A Note on TI and TI Defect Blocks

Jianbei An Department of Mathematics, University of Auckland Auckland, New Zealand Email: an@math.auckland.ac.nz

Charles W. Eaton School of Mathematics and Statistics, University of Birmingham Birmingham, England Email: eatonc@for.mat.bham.ac.uk

Abstract We show that every trivial intersection block of a finite group (as introduced by Alperin and Broué in [2]) has trivial intersection (TI) defect groups, but that the converse is not true in general. We then present some conditions equivalent to a block B being a TI block, generalizing the idea of a k-generated p-core to B-subgroups. In particular we give further weight to Olsson’s assertion that TI blocks are a better generalization of groups with TI Sylow p-subgroups than are TI defect blocks. Finally we describe the rôle of the generalized k-generated p-core in the control of fusion of subpairs.

1 Introduction and definitions

Let G be a finite group, p a prime and Blk(G) the set of all p-blocks of G. A subgroup H of G is called a trivial intersection (for short TI) set if H^{g} ∩ H = 1 for every g ∈ G\N_{G}(H). Let B be a p-block with a defect group D = D(B). Then B is called a TI defect block if D is a TI set.

A pair (R, b_{R}) of a p-subgroup R ≤ G and a block b_{R} ∈ Blk(RC_{G}(R)) is called a subpair of G. Given B ∈ Blk(G), a subpair (R, b_{R}) is called a B-subgroup (see [1]) if (1, B) ≤ (R, b_{R}), and is non-trivial if R ≠ 1. If moreover, R is a defect group D(B) of B, then a B-subpair (R, b_{R}) is called a Sylow B-subgroup. Following [2], B is called a trivial intersection (for short TI) block if each non-trivial B-subgroup is contained in a unique Sylow B-subgroup. Some properties of TI blocks can be found in the papers of Alperin and Broué [2] and Olsson [8].

Given a B-subgroup (Q, b_{Q}), we define

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

where m_{p}(R) is the p-rank of R. If (D, b_{D}) is a Sylow B-subgroup, then we call Γ_{(D,b_{D}),k}(G) the k-generated B-core, which is completely determined up to conjugacy in G. We say that a proper subgroup H of G is a strongly B-embedded subgroup of G if H ≤ Γ_{(D,b_{D}),1}(G), where (D, b_{D}) is a Sylow B-subgroup. If B_{0} is the principal p-block of G, then Brauer’s third main theorem tells us that Γ_{(D,b_{D}),k}(G) is the k-generated p-core Γ_{D,k}(G), and H is a strongly B_{0}-embedded subgroup if and only if it is a strongly p-embedded subgroup.

1991 Mathematics Subject Classification. Primary 20C20

The structure of this paper is as follows: in Section 2 we strengthen our characterisation of TI defect blocks in terms of the p-local rank of a block introduced in [3]. In Section 3 we show that if B is a TI block, then it must also be a TI defect block. We give an example (and means of generating more) of a TI defect block which is not TI. In Section 4 we give some equivalent definitions of a TI block, and finally in Section 5 we consider the k-generated B-core and describe the part it plays in the fusion of subpairs.

2 TI defect blocks

Before we state our result, we remind the reader of the concepts involved:

Given a chain of p-subgroups σ : Q_{0} < Q_{1} < ⋯ < Q_{n} of G, define the length |σ| = n, the final subgroup V^{σ} = Q_{n}, the initial subgroup V_{σ} = Q_{0}, the k-th initial subchain σ_{k} : Q_{0} < Q_{1} < ⋯ < Q_{k}, and the normalizer N_{G}(σ) = N_{G}(Q_{0})∩N_{G}(Q_{1})∩⋯∩N_{G}(Q_{n}). A chain σ is said to be radical if Q_{i} = O_{p}(N_{G}(σ_{i})) for each i. Denote by R = R(G) the set of all radical p-chains of G and write R(G, B) = {σ ∈ R(G) : Blk(N_{G}(σ)|B) ≠ ∅}, where for a subgroup H, Blk(H|B) = {b ∈ Blk(H) : b^{G} = B} (in the sense of Brauer). Following [3], the p-local rank plr(B) of B is defined to be the number plr(B) = max{|σ| : σ ∈ R(G, B)}.

