On the connectivity of the subpair complex of a block Charles W. Eaton School of Mathematics and Statistics University of Birmingham Birmingham B15 2TT England email: eatonc@for.mat.bham.ac.uk

Abstract

We demonstrate that the simplicial complex formed from the poset of nontrivial subpairs for a given p-block of a finite group G is disconnected if and only if G has the block analogue of a strongly p-embedded subgroup.

1 Introduction

Let G be a finite group and B be a p-block for G. Recall that a subpair is a pair (Q, b_{Q}) where Q is a p-subgroup and b_{Q} is a p-block for QC_{G}(Q). If further b_{Q}^{G} = B, then call (Q, b_{Q}) a B-subgroup. We refer the reader to [1] for further details and properties of subpairs. We say that (Q, b_{Q}) is trivial if Q = 1. Given B, the set of non-trivial B-subgroups (Q, b_{Q}) for which m_{p}(Q) ≥ k forms a poset S_{B,k} with respect to the usual inclusion of subpairs, and we may apply the constructions of [5] to obtain a simplicial complex |S_{B,k}| (the details of this are not important here - it suffices to consider the 1-skeleton of the complex, i.e., the partial order graph of the poset). The subpairs maximal with respect to the inclusion ordering are called the Sylow B-subgroups, and are precisely those (D, b_{D}) for which D is a defect group for B.

It is well-known that the simplex formed from the poset of p-subgroups of G is connected if and only if G has no strongly p-embedded subgroup, i.e., if the 1-embedded p-core is equal to G. Following [3], for a Sylow B-subgroup (D, b_{D}) define the kembedded B-core to be

Γ_{(D,b_{D}),k}(G) = ⟨N_{G}(R, b_{R}) : (R, b_{R}) ≤ (D, b_{D}), m_{p}(R) ≥ k⟩,

where m_{p}(R) is the p-rank of R. This is uniquely defined up to conjugacy in G. We say that G has a strongly B-embedded subgroup if the 1-generated B-core is a proper subgroup of G.

We show that |S_{B,k}| is connected if and only if G = Γ_{(D,b_{D}),k}(G) for a Sylow Bsubgroup (D, b_{D}). Note that if B_{0} is the principal p-block, then Γ_{(D,b_{D}),k}(G) is the k-embedded p-core and also the poset of B_{0}-subgroups is isomorphic to the poset of p-subgroups, so our result is a generalisation of the result for p-subgroups.

2 Conditions for the connectivity of the subpair complex

Throughout we assume that m_{p}(D) ≥ k when D is a defect group for the block B of G, for otherwise the results are vacuous.

Lemma 2.1 Let (D, b_{D}) be a Sylow B-subgroup and suppose that Γ_{(D,b_{D}),k}(G) ≤ H ≤ G. Then there is an unique block b of H such that b^{G} = B, (D, b_{D}) is a Sylow b-subgroup and every Sylow b-subgroup is a Sylow B-subgroup.

Proof: Since DC_{G}(D) ≤ N_{G}(D, b_{D}) ≤ Γ_{(D,b_{D}),k}(G) ≤ H, it follows that (D, b_{D}) is a subpair for the well-defined p-block b = b_{D}^{H} of H. Then D is contained in some defect group D_{1} of b. But C_{G}(D_{1}) ≤ C_{G}(D) ≤ H, so b^{G} is defined and must be B. Necessarily (D, b_{D}) is a Sylow b-subgroup. If (P, b_{P}) is another Sylow b-subgroup, then (P, b_{P}) must be conjugate to (D, b_{D}), and so is also a Sylow B-subgroup. □

We need a strengthening of a result of [3]. The proof is similar, but we give it here for completeness:

Lemma 2.2 Keeping the hypotheses of the previous lemma, let X(B, b, k) ⊆ S_{B,k} consist of those B-subgroups (Q, b_{Q}) which are contained in some Sylow b-subgroup (regarded as a B-subgroup). Then g ∈ H whenever (Q, b_{Q})^{g}, (Q, b_{Q}) ∈ X(B, n, k). In particular N_{G}(Q, b_{Q}) ≤ H for every (Q, b_{Q}) ∈ X(B, b, k).

Proof: Suppose that (Q, b_{Q}) ∈ X(B, b, k) and that g ∈ G satisfies (Q, b_{Q})^{g} ∈ X(B, b, k). Since the Sylow b-subgroups are conjugate in H, there are h_{1}, h_{2} ∈ H such that (Q, b_{Q})^{h_{1}} and (Q, b_{Q})^{gh_{2}} are both contained in (D, b_{D}). Let x = h_{1}^{−1}gh_{2}. It now follows from [2, 4,10] that there are B-subgroups (R_{i}, b_{i}), 1 ≤ i ≤ n, all contained in (D, b_{D}), and x_{i} ∈ N_{G}(R_{i}, b_{i}), 1 ≤ i ≤ n, such that (Q, b_{Q})^{h_{1}} ≤ (R_{1}, b_{1}), (Q, b_{Q})^{h_{1}x_{1}···x_{i}} ≤ (R_{i+1}, b_{i+1}) for i < n, x = x_{1} ⋯ x_{n} and m_{p}(R_{i}) ≥ k for every i. But by definition x_{i} ∈ N_{G}(R_{i}, b_{i}) ≤ H for each i, so x ∈ H. Hence g ∈ H. The last part follows immediately. □

Theorem 2.3 Let (D, b_{D}) be a Sylow B-subgroup, where B is a p-block of a finite group G. Then |S_{B,k}| is connected if and only if Γ_{(D,b_{D}),k}(G) = G.

