EXTENDED BLOCK INDUCTION

Wayne W. Wheeler Mathematics, I.A.S. Australian National University GPO Box 4 Canberra, ACT 2601

21 July 1992

Introduction

Two quite different definitions of block induction have been proposed and used in modular representation theory. The better-known definition is that given by Brauer in terms of central characters of the group algebra. Alperin and Burry, on the other hand, have considered a module-theoretic definition in [2]. But these two notions do not quite agree: if H is a subgroup of the group G and b is a block of H, then it is possible for the induced block b^{G} to be defined in only one of the two senses. Even so, it is known that the two definitions do agree in any situation where b^{G} is defined in both senses [9]. The purpose of this paper is to describe and study a single general way to define block induction that encompasses both the Brauer definition and the Alperin–Burry definition as special cases.

Throughout this paper let p be a prime, and let R be the ring of integers in a p-adic number field of characteristic zero. Fix a finite group G and a subgroup H of G, and assume that R contains a primitive |G|^{th} root of unity. We denote the residue field of R by R^{∗}. If x ∈ R, then x^{∗} denotes the image of x in R^{∗}; if x ∈ RG, then x^{∗} denotes the image of x in R^{∗}G. Finally, we let b be a block of H.

Recall that RG is an R[G × G]-module with the action α · (x, y) = x^{−1}αy for all α ∈ RG and x, y ∈ G. The blocks of RG are then the indecomposable summands of RG as an R[G×G]-module. The Alperin–Burry definition is that b^{G} is defined and b^{G} = B if B is the unique block of G such that b is a summand of the restriction B_{H×H}.

If O is a commutative ring, let Br_{H}^{G} : OG → OH be the O[H×H]-homomorphism defined on elements of G by setting

Br_{H}^{G}(x) = \{ \begin{matrix}x & if x ∈ H \\ 0 & if x ∈ G − H. \end{matrix}

Let ω_{b} : Z(R^{∗}H) → R^{∗} be the central character corresponding to the block b, and let ω_{b}^{G} : Z(R^{∗}G) → R^{∗} be the R^{∗}-linear function defined by setting ω_{b}^{G} = ω_{b} ∘ Br_{H}^{G}. Brauer’s definition of block induction states that b^{G} is defined and b^{G} = B if ω_{b}^{G} is an algebra homomorphism and ω_{b}^{G} = ω_{B}.

For any block B of a subgroup K of G, let e_{B} denote the corresponding idempotent of Z(RK), and let rank B denote the rank of B as a free R-module. In Section 1 we will prove the following result.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20.

Proposition 1. Suppose that b^{G} is defined either in the sense of Brauer or in the sense of Alperin–Burry, and set B = b^{G}. Then B is the unique block of G such that ω_{b}^{G}(e_{B}^{∗}) ≠ 0.

Proposition 1 suggests the following definition.

Definition 2. Suppose that there is a unique block B of G such that ω_{b}^{G}(e_{B}^{∗}) ≠ 0. Then we say that the induced block b^{G} is defined, and we set b^{G} = B. We will refer to the map b ↦ b^{G} as extended block induction.

In considering the question of whether b^{G} is defined, we will always be referring to this extended sense, unless we explicitly specify that b^{G} is defined in the sense of Brauer or in the sense of Alperin–Burry. Proposition 1.4 below gives a number of equivalent ways of describing extended block induction.

One problem studied by Blau in [3] is to find conditions under which block induction is defined in the sense of Brauer for a block of a normal subgroup. We consider the analogous problem for extended block induction in Section 2, where we prove the following result.

Theorem 3. Suppose that H ◁ G and b is a block of H with defect group D. Let b′ be a block of DC_{H}(D) with b′^{H} = b, and let T be the stabilizer of b′ in N_{G}(D). Then b^{G} is defined if and only if there is a unique T -conjugacy class of blocks B′ of DC_{T} (D) satisfying the following conditions:

(1) The size of the conjugacy class of B′ is not divisible by p.

(2) B′ covers b′.

(3) (rank B′)_{p} = |DC_{T} (D)|_{p}|DC_{H}(D)|_{p}/|D|.

Moreover, if these conditions are satisfied, then b^{G} = B′^{G}.

As an easy consequence we prove the following result on the behavior of the principal block.

