BLOCK INDUCTION FOR GROUP ALGEBRAS OVER SEMI-LOCAL RINGS

Wayne W. Wheeler Mathematics, IAS Australian National University GPO Box 4 Canberra, ACT 2601 Australia wheeler@pell.anu.edu.au

27 September 1992

Introduction

A natural generalization of p-block theory is to consider the situation that arises when we try to replace the prime p by a set of primes π, and a number of authors have studied the resulting π-blocks. For example, Slattery has developed the theory for π-separable groups in [9] and [10]. In [7] Robinson proved a version of Brauer’s Second Main Theorem for arbitrary groups, and Robinson and Staszewski studied subpairs in [8].

The current paper attempts to look at π-blocks from a somewhat more moduletheoretic perspective than has been done in the past. In particular, if G is a finite group, then we regard the π-blocks of G as G × G-modules. The fundamental difficulty in proving results about π-blocks with this approach is that it is necessary to work over a coefficient ring R such that the Krull–Schmidt Theorem fails for RG-modules. Because the blocks of RG are unique, and not just unique up to isomorphism, the absence of a Krull–Schmidt Theorem creates no problems in defining the π-blocks. It does mean, however, that if H is a subgroup of G and b is a π-block of H, then it is not at all clear whether there is a π-block B of G such that b | B_{H×H}. This complication highlights just one of the ways in which the study of π-blocks proves to be quite different from that of p-blocks.

We proceed by fixing a finite group G and a finite set of primes π. Let R be a principal ideal domain containing a primitive |G|^{th} root of unity and having a quotient field of characteristic zero. If π = {p_{1}, …, p_{r}}, then we will assume that R contains a unique prime ideal ℘_{i} such that p_{i} ∈ ℘_{i} for i = 1, …, r, and ℘_{1}, …, ℘_{r} are all of the nonzero prime ideals of R. For example, let ω = e^{2πi/|G|}, and choose a prime ideal ℘_{i} of ℤ[ω] with p_{i} ∈ ℘_{i} for i = 1, …, r. Then we obtain a suitable coefficient ring by setting R = \{\frac{a}{b} | a, b ∈ ℤ[ω] and b ∉ ℘_{i} for i = 1, …, r\}. It should be noted that although a slightly different ring is used in [7] and [8], the results proven there are still valid over R.

Typeset by AMS-TEX

The group ring RG can be considered as a right R[G × G]-module by setting α · (x, y) = x^{−1}αy for all α ∈ RG and x, y ∈ G. The π-blocks of G are then the indecomposable summands of the module RG. Each π-block corresponds to a primitive central idempotent of RG and to a set of irreducible characters of G, and it will occasionally be useful for us to think of π-blocks as sets of characters.

The approach we take is to try to obtain information about the blocks of G by choosing a prime ideal ℘ of R and working over either the localized ring R_{℘} or the ℘-adic completion R̂_{℘}. This procedure relies heavily on recent work by Ellers and Hill [3, 4] describing the conditions under which a p-block B of G and a p-block b of a subgroup H satisfy b | B_{H×H}. Using the theory of π-subpairs developed in [8], we show that if D ⊆ G is a nilpotent π-subgroup, H is a subgroup of G such that DC_{G}(D) ⊆ H ⊆ N_{G}(D), and b is a π-block of H, then there is an associated π-block B = b^{G} of G. This block has the property that if H = DC_{G}(D) or if G has nilpotent Hall π-subgroups, then b | B_{H×H}. For groups with a nilpotent Hall π-subgroup we also prove an analogue of the well-known result that if B is a p-block of G with defect group P and b is a p-block of P C_{G}(P) such that b^{G} = B, then B | b^{G×G}.

This research was done while the author was at MSRI, the University of Minnesota, and the University of Warwick. He would like to express his gratitude to the NSA for providing financial support at MSRI and to Professor J. L. Alperin for providing support of various kinds.

§1. Preliminaries

In this section we collect a number of the basic results on π-blocks that we will require for the later work.

If H is any subgroup of G, let η = η_{H}^{G} : RG → RH denote the R[H × H]- homomorphism determined by setting

η(x) = \{ \begin{matrix}x & if x ∈ H \\ 0 & if x ∈ G − H, \end{matrix}

and let (RG)^{H} denote the set of elements in RG that are invariant under conjugation by elements of H.

Proposition 1.1. Let p ∈ π, and suppose that ℘ is the prime ideal of R with p ∈ ℘. Let O denote R, R_{℘}, or R̂_{℘}. Suppose that H_{1}, …, H_{n} are subgroups of G, and let E ∈ Z(OG) and e_{i} ∈ Z(OH_{i}) be idempotents. Set B = (OG)E and b_{i} = (OH_{i})e_{i}. Then

(1) b_{1} | B_{H_{1}×H_{1}} if and only if there is an element z ∈ (OG)^{H_{1}}E with η(z) = e_{1}.

(2) B | ⨁_{i=1}^{n} b_{i}^{G×G} if and only if there are elements z_{i} ∈ (OG)^{H_{i}}e_{i} with ∑_{i} Tr_{H_{i}}^{G}(z_{i}) = E.

Proof. The proof is a straightforward extension of the proofs of Theorems 1 and 2 of [11]. □

The following lemma is well-known and can be obtained as an easy consequence of Lemma 76.28 and Corollary 76.9 of [2].

Lemma 1.2. Let H be a group such that the quotient field of R is a splitting field for H. Let ℘ be a prime ideal of R, and let U and V be R_{℘}H-modules. Then U | V if and only if R̂_{℘} ⊗ U | R̂_{℘} ⊗ V.

Proposition 1.3. Let H_{1}, …, H_{n} be subgroups of G, and let E ∈ Z(RG) and e_{i} ∈ Z(RH_{i}) be idempotents. Set B = (RG)E and b_{i} = (RH)e_{i}. Then

(1) b_{1} | B_{H_{1}×H_{1}} if and only if R̂_{℘} ⊗ b_{1} | R̂_{℘} ⊗ B_{H_{1}×H_{1}} for every prime ideal ℘ of R.

(2) B | ⨁_{i=1}^{n} b_{i}^{G×G} if and only if R̂_{℘} ⊗ B | ⨁_{i=1}^{n}(R̂_{℘} ⊗ b_{i}^{G×G}) for every prime ideal ℘ of R.

Proof. The proofs of (1) and (2) are similar, so we prove only the first statement.

Set H = H_{1}, e = e_{1}, and b = b_{1}. If b | B_{H×H}, then certainly R̂_{℘} ⊗b | R̂_{℘} ⊗B_{H×H} for all ℘. Conversely, suppose that R̂_{℘} ⊗ b | R̂_{℘} ⊗ B_{H×H} for every prime ideal ℘ of R. We may assume by Lemma 1.2 that π contains more than one prime. Let π = {p_{1}, …, p_{r}}, and let ℘_{i} be the prime ideal of R with p_{i} ∈ ℘_{i}. Then Lemma 1.2 implies that R_{℘_{i}} ⊗ b | R_{℘_{i}} ⊗ B_{H×H} for all i, and it follows from Proposition 1.1 that there is an element z_{i} ∈ (R_{℘_{i}}G)^{H}E such that η(z_{i}) = e. Let τ_{i} = π − {p_{i}}, and let n_{i} be the smallest positive integer such that n_{i}z_{i} ∈ RG. Then n_{i} is a τ_{i}- number. Since n_{1}, …, n_{r} are relatively prime, there are integers a_{1}, …, a_{r} such that a_{1}n_{1} + ⋯ + a_{r}n_{r} = 1. Let z = a_{1}n_{1}z_{1} + ⋯ + a_{r}n_{r}z_{r}. Then z ∈ (RG)^{H}E and η(z) = e, so b | B_{H×H} by Proposition 1.1. □

The following proposition is essentially an unpublished result of Robinson. The proof is not difficult and will be omitted.

Proposition 1.4. Suppose that D is a nilpotent π-subgroup of G with G = DC_{G}(D). Let \overline{G} = G/D, and let ϕ : RG → R\overline{G} denote the natural homomorphism. Then ϕ induces a one-to-one correspondence between the primitive central idempotents of RG and those of R\overline{G}.

Suppose that D is a nilpotent π-subgroup of G such that G = DC_{G}(D), and set \overline{G} = G/D. If D ⊆ H ⊆ G, we will write \overline{H} for H/D, and we will use an overline to indicate a block or a central idempotent of R\overline{H} corresponding to one in RH. Thus, for example, if \overline{b} is a block of R\overline{H} with central idempotent \overline{e}, then it is to be understood that b denotes the block of RH with idempotent e corresponding to \overline{e} as in Proposition 1.4. We will use this notational convention throughout the remaining sections, except when it is necessary to consider two such quotient groups simultaneously. In that case one of the quotients is denoted G̃, and a similar convention is used for the blocks and idempotents.

