A CHARACTERISATION OF NILPOTENT BLOCKS

RADHA KESSAR, MARKUS LINCKELMANN, AND GABRIEL NAVARRO

Abstract. Let B be a p-block of a finite group, and set m = ∑ χ(1)^{2}, the sum taken over all height zero characters of B. Motivated by a result of M. Isaacs characterising p-nilpotent finite groups in terms of character degrees, we show that B is nilpotent if and only if the exact power of p dividing m is equal to the p-part of |G : P |^{2}|P : R|, where P is a defect group of B and where R is the focal subgroup of P with respect to a fusion system F of B on P. The proof involves the hyperfocal subalgebra D of a source algebra of B. We conjecture that all ordinary irreducible characters of D have degree prime to p if and only if the F-hyperfocal subgroup of P is abelian.

1. Introduction

Let p be a prime number and let (K, O, k) be a p-modular system. We assume that k is algebraically closed and that K is a splitting field for all finite groups considered in the paper. Let G be a finite group and B a block algebra of OG. Let F = F_{(P,e_{P})}(G, B) be the fusion system of B with respect to a maximal B-Brauer pair (P, e_{P}). The focal subgroup of F is the subgroup 𝔣𝔬𝔠(F) of P generated by the commutators [Aut_{F}(Q), Q], where Q runs over the subgroups of P (see [1, I, Def. 7.1]). Let Irr(B) denote the set of K-valued irreducible characters of B and let Irr_{0}(B) denote the subset of height zero characters. Let i ∈ B^{P} be a source idempotent of B, and let S be a Sylow p-subgroup of G containing P.

Theorem 1.1. With the notation above, the following are equivalent.

(i) | ∑_{χ∈Irr_{0}(B)} χ(1)^{2}|_{p} = |S : P |^{2}|P : 𝔣𝔬𝔠(F)|.

(ii) | ∑_{χ∈Irr_{0}(B)} χ(i)^{2}|_{p} = |P : 𝔣𝔬𝔠(F)|.

(iii) | Irr_{0}(B)| = |P : 𝔣𝔬𝔠(F)|.

(iv) B is nilpotent.

Theorem 1.1 may be seen as a block-theoretic analogue of Isaacs’ result [9, Lemma 4] characterizing p-nilpotent groups via character degrees. The fact that statement (iv) in this theorem implies any of (i), (ii), (iii) is an immediate consequence of Puig’s structure theorem [16, Theorem 1.6] for source algebras of nilpotent blocks. It is known that in general the quantity on the right hand side of (i) or (ii) always divides the quantity on the left hand side of (i) or (ii), respectively. This is a consequence of the free action of P/𝔣𝔬𝔠(F) on the set of height zero characters via the ∗-construction (see [17, Introduction], [18]). The proof of Theorem 1.1 relies on the Clifford theoretic relationship between the representation theory of the source algebra iBi and its hyperfocal subalgebra D (a concept due to Puig; see [17, Theorem 1.8]). This relationship suggests the following ‘hyperfocal height zero’ version of Brauer’s height zero conjecture. Denote by Irr(D) a set of representatives of isomorphism classes of simple K ⊗_{O} D-modules and by Irr_{0}(D) the subset corresponding to simple modules of p′-degree. Let 𝔥𝔶𝔭(F) denote the hyperfocal subgroup of F. The focal subgroup 𝔣𝔬𝔠(F) is the product 𝔥𝔶𝔭(F)P′ of the hyperfocal subgroup of F and the derived subgroup P′ of P; see [1, Lemma 7.2].

2010 Mathematics Subject Classification. Primary 20C20; Secondary.

Key words and phrases. nilpotent block, height zero, hyperfocal subalgebra.

Conjecture 1.2. With the notation above, assume that K is a splitting field of D. Then, Irr(D) = Irr_{0}(D) if and only if 𝔥𝔶𝔭(F) is abelian.

Section 2 contains Clifford theoretic considerations regarding hyperfocal and source algebras. Theorem 1.1 is proved in Section 3. The last section contains some remarks around Conjecture 1.2.