Corollary 4. Suppose that H ◁ G and b_{0} is the principal block of H. If b_{0}^{G} is defined, then b_{0}^{G} is the principal block of G.

It is perhaps surprising that Corollary 4 is false without the assumption that H is normal in G. In Section 2 we give a counterexample due to L. Scott.

Finally, we let Irr(B) denote the set of all irreducible characters associated with the block B. If N ◁ G and ϕ is an irreducible character of N, then Irr(B|ϕ) denotes the set of characters in Irr(B) lying over ϕ. Other notation generally follows [7] and is intended to be standard.

§1. Basic Results on Extended Block Induction

In this section we will prove Proposition 1 and set down the elementary properties of extended block induction that will be needed to study the case in which H ◁ G in Section 2. Particularly useful is Proposition 1.4, which gives a number of equivalent ways to determine whether b^{G} is defined. We begin with an easy proposition about block induction in the sense of Brauer.

Proposition 1.1. Suppose that b^{G} is defined in the sense of Brauer and b^{G} = B. Then B is the unique block of G such that ω_{b}^{G}(e_{B}^{∗}) ≠ 0.

Proof. Let B′ be any block of G. Then ω_{b}^{G}(e_{B′}^{∗}) = ω_{B}(e_{B′}^{∗}), so ω_{b}^{G}(e_{B′}^{∗}) ≠ 0 if and only if B′ = B. □

Next we observe that there is always some block B of G such that ω_{b}^{G}(e_{B}^{∗}) ≠ 0. Indeed, let B_{1}, …, B_{n} denote all the blocks of G. Then

ω_{b}^{G}(e_{B_{1}}^{∗}) + ⋯ + ω_{b}^{G}(e_{B_{n}}^{∗}) = ω_{b}^{G}(1) = ω_{b}(1) = 1,

so ω_{b}^{G}(e_{B_{i}}^{∗}) ≠ 0 for some i. Moreover, if B is a block of G such that 0 ≠ ω_{b}^{G}(e_{B}^{∗}) = ω_{b}(Br_{H}^{G}(e_{B}^{∗})), then it follows that Br_{H}^{G}(e_{B})e_{b} ∉ J(Z(b)) and hence b | B_{H×H} by Theorem 1 of [10]. Thus we have proven the following result.

Proposition 1.2. Suppose that b^{G} is defined in the sense of Alperin–Burry and b^{G} = B. Then B is the unique block of G such that ω_{b}^{G}(e_{B}^{∗}) ≠ 0.

Proposition 1 now follows immediately from Propositions 1.1 and 1.2. In particular, these results show that whenever b^{G} is defined under one of the previous definitions of block induction, then extended block induction is also defined and agrees with the previous notion. Moreover, we have seen that b | B_{H×H} whenever ω_{b}^{G}(e_{B}^{∗}) ≠ 0, so it follows that a defect group of b is always contained in a defect group of b^{G}.

It is possible for extended block induction to be defined even when block induction is not defined in either of the other two senses. For example, take p = 2, G = S_{3}, and H = 1. Then b = RH is the only block of H, and every block B of G satisfies b | B_{H×H}. Since G has two blocks, b^{G} is not defined in the sense of Alperin–Burry. By (3F) of [4] or by direct computation, we see that it is also not defined in the sense of Brauer. But it is easy to check that extended block induction is defined, and b^{G} is the principal block of G.

Let ϕ ∈ Irr(H), and let B be a block of G. We recall that ϕ^{B} is defined by the equation

ϕ^{B} = ∑_{χ∈Irr(B)}(ϕ^{G}, χ) χ.

The next result is essentially proven in (9.4) of [7], although it is not explicitly stated in this form. It can also be obtained as an easy consequence of Lemma V.1.1 of [6].

Lemma 1.3. Let B be a block of G, and let ϕ be an irreducible character in b. Then

ω_{b}^{G}(e_{B}^{∗}) = [\frac{ϕ^{B}(1)}{ϕ^{G}(1)}]^{∗} .