§2. Results on p-Blocks

In [3] and [4] Ellers and Hill prove a theorem describing when a p-block B of G and a p-block b of a subgroup H satisfy b | B_{H×H}. In the next section we will require not only the theorem of Ellers and Hill but also an analogous result describing when B is a summand of b^{G×G}. Most of the current section is devoted to proving this proposition.

We begin by fixing a prime p ∈ π, and we let ℘ be the prime ideal of R with p ∈ ℘. Since all blocks considered in this section are p-blocks rather than π-blocks, we will work over the coefficient ring R̂_{℘} instead of R. The theorem of Ellers and Hill can now be stated as follows.

Proposition 2.1. Let G be a finite group and H a subgroup of G. Suppose b is a p-block of R̂_{℘}H and B is a p-block of R̂_{℘}G. Then b | B_{H×H} if and only if there exists a defect group P of b together with p-blocks b′ of L = P C_{H}(P) and B′ of K = P C_{G}(P) satisfying the following conditions:

(1) b′^{H} = b and B′^{G} = B.

(2) If \overline{K} = K/P, then \overline{b}′ | \overline{B}′_{\overline{L}×\overline{L}}.

Proof. When the coefficient ring is an algebraically closed field of characteristic p instead of R̂_{℘}, the proposition is a restatement of Theorem 1 of [4]. If k̂ denotes the residue field of R̂_{℘}, then the result is also true over k̂ since it is a splitting field for all sections of G × G. Because blocks are trivial source modules, Theorem II.12.4 of [6] implies that the result also holds over R̂_{℘}. □

It should be noted that the condition \overline{b}′ | \overline{B}′_{\overline{L}×\overline{L}} merely says that the canonical character θ of b′ is a constituent of χ_{L} for some character χ in B′.

We will prove the following similar result.

Proposition 2.2. Let G be a finite group and H a subgroup of G. Suppose b is a p-block of R̂_{℘}H and B is a p-block of R̂_{℘}G. Then B | b^{G×G} if and only if there exists a defect group P ⊆ H of B together with p-blocks b′ of L = P C_{H}(P) and B′ of K = P C_{G}(P) satisfying the following conditions:

(1) b′^{H} = b and B′^{G} = B.

(2) If \overline{K} = K/P, then \overline{B}′ | \overline{b}′^{\overline{K}×\overline{K}}.

Although the details of the proof are sometimes quite different, the strategy we follow is the same as that used in [3] and [4], and the ideas are essentially those of Ellers and Hill. We begin with some lemmas describing the situation when G = P C_{G}(P).

Let δ : G → G × G denote the diagonal homomorphism.

Lemma 2.3. Let P be a central p-subgroup of G. Suppose that H is a subgroup containing P, and set \overline{G} = G/P. Let B be a p-block of G with defect group P, and

let b be any p-block of H. If \overline{B} | \overline{b}^{\overline{G}×\overline{G}}, then B | b^{G×G}.

Proof. Since P is central in G, both b^{G×G} and B are (G × G)/δP -modules. By Lemma 4.3 of [3], B is the unique projective indecomposable module lying in a block of (G × G)/δP with defect group (P × P)/δP. Write b^{G×G} = U_{1} ⊕ U_{2}, where U_{1} is the summand lying in the same block of (G × G)/δP as B, and U_{2} is the summand lying in other blocks.

We begin by showing that U_{1} ≠ 0. Let χ be the irreducible character of \overline{G} lying in the block \overline{B}. Then \overline{B} affords the character \overline{χ} ⊗ χ, so it follows that \overline{χ} ⊗ χ

is an irreducible constituent of the character afforded by \overline{b}^{\overline{G}×\overline{G}}. Thus there is an irreducible character θ of \overline{H} lying in the block \overline{b} such that (\overline{θ}\overline{G} ⊗θ\overline{G}, \overline{χ}⊗χ)_{\overline{G}×\overline{G}} ≠ 0. If we regard χ as a character of G and θ as a character of H, then we conclude that (\overline{θ}^{G} ⊗ θ^{G}, \overline{χ} ⊗ χ)_{G×G} ≠ 0. Since \overline{χ} ⊗ χ is a constituent of the character of B and \overline{θ}^{G} ⊗ θ^{G} is a constituent of the character of b^{G×G}, it follows that B and b^{G×G} have a common composition factor as (G × G)-modules and hence also as (G × G)/δP -modules. Thus U_{1} ≠ 0.

Now let V be an indecomposable summand of U_{1}. Since V lies in a block with defect group (P × P)/δP, it is relatively (P × P)/δP -projective. Hence V is a summand of ((b^{G×G})_{(P ×P)/δP} )^{(G×G)/δP}. But b^{G×G} is a summand of (R̂_{℘}H)^{G×G} ≅ Ind_{δH}^{G×G} R̂_{℘}. Since (P × P)/δP acts freely on the right cosets of δH in G × G, it follows that (b^{G×G})_{(P ×P)/δP} is projective and hence so is V. Thus V ≅ B as (G × G)/δP -modules, and it follows that B | b^{G×G} as G × G-modules. □

Lemma 2.4. Let G be a group with a p-subgroup P such that G = P C_{G}(P), and let H be a subgroup of G containing P.

(1) If U is an indecomposable R̂_{℘}[G × G]-module with U | (R̂_{℘}H)^{G×G}, then U_{C_{G}(P)×C_{G}(P)} ≅ |P : Z(P)| M for some indecomposable R̂_{℘}[C_{G}(P)×C_{G}(P)]- module M.

(2) If U and V are indecomposable direct summands of (R̂_{℘}H)^{G×G} such that U_{C_{G}(P)×C_{G}(P)} ≅ V_{C_{G}(P)×C_{G}(P)}, then U ≅ V.

Proof. Since R̂_{℘}H ≅ Ind_{δH}^{H×H} R̂_{℘}, we have (R̂_{℘}H)^{G×G} ≅ Ind_{δH}^{G×G} R̂_{℘}. Let W be an indecomposable summand of Ind_{δH}^{(δP)(C_{G}(P)×C_{G}(P))} R̂_{℘}. Then W is a trivial δP -module, so W_{C_{G}(P)×C_{G}(P)} is indecomposable. Green’s Indecomposability Criterion now implies that (W^{G×G})_{G×C_{G}(P)} ≅ (W_{C_{G}(P)×C_{G}(P)})^{G×C_{G}(P)} is indecomposable and hence so is W^{G×G}. It follows that every indecomposable summand of (R̂_{℘}H)^{G×G} is of the form W^{G×G} for some such W. Moreover, Mackey’s Theorem gives

(W^{G×G})_{C_{G}(P)×C_{G}(P)} ≅ ∑_{x∈P/Z(P)} (W^{(x,1)})_{C_{G}(P)×C_{G}(P)}
≅ |P : Z(P)| W_{C_{G}(P)×C_{G}(P)},

and (1) and (2) follow. □

Proposition 2.5. Let P be a p-subgroup of G such that G = P C_{G}(P). Suppose that H is a subgroup containing P, and set \overline{G} = G/P. Let B be a p-block of G with defect group P, and let b be any p-block of H. Then B | b^{G×G} if and only if

\overline{B} | \overline{b}^{\overline{G}×\overline{G}}.

Proof. If B | b^{G×G}, then it follows easily from Proposition 1.1 that \overline{B} | \overline{b}^{\overline{G}×\overline{G}}.

Conversely, suppose that \overline{B} | \overline{b}^{\overline{G}×\overline{G}}. Let E be the primitive central idempotent of R̂_{℘}G corresponding to B. Then E ∈ R̂_{℘}C_{G}(P), so B^{∗} = (R̂_{℘}C_{G}(P))E is a p-block of C_{G}(P). The map B ↦ B^{∗} is a one-to-one correspondence between the blocks of G and those of C_{G}(P), and B^{∗} | B_{C_{G}(P)×C_{G}(P)} by Proposition 1.1. Since B has defect group P and P ⊆ H, Corollary 6 of [11] implies that B | (R̂_{℘}H)^{G×G}. Thus we see by Lemma 2.4 that B_{C_{G}(P)×C_{G}(P)} ≅ |P : Z(P)| B^{∗}. Moreover, the block B^{∗} has defect group Z(P). Similarly, the map b ↦ b^{∗} gives a one-to-one correspondence between blocks of H and blocks of C_{H}(P) such that b_{C_{H}(P)×C_{H}(P)} ≅ |P : Z(P)| b^{∗}.

Let C = C_{G}(P), and set C̃ = C_{G}(P)/Z(P) so that C̃ ≅ \overline{G}. Since \overline{B} | \overline{b}^{\overline{G}×\overline{G}}, we also have B̃^{∗} | (b̃^{∗})^{C̃×C̃}. Then Lemma 2.3 implies that B^{∗} | (b^{∗})^{C×C}, so B_{C×C} ≅ |P : Z(P)| B^{∗} is a summand of |P : Z(P)| (b^{∗})^{C×C} ≅ (b_{C_{H}(P)×C_{H}(P)})^{C×C} ≅ (b^{G×G})_{C×C}. It now follows from Lemma 2.4 that B | b^{G×G}. This completes the proof. □

We now proceed to study the situation of Proposition 2.2 for an arbitrary finite group G.