We improve a result of [3] in order to consider the p-local rank of a TI block later.

Proposition 2.1 A block B ∈ Blk(G) is a TI defect block if and only if either plr(B) = 0 or plr(B) = 1 and O_{p}(G) = 1.

Proof: Suppose B is a TI defect block with a defect group D = D(B). Then D is a radical subgroup of G and so O_{p}(G) ≤ D, where a subgroup R of G is radical if R = O_{p}(N_{G}(R)). Let \overline{G} = G/O_{p}(G) and denote by \overline{x} the image of an element x ∈ G in \overline{G}. Then \overline{N_{G}(D)} = N\overline{G}(\overline{D}) and \overline{x} = xO_{p}(G) ∈ \overline{G}\N\overline{G}(\overline{D}) for some x ∈ G if and only if x ∈ G\N_{G}(D). It follows that if D ≠ O_{p}(G), then D/O_{p}(G) is a TI set in G/O_{p}(G) and by [3, Proposition 5.1], plr(B) = 1. Thus either plr(B) = 0 or 1. In the latter case, O_{p}(G) = 1 since O_{p}(G) ≤ D ∩ D^{x} for each x ∈ G.

Suppose plr(B) = 0. Since D is radical in G, it follows that D = O_{p}(G), and since D is normal in G, it is a TI set and B is a TI defect block. Suppose plr(B) = 1 and O_{p}(G) = 1. Then D ≠ 1 and again by [3, Proposition 5.1], D = D/O_{p}(G) is a TI set of G = G/O_{p}(G), so that B is a TI defect block. □

3 TI and TI defect blocks

Lemma 3.1 Let B be a TI block, (D, b_{D}) a Sylow B-subgroup and (Q, b_{Q}) ≤ (D, b_{D}). Then there is a B-subgroup (P, b_{P}) with (Q, b_{Q})◁(P, b_{P}) ≤ (D, b_{D}) such that N_{G}(Q, b_{Q}) ≤ N_{G}(P, b_{P}).

Proof: Let P = N_{D}(Q) > Q. Then there is an unique (P, b_{P}) such that

(Q, b_{Q}) ◁ (P, b_{P}) ≤ (D, b_{D}).

Since B is TI, we have N_{G}(Q, b_{Q}) ≤ N_{G}(D, b_{D}) ≤ N_{G}(D). So N_{G}(Q, b_{Q}) ≤ N_{G}(P). Let x ∈ N_{G}(Q, b_{Q}). Then (P, b_{P})^{x} = (P, b_{P}^{x}). But x ∈ N_{G}(D, b_{D}), so both (P, b_{P}) and (P, b_{P}^{x}) are contained in (D, b_{D}). Hence x ∈ N_{G}(P, b_{P}) and we are done. □

Proposition 3.2 Let B be a TI block, (Q, b_{Q}) a non-trivial B-subgroup. Suppose Q is a subgroup of a defect group D = D(B). Then (Q, b_{Q}) is contained in a unique Sylow B-subgroup (D, b_{D}).

Proof: It suffices to show the existence of the Sylow B-subgroup (D, b_{D}).

Let Q be maximal in D such that there is a B-subgroup (Q, b_{Q}) with no Sylow B-subgroup of the form (D, b_{D}) containing it. Clearly Q < D.

Let (D, b′_{D}) be a fixed Sylow B-subgroup. This defines a B-subgroup (Q, b_{0}) contained in it. Then by Lemma 3.1 there is a B-subgroup (P, b_{1}) with (Q, b_{0}) ◁ (P, b_{1}) ≤ (D, b′_{D}) and N_{G}(Q, b_{0}) ≤ N_{G}(P, b_{1}).

Now C_{G}(Q)Q ≤ N_{G}(Q, b_{0}) and N_{G}(P, b_{1}) ≤ N_{G}(P), so C_{G}(Q)Q ≤ N_{G}(P) and C_{G}(Q)P ≤ N_{G}(P). Since C_{G}(P) ≤ C_{G}(Q) and C_{G}(P)P ⊴ N_{G}(P), it follows that both C_{G}(P)P and P are normal subgroups of C_{G}(Q)P. If H = C_{G}(Q)P, then C_{H}(Q) = C_{G}(Q), (Q, b_{Q}) is a subpair of H and b_{Q}^{H} is a well-defined block of H. Set B_{H} = b_{Q}^{H}. Then B_{H}^{G} = B as b_{Q}^{G} = B, and hence B_{H} ∈ Blk(H|B).