Proof. Since

e_{B} = ∑_{χ∈Irr(B)} \frac{χ(1)}{|G|} ∑_{g∈G} χ(g^{−1}) g,

we have

Br_{H}^{G}(e_{B})e_{ϕ} = ∑_{χ∈Irr(B)} \frac{χ(1)ϕ(1)}{|G| |H|} ∑_{h,x∈H} χ(h^{−1})ϕ(x^{−1}) hx

= ∑_{χ∈Irr(B)} \frac{χ(1)ϕ(1)}{|G| |H|} ∑_{h,x∈H} χ(h^{−1})ϕ(x^{−1}h) x

= ∑_{χ∈Irr(B)} \frac{χ(1)}{|G|} ∑_{x∈H} ∑_{θ∈Irr(H)}(χ_{H}, θ) [\frac{ϕ(1)}{|H|} ∑_{h∈H} ϕ(x^{−1}h)θ(h^{−1})] x

= ∑_{χ∈Irr(B)} \frac{χ(1)}{|G|} ∑_{x∈H}(χ_{H}, ϕ)ϕ(x^{−1}) x

= ∑_{χ∈Irr(B)} \frac{(ϕ^{G}, χ)χ(1)}{|G : H| ϕ(1)} e_{ϕ}

= \frac{ϕ^{B}(1)}{ϕ^{G}(1)} e_{ϕ}.

Thus

ω_{b}^{G}(e_{B}^{∗}) = ω_{ϕ}(Br_{H}^{G}(e_{B})e_{b})^{∗} = ∑_{θ∈Irr(b)} ω_{ϕ} (\frac{θ^{B}(1)}{θ^{G}(1)} e_{θ})^{∗} = [\frac{ϕ^{B}(1)}{ϕ^{G}(1)}]^{∗} ,

and this completes the proof. □

Let I(G) denote the subalgebra of Z(R^{∗}G) spanned by the idempotents in Z(R^{∗}G).

Proposition 1.4. The following statements are equivalent:

(1) b^{G} is defined and b^{G} = B.

(2) There is a unique block B of G such that Br_{H}^{G}(e_{B})e_{b} does not lie in the radical J(Z(b)) of Z(b).

(3) There is a unique block B of G such that the composite of the R[H × H]- homomorphisms ξ : b → B_{H×H} and η : B_{H×H} → b given by ξ(x) = xe_{B} and η(y) = Br_{H}^{G}(y)e_{b} is an automorphism of b.

(4) Let ϕ be an irreducible character in b. Then B is the unique block of G such that ϕ^{B}(1)_{p} = ϕ^{G}(1)_{p}.

(5) ω_{b}^{G} = ω_{B} on I(G).

Proof. It is clear that (1) and (2) are equivalent. In (3) we have ηξ(x) = η(xe_{B}) = x Br_{H}^{G}(e_{B})e_{b}, so (2) and (3) are also equivalent. The equivalence of (1) and (4) follows immediately from Lemma 1.3.

Finally, we show that (1) and (5) are equivalent. Let B = B_{1}, …, B_{n} denote all the blocks of G. If (1) holds, then ω_{b}^{G}(e_{B_{1}}^{∗}) = 1 and ω_{b}^{G}(e_{B_{i}}^{∗}) = 0 for i > 1. Thus

ω_{b}^{G}((s_{1}e_{B_{1}}^{∗} + … + s_{n}e_{B_{n}}^{∗})(t_{1}e_{B_{1}}^{∗} + ⋯ + t_{n}e_{B_{n}}^{∗}))
= ω_{b}^{G}(s_{1}t_{1}e_{B_{1}}^{∗} + ⋯ + s_{n}t_{n}e_{B_{n}}^{∗})
= s_{1}t_{1}
= ω_{b}^{G}(s_{1}e_{B_{1}}^{∗} + ⋯ + s_{n}e_{B_{n}}^{∗})ω_{b}^{G}(t_{1}e_{B_{1}}^{∗} + ⋯ + t_{n}e_{B_{n}}^{∗}),

and (5) follows. Conversely, if (5) holds, then ω_{b}^{G}(e_{B}^{∗}) ≠ 0. Moreover, if i > 1, then 0 = ω_{b}^{G}(e_{B}^{∗}e_{B_{i}}^{∗}) = ω_{b}^{G}(e_{B}^{∗})ω_{b}^{G}(e_{B_{i}}^{∗}), so we must have ω_{b}^{G}(e_{B_{i}}^{∗}) = 0. Thus (1) holds, and this completes the proof. □

We end this section with a proposition that generalizes a result of Brauer on the transitivity of block induction.

