The p-local Rank of a Block

Jianbei An and Charles W. Eaton ^{∗}

August 19, 1999

Abstract We generalize the p-local rank of a finite group, introduced by G. Robinson [7], to a p-block of a finite group and show that this has properties analagous to most of those given in [7].

1 Introduction

Let G be a finite group and p a prime. Given a chain of p-subgroups

σ : Q_{0} < Q_{1} < ⋯ < Q_{n}  (1.1)

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

G_{σ} = N_{G}(σ) = N_{G}(Q_{0}) ∩ N_{G}(Q_{1}) ∩ ⋯ ∩ N_{G}(Q_{n}).  (1.2)

Write C(G|Q) for the set of those chains with initial subgroup Q and Blk(G) for the set of all p-blocks of G.

Let (K, O, k) be a p-modular system which splits for G, and consider a p-block B of G with defect group D(B). Given a subgroup H of G, write Blk(H|B) = {b ∈ Blk(H) : b^{G} = B} (in the sense of Brauer).

The Knörr-Robinson reformulation of Alperin’s weight conjecture (see [5] and [1]) states that if B has positive defect then ∑_{σ∈C(G|1)} l(G_{σ}, B) = 0, where l(G_{σ}, B) denotes the number of simple kG_{σ}-modules belonging to p-blocks of G_{σ} whose Brauer correspondent is B. Clearly if l(G_{σ}, B) ≠ 0 then there is some p-block b of G_{σ} with b^{G} = B. It thus makes sense to consider the set

C(G, B) = {σ ∈ C(G) : Blk(G_{σ}|B) ≠ ∅}.

∗Research carried out whilst the second author was visiting the University of Auckland, supported by the Marsden Fund and a Royal Society Study Visit award.

1991 Mathematics Subject Classification. Primary 20C20

We say that σ is a p-chain for B, or a B-chain (as we will see we may associate with σ a chain of B-subgroups in the sense of Alperin [2]) if σ ∈ C(G, B). This is in part the motivation for this paper.

As demonstrated in [5], when considering this and similar conjectures we may restrict our attention to certain subsets of C(G), such as the following. A p-subgroup R of G is radical if O_{p}(N_{G}(R)) = R, where O_{p}(N_{G}(R)) is the maximal normal psubgroup of the normalizer N_{G}(R). A p-chain σ given by (1.1) 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) ≠ ∅}.

By [7], the p-local rank plr(G) of G is the number plr(G) = max{|σ| : σ ∈ R(G)}. We define the p-local rank plr(B) of B to be the number

plr(B) = max{|σ| : σ ∈ R(G, B)}.

It is clear that plr(B) ≤ plr(G) for all B ∈ Blk(G). Also we can recover the p-local rank of G by considering the principal p-block. For suppose plr(G) = n and σ ∈ R(G) with |σ| = n. If b_{0} = B_{0}(N_{G}(σ)) is the principal block of N_{G}(σ), then by [5, 3.2], b_{0}^{G} is well-defined and by Brauer’s Third Main Theorem, b_{0}^{G} = B_{0} = B_{0}(G), so that σ ∈ R(G, B_{0}) and

plr(B_{0}) = plr(G).

The paper is organized as follows. In Section 2 we consider our definitions in terms of subpairs, and give the lemma which is the engineroom of this paper. In Sections 3 and 4 we prove that if B′ is a p-block of a section of G corresponding to B under the Brauer correspondence and the block correspondence outlined in Section 4, then plr(B′) ≤ plr(B), and give some conditions for equality/inequality. Finally, in Section 5 we give necessary and sufficient conditions for a p-block to have p-local rank one. We would like to draw the readers’ attention to the fact that all proofs given here are direct, so the p-local structure is transparent(!) at each stage.

2 B-subgroups

Let R be a p-subgroup of G and b_{R} ∈ Blk(C_{G}(R)R). Then the pair (R, b_{R}) is called a subpair for G. If R is radical in G, then (R, b_{R}) is called a radical subpair for G. Following Alperin [2], for B ∈ Blk(G), the pair (R, b_{R}) is called a B-subgroup if (1, B) ≤ (R, b_{R}).