Let b_{P} be a block of C_{G}(P)P covered by B_{H}. By [1, Theorem 15.1], b_{P}^{H} = B_{H} = b_{Q}^{H} and so (Q, b_{Q}) ◁ (P, b_{P}).

By the maximality of Q, since Q < P there is a Sylow B-subgroup (D, b_{D}) containing (P, b_{P}), and so also (Q, b_{Q}), a contradiction. □

Proposition 3.3 A TI block B is a TI defect block. In particular, plr(B) = 0 or 1, and in the latter case O_{p}(G) = 1.

Proof: Let B be a TI block and suppose D = D(B) is not a TI set. Then 1 ≠ D ∩ D^{x} for some x ∈ G\N_{G}(D). Let (D, b_{D}) be a fixed Sylow B-subgroup and Q = D ∩ D^{x}. Then (Q, b_{Q}) ≤ (D, b_{D}) for a unique block b_{Q} ∈ Blk(C_{G}(Q)Q), so that (Q, b_{Q}) is a non-trivial B-subgroup. Since D^{x} is also a defect group of B and Q ≤ D^{x}, it follows by Proposition 3.2 that (Q, b_{Q}) < (D^{x}, b_{x}) for some Sylow B-subgroup (D^{x}, b_{x}), which is impossible since (D, b_{D}) ≠ (D^{x}, b_{x}). Thus B is a TI defect group, and the last assertion follows by Proposition 2.1. □

Suppose B = B_{0}(G) is the principal block of G. If B is a TI defect block and if a non-trivial B-subgroup (R, b_{R}) is contained in Sylow B-subgroups (D, b_{D}) and (D′, b_{D′}), then D′ = D^{x} for some x ∈ G and R ≤ D ∩ D′ = D ∩ D^{x}. Thus x ∈ N_{G}(D) and D = D′ as D is a TI set and R ≠ 1. By Brauer’s Third Main Theorem, b_{D} = b_{D′} is the principal block of C_{G}(D)D, so that B is a TI block. It follows that if a principal block is a TI defect block, then it is a TI block. But this is not true in general.

We describe where examples of TI defect blocks which are not TI may be found, and then give a specific example. Suppose that G has TI Sylow p-subgroups, and that these Sylow p-subgroups are abelian and have p-rank greater than one. Then G = O_{p′}(G).(N.A), where N is a non-abelian simple group and A is a finite group such that N.A acts faithfully on N (see for example [5, 1.2]). Let P ∈ Syl_{p}(G) (and note that P C_{G}(P) = C_{G}(P)). Then O_{p′}(G) ≤ N_{G}(P), and so O_{p′}(G) ≤ C_{G}(P). Hence

N ≤ X = ⟨P : P ∈ Syl_{p}(G)⟩ ⊴ G,

and O_{p′}(G) centralizes N. So G has the form (O_{p′}(G)⋆M).A for some quasisimple group M with TI Sylow p-subgroups, where (O_{p′}(G)⋆M) is a central product of O_{p′}(G) and M over O_{p′}(G)∩M. To produce examples of TI defect blocks which are not TI, it suffices to find non-abelian simple groups M with TI Sylow p-subgroups and outer automorphism groups A of M, such that for a non-trivial subgroup Q < P we have C_{A}(Q) ≠ C_{A}(P). Then we may choose O_{p′}(G) such that C_{A}(Q) and C_{A}(P) act nontrivially and trivially on O_{p′}(G) respectively. Choose μ_{1}, μ_{2} ∈ Irr(O_{p′}(G)) fixed by C_{A}(P) but conjugate C_{A}(Q), and consider b_{P} ∈ Blk(C_{G}(P)) covering μ_{1} (after identifying μ_{i} with the block of defect zero containing it), and its Brauer correspondent b_{Q} in C_{G}(Q). But b_{Q} must also cover μ_{2}, so there must be a block b′_{P} ≠ b_{P} of C_{G}(P) covering μ_{2} which is also a Brauer correspondent of b_{Q} (consider the conjugate of b_{P} by the element of x ∈ C_{G}(Q) taking μ_{1} to μ_{2}). This would give the counterexample we require.