Proposition 1.5. Suppose H ⊆ K ⊆ G and b is a block of H such that b^{K} is defined in the sense of Brauer. Then b^{G} is defined if and only if (b^{K})^{G} is defined, and in this case (b^{K})^{G} = b^{G}.

Proof. Let β = b^{K}, and let B be any block of G. Then

ω_{β}^{G}(e_{B}^{∗}) = ω_{β}(Br_{K}^{G}(e_{B}^{∗})) = ω_{b}^{K}(Br_{K}^{G}(e_{B}^{∗})) = ω_{b}(Br_{H}^{G}(e_{B}^{∗})) = ω_{b}^{G}(e_{B}^{∗}),

and the desired result follows. □

§2. Normal Subgroups

Throughout this section we will assume that H ◁ G, and we will consider the behavior of extended block induction from H. Our main goal is to prove Theorem 3 and Corollary 4. We begin with a number of preliminary lemmas that will allow us to reduce the proof to considering a special case.

If C is any subset of G, let C^{+} denote the sum of all elements in C. Recall that a block B of G is said to be weakly regular with respect to the normal subgroup H if there is a G-conjugacy class C ⊆ H such that ω_{B}(C^{+}) ≠ 0 and the defect of C is equal to the defect of B. The basic theory of weakly regular blocks appears in [6].

Proposition 2.1. Suppose that B is a block of G such that ω_{b}^{G}(e_{B}^{∗}) ≠ 0. Then B is a weakly regular block covering b.

Proof. Since ω_{b}^{G}(e_{B}^{∗}) ≠ 0, we know that b | B_{H×H} so that B covers b. Hence it suffices to show that B is weakly regular.

Because B covers b and Br_{H}^{G}(e_{B}^{∗}) ∈ Z(R^{∗}G) ∩ Z(R^{∗}H), we have ω_{B}(Br_{H}^{G}(e_{B}^{∗})) = ω_{b}(Br_{H}^{G}(e_{B}^{∗})) ≠ 0. It follows that there is a conjugacy class C of G such that C ⊆ H, C^{+} occurs with nonzero coefficient in e_{B}^{∗}, and ω_{B}(C^{+}) ≠ 0. Thus a defect group of C is also a defect group of B, and B is weakly regular. □

The following result is an immediate consequence of Proposition 2.1.

Corollary 2.2. If there is a unique weakly regular block B covering b, then b^{G} is defined and b^{G} = B.

Now suppose that B is a block of G covering b, and let P be a defect group of B such that P ∩H is a defect group of b. Since H ◁ G, we have N_{G}(P) ⊆ N_{G}(P ∩H). Thus there is a unique block B′ of N_{G}(P ∩ H) with defect group P such that B′^{G} = B. Moreover, if b′ is the block of N_{H}(P ∩ H) such that b′^{H} = b, then we know that B′ covers b′ (See, for example, [1] or [8]).

Lemma 2.3. Let B be a block of G covering b, and let P be a defect group of B such that D = P ∩H is a defect group of b. Let B′ be the block of N_{G}(D)with defect group P such that B′^{G} = B, and let b′ be the block of N_{H}(D) with b′^{H} = b. Then b^{G} = B if and only if b′^{N_{G}(D)} = B′.

Proof. If B_{1} is any block of G, then Br_{N_{G}(D)}^{G}(e_{B_{1}}^{∗}) is the sum of all block idempotents e_{b_{1}}^{∗} corresponding to blocks b_{1} with b_{1}^{G} = B_{1}. Moreover,

ω_{b}^{G}(e_{B_{1}}^{∗}) = ω_{b}(Br_{H}^{G}(e_{B_{1}}^{∗})) = ω_{b′}^{H}(Br_{H}^{G}(e_{B_{1}}^{∗})) = ω_{b′}^{G}(e_{B_{1}}^{∗}) = ω_{b′}^{N_{G}(D)}(Br_{N_{G}(D)}^{G}(e_{B_{1}}^{∗})).

If b′^{N_{G}(D)} = B′, then it follows that ω_{b}^{G}(e_{B_{1}}^{∗}) = 0 for B_{1} ≠ B, and hence b^{G} = B.