If (R, b_{R}) is a B-subgroup, then b_{R}^{G} = B and b_{R}^{N_{G}(R)} is a well-defined block of N_{G}(R). In particular, b_{R}^{N_{G}(R)} ∈ Blk(N_{G}(R)|B) and Blk(N_{G}(R)|B) ≠ ∅. Conversely, if Blk(N_{G}(R)|B) ≠ ∅, then R is conjugate to a subgroup of D(B) and there is a B-subgroup (R, b_{R}).

A B-subgroup (R, b_{R}) is called a radical B-subgroup if R is radical in G.

Suppose that σ ∈ R(G, B). Then there is a block b ∈ Blk(N_{G}(σ)|B) and we may suppose that D(b) ≤ D(B). Since Q_{0}, …, Q_{n} ⊴ N_{G}(σ), it follows that Q_{i} ≤ D(b) and there is a B-subgroup (Q_{i}, b_{i}) ≤ (D(B), b_{D}) for each i, where (D(B), b_{D}) is a fixed Sylow B-subgroup. Conversely if we can associate to σ ∈ R(G) such a chain of B-subgroups, then σ ∈ R(G, B). A radical p-chain σ is called a radical B-chain if σ ∈ R(G, B).

Lemma 2.1 Let B be a p-block of G and (Q, b_{Q}) a B-subgroup. Then there is a radical B-subgroup (P, b_{P}) such that

(Q, b_{Q}) ≤ (P, b_{P})

and N_{G}(Q) ≤ N_{G}(P). In particular, (Q, b_{Q}) and (P, b_{P}) are subpairs for N_{G}(P). Moreover, suppose that Q ≤ H ≤ G and Q is radical in H. Then Q = P ∩ N_{H}(Q) = P ∩ H and N_{H}(Q) = H ∩ N_{G}(P).

Proof: Let M_{1} = N_{G}(Q) and M_{i} = N_{G}(O_{p}(M_{i−1})) for all i ≥ 2. Then

N_{G}(Q) = M_{1} ≤ M_{2} ≤ … ≤ M_{m}  (2.1)

for all m ≥ 2 and since Q ≤ O_{p}(M_{1}) and O_{p}(M_{i−1}) ≤ O_{p}(M_{i}), it follows that

Q ⊴ O_{p}(M_{1}) ⊴ O_{p}(M_{2}) ⊴ … ⊴ O_{p}(M_{m}).  (2.2)

Thus there is a minimal m ≥ 1 such that M_{m} = M_{m+1} = N_{G}(O_{p}(M_{m})), so O_{p}(M_{m}) = O_{p}(M_{m+1}) and O_{p}(M_{m}) is radical in G. Let L = M_{m} and P = O_{p}(L), so that N_{G}(Q) ≤ L = N_{G}(P). Since C_{G}(Q) ≤ N_{G}(Q) ≤ L, it follows that C_{G}(Q) = C_{L}(Q), (Q, b_{Q}) is a subpair for L and b_{Q}^{L} is a well-defined block of L. Thus (Q, b_{Q}) is contained in a Sylow b_{Q}^{L}-subgroup (D_{1}, b_{D_{1}}), where D_{1} =_{L} D(b_{Q}^{L}). Since P ≤ D_{1}, it follows that there is a unique b_{Q}^{L}-subgroup (P, b_{P}) such that

(Q, b_{Q}) ≤ (P, b_{P}) ⊴ (D_{1}, b_{D_{1}}).

But C_{G}(P) ≤ C_{G}(Q) ≤ L, so (P, b_{P}) is also a subpair for G and (1, B) ≤ (Q, b_{Q}) ≤ (P, b_{P}). Thus (P, b_{P}) is a B-subgroup.

Suppose that Q ≤ H ≤ G and Q = O_{p}(K), where K = N_{H}(Q) ≤ L. Then Q ≤ K∩P ⊴ K and Q = K∩P. Since Q ⊴ N_{H∩P} (Q) ≤ K and N_{H∩P} (Q) ≤ K∩P = Q, it follows that

Q = P ∩ K = P ∩ H.

If h ∈ L ∩ H, then h normalizes O_{p}(L) ∩ H = Q, so that h ∈ K, K ≤ L ∩ H ≤ K and K = L ∩ H.