Remark 1.3. For χ ∈ Irr(B) denote by e_{χ} the corresponding central primitive idempotent in Z(K ⊗_{O} B). If e is an idempotent in Z(K ⊗_{O} B), then multiplication by e induces a surjective O-algebra homomorphism B → Be; note that Be is an O-free quotient algebra of B and an O-subalgebra of K ⊗_{O} B but not necessarily a subalgebra of B. The integer ∑_{χ∈Irr_{0}(B)} χ(1)^{2} is equal to the O-rank of the quotient algebra Be of B, where e = ∑_{χ∈Irr_{0}(B)} e_{χ}. If B is nilpotent, then this algebra is Morita equivalent to the commutative symmetric algebra OP/P′, where P′ is the derived subgroup of P. Indeed, through a Morita equivalence between B and OP, the height zero characters in B correspond to the characters with degree one of P, and the intersection of the kernels of these is P′. Okuyama and Tsushima proved in [14] that B is Morita equivalent to a commutative (and necessarily symmetric) algebra if and only if B is nilpotent with abelian defect groups. In a similar spirit one may ask whether there is a characterisation of blocks whose quotient Be as defined above is Morita equivalent to a commutative symmetric O-algebra.

Remark 1.4. If |P : 𝔣𝔬𝔠(F)|·|S : P |^{2} is the exact power of p dividing ∑_{χ∈Irr_{0}(B)} χ(1)^{2}, then the above theorem implies that B is nilpotent, hence 𝔣𝔬𝔠(F) is the derived subgroup P′ of P. If B is not nilpotent, then the highest power of p dividing ∑_{χ∈Irr_{0}(B)} χ(1)^{2} is strictly bigger than |P : 𝔣𝔬𝔠(P)| · |S : P |^{2}, and one might wonder whether it is always at least |P : P′| · |S : P |^{2}. This is, however, not the case. Let p = 3. The group H = A_{4} ×C_{4} has a faithful irreducible 𝔽_{3}H-module V of dimension 6. Let G = V H be the corresponding semidirect product. The group algebra OG has a unique block. If P is a Sylow-3-subgroup of G, then |P : P′| = 27. This does not divide the sum of the squares of the 3′-degree characters, which is equal to 1548 = 2^{2} · 3^{2} · 43.

2. Extending characters of the hyperfocal subalgebra

Background material on focal and hyperfocal subgroups of fusion systems as well as the fusion subsystem O^{p}(F) of the fusion system F on P, can be found in [3, §7.5]. We refer to [17] for the notion and basic properties of hyperfocal subalgebras of source algebras of blocks. We show in this section that ordinary irreducible characters of the hyperfocal subalgebra of degree prime to p extend to the source algebra in precisely |P : 𝔣𝔬𝔠(F)| ways. The key ingredient is a special case of a result of Diaz, Glesser, Park and Stancu [4] which extends to fusion systems work of Gagola and Isaacs [6] on the transfer homomorphism. For a saturated fusion system G on a finite p-group S, we denote by τ_{S,Ω}^{G} : S → S/[S, S] the transfer map with respect to some characteristic element Ω for F (see [4, Definition 2.5]), and by T_{G} the subgroup of S containing [S, S] such that T_{G}/[S, S] = Im(τ_{S,Ω}^{G}). By [4, Lemma 2.6], T_{G} is independent of the choice of Ω. Further, if U is a subgroup of S containing 𝔥𝔶𝔭(G), we denote by G_{U}, the unique saturated subsytem of G on U of p-power index (see [1, Theorem I.7.4]).

Proposition 2.1. [4, Proposition 5.3] Let G be a saturated fusion system on a finite p-group S and let U be a normal subgroup of S containing 𝔥𝔶𝔭(G). If V is a subgroup of U containing T_{G_{U}} [U, S], then S/V ≅ U/V × T_{G}V/V.

Corollary 2.2. Let G be a saturated fusion system on a finite p-group S. Then 𝔥𝔶𝔭(G)/[𝔥𝔶𝔭(G), S] is a direct factor of S/[𝔥𝔶𝔭(G), S].