Lemma 2.6. Let G be a finite group and H a subgroup of G. Suppose b is a p-block of R̂_{℘}H and B is a p-block of R̂_{℘}G with B | b^{G×G}. Then there exists a defect group P ⊆ H of B together with p-blocks b′ of L = P C_{H}(P) and B′ of K = P C_{G}(P) such that b′^{H} = b, B′^{G} = B, and B′ | b′^{K×K}.

Proof. Since b is δH-projective and B | b^{G×G}, we know that B has a defect group P with P ⊆ H. Let B′ be any p-block of K = P C_{G}(P) with B′^{G} = B. Then B′ | (b^{G×G})_{K×K} and hence B′ | ((R̂_{℘}H)^{G×G})_{K×K}. Choose a set T of (δH, K ×K)- double coset representatives (x, y) in G × G with the property that if x^{−1}y ∈ K, then y = x. Since R̂_{℘}H ≅ Ind_{δH}^{H×H} R̂_{℘}, Mackey’s Theorem gives

((R̂_{℘}H)^{G×G})_{K×K} ≅ ⨁_{(x,y)∈T} Ind_{(δH)^{(x,y)}∩(K×K)}^{K×K} R̂_{℘}.

Choose a pair (x, y) ∈ T such that B′ is a summand of Ind_{(δH)^{(x,y)}∩(K×K)}^{K×K} R̂_{℘}, and set L_{x} = H^{x} ∩ K. Then δP is conjugate in K × K to a subgroup of (δH)^{(x,y)} ∩ (K × K), so there are elements s_{1}, s_{2} ∈ P such that (s_{1}, s_{2})^{−1}(δP)(s_{1}, s_{2}) ⊆ (x, y)^{−1}(δH)(x, y). It follows that s_{1}x^{−1}ys_{2}^{−1} ∈ C_{G}(P) ⊆ K so that x^{−1}y ∈ K, and the choice of double coset representatives gives y = x. Moreover, (δH)^{(x,x)} ∩ (K × K) = δL_{x}, so we see that P ⊆ L_{x} and B′ is a summand of Ind_{δL_{x}}^{K×K} R̂_{℘} ≅ Ind_{L_{x}×L_{x}}^{K×K}(R̂_{℘}L_{x}). Thus if S = {x ∈ G | (x, x) ∈ T and P ⊆ H^{x}}, then we can write

(1)  ((R̂_{℘}H)^{G×G})_{K×K} ≅ ⨁_{x∈S}(R̂_{℘}L_{x})^{K×K} ⊕ U,

and B′ is not a summand of U.

Let B_{x} denote the set of all p-blocks β of L_{x} such that β^{H^{x}} = b^{x}. Since {(x, x) | x ∈ S} is contained in a set of representatives for the (H ×H, K ×K)-double cosets in G × G, there is a (K × K)-module U′ such that

(b^{G×G})_{K×K} ≅ ⨁_{x∈S} ( ⨁_{β∈B_{x}} β^{K×K} ) ⊕ U′,

and (1) implies that B′ is a summand of β^{K×K} for some β occurring in this sum. If x ∈ S and b′ ∈ B_{x} is a p-block such that B′ | b′^{K×K}, then the conclusions of the lemma are satisfied by the defect group P^{x^{−1}} and the p-blocks b′^{x^{−1}} and B′^{x^{−1}}. □

Lemma 2.7. Let G be a finite group and H a subgroup of G. Suppose that B is a p-block of G with a defect group P ⊆ H, and let B′ be a p-block of K = P C_{G}(P) with B′^{G} = B. Let b be any p-block of H. If there is a p-block b′ of P C_{H}(P) such that B′ | b′^{K×K} and b′^{H} = b, then B | b^{G×G}.

Proof. Set M = N_{G}(P) and N = N_{H}(P). Let B′′ = B′^{M} and b′′ = b′^{N}. If we can show that B′′ | b′′^{M×M}, then it follows that B′′ | (b_{N×N})^{M×M} and hence B′′ | (b^{G×G})_{M×M}. But B′′ is the Green correspondent of B, so the Burry–Carlson– Puig Theorem implies that B | b^{G×G}, and we get the desired result. Thus we may assume that P ◁ G so that M = G and N = H. Moreover, if we let B̂ = B′^{HK}, then B | B̂^{G×G} by Corollary 5 of [11] so that we need only show that B̂ | b^{HK×HK}. Thus we may also assume that G = HK.

Let L = P C_{H}(P), and let T be the stabilizer of B′ in G. If B̃ denotes the sum in R̂_{℘}K of all the G-conjugates of B′, then B̃ is a (K × K)δG-module with B̃ ≅ Ind_{(K×K)δT}^{(K×K)δG} B′ = Ind_{(K×K)δ(H∩T)}^{(K×K)δH} B′. By Lemma 15.2 of [1], B = B′^{G} ≅ Ind_{(K×K)δT}^{G×G} B′ so that B ≅ Ind_{(K×K)δH}^{G×G} B̃. Similarly, if we let b̃ denote the sum in R̂_{℘}L of all the H-conjugates of b′, then b ≅ Ind_{(L×L)δH}^{H×H} b̃. Thus it suffices to show that B̃ | Ind_{(L×L)δH}^{(K×K)δH} b̃, for then B ≅ Ind_{(K×K)δH}^{G×G} B̃ is a summand of Ind_{(L×L)δH}^{G×G} b̃ ≅ b^{G×G}. We will complete the proof by showing that there are homomorphisms ϕ : Ind_{(L×L)δH}^{(K×K)δH} b̃ → B̃ and θ̃: B̃ → Ind_{(L×L)δH}^{(K×K)δH} b̃ such that ϕθ̃ = 1B̃.

Let e′ ∈ Z(R̂_{℘}L) and E′ ∈ Z(R̂_{℘}K) be the primitive idempotents corresponding to b′ and B′ respectively. Since B′ | b′^{K×K}, we know by Proposition 1.1 that there is an element z ∈ (R̂_{℘}K)^{L}e′ such that Tr_{L}^{K}(z) = E′. Because B and B′ both have defect group P, it follows that |T : K| is a p′-number. Let z̃ = \frac{1}{|T : K|} Tr_{L}^{H∩T} (z),

and let ẽ ∈ Z(R̂_{℘}L) be the idempotent with b̃ = (R̂_{℘}L)ẽ. Then z̃ ∈ (R̂_{℘}K)ẽ is (H ∩ T)-invariant, and Tr_{L}^{K}(z̃) = E′.

Let Ẽ = Tr_{T}^{G}(E′) so that B̃ = (R̂_{℘}K)Ẽ, and let T = {x_{1}, …, x_{n}} be a set of representatives for the right cosets of L in K. Then T ×T is a set of representatives for the right cosets of (L × L)δH in (K × K)δH, so any element of Ind_{(L×L)δH}^{(K×K)δH} b̃ can be written uniquely in the form ∑_{i,j} a_{ij} ⊗ (x_{i}, x_{j}) with a_{ij} ∈ b̃. Define a map ϕ : Ind_{(L×L)δH}^{(K×K)δH} b̃ → B̃ by

ϕ(∑_{i,j} a_{ij} ⊗ (x_{i}, x_{j})) = (∑_{i,j} x_{i}^{−1}a_{ij}x_{j})Ẽ.

It is easy to check that ϕ is a (K × K)δH-homomorphism.

Now let η = η_{L}^{K} : R̂_{℘}K → R̂_{℘}L be the map defined in Section 1, and note that η is an (L × L)δH-homomorphism. Let ρ : R̂_{℘}G → R̂_{℘}[G × G] be the algebra homomorphism defined on elements of G by setting ρ(g) = (1, g). Define θ : B′ → Ind_{(L×L)δH}^{(K×K)δH} b̃ by

θ(y) = ∑_{i,j} η(x_{i}yx_{j}^{−1})ẽ ⊗ ρ(z̃)(x_{i}, x_{j}) = ∑_{i,j} η(x_{i}yx_{j}^{−1}) ⊗ ρ(z̃)(x_{i}, x_{j}).

Then it is easy to check that θ is a (K × K)-homomorphism. Using the fact that T × T is a set of representatives for the right cosets of (L × L)δ(H ∩ T) in (K×K)δ(H∩T), one can show that θ is actually a (K×K)δ(H∩T)-homomorphism. Since B̃ ≅ Ind_{(K×K)δ(H∩T)}^{(K×K)δH} B′, it follows that θ extends uniquely to a (K × K)δH-

homomorphism θ̃: B̃ → Ind_{(L×L)δH}^{(K×K)δH} b̃.