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3 Subgoups and the p-local rank

Proposition 3.1 Let H be a subgroup of G and b ∈ Blk(H|B). Suppose that

σ : O_{p}(H) = Q_{0} < Q_{1} < … < Q_{n}

is a radical b-chain with n ≥ 1. Set K_{i} = N_{H}(σ_{i}) for i ≥ 0. Then there is a radical B-chain

σ^{∗} : O_{p}(G) < P_{1} < … < P_{n}

of G such that for each i ≥ 1, K_{i} ≤ N_{L_{i−1}}(Q_{i}) ≤ L_{i}, K_{i} = L_{i} ∩ H, C_{G}(Q_{i}) = C_{L_{i}}(Q_{i}) and Q_{i} = P_{i} ∩ K_{i} = P_{i} ∩ H, where L_{i} = N_{G}(σ_{i}^{∗}).

Proof: We may suppose D(b) ≤ D(B). Since σ is a radical b-chain, it follows that Blk(N_{H}(σ)|b) ≠ ∅ and we may suppose that

O_{p}(H) = Q_{0} < Q_{1} < … < Q_{n} ≤ D(b) ≤ D(B).  (3.1)

Let (D(B), b_{D}) be a Sylow B-subgroup. For each i ≥ 0, there is a B-subgroup (Q_{i}, b_{i}) such that (Q_{i}, b_{i}) ≤ (D(B), b_{D}).

By Lemma 2.1, there is a radical B-subgroup (P_{1}, B_{1}) such that (Q_{1}, b_{1}) ≤ (P_{1}, B_{1}), K_{1} ≤ N_{G}(Q_{1}) ≤ L_{1} = N_{G}(P_{1}), C_{G}(Q_{1}) = C_{L_{1}}(Q_{1}), Q_{1} = P_{1} ∩ K_{1} = P_{1} ∩ H and K_{1} = H ∩ L_{1}. Thus O_{p}(G) ≤ P_{1} and by (3.1),

O_{p}(G) ∩ H ≤ O_{p}(H) < Q_{1} ≤ P_{1} ∩ H.

In particular, P_{1} ≠ O_{p}(G),

σ_{1}^{∗} : O_{p}(G) < P_{1}

is a radical B-chain and L_{1} = N_{G}(σ_{1}^{∗}).

Suppose that n ≥ 2 and there is a radical B-chain

σ_{t}^{∗} : O_{p}(G) < P_{1} < … < P_{t}

with 1 ≤ t < n satisfying the conditions of the Proposition. Since σ is radical, it follows that Q_{t+1} ≤ Q_{n} ≤ K_{n} ≤ K_{t} ≤ L_{t}, C_{G}(Q_{t+1}) ≤ C_{G}(Q_{t}) = C_{L_{t}}(Q_{t}) and (Q_{t+1}, b_{t+1}) is a subpair for L_{t}.

Apply Lemma 2.1 for Q = Q_{t+1}, G = L_{t} and H = K_{t}. There is a radical subpair (P_{t+1}, B_{t+1}) for L_{t} such that (Q_{t+1}, b_{t+1}) ≤ (P_{t+1}, B_{t+1}), K_{t} ≤ N_{L_{t}}(Q_{t+1}) ≤ L_{t+1} = N_{L_{t}}(P_{t+1}), Q_{t+1} = P_{t+1} ∩ K_{t} = P_{t+1} ∩ K_{t+1}, K_{t+1} = L_{t+1} ∩ K_{t} and C_{L_{t}}(Q_{t+1}) = C_{L_{t+1}}(Q_{t+1}).

Since K_{t} = L_{t} ∩ H, it follows that K_{t+1} = L_{t+1} ∩ L_{t} ∩ H = L_{t+1} ∩ H and Q_{t+1} = P_{t+1} ∩ K_{t} = P_{t+1} ∩ H. Similarly, since C_{G}(P_{t+1}) ≤ C_{G}(Q_{t}) = C_{L_{t}}(Q_{t}) ≤ L_{t}, it follows by Lemma 2.1 that

C_{G}(P_{t+1}) = C_{L_{t}}(P_{t+1}) = C_{L_{t+1}}(P_{t+1}),

so that (P_{t+1}, B_{t+1}) is a subpair for G. But (1, B) ≤ (Q_{t+1}, b_{t+1}) ≤ (P_{t+1}, B_{t+1}), so (P_{t+1}, B_{t+1}) is a B-subgroup. Since Q_{t} = P_{t} ∩ H < Q_{t+1} = P_{t+1} ∩ H, it follows that P_{t} < P_{t+1},

σ_{t+1}^{∗} : O_{p}(G) < P_{1} < … < P_{t} < P_{t+1}

is a radical B-chain of G and L_{t+1} = N_{G}(σ_{t+1}^{∗}). The result now follows by induction on n.