Conversely, suppose b^{G} = B. If B̃ is any block of N_{G}(D) such that B̃^{G} = B and ω_{b′}^{N_{G}(D)}(e∗ ; B̃) ≠ 0, then Proposition 2.1 shows that B̃ is a weakly regular block covering b′. By Theorem V.3.12 of [6] a defect group P̃ of B̃ contains a defect group of every block of N_{G}(D) covering b′. In particular, since B′ covers b′, it follows that P is conjugate to a subgroup of P̃ in N_{G}(D). But B̃^{G} = B, so P must actually be conjugate to P̃ and hence B̃ = B′. By Proposition 1.5 we have B = b^{G} = (b′^{H})^{G} = b′^{G}, so it follows that

0 ≠ ω_{b′}^{G}(e_{B}^{∗}) = ω_{b′}^{N_{G}(D)}(Br_{N_{G}(D)}^{G}(e_{B}^{∗})) = ω_{b′}^{N_{G}(D)}(e_{B′}^{∗}).

In particular, B′ is weakly regular with respect to N_{H}(D). Thus Theorem V.3.12 of [6] shows that any weakly regular block of N_{G}(D) covering b′ must have defect group P.

Let B′_{1} be a block of N_{G}(D) such that ω_{b′}^{N_{G}(D)}(e_{B′_{1}}^{∗}) ≠ 0, and let B_{1} = (B′_{1})^{G}. Then B′_{1} and B_{1} have defect group P. Any block B̃_{1} of N_{G}(D) such that B̃_{1}^{G} = B_{1} and ω_{b′}^{N_{G}(D)}(e∗ ; B̃_{1}) ≠ 0 must also have defect group P, so B̃_{1} = B′_{1}. Thus

0 ≠ ω_{b′}^{N_{G}(D)}(e_{B′_{1}}^{∗}) = ω_{b′}^{N_{G}(D)}(Br_{N_{G}(D)}^{G}(e_{B_{1}}^{∗})) = ω_{b′}^{G}(e_{B_{1}}^{∗}).

Since b′^{G} = B, we conclude that B_{1} = B and B′_{1} = B′. Hence b′^{N_{G}(D)} = B′, and this completes the proof. □

Lemma 2.4. Let ϕ be an irreducible character in the block b, and set T = I_{G}(ϕ). If B is any block of G, then

\frac{ϕ^{B}(1)}{ϕ^{G}(1)} = \frac{1}{|G : H| |G : T | ϕ(1)^{2}} ∑_{χ∈Irr(B|ϕ)} χ(1)^{2}.

Proof. If B does not cover b, then both sides of the above equation are zero. Suppose, therefore, that B is any block of G covering b. If χ ∈ Irr(B|ϕ), then there is an integer e such that χ_{H} = e ∑_{x∈T \G} ϕ^{x}, and

(ϕ^{G}, χ) = (ϕ, χ_{H}) = e = \frac{χ(1)}{|G : T | ϕ(1)}.

Hence

\frac{ϕ^{B}(1)}{ϕ^{G}(1)} = \frac{1}{|G : H| ϕ(1)} ∑_{χ∈Irr(B|ϕ)}(ϕ^{G}, χ)χ(1) = \frac{1}{|G : H| |G : T | ϕ(1)^{2}} ∑_{χ∈Irr(B|ϕ)} χ(1)^{2},

as desired. □

Next we collect some information about the case in which G contains a psubgroup D such that G = DC_{G}(D). Set \overline{G} = G/D. Then it is well-known that the canonical map RG → R\overline{G} induces a one-to-one correspondence between the blocks of G and those of \overline{G}. For any block B of G we always let \overline{B} denote the corresponding block of \overline{G}. Lemma 2.5. Suppose that D is a p-subgroup of H such that G = DC_{G}(D), and set \overline{G} = G/D. If b has defect group D, then for any block B of G we have ω_{b}^{G}(e_{B}^{∗}) =

ω\overline{G} ; \overline{b}(e∗ ; \overline{B}).

