Q)O_{p′}(C_{G}(Q)) as the Z^{∗}-theorem holds true for G_{1}. Note that G = HC_{G}(Q) as H strongly controls p-fusion in G, so it follows that G = HO_{p′}(C_{G}(Q)) = HO_{p′}(G). In any case G = HO_{p′}(G) so H is weakly p-embedded in G (a general fact depending on the Schur–Zassenhaus Theorem).

Thus we can assume that O_{p}(G) = 1. Let Q be a nontrivial p-subgroup of H. Set G_{1} = N_{G}(Q) and H_{1} = N_{H}(Q). Then S ≤ H_{1} ≤ G_{1} and C_{G_{1}}(S) = G_{1} ∩ H = H_{1}. As above it follows that H_{1} strongly controls p-fusion in G_{1}. Since the Z^{∗}-theorem holds true for G_{1} we obtain G_{1} = H_{1}O_{p′}(G_{1}). This means that H is weakly p-embedded in G, and the proof is complete.

Deduction of the Z^{∗}-theorem Suppose the Z^{∗}-theorem holds true for all groups of smaller order than G, and let H < G. Suppose that H strongly controls 2-fusion in G and that H is the centralizer in G of a nontrivial 2subgroup S of G. Let t be an involution in S. By Remark 5, H is a weakly 2-embedded subgroup of G, so C_{G}(t) = N_{G}(⟨t⟩) = HO_{2′}(C_{G}(t)). Thus we are done if C_{G}(t) = G. Otherwise, C_{G}(t) is a weakly 2-embedded subgroup of G by Remark 5, whence G = C_{G}(t)O_{2′}(G) by Theorem 4 and O_{2′}(C_{G}(t)) ≤ O_{2′}(G). It follows again that G = HO_{2′}(G), so the Z^{∗}-theorem is proved by induction on the order of G.

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