□

Proposition 3.2 Let H be a subgroup of G and b ∈ Blk(H|B). Then

plr(b) ≤ plr(B).

Proof: We may suppose plr(b) ≥ 1. Suppose that σ is a radical b-chain given by (1.1) such that |σ| = plr(b) = n. Since Q_{0} is radical in H, it follows that O_{p}(H) ≤ Q_{0}. If O_{p}(H) < Q_{0}, then

σ′ : O_{p}(H) < Q_{0} < Q_{1} < … < Q_{n}

is also a radical b-chain of H and so plr(b) ≥ |σ′| = plr(b) + 1, which is impossible. Thus O_{p}(H) = Q_{0}. By Proposition 3.1, there is a radical B-chain σ^{∗} of length n, so that plr(b) ≤ plr(B).

□

We examine the situation where H contains O^{p′}(G), the minimal normal subgroup with p-regular index in G. We first need the following in order to locate b-subgroups when moving from G to H:

Lemma 3.3 Let N ⊴ G and N ≤ H ≤ G. Suppose that B ∈ Blk(G) and b ∈ Blk(H|B). Then

(i) there is B′ ∈ Blk(N) such that B′ is covered by both B and b,

(ii) B and b share a defect group D.

Proof: (i) By [3, 58.17] b|B_{H×H}, and by [3, 61.4] B covers B′ ∈ Blk(N) if and only if B′|B_{N×N} (and similarly for b). Suppose that B covers B′ ∈ Blk(N) and b covers B′′ ∈ Blk(N). Then B′′|b_{N×N} and b|B_{H×H}, so B′′|B_{N×N}, and B′ is conjugate in G to B′′. Hence we can take B′ = B′′.

(ii) Let B′ ∈ Blk(N) be covered by both B and b as in part (i), and let D(G) be a defect group for B, D(H) a defect group for b such that D(G) ∩ N =_{G} D′ for some defect group D′ of B′ and D(H) ∩ N =_{H} D′′ for some other defect group D′′ of B′ (see [3, 61.2]). We may assume that D′′ = D′. But D(G) ∩ N = D(G) and D(H) ∩ N = D(H), so D(G) =_{G} D′ =_{H} D(H), and we may take D(G) = D(H).

□

Proposition 3.4 Suppose that O^{p′}(G) ≤ H ≤ G and b ∈ Blk(H|B). Then R(G, B) = R(H, b), and in particular plr(B) = plr(b).

Proof: Write N = O^{p′}(G), and let D be a defect group for B and b as in Lemma 3.3.

Note that if X and Y are finite groups with X ⊴ Y and [Y : X]_{p} = 1, then O_{p}(X) = O_{p}(Y), since O_{p}(X) is the unique maximal normal p-subgroup of X. Also note that every p-subgroup of Y is contained in X.

Suppose that σ ∈ R(G, B). Then we may take V^{σ} ≤ D ≤ H. Now

O^{p′}(N_{G}(σ)) ≤ N_{H}(σ) ≤ N_{G}(σ),

so O_{p}(N_{H}(σ)) = O_{p}(O^{p′}(N_{G}(σ))) = O_{p}(N_{G}(σ)) = V^{σ}. Hence σ ∈ R(H). But V^{σ} ≤ D, a defect group for b, so there is a b-subgroup (V^{σ}, b_{V^{σ}}). But b_{V^{σ}}^{N_{H}(σ)} is a well-defined block, as is (b_{V^{σ}}^{N_{H}(σ)})^{H}, and b_{V^{σ}}^{H} = b, so σ ∈ R(H, b).

Ther argument that R(H, b) ⊆ R(G, B) is similar.

□

Proposition 3.5 Let G_{1} and G_{2} be finite groups and B = B_{1} × B_{2} a p-block of G = G_{1} × G_{2}, where each B_{i} ∈ Blk(G_{i}). Then

plr(B) = plr(B_{1}) + plr(B_{2}).