Now suppose that u ∈ K. For each j there is a unique i such that x_{i}ux_{j}^{−1} ∈ L, so it follows that ∑_{i} x_{i}^{−1}η(x_{i}ux_{j}^{−1}) = ux_{j}^{−1}. Thus if y ∈ B′, then we see that

ϕθ̃(y) = ϕ(∑_{i,j} η(x_{i}yx_{j}^{−1}) ⊗ ρ(z̃)(x_{i}, x_{j}))

= (∑_{i,j} x_{i}^{−1}η(x_{i}yx_{j}^{−1})z̃x_{j})Ẽ

= y Tr_{L}^{K}(z̃)Ẽ

= y.

Since ϕθ̃ is a (K × K)δH-homomorphism, ϕθ̃ must be the identity on all of B̃. Hence B̃ | Ind_{(L×L)δH}^{(K×K)δH} b̃, and this completes the proof. □

We remark that Lemmas 2.6 and 2.7 have also been proven independently by Yun Fan [5].

Proof of Proposition 2.2. First suppose that B | b^{G×G}. Then Lemma 2.6 implies that there is a defect group P ⊆ H of B together with p-blocks b′ of L = P C_{H}(P) and B′ of K = P C_{G}(P) such that b′^{H} = b, B′^{G} = B, and B′ | b′^{K×K}. It follows

from Proposition 2.5 that \overline{B}′ | \overline{b}′^{\overline{K}×\overline{K}}, as desired.

Conversely, suppose that P ⊆ H is a defect group of B and that there are p-

blocks b′ of L and B′ of K such that b′^{H} = b, B′^{G} = B, and \overline{B}′ | \overline{b}′^{\overline{K}×\overline{K}}. Then Proposition 2.5 shows that B′ | b′^{K×K}, and Lemma 2.7 implies that B | b^{G×G}. This completes the proof. □

Our next goal is to strengthen the result of Proposition 2.5 by eliminating the requirement that the block B have defect group P. This generalization will be important in the next section.

Lemma 2.8. Let p ∈ π, and let P be a p-group with Z ⊆ P ⊆ H ⊆ G. Suppose that G = ZC_{G}(Z) = P C_{G}(\overline{P}), where \overline{G} = G/Z. Let b be a p-block of R̂_{℘}H, and let

B be a p-block of R̂_{℘}G with defect group P. If \overline{B} | \overline{b}^{\overline{G}×\overline{G}}, then B | b^{G×G}.

Proof. Set K = P C_{G}(P) and L = P C_{H}(P). Let B′ be a p-block of K with B′^{G} = B. Then B | B′^{G×G}, and it is easy to see that Proposition 1.1 implies that \overline{B} | \overline{B}′^{\overline{G}×\overline{G}}. Set G̃ = \overline{G}/\overline{P}. Then \overline{B} and \overline{B}′ in turn give p-blocks B̃ and B̃′ with B̃ | (B̃′)^{G̃×G̃}. Let χ be the unique irreducible character in B̃, and let χ′ be

the unique irreducible character in B̃′. Since \overline{B} | \overline{b}^{\overline{G}×\overline{G}}, Proposition 2.5 implies that B̃ | b̃^{G̃×G̃} so that there is an irreducible character θ in b̃ with (θ, χH̃)H̃ ≠ 0. Since (χ′G̃, χ)G̃ ≠ 0, it follows from Mackey’s Theorem that there is an x ∈ G such that θ is a constituent of ((χ′x̃)L̃)H̃. Replacing B′ by a conjugate if necessary, we may assume that x = 1. Thus there is a θ′ ∈ Irr(L̃) such that (θ′H̃, θ)H̃ ≠ 0 and

(χ′ ; L̃, θ′)L̃ ≠ 0.

We can regard θ′ as a character of \overline{L} with \overline{P} ⊆ Ker θ′ or as a character of L with P ⊆ Ker θ′. Let b′ denote the p-block of L containing θ′. If we identify the isomorphic groups \overline{K}/\overline{P} and K/P, then it follows from Proposition 2.5 that B′ | b′^{K×K}. In order to show that B | b^{G×G}, it now suffices to show that b covers b′, for then b′^{H} = b and the result follows by Lemma 2.7. But if we regard θ as a character of H, then θ lies in b and (θ_{L}, θ′)_{L} = (θL̃, θ′)L̃ ≠ 0, so we are done. □

Proposition 2.9. Let H be a subgroup of G. Suppose p ∈ π, b is a p-block of R̂_{℘}H, and B is a p-block of R̂_{℘}G. Let Z be a p-subgroup of H such that G = ZC_{G}(Z),

and let \overline{G} = G/Z. Then B | b^{G×G} if and only if \overline{B} | \overline{b}^{\overline{G}×\overline{G}}.

Proof. If B | b^{G×G}, then it follows easily from Proposition 1.1 that \overline{B} | \overline{b}^{\overline{G}×\overline{G}}.

Conversely, suppose that \overline{B} | \overline{b}^{\overline{G}×\overline{G}}. Then Lemma 2.6 gives a defect group \overline{P} ⊆ \overline{H}

of \overline{B} and p-blocks \overline{B}′ of \overline{P} C\overline{G}(\overline{P}) and \overline{b}′ of \overline{P} C\overline{H}(\overline{P}) such that \overline{B}′\overline{G} = \overline{B}, \overline{b}′\overline{H} = \overline{b},

and \overline{B}′ | \overline{b}′^{\overline{P} C\overline{G}(\overline{P})×\overline{P} C\overline{G}(\overline{P})}. Let b′ be the block of P C_{H}(\overline{P}) corresponding to \overline{b}′.

Since b′^{H} and \overline{b}′\overline{H} are defined, it follows that b′^{H} = b. Similarly, we get B′^{G} = B.

Since \overline{B}′ | \overline{b}′^{\overline{P} C\overline{G}(\overline{P})×\overline{P} C\overline{G}(\overline{P})}, Lemma 2.8 shows that B′ | b′^{P C_{G}(\overline{P})×P C_{G}(\overline{P})}. Because B and B′ have defect group P, it is now easy to see that Proposition 2.2 implies B | b^{G×G}. This completes the proof. □

Finally, we have the following analogous result for restriction. The proof is omitted since it is very similar to that of Proposition 2.9.

Proposition 2.10. Let H be a subgroup of G. Suppose p ∈ π, b is a p-block of R̂_{℘}H, and B is a p-block of R̂_{℘}G. Let Z be a p-subgroup of H such that G = ZC_{G}(Z), and let \overline{G} = G/Z. Then b | B_{H×H} if and only if \overline{b} | \overline{B}_{\overline{H}×\overline{H}}.

§3. Applications to π-Blocks

The theory of π-subpairs developed in [8] gives an implicit definition of π-block induction applying to any block b of a subgroup H satisfying DC_{G}(D) ⊆ H ⊆ N_{G}(D) for a nilpotent π-subgroup D. In this section we make this definition explicit and apply the previous results to obtain certain module-theoretic connections between the block b and the induced block b^{G}.

Suppose that H ◁ G and e is a primitive central idempotent of RH. For any x ∈ G it follows that e^{x} is also a primitive central idempotent of RH. If T denotes the stabilizer of e in G, then Tr_{T}^{G}(e) must be an idempotent of Z(RG), so it follows that there are primitive idempotents E_{1}, …, E_{n} ∈ Z(RG) with Tr_{T}^{G}(e) = E_{1} + ⋯ + E_{n}. If b = (RH)e and B_{i} = (RG)E_{i} for i = 1, …, n, then we will say that B_{i} covers b.

Note that if E is a primitive central idempotent of RG, then E is a sum of π′-elements by Lemma 1 of [7]. Moreover, if D is a normal π-subgroup of G, then E ∈ RC_{G}(D) by Theorem 9 of [7].

We now recall some of the relevant definitions and results from [8]. A π-subpair of G is defined to be a pair (D, b^{∗}), where D is a nilpotent π-subgroup of G and b^{∗} is a π-block of C_{G}(D). Since we will often find it more convenient to consider blocks of DC_{G}(D), we will let b denote the unique block of DC_{G}(D) covering b^{∗}.

Suppose that B is a π-block of G and x is a p-element for some p ∈ π. Let ℘ be the prime ideal of R with p ∈ ℘, and let E ∈ Z(RG) be the primitive idempotent corresponding to B. If Br_{⟨x⟩}^{G}(E) denotes the sum of all p-block idempotents e_{p} ∈ Z(R_{℘}C_{G}(x)) corresponding to p-blocks b_{p} with b_{p}^{G} contained in B, then Br_{⟨x⟩}^{G}(E) is actually a sum of π-block idempotents in Z(RG) by Lemma 1.3 and Theorem 1.4 of [8].