Proof. Since G = DC_{G}(D) and D ⊆ H, it follows that H = DC_{H}(D). Let ϕ be the canonical character of b, and set T = I_{G}(ϕ). Since D ⊆ Ker ϕ, any irreducible character of B that lies over ϕ is actually a character of \overline{B}. Thus by Lemma 2.4 we get

\frac{ϕ^{B}(1)}{ϕ^{G}(1)} = \frac{1}{|G : H| |G : T | ϕ(1)^{2}} ∑_{χ∈Irr(B|ϕ)} χ(1)^{2}

= \frac{1}{|\overline{G} : \overline{H}| |\overline{G} : \overline{T} | ϕ(1)^{2}} ∑_{χ∈Irr(\overline{B}|ϕ)} χ(1)^{2}

= \frac{ϕ\overline{B}(1)}{ϕ\overline{G}(1)},

and the result follows by Lemma 1.3. □ Lemma 2.6. Suppose B is a block of G with defect group D. Then (rank B)_{p} = |G : D|_{p}|G|_{p}. Proof. See Theorem 1 of [5]. □ Lemma 2.7. Suppose that D is a p-subgroup of G such that G = DC_{G}(D), and set \overline{G} = G/D. If B is any block of G, then (rank B)_{p} = |D|(rank \overline{B})_{p}. Proof. Let P be a defect group of B. Then D ⊆ P, and by Lemma 2.6 we have

(rank B)_{p} = |G : P |_{p}|G|_{p} = |\overline{G} : \overline{P} |_{p}|\overline{G}|_{p}|D| = |D|(rank \overline{B})_{p},

as desired. □

The following well-known result relates blocks of G covering b to blocks of the stabilizer I(b) covering b.

Lemma 2.8. The map sending B′ to B′^{G} defines a one-to-one correspondence between the set of all blocks of I(b) covering b and the set of all blocks of G covering b. Moreover, the map θ ↦ θ^{G} defines a one-to-one correspondence between Irr(B′) and Irr(B′^{G}), and B′ and B′^{G} have a defect group in common.

Proof. See Theorem V.2.5 of [6]. □

In the situation described by Lemma 2.8 there is an irreducible character θ in B′ such that θ^{G} is irreducible, so B′^{G} is actually defined in the sense of Brauer.

Lemma 2.9. Suppose that b is a block of H, B′ is a block of I(b) covering b, and B = B′^{G}. Then b^{G} = B if and only if b^{I(b)} = B′.

Proof. Let ϕ be an irreducible character in b, and set T = I(ϕ). Then T ⊆ I(b), and by Lemmas 2.4 and 2.8 we have

\frac{ϕ^{B′}(1)}{ϕ^{I(b)}(1)} = \frac{1}{|I(b) : H| |I(b) : T | ϕ(1)^{2}} ∑_{θ∈Irr(B′|ϕ)} θ(1)^{2}

= \frac{1}{|G : H| |G : T | ϕ(1)^{2}} ∑_{θ∈Irr(B′|ϕ)} θ^{G}(1)^{2}

= \frac{1}{|G : H| |G : T | ϕ(1)^{2}} ∑_{χ∈Irr(B|ϕ)} χ(1)^{2}

= \frac{ϕ^{B}(1)}{ϕ^{G}(1)}.

It now follows from Lemma 1.3 that b^{G} = B if and only if b^{I(b)} = B′. □

We are now ready to consider Theorem 3. Note that under the assumptions of the theorem the blocks B′^{T}, B′^{N_{G}(D)} = (B′^{T})^{N_{G}(D)}, and B′^{G} = (B′^{N_{G}(D)})^{G} are all defined in the sense of Brauer.

Proof of Theorem 3. We know that if B is a block of G that does not cover b, then ω_{b}^{G}(e_{B}^{∗}) = 0. Choose a block B of G covering b, and let P be a defect group of B such that P ∩ H is a defect group of b. We may assume without loss of generality that P ∩ H = D. Let B̃ be the unique block of N_{G}(D) with defect group P such that B̃^{G} = B, and let b̃ be the unique block of N_{H}(D) with b̃^{H} = b. Then Lemma 2.3 implies that b^{G} = B if and only if b̃^{N_{G}(D)} = B̃. Hence we may assume that D ◁ G. By Proposition 1.5 we know that b^{G} is defined if and only if b′^{G} is defined, so we may also assume that H = DC_{H}(D) and hence b′ = b. Finally, under these assumptions Lemma 2.9 shows that b^{G} is defined if and only if b^{T} is defined, so we may suppose in addition that b is G-invariant. For the remainder of the proof, then, we assume that G = T and H = DC_{H}(D).