Proof: Let π_{i} be the natural projection of G onto G_{i} for i = 1, 2. If σ ∈ R(G) is given by (1.1), then let Ω_{i} = {π_{i}(Q_{j}) : 0 ≤ j ≤ n} and |Ω_{i}| = u_{i}. Relabel the subgroups in Ω_{i} such that

Ω_{i} = {W_{0} < W_{1} < … < W_{u_{i}}}.

It follows by [7, 4.6] that each radical subgroup R of G is of the form R = R_{1} × R_{2}, where R_{1} = R ∩ G_{1} and R_{2} = R ∩ G_{2}. Thus N_{G}(R) = N_{G_{1}}(R_{1}) × N_{G_{2}}(R_{2}), and it follows that π_{i}(σ) : W_{0} < W_{1} < … < W_{u_{i}} is a radical chain of G_{i} and

N_{G}(σ) = N_{G_{1}}(π_{1}(σ)) × N_{G_{2}}(π_{2}(σ)).

In addition, if σ is a radical B-chain, then π_{i}(σ) is a radical B_{i}-chain, so that plr(B) ≤ plr(B_{1}) + plr(B_{2}).

Conversely, suppose that σ_{1} : W_{0} < W_{1} < … < W_{u_{1}} and σ_{2} : P_{0} < P_{1} < … < P_{u_{2}} are radical B_{1}-chain and B_{2}-chain, respectively. Define

σ : Q_{0} < Q_{1} < … < Q_{u_{1}+u_{2}},

such that Q_{i} = W_{i} × P_{0} for 0 ≤ i ≤ u_{1} and Q_{i} = W_{u_{1}} × P_{i−u_{1}} for u_{1} ≤ i ≤ u_{1} + u_{2}. Then σ is radical B-chain and |σ| = |σ_{1}| + |σ_{2}|. Thus plr(B) ≥ plr(B_{1}) + plr(B_{2}).

□

4 Quotient groups and the p-local rank

Let N be a normal subgroup of G and V an indecomposable OG-module which is indecomposable as an ON-module V_{N}, where O is O or k. Following [6], a block B of G dominates a block \overline{B} of \overline{G} = G/N through V if there is an O\overline{G}-module X such that V ⊗ Inf(X) is a B-module, where Inf(X) is the inflation of X to G. This is equivalent, by [6], to the fact that B covers the block b of N containing V_{N}. In addition, by [6], \overline{B} has a defect group QN/N such that

D ∩ N ≤ Q ≤ D,

where D = D(B).

Let \overline{H} ≤ \overline{G} = G/N and \overline{b} ∈ Blk(\overline{H}|\overline{B}). We may suppose that D(\overline{b}) ≤ D(\overline{B}) = QN/N. Since QN/N ≃ Q/(Q ∩ N), there is a subgroup W of Q such that W N/N = D(\overline{b}) and

D ∩ N ≤ Q ∩ N ≤ W ≤ Q ≤ D.  (4.1)

Moreover, suppose that W_{1} and W_{2} are subgroups of Q such that Q∩N ≤ W_{1} ≤ W_{2} ≤ Q and W_{1}N/N ⊴ W_{2}N/N. Then W_{1}N/N ≃ η(W_{1}) ⊴ η(W_{2}) ≃ W_{2}N/N, where η is the natural group homomorphism from Q onto Q/(Q ∩ N). Since Q ∩ N = ker(η) ≤ W_{1}, it follows that W_{1} ⊴ W_{2}.

Proposition 4.1 Let N be a normal subgroup of G and \overline{B} ∈ Blk(G/N) which is dominated by a block B of G through V. Then

plr(\overline{B}) ≤ plr(B).

Moreovr, suppose that b is the block of N containing V_{N} and plr(b) ≥ 1. Then plr(\overline{B}) < plr(B).

Proof: Let \overline{G} = G/N and σ ∈ R(\overline{G}, \overline{B}) with |σ| = n ≥ 1. Then there is a block \overline{b} ∈ Blk(N\overline{G}(σ)|\overline{B}), so we may suppose that W N/N = D(\overline{b}) for some subgroup W ≤ Q satisfying (4.1). Thus we may suppose that there are subgroups Q_{0}, Q_{1}, …, Q_{n} of Q such that

D ∩ N ≤ Q ∩ N ≤ Q_{0} < Q_{1} < … < Q_{n} ≤ Q ≤ D  (4.2)

and σ : Q_{0}N/N < Q_{1}N/N < … < Q_{n}N/N. Thus for each i with 0 ≤ i ≤ n there is a B-subgroup (Q_{i}, b_{i}) for G. Since σ is radical, it follows by the remark above that Q_{i} ⊴ Q_{n} for all 0 ≤ i ≤ n.