Let (D, b^{∗}) and (D_{1}, b_{1}^{∗}) be π-subpairs of G such that D is a normal subgroup of D_{1} of prime index p ∈ π, and suppose that e and e_{1} are the central idempotents corresponding to b and b_{1} respectively. Let x ∈ D_{1} be a p-element with D_{1} = D⟨x⟩. If b^{∗} is invariant under conjugation by D_{1} and Br_{⟨x⟩}^{D_{1}C_{G}(D)}(e)e_{1} = e_{1}, then we write (D, b^{∗}) ◁ (D_{1}, b_{1}^{∗}). If (D, b^{∗}) ◁ (D_{1}, b_{1}^{∗}) ◁ ⋯ ◁ (D_{n}, b_{n}^{∗}), then we write (D, b^{∗}) ⊆ (D_{n}, b_{n}^{∗}). If (1, B) ⊆ (D, b^{∗}), then b^{∗} is said to be a B-subpair; if (D, b^{∗}) is a maximal B-subpair, then D is said to be a defect group of B.

The following proposition summarizes the results of [8] that we will need.

Proposition 3.1. Let (D, b^{∗}) be a π-subpair of G, and let D̃ be a nilpotent πsubgroup of G.

(1) If D̃ ⊆ D, then there is a unique π-subpair (D̃, b̃^{∗}) with (D̃, b̃^{∗}) ⊆ (D, b^{∗}).

(2) Suppose that D is a p-subgroup for some p ∈ π and (D̃, b̃^{∗}) is a π-subpair with D̃ ⊆ D. Then (D̃, b̃^{∗}) ⊆ (D, b^{∗}) if and only if there are p-blocks b̃_{p} ⊆ b̃ and b_{p} ⊆ b such that (D̃, b̃_{p}^{∗}) ⊆ (D, b_{p}^{∗}) as p-subpairs.

(3) Let H be a subgroup such that D̃C_{G}(D̃) ⊆ H. Then (D̃, b̃^{∗}) is a π-subpair of G if and only if it is a π-subpair of H.

(4) Let H be a subgroup such that DC_{G}(D) ⊆ H and D̃C_{G}(D̃) ⊆ H. Then (D̃, b̃^{∗}) ⊆ (D, b^{∗}) in G if and only if (D̃, b̃^{∗}) ⊆ (D, b^{∗}) in H.

(5) If G = D̃C_{G}(D̃), then D̃ is contained in every defect group of every block of G.

Proof. See Theorem 1.6, Corollary 1.7, Remark 1.8, and Proposition 1.14 of [8]. □

Proposition 3.2. Let D_{1} and D_{2} be nilpotent π-subgroups of G, and let H be a subgroup such that D_{i}C_{G}(D_{i}) ⊆ H ⊆ N_{G}(D_{i}) for i = 1, 2. Let β be a block of H, and let b_{i} be a block of D_{i}C_{G}(D_{i}) covered by β. Then there is a block B of G such that (1, B) ⊆ (D_{i}, b_{i}^{∗}) for i = 1, 2.

Proof. If D_{1} ⊆ D_{2}, then C_{G}(D_{2}) ◁ C_{G}(D_{1}) ◁ H, and β covers both b_{1}^{∗} and b_{2}^{∗}. Hence some H-conjugate of b_{1}^{∗} covers b_{2}^{∗}. Replacing b_{1} by a conjugate if necessary, we may assume that b_{1}^{∗} covers b_{2}^{∗}. But then it is easy to see that (D_{1}, b_{1}^{∗}) ⊆ (D_{2}, b_{2}^{∗}), and the result follows.

Now assume that D_{1} ⊈ D_{2}, and set D̃ = D_{1}D_{2}. Then D̃ is nilpotent, and D̃C_{G}(D̃) ⊆ H ⊆ N_{G}(D̃). Let b̃ be a block of D̃C_{G}(D̃) covered by β, and let B be the block of G with (1, B) ⊆ (D̃, b̃^{∗}). Since D_{1} ⊆ D̃ and D_{2} ⊆ D̃, it follows from the previous paragraph that (1, B) ⊆ (D_{i}, b_{i}^{∗}) for i = 1, 2. This completes the proof. □

Let H be a subgroup of G and β a block of H. Suppose that there is a nilpotent π-subgroup D such that DC_{G}(D) ⊆ H ⊆ N_{G}(D). Choose a block b of DC_{G}(D) covered by β, and let B be the block of G with (1, B) ⊆ (D, b^{∗}). Define β^{G} = B. Then the previous proposition shows that this definition is independent of the choice of the subgroup D and the block b.

Our first goal is to prove Theorem 3.8 and Proposition 3.9, which describe two situations in which the induced block B satisfies β | B_{H×H}. We begin by considering the case in which D ◁ G and H = DC_{G}(D).

Proposition 3.3. Suppose that D is a normal π-subgroup of G and H = DC_{G}(D). Let e be a primitive idempotent of Z(RH), and let T be the stabilizer of e in G. Then E = Tr_{T}^{G}(e) is a primitive idempotent in Z(RG), and every primitive idempotent of Z(RG) arises in this way. Moreover, if b = (RH)e and B = (RG)E, then B is the unique block of RG with b | B_{H×H}.

Proof. We begin by showing that there is a one-to-one correspondence between primitive idempotents of Z(RG) and G-conjugacy classes of primitive idempotents of Z(RH). Let E be any primitive idempotent of Z(RG). Then E ∈ Z(RH), so there are primitive idempotents e_{1}, …, e_{n} of Z(RH) with E = e_{1} + ⋯ + e_{n}. Let x ∈ G. Because E = E^{x}, it follows that e_{1}^{x} = e_{i} for some i. If T_{1} denotes the stabilizer of e_{1} in G, then E′ = Tr_{T_{1}}^{G}(e_{1}) is an idempotent of Z(RG). But E is primitive and E′E = E′, so it follows that E = E′. Thus {e_{1}, …, e_{n}} is a single G-conjugacy class of idempotents.

Conversely, suppose that {e_{1}, …, e_{n}} is a conjugacy class of primitive idempotents of Z(RH). If T_{1} is the stabilizer of e_{1} in G, then E = e_{1}+⋯+e_{n} = Tr_{T_{1}}^{G}(e_{1}) is an idempotent of Z(RG). Moreover, E must be primitive since we have just seen that all primitive idempotents of Z(RG) have this form. Thus we get the desired correspondence.

Now let η = η_{H}^{G}, and let e ∈ Z(RH) be the primitive idempotent corresponding to the block b = (RH)e. Then there is a unique primitive idempotent E ∈ Z(RG) with Ee = e, and e ∈ (RG)^{H}E satisfies η(e) = e. Thus if B = (RG)E, then b | B_{H×H} by Proposition 1.1.

To prove the uniqueness, suppose that E′ ∈ Z(RG) is a primitive idempotent such that if B′ = (RG)E′, then b | B′_{H×H}. By Proposition 1.1 there is an element z ∈ (RG)^{H}E′ such that η(z) = e. Then e = η(z)e = η(ze), so ze ≠ 0 and hence E′e ≠ 0. It follows that E′ = E, as desired. □

Lemma 3.4. Suppose that p ∈ π and P is a p-subgroup of G. Let H = P C_{G}(P), and let b be a π-block of H. Set B = b^{G}, and let ℘ be the prime ideal of R with p ∈ ℘. Then R̂_{℘} ⊗ b | R̂_{℘} ⊗ B_{H×H}.

Proof. Let b_{p} be a p-block of H with b_{p} | R̂_{℘} ⊗ b. If B_{p} = b_{p}^{G}, then (1, B_{p}) ⊆ (P, b_{p}^{∗}) and hence B_{p} is contained in B by Proposition 3.1. Thus b_{p} | R̂_{℘} ⊗ B_{H×H}, and the result follows. □

Lemma 3.5. Let D be a nilpotent π-subgroup of G, and let Z be a subgroup of D with G = ZC_{G}(Z). Set \overline{G} = G/Z, and suppose that H is a subgroup with

DC_{G}(\overline{D}) ⊆ H ⊆ N_{G}(D). If b is any π-block of H, then \overline{b}\overline{G} = \overline{b^{G}}.

Proof. By induction on the number of prime divisors of |Z|, it suffices to consider the case in which Z is a p-group for some p ∈ π. Let b′ be any block of DC_{G}(D) covered by b, and let β = b′^{DC_{G}(\overline{D})}. Suppose that e and ε are the primitive central idempotents corresponding to b and β respectively. Then Proposition 3.3 implies

that εe = ε, and it follows that \overline{b} covers \overline{β}. If we can show that \overline{β}\overline{G} = \overline{β^{G}}, then

\overline{b^{G}} = \overline{b′^{G}} = \overline{β^{G}} = \overline{β}\overline{G} = \overline{b}\overline{G}, as desired. Hence we may assume that H = DC_{G}(\overline{D}) so that b = β.