Let K = DC_{G}(D). Since K is normal in G, there is a unique conjugacy class of blocks B_{1}, …, B_{n} of K covered by B, and B_{i}^{G} = B for all i. Then

ω_{b}^{G}(e_{B}^{∗}) = ω_{b}^{K}(Br_{K}^{G}(e_{B}^{∗})) = ∑_{i=1}^{n} ω_{b}^{K}(e_{B_{i}}^{∗}).

Let ϕ denote the canonical character of b. Since ϕ is G-invariant and B_{1}, …, B_{n} are conjugate in G, it follows that ϕ^{B_{i}}(1) = ϕ^{B_{1}}(1) for all i. Let \overline{K} = K/D. Then Lemmas 1.3 and 2.5 give

ω_{b}^{G}(e_{B}^{∗}) = nω_{b}^{K}(e_{B_{1}}^{∗}) = nω\overline{K} ; \overline{b}(e_{\overline{B}_{1}}^{∗}).

Since B covers b, we know that B_{1} must also cover b and hence \overline{B}_{1} covers \overline{b}. But ϕ is the unique irreducible character in \overline{b}, so it follows that every character in \overline{B}_{1} lies over ϕ. Thus Lemma 2.4 gives

\frac{ϕ^{\overline{B}^{1}}(1)}{ϕ\overline{K}(1)} = \frac{rank \overline{B}_{1}}{|\overline{K} : \overline{H}|ϕ(1)^{2}} ,

and

ω_{b}^{G}(e_{B}^{∗}) = |G : I(B_{1})| · \frac{rank \overline{B}_{1}}{|K : H| ϕ(1)^{2}} .

Hence by Lemma 2.7 we see that ω_{b}^{G}(e_{B}^{∗}) ≠ 0 if and only if |G : I(B_{1})| is a p′-number and

(rank B_{1})_{p} = |D|(rank \overline{B}_{1})_{p}

= |D||K : H|_{p}ϕ(1)_{p}^{2}

= |D||DC_{G}(D) : H|_{p}|H : D|_{p}^{2}

= \frac{|DC_{G}(D)|_{p}|H|_{p}}{|D|} .

Finally, if b^{G} is defined and ω_{b}^{G}(e_{B}^{∗}) ≠ 0, then b^{G} = B = B_{1}^{G}. This completes the proof. □

Proof of Corollary 4. If we take b = b_{0} in Theorem 3, then D is a Sylow p-subgroup of H, b′ is the principal block of DC_{H}(D), and T = N_{G}(D). The principal block B′_{0} of DC_{G}(D) clearly satisfies conditions (1) and (2). Moreover, by Lemma 2.6 we have

(rank B′_{0})_{p} = |DC_{G}(D)|_{p} = \frac{|DC_{T} (D)|_{p}|DC_{H}(D)|_{p}}{|D|} ,

so it also satisfies (3). Thus b_{0}^{G} = B′_{0}^{G} is the principal block of G, as desired. □

Finally, we conclude with an example of L. Scott showing that Corollary 4 is false without the assumption that H is normal in G.

Example 2.10. Let G = GL(3, 2) be the simple group of order 168, and let p = 3. Take H to be a Sylow 2-subgroup of G. The group G has a unique irreducible character χ of degree 8 and a unique irreducible character θ of degree 6. Moreover, χ lies in the principal block B_{0} of G, whereas θ lies in a block B_{1} of defect zero. It is easy to check that (1_{H})^{G} = 1_{G} +χ+2θ, so Proposition 1.4 implies that b_{0}^{G} = B_{1}. In particular, b_{0}^{G} is not the principal block.

Acknowledgements

Portions of the research in this paper were conducted while I was visiting the University of Warwick and the University of Virginia, and I would like to thank those two institutions for their hospitality. I am indebted to Professor J. L. Alperin for suggesting that there might be a definition of block induction that simultaneously generalizes both of the previous definitions. Finally, I would like to thank Professor Leonard Scott for a number of interesting and helpful conversations and for providing me with Example 2.10 above.

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