By Lemma 2.1, there is a radical B-subgroup (P_{1}, b_{P_{1}}) containing (Q_{1}, b_{1}) and N_{G}(Q_{1}) ≤ L_{1} = N_{G}(P_{1}), so that O_{p}(G) ≤ P_{1} and Q_{n} ≤ L_{1} as Q_{1} ⊴ Q_{n}. Since \overline{O_{p}(G)} = O_{p}(G)N/N ⊴ \overline{G} and \overline{Q}_{0} is radical in \overline{G}, it follows that

\overline{O_{p}(G)} ≤ \overline{Q}_{0} < \overline{Q}_{1} ≤ \overline{P}_{1},

and in particular, O_{p}(G) < P_{1}. Thus σ_{1}^{∗} : O_{p}(G) < P_{1} is a radical B-chain. In addition, since N_{G}(Q_{1}) ≤ L_{1}, it follows that \overline{N_{G}(Q_{1})} = N\overline{G}(\overline{Q}_{1}) ≤ \overline{L}_{1}, \overline{Q}_{1} is a radical subgroup of \overline{L}_{1}, \overline{P}_{1} ≤ \overline{Q}_{1} and \overline{P}_{1} = \overline{Q}_{1}. Thus N\overline{G}(\overline{P}_{1}) = N\overline{G}(\overline{Q}_{1}) and \overline{L}_{1} = N\overline{G}(\overline{Q}_{1}).

Suppose that there is a radical B-chain

σ_{t}^{∗} : O_{p}(G) < P_{1} < … < P_{t}

with 1 ≤ t < n such that Q_{n} ≤ N_{L_{t−1}}(Q_{t}) ≤ L_{t}, Q_{t} ≤ P_{t}, \overline{Q}_{t} = \overline{P}_{t}, C_{G}(Q_{t}) = C_{L_{t}}(Q_{t}) and N\overline{G}(σ_{t}) = \overline{L}_{t}, where L_{0} = G. Thus Q_{t+1} ≤ L_{t}, C_{G}(Q_{t+1}) ≤ C_{G}(Q_{t}) = C_{L_{t}}(Q_{t}) ≤ L_{t} and C_{G}(Q_{t+1}) = C_{L_{t}}(Q_{t+1}). So (Q_{t+1}, b_{t+1}) is a subpair for L_{t} and by Lemma 2.1, (Q_{t+1}, b_{t+1}) ≤ (P_{t+1}, B_{t+1}) for some radical subpair (P_{t+1}, B_{t+1}) for L_{t} and N_{L_{t}}(Q_{t+1}) ≤ L_{t+1} = N_{L_{t}}(P_{t+1}). Since C_{G}(P_{t+1}) ≤ C_{G}(Q_{t+1}) ≤ L_{t}, it follows that C_{G}(P_{t+1}) = C_{L_{t}}(P_{t+1}) ≤ L_{t+1}, so that C_{G}(P_{t+1}) = C_{L_{t+1}}(P_{t+1}) and (P_{t+1}, B_{t+1}) is a subpair for G. Since (1, B) ≤ (Q_{t+1}, b_{t+1}) ≤ (P_{t+1}, B_{t+1}), it follows that (P_{t+1}, B_{t+1}) is a B-subgroup. Since Q_{n} ≤ L_{t} and Q_{t+1} ⊴ Q_{n}, it follows that Q_{n} ≤ L_{t+1}.