In particular, consider the following subgroup of S_{10}. Take M = A_{6} ≤ S_{10} permuting {1, …, 6} and O_{p′}(G) = ⟨(7, 8, 9, 10)⟩. Take P = ⟨(1, 2, 3), (4, 5, 6)⟩ and Q = ⟨(4, 5, 6)⟩. Let A = ⟨(1, 2)(7, 10)(8, 9)⟩, and define G = (M × O_{p′}(G)).A. Note that A_{6} has TI Sylow 3-subgroups, hence so has G, and every 3-block of G is TI defect. Then C_{G}(P) = P × O_{p′}(G), and so has four 3-blocks. C_{G}(Q) = (P × O_{p′}(G)).A, and has three 3-blocks. Now every 3-block of C_{G}(P) has a Brauer correspondent in C_{G}(Q), and so there must be two distinct 3-blocks b_{P}, b′_{P} of C_{G}(P) sharing a 3-block b_{Q} of C_{G}(Q). Hence (Q, b_{Q}) is contained in distinct subpairs (P, b_{P}) and (P, b′_{P}), so b_{Q}^{G} has TI defect groups but is not TI.

4 Some alternative characterisations of TI blocks

A B-subgroup (Q, b_{Q}) of G is special if it is maximal amongst those N_{G}(Q, b_{Q})-stable subpairs for N_{G}(Q, b_{Q}). (Recall that if (R, b_{R}) is a subpair for N_{G}(Q, b_{Q}) containing (Q, b_{Q}), then it is necessarily a subpair for G, and a B-subgroup). Note that (1, B) and the Sylow B-subgroups are special. Note also that if B_{0} is the principal p-block, then the special B-subgroups correspond to the radical p-subgroups of G.

Before proving our main result we give a well-known but useful lemma:

Lemma 4.1 Let (Q, b_{Q}) be a B-subpair, where B is a p-block of a finite group G, and let (P, b_{P}) be a B-subpair properly containing (Q, b_{Q}). Then Q < N_{P} (Q, b_{Q}).

Proof: Write R = N_{P} (Q) > Q. Then there is a B-subpair (R, b_{R}) with (Q, b_{Q}) ≤ (R, b_{R}) ≤ (P, b_{P}). Since Q ◁ R, it follows that (Q, b_{Q}) ◁ (R, b_{R}), where by definition Q < R ≤ N_{G}(Q, b_{Q}), and the result follows. □

Proposition 4.2 Let B be a p-block of a finite group G with O_{p}(G) = 1. Then the following are equivalent:

(i) B is a TI block.

(ii) N_{G}(D, b_{D}) is a strongly B-embedded subgroup for each Sylow B-subgroup (D, b_{D}). (iii) The only non-trivial special B-subgroups are the Sylow B-subgroups.

Proof: (i) ⇒ (ii) is immediate.

(ii) ⇒ (iii): Suppose that (Q, b_{Q}) is a non-trivial special B-subgroup. Then (Q, b_{Q}) ≤ (D, b_{D}) for some Sylow B-subgroup (D, b_{D}). Suppose that Q ≠ D. Then R = N_{D}(Q, b_{Q}) > Q by Lemma 4.1. Let b_{R} be the unique p-block of RC_{G}(R) such that (Q, b_{Q}) < (R, b_{R}) ≤ (D, b_{D}). Then C_{G}(R) ≤ C_{G}(Q) ≤ N_{G}(Q, b_{Q}) and R ≤ N_{G}(Q, b_{Q}), so (R, b_{R}) is a subpair for N_{G}(Q, b_{Q}). But N_{G}(R, b_{R}) ≤ N_{G}(D, b_{D}) by our assumption, so (R, b_{R}) is N_{G}(Q, b_{Q})-stable, since N_{G}(Q, b_{Q}) ≤ N_{G}(D, b_{D}). Hence (R, b_{R}) is a N_{G}(Q, b_{Q})-stable subpair for N_{G}(Q, b_{Q}) which strictly contains (Q, b_{Q}), contradicting our assumption that (Q, b_{Q}) is special. So Q = D and we are done.