The general idea for the proof derives from [4].

Proof: As mentioned earlier, it suffices to consider just the partial order graph for the poset S_{B,k} (and we use S_{B,k} to denote this graph). Write Γ = Γ_{(D,b_{D}),k}(G). Let b and X(B, b, k) be as above. We may assume that k > 0, for otherwise the result holds trivially.

Let Δ be the connected component containing (D, b_{D}). Suppose (Q, b_{Q}) ≤ (D, b_{D}) with m_{p}(Q) ≥ k. Since G permutes the connected components transitively (recall that the Sylow B-subgroups are permuted transitively, and every B-subgroup is contained in some Sylow B-subgroup), it follows that N_{G}(Q, b_{Q}) ≤ N_{G}(Δ). Hence Γ ≤ N_{G}(Δ).

Now let (P, b_{P}) be a Sylow b-subgroup (so (P, b_{P}) ∈ X(B, b, k)). Then there is h ∈ Γ such that (P, b_{P}) = (D, b_{D})^{h}. But h ∈ N_{G}(Δ), and so (P, b_{P}) ∈ Δ. It follows that X(B, b, k) ⊆ Δ.

We show that X(B, b, k) is a union of connected components of S_{B,k}: suppose (Q, b_{Q}) ≤ (P, b_{P}) are elements of S_{B,k}. It suffices to assume that (Q, b_{Q}) ◁ (P, b_{P}). If (P, b_{P}) ∈ X(B, b, k), then it is immediate that (Q, b_{Q}) ∈ X(B, b, k). If (Q, b_{Q}) ∈ X(B, b, k), then N_{G}(Q, b_{Q}) ≤ Γ by Lemma 2.2. But P ≤ N_{G}(Q, b_{Q}) and C_{G}(P) ≤ C_{G}(Q) ≤ N_{G}(Q, b_{Q}), so (P, b_{P}) is a subpair for Γ, and necessarily a b-subgroup. Hence (P, b_{P}) ∈ X(B, b, s) and it follows that X(B, b, k) is a union of connected components as required. Hence Δ ⊆ X(B, b, k) (since (D, b_{D}) ∈ Δ∩X(B, b, k)), and so X(B, b, k) = Δ.

It now suffices to show that N_{G}(Δ) ≤ Γ. For then Γ = N_{G}(Δ), and S_{B,k} is connected if and only if G = N_{G}(Δ) (since the connected components are permuted transitively by G).

Since Δ = X(B, b, k), it follows that Δ^{h} = Δ for every h ∈ H, and it suffices to show that Δ^{g} ∩ Δ = ∅ for g ∈ G \ Γ.

Suppose that (Q, b_{Q}) ∈ Δ^{g} ∩ Δ. Then (Q, b_{Q})^{g^{−1}}, (Q, b_{Q}) ∈ Δ = X(B, b, k). But then by Lemma 2.2 g ∈ Γ and we are done. □

Remark In analogy with p-groups, we may define

Γ_{(D,b_{D}),k}^{0}(G) = ⟨N_{G}(R, b_{R}) : (R, b_{R}) ≤ (D, b_{D}), m_{p}(R) ≥ k, m_{p}(RC_{G}(R)) > k⟩,

and consider the poset S_{B,k}^{0} ⊆ S_{B,k} consisting of B-subgroups (Q, b_{Q}) in S_{B,k} with the further condition that m_{p}(QC_{G}(Q)) > k. Only slight modification of the above proof is needed to show that the partial order graph of S_{B,k}^{0} is connected if and only if Γ_{(D,b_{D}),k}^{0}(G) = G, where (D, b_{D}) is a Sylow B-subgroup.

References

[1] J.L.Alperin, “Local representation theory,” Cambridge University Press, 1986.

[2] J.L.Alperin and M.Broué, Local method in block theory, Ann. of Math. 110(1979), 143-157.

[3] J.An and C.W.Eaton, A note on TI and TI defect blocks, to appear, J. Algebra.

[4] M.Aschbacher, “Finite Group Theory,” Cambridge University Press, 1986.

[5] D.Quillen, Homotopy properties of the poset of non-trivial p-subgroups of a group, Adv. in Math. 28(1978), 101-128.