Since P_{t+1} is radical in L_{t} and N_{L_{t}}(Q_{t+1}) ≤ L_{t+1}, it follows that P_{t} = O_{p}(L_{t}) ≤ P_{t+1},

\overline{N_{L_{t}}(Q_{t+1})} = N_{\overline{L}_{t}}(\overline{Q}_{t+1}) = N\overline{G}(σ_{t+1}) ≤ \overline{L}_{t+1},

\overline{Q}_{t+1} is radical in \overline{L}_{t+1} and \overline{P}_{t+1} ≤ \overline{Q}_{t+1}. Thus \overline{P}_{t+1} = \overline{Q}_{t+1} and N\overline{G}(σ_{t+1}) = \overline{L}_{t+1}. In particular, \overline{Q}_{t} = \overline{P}_{t} < \overline{Q}_{t+1} = \overline{P}_{t+1} and P_{t} ≠ P_{t+1}. Thus

σ_{t+1}^{∗} : O_{p}(G) < P_{1} < … < P_{t} < P_{t+1}

is a radical B-chain and by induction on n, plr(\overline{B}) ≤ plr(B).

Suppose that plr(b) > 0, so that B covers b by [6]. We may choose above D so that D(b) = D ∩ N. Thus

O_{p}(G) ∩ N ≤ O_{p}(N) < D ∩ N ≤ Q_{0}

as plr(b) ≥ 1. By Lemma 2.1, there is a radical B-subgroup (W, b_{W}) containing the B-subgroup (Q_{0}, b_{0}) and N_{G}(Q_{0}) ≤ N_{G}(W). Thus O_{p}(G)∩N < D ∩N ≤ Q_{0} ≤ W ∩N and O_{p}(G) < W. Repeat the proof above with G replaced by N_{G}(W). Then there is a radical B-chain O_{p}(Q) < W < P_{1} < … < P_{n} of length n + 1, so that plr(\overline{B}) < plr(B).

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5 Blocks of p-local rank one

We say that a subgroup H of G is trivial intersection (TI) if H^{g} ∩ H = 1 for every g ∈ G − N_{G}(H). By [7, 7.1] if plr(G) > 0, then plr(G) = 1 if and only if G/O_{p}(G) has TI Sylow p-subgroups. Hence if B is the principal p-block, then plr(B) = 1 if and only if B has defect groups intersecting pairwise in O_{p}(G). We show that this may be generalized to any p-block.

We then highlight the consequence of this to conjectures such as those of Alperin [1], Dade [4] and of Robinson [7], although the consequence to Alperin’s conjecture has been proved in [8].

Proposition 5.1 Let B ∈ Blk(G) have defect group D ≠ O_{p}(G). Then plr(B) = 1 if and only if D/O_{p}(G) is a TI set in G/O_{p}(G). Further, in this case O_{p}(G) is the only radical p-subgroup of G contained in D.

Proof: Suppose first that D/O_{p}(G) is a TI subgroup of G/O_{p}(G), and that Q ≠ O_{p}(G) is a radical p-subgroup of G such that Blk(N_{G}(Q)|B) ≠ ∅. Then Q is conjugate to a subgroup of D, and we may suppose that Q ≤ D. Suppose that Q ≠ D. Since N_{G}(Q) ≤ N_{G}(D), it follows that N_{D}(Q) ⊴ N_{G}(Q), and so Q < N_{D}(Q) ≤ O_{p}(N_{G}(Q)), contradicting our assumption that Q is radical. Hence a radical B-chain σ of maximal length must satisfy σ_{1} : O_{p}(G) < D, since D is a radical p-subgroup such that Blk(N_{G}(D)|B) ≠ ∅. But V^{σ} is G-conjugate to a subgroup of D, so O_{p}(G) < D is a radical p-chain for B of maximal length, and plr(B) = 1.

Now suppose that plr(B) = 1, but that D/O_{p}(G) is not TI. We show first that D cannot contain any radical p-subgroups of G other than O_{p}(G) and itself.

Suppose that R is a radical p-subgroup of G with O_{p}(G) < R < D, and let (D, b_{D}) be a Sylow B-subgroup. Then there is a unique radical B-subgroup (R, b_{R}) such that (1, B) < (R, b_{R}) < (D, b_{D}). Now b_{R}^{N_{G}(R)} is a well-defined block of N_{G}(R) and has defect group R, since otherwise O_{p}(G) < R < D(b_{R}^{N_{G}(R)}) is a radical B-chain and plr(B) > 1, contradicting our original assumption. But then (b_{R}^{N_{G}(R)})^{G} must have defect group R ≠ D by Brauer’s First Main Theorem, a contradiction since (b_{R}^{N_{G}(R)})^{G} = B and D(B) = D.