(iii) ⇒ (i): Suppose that B is not TI, but that condition (iii) holds. Choose Q maximal for which there is some b_{Q} ∈ Blk(QC_{G}(Q)|B) such that (Q, b_{Q}) is contained in two distinct Sylow B-subgroups (D, b_{D}) and (D_{1}, b_{D_{1}}). Then (Q, b_{Q}) is non-trivial, a B-subgroup and not special. Consider a chain

(Q, b_{Q}) = (Q_{0}, b_{0}) ≤ (Q_{1}, b_{1}) ≤ (Q_{2}, b_{2}) ≤ ⋯

of B-subgroups, where (Q_{i+1}, b_{i+1}) is a maximal N_{G}(Q_{i}, b_{i})-stable subpair for N_{G}(Q_{i}, b_{i}) containing (Q_{i}, b_{i}). Since G is finite eventually there must be an n such that

(Q_{n+j}, Q_{n+j}) = (Q_{n}, b_{n})

for every j ≥ 0. But then (Q_{n}, b_{n}) is special, and so a Sylow B-subgroup, say (D′, b_{D′}). Note that by definition N_{G}(Q_{0}, b_{0}) ≤ N_{G}(Q_{1}, b_{1}) ≤ ⋯ ≤ N_{G}(Q_{n}, b_{n}), so N_{G}(Q, b_{Q}) ≤ N_{G}(D′, b_{D′}). Without loss of generality we may take (D′, b_{D′}) = (D, b_{D}).

Let R = N_{D_{1}}(Q, b_{Q}), so Q < R ≤ D_{1} by Lemma 4.1. Let b_{R} be the unique block of Blk(RC_{G}(R)) such that (R, b_{R}) ≤ (D_{1}, b_{D_{1}}). We have

C_{G}(R) ≤ C_{G}(Q) ≤ N_{G}(Q, b_{Q}) ≤ N_{G}(D, b_{D}),

and also R ≤ N_{G}(Q, b_{Q}) ≤ N_{G}(D, b_{D}), so (R, b_{R}) is a subpair for N_{G}(D, b_{D}) and b_{R}^{N_{G}(D,b_{D})} is defined. But (b_{R}^{N_{G}(D,b_{D})})^{G} = b_{R}^{G} = B, so (D, b_{D}) is the unique Sylow b_{R}^{N_{G}(D,b_{D})}-subpair and so contains (R, b_{R}). Hence (R, b_{R}) is a B-subgroup contained in distinct Sylow B-subgroups (D, b_{D}), (D_{1}, b_{D_{1}}), contradicting our choice of Q. □

Remarks (i) It is easy to see that if B is a TI defect block, then G possesses a strongly B-embedded subgroup (for if (D, b_{D}) is a Sylow B-subpair for a TI defect block B, and (Q, b_{Q}) ≤ (D, b_{D}), then N_{G}(Q, b_{Q}) ≤ N_{G}(Q) ≤ N_{G}(D) < G). However, this does not lead to a strategy for proving that every TI block is TI defect by way of Proposition 4.2, for it is not necessarily the case that if G possesses a strongly B-embedded subgroup then B is TI. For example, consider the principal 5-block of M_{22} (see [7, 24.2]).

(ii) Special subpairs may be seen as a generalisation of radical p-subgroups. Define a special chain of subpairs and the special local rank analogously to the p-local rank, but using special subpairs instead of radical p-subgroups (so a chain σ : (Q_{0}, b_{0}) < ⋯ < (Q_{n}, b_{n}) of B-subgroups is special if (Q_{0}, b_{0}) is special and (Q_{i+1}, b_{i+1}) is a special subpair of N_{G}(σ_{i}) for each i). It follows from Proposition 2.1 and Proposition 4.2 that if B is a p-block of special local rank one of a finite group with O_{p}(G) = 1, then plr(B) = 1.