If D is a p-group, then the desired conclusion follows easily from the corresponding result for p-blocks. Write D = P × Q, where P is a Sylow p-subgroup of D,

and let L = P C_{G}(\overline{P}). If L < G, then it follows by induction that \overline{b^{L}} = \overline{b}\overline{L}, and it is easy to see that

\overline{b}\overline{G} = (\overline{b}\overline{L})\overline{G} = (\overline{b^{L}})\overline{G} = \overline{(b^{L})^{G}} = \overline{b^{G}}.

Thus we may assume that L = G. Then H = DC_{G}(\overline{D}) = QC_{G}(Q), so we may also assume by induction on the number of prime divisors of |Q| that Q is a q-group for some prime q ∈ π − {p}.

Set B = b^{G}, and let ℘ be the prime ideal of R with q ∈ ℘. Then R̂_{℘} ⊗ b | R̂_{℘} ⊗ B_{H×H} by Lemma 3.4, and it is easy to see that R̂_{℘} ⊗ \overline{b} | R̂_{℘} ⊗ \overline{B}_{\overline{H}×\overline{H}} by Proposition 1.1. Since q-block induction from \overline{H} to \overline{G} is defined for all q-blocks, it

follows that \overline{b}\overline{G} = \overline{B}, and this completes the proof. □

Lemma 3.6. Let p, q ∈ π with p ≠ q. Suppose that P is a p-subgroup of G and Q is a q-subgroup of C_{G}(P). Let B be a block of P C_{G}(P) and b a block of P QC_{G}(P Q). Then (1, B) ⊆ (Q, b^{∗}) in P C_{G}(P) if and only if (P, B^{∗}) ⊆ (P Q, b^{∗}) in G.

Proof. Note that (P, B^{∗}) ⊆ (P Q, b^{∗}) in G if and only if (1, B) ⊆ (P, B^{∗}) ⊆ (P Q, b^{∗}) in P C_{G}(P). But this condition and the condition that (1, B) ⊆ (Q, b^{∗}) in P C_{G}(P) merely assert that b^{P C_{G}(P)} = B, so the desired result follows. □

Lemma 3.7. Suppose that p ∈ π and P is a p-subgroup of G. Let H = P C_{G}(P), and let b be a π-block of H. Then B = b^{G} satisfies b | B_{H×H}.

Proof. We will prove that b | B_{H×H} by induction on |G|. Let q ∈ π − {p}, and suppose that ℘ is the prime ideal of R with q ∈ ℘. Let b_{q} be a q-block of H with b_{q} | R̂_{℘} ⊗ b. It suffices by Proposition 1.3 and Lemma 3.4 to show that b_{q} | R̂_{℘} ⊗ B_{H×H}. Let Q be a q-defect group of b_{q}, and let b′_{q} be a q-block of L = QC_{H}(Q) with (b′_{q})^{H} = b_{q}. Suppose that b′ is the block of L with b′_{q} | R̂_{℘} ⊗ b′. Because (1, b_{q}) ⊆ (Q, b′_{q}^{∗}) in H, it follows by Proposition 3.1 that (1, b) ⊆ (Q, b′^{∗}) in H. Then (1, B) ⊆ (P, b^{∗}) ⊆ (P Q, b′^{∗}) in G by Lemma 3.6, so there is a block B′ of K = QC_{G}(Q) with (1, B) ⊆ (Q, B′^{∗}) ⊆ (P Q, b′^{∗}). Since R̂_{℘} ⊗ B′ | R̂_{℘} ⊗ B_{K×K} by Lemma 3.4, it suffices by Proposition 2.1 to show that b′_{q} | R̂_{℘} ⊗ B′_{L×L}.

Since (Q, B′^{∗}) ⊆ (P Q, b′^{∗}) in G, it follows by Lemma 3.6 that (1, B′) ⊆ (P, b′^{∗}) in K. If K < G, then we conclude by induction that b′ | B′_{L×L}, and we are finished. Thus we may assume that K = G. Then L = H, b′ = b, and B′ = B. Set \overline{G} = G/Q. Since (1, B) ⊆ (P, b^{∗}) in G, we have (1, \overline{B}) ⊆ (\overline{P} , \overline{b}^{∗}) in \overline{G} by Lemma 3.5. If |\overline{G}| < |G|, then by induction it follows that \overline{b} | \overline{B}_{\overline{H}×\overline{H}}. Hence \overline{b}_{q} | R̂_{℘} ⊗ \overline{B}_{\overline{H}×\overline{H}}, and b_{q} | R̂_{℘} ⊗ B_{H×H} by Proposition 2.1.

Finally, if |\overline{G}| = |G|, then Q = 1 so that there is a unique irreducible character θ lying in b_{q}. Since θ lies in some p-block b_{p} with b_{p}^{G} contained in B, it follows that there is a character χ in B with (χ_{H}, θ) ≠ 0. Hence the character \overline{θ} ⊗ θ afforded by b_{q} is a constituent of the character afforded by B_{H×H}, and it follows that b_{q} | R̂_{℘} ⊗ B_{H×H}. This completes the proof. □

Theorem 3.8. Let D be a nilpotent π-subgroup of G, and let b be a π-block of L = DC_{G}(D). Then b^{G} is the unique block of G satisfying the following conditions:

(1) b | (b^{G})_{L×L}.

(2) If p ∈ π and P is a Sylow p-subgroup of D, then (b^{P C_{G}(P)})^{G} = b^{G}.

Proof. Set B = b^{G}. We begin by verifying that B satisfies (1). If L = G, then B = b and the result is obvious, so assume that L < G. Then there is a prime p ∈ π and a Sylow p-subgroup P of D such that P C_{G}(P) < G. Set H = P C_{G}(P) and B′ = b^{H}. It follows by induction on |G| that b | B′_{L×L}, and we know that B′ | B_{H×H} by Lemma 3.7. Hence b | B_{L×L}, as desired.

It is easy to see that block induction satisfies the required transitivity condition and that the conditions (1) and (2) characterize the block b^{G} uniquely. □

Proposition 3.9. Assume that G has nilpotent Hall π-subgroups. Let D be a πsubgroup of G, and let H be a subgroup with DC_{G}(D) ⊆ H ⊆ N_{G}(D). Suppose that b is a π-block of H, and set B = b^{G}. Then b | B_{H×H}.

Proof. We begin by proving the theorem when D is a p-group for some p ∈ π. Let ℘ be a prime ideal of R. We will show that R̂_{℘} ⊗ b | R̂_{℘} ⊗ B_{H×H}. Since p-block induction is defined for p-blocks of DC_{G}(D) or of H, this result follows easily if p ∈ ℘. Moreover, we also see that if θ is any character in b, then there is a character χ in B with (θ^{G}, χ) ≠ 0.

Now suppose that q ∈ ℘ for some q ∈ π − {p}, and let b_{q} be a q-block that is a summand of R̂_{℘} ⊗ b. We must show that b_{q} | R̂_{℘} ⊗ B_{H×H}. Let Q be a defect group of b_{q}. Since G has nilpotent Hall π-subgroups, it follows that Q ⊆ C_{G}(D), and DC_{G}(D) ∩ QC_{G}(Q) = DQC_{G}(DQ). Suppose that b′_{q} is a q-block of QC_{H}(Q) with (b′_{q})^{H} = b_{q}. If b′ is the π-block of QC_{H}(Q) with b′_{q} | R̂_{℘}⊗b′, then we have (b′)^{H} = b. Now let β′ be a block of DQC_{G}(DQ) covered by b′, and let β = β′^{DC_{G}(D)}. Then b = (b′)^{H} = (β′^{QC_{H}(Q)})^{H} = β′^{H} = (β′^{DC_{G}(D)})^{H} = β^{H}. Hence b covers β, and it follows that B = β^{G}.

Let B′ = β′^{QC_{G}(Q)}. Then (B′)^{G} = (β′)^{G} = (β′^{DC_{G}(D)})^{G} = β^{G} = B. If QC_{G}(Q) < G, then it follows by induction that b′ | B′_{QC_{H}(Q)×QC_{H}(Q)} and hence b′_{q} | R̂_{℘} ⊗ B′_{QC_{H}(Q)×QC_{H}(Q)}. By Proposition 2.1 we conclude that b_{q} | R̂_{℘} ⊗ B_{H×H}. Suppose, then, that QC_{G}(Q) = G, and let θ be the canonical character in b_{q}. Since there is a character χ in B with (θ^{G}, χ) ≠ 0, we again conclude from Proposition 2.1 that b_{q} | R̂_{℘} ⊗ B_{H×H}. Then R̂_{℘} ⊗ b | R̂_{℘} ⊗ B_{H×H} for all ℘, and hence b | B_{H×H}. Thus we are done if D is a p-group.