Since D/O_{p}(G) is not TI, we can choose a defect group D_{1} ≠ D for B with |D_{1} ∩D| maximized. Write Q = D_{1} ∩ D > O_{p}(G). As in the above argument, there is a B-subgroup (Q, b_{Q}) such that (Q, b_{Q}) < (D, b_{D}). By Lemma 2.1, there is radical Bsubgroup (P, b_{P}) strictly containing (Q, b_{Q}) with N_{G}(Q) ≤ N_{G}(P) (since Q is not itself a radical B-subgroup).

Now we have seen that the only radical p-subgroups for B are either O_{p}(G) or conjugate to D. By maximality, Q = P ∩ D = P ∩ D_{1}, hence we may take P = D, and N_{G}(Q) ≤ N_{G}(D).

Let (D_{1}, b_{D_{1}}) be a Sylow B-subgroup and R = N_{D_{1}}(Q). Then Q < R ≤ D_{1}, so there is a B-subgroup (R, b_{R}). Since

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

(R, b_{R}) is a subpair for N_{G}(D), and b_{R}^{N_{G}(D)} is defined.

Let D(b_{R}^{N_{G}(D)}) be a defect group for b_{R}^{N_{G}(D)} containing R. Then D ≤ D(b_{R}^{N_{G}(D)}) as D ⊴ N_{G}(D). Since (b_{R}^{N_{G}(D)})^{G} = b_{R}^{G} = B, it follows that D(b_{R}^{N_{G}(D)}) is G-conjugate to a subgroup of D, so D = D(b_{R}^{N_{G}(D)}). Hence Q < R = N_{D_{1}}(Q) ≤ D(b_{R}^{N_{G}(D)}) = D, and Q < |N_{D_{1}}(Q)| < |D_{1} ∩ D|, contradicting our choice of Q.

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Remark We note that there is already a notion of a TI block (see [1]), where B is TI if every nontrivial B-subgroup (Q, b_{Q}) is contained in a unique Sylow B-subgroup. However, it is not clear whether this is equivalent to B having TI defect groups. To distinguish between the two definitions, we say that a block with TI defect groups is a TI defect block.

Corollary 5.2 Suppose that B is a p-block of G. Let f : R(G, B) → A be a G-stable function dependent only on N_{G}(σ) and V_{σ}, where A is an abelian group. If D ≠ O_{p}(G) is a defect group for B, plr(G) > 0 and any two defect groups for B intersect in O_{p}(G), then

∑_{σ∈R(G,B)/G}(−1)^{|σ|}f(N_{G}(σ), V_{σ}) = f(G, O_{p}(G)) + f(N_{G}(D), D) − f(N_{G}(D), O_{p}(G)).

Remark With analogy to the remarks in [5] we may replace R(G, B) by D_{max} in any alternating sum defined by such a function f, where D_{max} is the set of p-chains of G which are intersections of defect groups of B (here we do not make any assumptions on plr(B). The proof is similar to that given in [5], so we do not include it here.

Acknowlegment

The authors would like to thank the Marsden Fund committee and The Royal Society for their support, and the University of Auckland for its generous hospitality.

References

[1] J.L. Alperin. Weights for finite groups. In The Arcata Conference on Representations of Finite Groups, Proc. of Symposia in Pure Math. 47 (1987), 369-379.

[2] J.L. Alperin. Local representation theory (Cambridge University Press, 1986).

[3] C.W. Curtis and I. Reiner. Methods of representation theory II (Wiley-Interscience, 1987).

[4] E.C. Dade. Counting Characters in Blocks, II.9. In Representation theory of finite groups, Ohio State Univ. Math. Res. Inst. Publ., 6 (de Gruyter, Berlin, 1997), 45-59.

[5] R. Knorr and G. Robinson. Some remarks on a conjecture of Alperin. J. London Math. Soc. (2) 39 (1989), 48-60.

[6] M. Murai. Normal subgroups and heights of character. J. Math. Kyoto Univ. 36 (1996), 31-43.

[7] G. Robinson. Local structure, vertices and Alperin’s conjecture. Proc. London Math. Soc. (3) 72 (1996), 312-330.

[8] Zeng Jiwen. The problem of the Alperin conjecture for a block with a TI defect group. Chinese Ann. Math. Ser. A 19 (1998), no.4, 519-524.

Department of Mathematics University of Auckland Auckland, New Zealand