Proof. Let U = 𝔥𝔶𝔭(G). By the minimality of G_{U} as normal saturated subsytem of G of p-power index, we have that 𝔣𝔬𝔠(G_{U}) = 𝔥𝔶𝔭(G_{U}) = U (see [3, Theorem 7.53] and [1, Lemma I.7.2]). By Proposition 2.7 of [4] and the first line of its proof we have that

U/[U, U] ≅ 𝔣𝔬𝔠(G_{U})/[U, U] × T_{G_{U}} /[U, U].

Since 𝔣𝔬𝔠(G_{U}) = U, it follows that T_{G_{U}} = [U, U]. The corollary follows from Proposition 2.1 applied with V = [U, S]. □

Keep the previous notation. In addition, set Q = 𝔥𝔶𝔭(F) and R = 𝔣𝔬𝔠(F), and set A = iBi, the source algebra of B corresponding to the defect group P of B and the source idempotent i. By [17, Theorem 1.8] the hyperfocal subalgebra D is the unique (up to (A^{P})^{×}-conjugation) P -stable unitary subalgebra of A such that

D ∩ P i = Qi and A = D ⊗_{OQ} OP = ⊕_{u∈P/Q}Du.

In particular, D is a Q-interior P -algebra. We denote by Â = K⊗_{O}A and D̂ = K⊗_{O}A the K-algebras obtained by extensions of scalars to K. The K-algebras Â and D̂ are semi-simple; indeed, for Â this follows from the Morita equivalence between A and B, and for D̂ this is a consequence of the fact that J(D̂)Â is an ideal in Â, hence nilpotent, hence zero. By replacing K by a suitable finite extension, we may and will assume that Â and D̂ are also split. We denote by Irr(A) (respectively (Irr(D)) a set of representatives of isomorphism classes of simple Â-modules (respectively D̂-modules).

The Q-interior P -algebra structure of D allows the usual notions of Clifford theory to be carried over to the inclusion of D̂ in Â (see [5]). In particular, P acts on Irr(D). For W ∈ Irr(D) and u ∈ P we denote by ^{u}W ∈ Irr(D) the image of W under the action of u and by I_{P} (W) the stabiliser of W in P. Note that Q ≤ I_{P} (W). We say that a simple Â-module covers W if W appears as a composition factor of ResD̂(V) and denote by Irr(A|W) the subset of Irr(A) which cover W. For W ∈ Irr(D) denote by e_{V} the primitive central idempotent of D̂ corresponding to W and similarly for Â. Note that D̂e_{W} ≅ Mat_{dim_{K}(W)}(K).

Suppose that W ∈ Irr(D) is P -stable. Then D̂e_{W} ≅ Mat_{dim_{K}(W)}(K) is a P -algebra and by the Noether-Skolem theorem, every automorphism of D̂e_{W} is inner. Let I be a set of coset representatives of Q in P containing 1. For each y ∈ I, choose an element s_{y} ∈ (D̂e_{W})× such that s_{y}ds_{y^{−1}} = ^{y}d for all d ∈ D̂. For each x ∈ Q and y ∈ I, set s_{xy} = xs_{y}. Then, for all x, y ∈ P, we have s(x)s(y)s(xy)^{−1} ∈ Z(D̂e_{W})^{×}. Identifying Z(D̂e_{W}) with K, the map β : P × P → K^{×} given by β(y, y′) = s_{yy′}^{−1}s_{y}s_{y′}, where y, y′ ∈ P, is a 2-cocycle such that β(xy, x′y′) = β(y, y′) for all x, x′ ∈ Q. So, β is the restriction to P along the canonical map P → P/Q of a 2 -cocycle β̄ : P/Q × P/Q → K^{×}. We denote by α_{W} the image of β̄ in H^{2}(P/Q, K^{×}). Then α_{W} is independent of the choice of I and the choice of elements s_{y}, y ∈ I.

For L a finite group and α ∈ H^{2}(L, K^{×}), we denote by K_{α}L a twisted group algebra of L over K corresponding to α. The following proposition is a rewording of some of the results in [5].

Proposition 2.3. Let V ∈ Irr(A) and W ∈ Irr(D).