Now suppose that D is any π-subgroup of G, and let β be a block of DC_{G}(D) covered by b. Assume that there is a prime p ∈ π for which D has a Sylow psubgroup P with HC_{G}(P) < G, and let b′ = b^{HC_{G}(P)}. Because HC_{G}(P) has nilpotent Hall π-subgroups, we conclude by induction that b | b′_{H×H}. If β′ = β^{P C_{G}(P)}, then b′ covers β′ and B = (β′)^{G}. Since P C_{G}(P) ⊆ HC_{G}(P) ⊆ N_{G}(P), we conclude by the previous paragraph that b′ | B_{HC_{G}(P)×HC_{G}(P)}. Hence b | B_{H×H}, and we are done in this case.

Thus we may assume that HC_{G}(P) = G for every such P. Then D is normal in G, so B is the unique block of G covering β. Let e and E be the primitive central idempotents corresponding to b and B respectively. Then Proposition 3.3 implies that Ee = e, and e ∈ (RG)^{H}E satisfies η_{H}^{G}(e) = e. Hence b | B_{H×H}, and this completes the proof. □

The remainder of this section is devoted to proving Theorem 3.12, an analogue of the proposition that if B is a p-block of G with defect group P and b is a p-block of P C_{G}(P) with b^{G} = B, then B | b^{G×G}. The corresponding statement for π-blocks is slightly more complicated, however, because the defect groups of a π-block do not necessarily form a single conjugacy class (Example 1.13 of [8]).

The use of subpairs in what follows will often lead us to consider subgroups of the form H = D_{1}C_{G}(D), where D_{1} is nilpotent and D is a normal subgroup of D_{1} of index p ∈ π. Suppose that P and P_{1} are Sylow p-subgroups of D and D_{1} respectively. Then it will be useful to observe that D_{1}C_{G}(D_{1}) = P_{1}C_{H}(P_{1}) and DC_{G}(D) = P C_{H}(P). Indeed, suppose that Q is a Hall p′-subgroup of D so that D = P × Q and D_{1} = P_{1} × Q. Then we have

C_{G}(P_{1}) ∩ P_{1}C_{G}(D) = C_{G}(P_{1}) ∩ P_{1}(C_{G}(P) ∩ C_{G}(Q))
= C_{G}(P_{1}) ∩ P_{1}C_{G}(P) ∩ C_{G}(Q)
= C_{G}(D_{1}) ∩ P_{1}C_{G}(P)
= C_{G}(D_{1}).

Hence

P_{1}C_{H}(P_{1}) = P_{1}(C_{G}(P_{1}) ∩ D_{1}C_{G}(D)) = D_{1}(C_{G}(P_{1}) ∩ P_{1}C_{G}(D)) = D_{1}C_{G}(D_{1}).

Similarly, DC_{G}(D) = P C_{H}(P).

One can easily show that (D, b^{∗}) ◁ (D_{1}, b_{1}^{∗}) if and only if D ◁ D_{1}, the index |D_{1} : D| is a prime in π, the block b^{∗} is D_{1}-invariant, and b^{D_{1}C_{G}(D)} = b_{1}^{D_{1}C_{G}(D)}. In fact, the condition that b^{∗} be D_{1}-invariant is unnecessary, as the next lemma shows.

Lemma 3.10. Suppose that (D, b^{∗}) and (D_{1}, b_{1}^{∗}) are π-subpairs such that D ◁ D_{1}, the index |D_{1} : D| is a prime p ∈ π, and b^{D_{1}C_{G}(D)} = b_{1}^{D_{1}C_{G}(D)}. Then (D, b^{∗}) ◁ (D_{1}, b_{1}^{∗}).

Proof. We must show that the block b is D_{1}-invariant. Set H = D_{1}C_{G}(D), and let P and P_{1} be the Sylow p-subgroups of D and D_{1} respectively. Suppose that ℘ is the prime ideal of R with p ∈ ℘. Let b_{1,p} be a p-block of P_{1}C_{H}(P_{1}) = D_{1}C_{G}(D_{1}) with b_{1,p} | R̂_{℘} ⊗ b_{1}, and let B_{p} = b_{1,p}^{H}. Then P_{1} is contained in a p-defect group of B_{p}, so there is a p-block β_{p} of P C_{H}(P) = DC_{G}(D) such that β_{p}^{H} = B_{p}. Since |H : P C_{H}(P)| is either 1 or p, the block β_{p} is H-invariant if and only if it is P_{1}- invariant. Suppose that it is not H-invariant. Then a defect group of β_{p} is also a defect group of B_{p}, and it follows that P_{1} ⊆ DC_{G}(D). Hence H = DC_{G}(D), and we have a contradiction. Thus β_{p} is H-invariant.

Now let B and β be the π-blocks such that B_{p} | R̂_{℘} ⊗ B and β_{p} | R̂_{℘} ⊗ β. Then b_{1}^{H} = B = β^{H}, and β must be H-invariant since β_{p} is. But b is also a π-block of DC_{G}(D) such that b^{H} = B. Since D ◁ H, it follows that b is conjugate to β in H and hence b = β. Thus b is D_{1}-invariant, as desired. □

Proposition 3.11. Suppose that Z is a nilpotent π-subgroup of G such that G = ZC_{G}(Z), and set \overline{G} = G/Z. Let B be any π-block of G, and suppose that S_{G}(B) = {(D_{i}, b_{i}^{∗}) | i ∈ I} is a set of representatives for the conjugacy classes of maximal B-subpairs in G. Let β_{i} = b_{i}^{D_{i}C_{G}(\overline{D}_{i})}. Then S\overline{G}(\overline{B}) = \{(\overline{D}_{i}, \overline{β}_{i}^{∗}) | i ∈ I\} is a set of representatives for the conjugacy classes of maximal \overline{B}-subpairs in \overline{G}. In particular, suppose that D is a nilpotent π-subgroup of G with Z ⊆ D. Then D is a defect group of B if and only if \overline{D} is a defect group of \overline{B}.

Proof. First note that Z ⊆ D_{i} for all i by Proposition 3.1. Thus the second assertion follows immediately from the first.

If (D_{i}, b_{i}^{∗}) is a B-subpair, then Lemma 3.5 implies that (\overline{D}_{i}, \overline{β}_{i}^{∗}) is a \overline{B}-subpair. Hence it is only necessary to show that if (D, b^{∗}) is a B-subpair with Z ⊆ D and β = b^{DC_{G}(\overline{D})}, then (D, b^{∗}) is maximal in G if and only if (\overline{D}, \overline{β}^{∗}) is maximal in \overline{G}.

Suppose first that (D, b^{∗}) ◁ (D_{1}, b_{1}^{∗}). Set H = D_{1}C_{G}(\overline{D}) and β_{1} = b_{1}^{D_{1}C_{G}(\overline{D}_{1})}, and let B_{1} = b_{1}^{H} = β_{1}^{H}. Since H contains D_{1}C_{G}(D_{1}) and DC_{G}(D), it follows that (D, b^{∗}) ⊆ (D_{1}, b_{1}^{∗}) in H, and (D, b^{∗}) must be a B_{1}-subpair of H because (D_{1}, b_{1}^{∗})

is. Thus β^{H} = B_{1} = β_{1}^{H} so that \overline{β}\overline{H} = \overline{β}\overline{H} ; 1 by Lemma 3.5, and (\overline{D}, \overline{β}^{∗}) ◁ (\overline{D}_{1}, \overline{β}_{1}^{∗}) by Lemma 3.10.

Conversely, suppose that (D, b^{∗}) is a B-subpair with Z ⊆ D such that the corresponding \overline{B}-subpair (\overline{D}, \overline{β}^{∗}) is not maximal in \overline{G}. We must show that (D, b^{∗}) is not maximal in G. Let (\overline{D}_{1}, \overline{β}_{1}^{∗}) be a \overline{B}-subpair such that (\overline{D}, \overline{β}^{∗}) ◁ (\overline{D}_{1}, \overline{β}_{1}^{∗})

and D ⊆ D_{1}. Set H = D_{1}C_{G}(\overline{D}), and let B_{1} = b^{H} = β^{H}. Then \overline{B}_{1} = \overline{β}\overline{H} = \overline{β}\overline{H} ; 1, so we must have β_{1}^{H} = B_{1}. Let b_{1} be a π-block of D_{1}C_{G}(D_{1}) covered by β_{1} so that b_{1}^{D_{1}C_{G}(\overline{D}_{1})} = β_{1} and b_{1}^{H} = B_{1}. Since D ⊆ D_{1}, there is a unique π-subpair (D, b̃^{∗}) of D_{1}C_{G}(D) such that (D, b̃^{∗}) ⊆ (D_{1}, b_{1}^{∗}) in D_{1}C_{G}(D). Moreover, we also have (D, b̃^{∗}) ⊆ (D_{1}, b_{1}^{∗}) in H and in G because D_{1}C_{G}(D) contains both D_{1}C_{G}(D_{1}) and DC_{G}(D). Since (D_{1}, b_{1}^{∗}) is a B_{1}-subpair of H, so is (D, b̃^{∗}). Thus b^{H} = b̃^{H}, and the fact that D ◁ H implies that b is conjugate to b̃ in H. Hence the subpairs (D, b̃^{∗}) and (D, b^{∗}) are conjugate. Since (D, b̃^{∗}) is not maximal in G, neither is (D, b^{∗}). This completes the proof. □

Theorem 3.12. Suppose that G is a group with nilpotent Hall π-subgroups. Let B be a block of G, and let S_{G}(B) = {(D_{i}, b_{i}^{∗}) | i ∈ I} be a set of representatives for the conjugacy classes of maximal B-subpairs of G. Then B | ⨁_{i∈I} b_{i}^{G×G}.