5 The strongly k-embedded B-core

Just as Γ_{P,k}(G) plays a rôle in the fusion of p-subgroups (see [4]), we demonstrate that Γ_{(D,b_{D}),k}(G) plays a similar part in the fusion of certain B-subgroups:

Lemma 5.1 Suppose that H ≤ G, b ∈ Blk(H|B) and that (D, b_{D}) is a Sylow bsubgroup of H which is also a B-subgroup of G. If Γ_{(D,b_{D}),k}(G) ≤ H, then (D, b_{D}) is a Sylow B-subgroup of G.

Proof: Suppose that (D, b_{D}) is not a Sylow B-subgroup. Since (D, b_{D}) is a Bsubgroup, it follows from Lemma 4.1 that there is a B-subgroup (P, b_{P}) with (D, b_{D}) ◁ (P, b_{P}) and D ≠ P, where P is contained in a defect group for B. But P C_{G}(P) ≤ P C_{G}(D) ≤ N_{G}(D, b_{D}) ≤ H, and so (P, b_{P}) is a subpair for H, and necessarily a bsubgroup, contradicting our choice of (D, b_{D}) as a Sylow b-subgroup. □

Proposition 5.2 Suppose that H ≤ G, b ∈ Blk(H|B) and that (D, b_{D}) is a B-subgroup which is also a Sylow b-subgroup. Write S_{B} for the poset of B-subgroups, and

S_{B}(b, k) = {(Q, b_{Q}) ∈ S_{B} : QC_{G}(Q) ≤ H, b_{Q}^{H} = b, m_{p}(Q) ≥ k}.

Then Γ_{(D,b_{D}),k}(G) ≤ H if and only if H controls fusion in S_{B}(b, k) and N_{G}(Q, b_{Q}) ≤ H for every (Q, b_{Q}) ∈ S_{B}(b, k) and every B-subgroup (Q, b_{Q}) ≤ (D, b_{D}).

Proof: Suppose that Γ_{(D,b_{D}),k}(G) ≤ H. Then by Lemma 5.1 (D, b_{D}) is a Sylow Bsubgroup. Let (Q, b_{Q}) ∈ S_{B}(b, k), and suppose that g ∈ G satisfies (Q, b_{Q})^{g} ∈ S_{B}(b, k). We show that g ∈ H, from which it follows immediately that H controls fusion in S_{B}(b, k) and N_{G}(Q, b_{Q}) ≤ H for every (Q, b_{Q}) ∈ S_{B}(b, k). Clearly also N_{G}(Q, b_{Q}) ≤ H for every (Q, b_{Q}) ≤ (D, b_{D}).

Since all Sylow b-subgroups are conjugate in H, there are h_{1}, h_{2} ∈ H such that (Q, b_{Q})^{h_{1}}, (Q, b_{Q})^{gh_{2}} ≤ (D, b_{D}). Let x = h_{1}^{−1}gh_{2} ∈ G. We may now apply [2, 4.10] (note that we may use the results of [2] here despite the differing definitions of a subpair, since there is a 1-1 correspondence between p-blocks of QC_{G}(Q) and p-blocks of C_{G}(Q) which respects the Brauer correspondence and conjugacy in G): there are B-subgroups

(R_{1}, b_{1}), …, (R_{n}, b_{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, and x = x_{1} ⋯ x_{n}. Manifestly m_{p}(R_{i}) ≥ m_{p}(Q) ≥ k for each i, so each x_{i} ∈ N_{G}(R_{i}, b_{i}) ≤ Γ_{(D,b_{D}),k}(G) ≤ H, and so x ∈ H. Hence g ∈ H as required.

Now suppose that H controls fusion in S_{B}(b, k) and N_{G}(Q, b_{Q}) ≤ H for every B-subgroup (Q, b_{Q}) ≤ (D, b_{D}). Then (Q, b_{Q}) ∈ S_{B}(b, k) for all such (Q, b_{Q}), and H controls fusion of B-subgroups contained in (D, b_{D}). Then it follows immediately from [2, 4.24] that N_{G}(Q, b_{Q}) = H ∩ N_{G}(Q, b_{Q}) for each such (Q, b_{Q}), and so by definition Γ_{(D,b_{D}),k}(G) ≤ H. □

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