Proof. We prove the result by induction on |G|. Let p ∈ π, and let ℘ be the prime ideal of R with p ∈ ℘. Suppose B_{p} is a p-block with B_{p} | R̂_{℘} ⊗ B. Then it suffices by Proposition 1.3 to show that B_{p} | ⨁_{i∈I}(R̂_{℘} ⊗ b_{i}^{G×G}). Let P be a p-defect group of B_{p}.

First suppose that P ≠ 1. Let b′_{p} be a p-block of P C_{G}(P) with b′_{p}^{G} = B_{p}, and let b′ be the π-block with b′_{p} | R̂_{℘} ⊗ b′. Since (P, b′_{p}^{∗}) is a B_{p}-subpair, we know by Proposition 3.1 that (P, b′^{∗}) is a B-subpair, and B_{p} | R̂_{℘} ⊗ b′^{G×G}. Now choose a B-subpair (D, b^{∗}) such that (P, b′^{∗}) ⊆ (D, b^{∗}), the p-block B_{p} is a summand of R̂_{℘} ⊗ b^{G×G}, and |D| is as large as possible. We will show that (D, b^{∗}) is a maximal B-subpair so that it is conjugate to (D_{i}, b_{i}^{∗}) for some i ∈ I.

Suppose, on the contrary, that there is a subpair (D̃, b̃^{∗}) with (D, b^{∗}) ◁ (D̃, b̃^{∗}). Then there is an element x ∈ D̃ of prime-power order such that D̃ = D⟨x⟩, and x stabilizes the primitive central idempotent e of RDC_{G}(D) corresponding to b. Hence e is also a primitive central idempotent of H = D̃C_{G}(D). Let E_{p} ∈ Z(R̂_{℘}G) be the primitive idempotent corresponding to B_{p}. Since B_{p} | R̂_{℘} ⊗ b^{G×G}, it follows from Proposition 1.1 that there is an element z ∈ (R̂_{℘}G)^{DC_{G}(D)}e such that E_{p} = Tr_{DC_{G}(D)}^{G}(z). Set z′ = Tr_{DC_{G}(D)}^{H}(z) and β = b^{H}. Then z′ ∈ (R̂_{℘}G)^{H}e and Tr_{H}^{G}(z′) = E_{p}, so it follows that B_{p} | R̂_{℘} ⊗ β^{G×G}. Let β_{p} be a p-block such that β_{p} | R̂_{℘} ⊗ β and B_{p} | β_{p}^{G×G}.

Let P_{0} be a Sylow p-subgroup of D. If x is a p-element, then it must centralize some nonidentity subgroup Z ⊆ Z(P_{0}). If x is not a p-element, then x centralizes Z(P_{0}) since D̃ is nilpotent. Thus in either case we get a nontrivial p-subgroup Z ⊆ Z(H). Set \overline{H} = H/Z, and let S_{H}(β) = \{(Δ_{j}, δ_{j}^{∗}) | j ∈ J\} be a set of representatives

for the conjugacy classes of maximal β-subpairs of H. Let γ_{j} = δ_{j}^{Δ_{j}C_{H}(\overline{Δ}_{j})}. Then by Proposition 3.11 the set S\overline{H}(\overline{β}) = \{(\overline{Δ}_{j}, \overline{γ}_{j}^{∗}) | j ∈ J\} is a set of representatives for the conjugacy classes of maximal \overline{β}-subpairs of \overline{H}. Since |\overline{H}| < |G| and \overline{H} has nilpotent Hall π-subgroups, we conclude by induction that \overline{β} | ⨁_{j∈J} \overline{γ}_{j}^{\overline{H}×\overline{H}}. In

particular, there is a j ∈ J such that \overline{β}_{p} | R̂_{℘} ⊗ \overline{γ}_{j}^{\overline{H}×\overline{H}}, and Proposition 2.9 implies that β_{p} | R̂_{℘} ⊗ γ_{j}^{H×H}.

Let L = Δ_{j}C_{H}(\overline{Δ}_{j}). Write Δ_{j} = P_{j} × Q_{j}, where P_{j} is a Sylow p-subgroup of Δ_{j}. Then

C_{L}(P_{j}) = C_{H}(P_{j}) ∩ Δ_{j}C_{H}(\overline{Δ}_{j})
= C_{H}(P_{j}) ∩ Δ_{j}(C_{H}(\overline{P}_{j}) ∩ C_{H}(Q_{j}))
= C_{H}(P_{j}) ∩ Q_{j}(P_{j}C_{H}(\overline{P}_{j}) ∩ C_{H}(Q_{j}))
= Q_{j}(C_{H}(P_{j}) ∩ P_{j}C_{H}(\overline{P}_{j}) ∩ C_{H}(Q_{j}))
= Q_{j}(C_{H}(Δ_{j}) ∩ P_{j}C_{H}(\overline{P}_{j}))
⊆ Δ_{j}C_{H}(Δ_{j}),

so it follows that Δ_{j}C_{H}(Δ_{j}) = P_{j}C_{L}(P_{j}). Since (Δ_{j}, δ_{j}^{∗}) is a maximal β-subpair of H, it must also be a maximal γ_{j}-subpair of L. Because L has nilpotent Hall π-subgroups, every p-block that is a summand of R̂_{℘} ⊗ δ_{j} has an inertial index in L that is not divisible by p. Thus R̂_{℘} ⊗ γ_{j} | R̂_{℘} ⊗ δ_{j}^{L×L}, and it follows that B_{p} | R̂_{℘} ⊗ δ_{j}^{G×G}.

Set (D̂, b̂^{∗}) = (Δ_{j}, δ_{j}^{∗}) so that (D̂, b̂^{∗}) is a maximal β-subpair of H with B_{p} | R̂_{℘} ⊗ b̂^{G×G}. We claim that D ⊆ D̂. Let ê be the primitive central idempotent of RD̂C_{H}(D̂) corresponding to b̂. Then ê is a sum of π′-elements in C_{H}(D̂). Since all π′-elements of H lie in C_{G}(D), we see that ê is a sum of elements in C_{H}(D̂) ∩ C_{G}(D) = C_{H}(DD̂). If ê_{1} is a primitive idempotent of Z(RDD̂C_{H}(DD̂)) with êê_{1} = ê_{1} and b̂_{1} is the corresponding π-block, then we see that (D̂, b̂^{∗}) ⊆ (DD̂, b̂_{1}^{∗}) in H. Since (D̂, b̂^{∗}) is maximal, it follows that D ⊆ D̂.

By Proposition 3.1 there is a unique subpair (D, b′′^{∗}) of H such that (D, b′′^{∗}) ⊆ (D̂, b̂^{∗}), and (D, b′′^{∗}) must be a β-subpair. Since e is the primitive central idempotent corresponding to both b and β, it follows that b is the unique block of DC_{G}(D) such that b^{H} = β, and hence b′′ = b. Moreover, since DC_{G}(D) ⊆ H and D̂C_{G}(D̂) ⊆ H, the inclusion of subpairs (D, b^{∗}) ⊆ (D̂, b̂^{∗}) in H implies that the same inclusion holds in G. By the choice of the pair (D, b^{∗}), it now follows that (D, b^{∗}) = (D̂, b̂^{∗}). Thus (D, b^{∗}) is maximal in H, and this is a contradiction. Hence (D, b^{∗}) is maximal in G, as desired.

Now consider the case in which P = 1, and let χ be the unique irreducible character lying in the block B_{p}. Assume that there is a prime q ∈ π − {p} such that χ lies in a q-block B_{q} of positive q-defect, and let ℘′ be the prime ideal of R with q ∈ ℘′. Then the previous argument shows that B_{q} | R̂_{℘′} ⊗ b_{i}^{G×G} for some i ∈ I. Hence there is a character θ in b_{i} such that (χ, θ^{G})_{G} ≠ 0, and it follows that B_{p} | R̂_{℘} ⊗ b_{i}^{G×G}.

Finally, if P = 1 and there is no such prime q, then there is a unique irreducible character in B and S_{G}(B) = {(1, B)} so that the theorem is trivial in this case. This completes the proof. □

In the situation described by Theorem 3.12, it would be quite interesting to know whether there are blocks B satisfying the condition that B | ⨁_{j∈J} b_{j}^{G×G} for a proper subset J of I.

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