(i) ResD̂(V) is semi-simple, and if V covers W, then

ResD̂(V) ≅ m_{V} ∑_{u∈P/I_{P} (W)} ^{u}W.

(ii) V covers W if and only if e_{W} e_{V} ≠ 0.

(iii) Suppose that W ∈ Irr(A) is P -stable. Then

Âe_{W} ≅ D̂e_{W} ⊗_{K} K_{α_{W} −1}(P/Q)

as K-algebras. In particular, there is a bijection V → V_{0} between Irr(A|W) and Irr(K_{α_{W}} (P/Q)) such that dim_{K}(V) = dim_{K}(W) dim_{K}(V_{0}).

(iv) Suppose that V covers W. If dim_{K}(V) is prime to p then W is P -stable, dim_{K}(W) = dim_{K}(V) is prime to p and α_{W} is trivial.

(v) Suppose that W is P -stable, dim_{K}(W) is prime to p, and α_{W} is trivial. Then there are [P : R] elements in Irr(A|W) whose dimension is prime to p and they all have the same degree dim_{K}(W).

Proof. See the proof of [5, Theorem 1.6]. □

Corollary 2.2 provides the following key extendibility result.

Proposition 2.4. Suppose that W ∈ Irr(D) is P -stable and that dim_{K}(W) is prime to p. Then α_{W} is trivial. Equivalently, W extends to |P : R| pairwise nonisomorphic simple Â-modules.

Proof. Let δ : Q → K^{×} be defined by δ(x) = det(xe_{W}), where x ∈ Q. Then δ is a P -stable group homomorphism. Since every element of P acts as an inner automorphism of D̂e_{W}, [Q, P] ≤ Ker(δ). Thus, by Corollary 2.2, δ extends to a group homomorphism from P to K^{×}. Choose such an extension, and by abuse of notation, denote it again by δ. Let I be a set of coset representatives of Q in P. For each y ∈ I, choose the element s_{y} in (D̂e_{W})^{×} as in the definition of α_{W} such that det(s_{y}) = δ(y). This is possible as the dimension of W is prime to p and δ(y) is a |P |-th root of unity (and since we may assume that K contains a |P |-th root of unity). Then for all x ∈ P, det(s_{x}) = δ(x) and consequently, det(s_{xy}) = det(s_{x})det(s_{y}) for all x, y ∈ P. By the definition of β, we have s_{x}s_{y} = β(x, y)s_{xy} for all x, y ∈ P. Hence , for all x, y ∈ P, we have β(x, y)^{n} = 1, where n = dim_{K}(W). Consequently, α_{W} ∈ H^{2}(P/Q, K^{×}) has order dividing n. Since n is prime to p and H^{2}(P/Q, K^{×}) is a p-group, it follows that α_{W} = 1, as claimed. □

3. Proof of Theorem 1.1

Proof of Theorem 1.1. As before, we denote by A = iBi the source algebra of B with respect to the defect group P and the source idempotent i ∈ B^{P}. We denote by F the fusion system of B on P determined by i. We set R = 𝔣𝔬𝔠(F) and Q = 𝔥𝔶𝔭(F). We first prove the equivalence of (i) and (ii). Recall from [17, Introduction] or [18] that the Broué-Puig ∗-construction from [2] on characters yields a free action of P/R on Irr_{0}(B). Let χ, χ′ ∈ Irr_{0}(B). If χ, χ′ lie in the same orbit of P/R, then from the definition of the ∗-construction, it is immediate that χ(x) = χ′(x) for all p-regular elements of G, hence χ(j) = χ′(j) for any idempotent j of OG (as χ and χ′ have the same decomposition numbers). In particular, if χ and χ′ are in the same orbit of P/R on Irr_{0}(B), then χ(1) = χ′(1) and χ(i) = χ′(i). Thus, letting S denote a set of representatives of the P/R-orbits on Irr_{0}(B), we have that

∑_{χ∈Irr_{0}(B)} χ(1)^{2} = |P : R| ∑_{χ∈S} χ(1)^{2};

∑_{χ∈Irr_{0}(B)} χ(i)^{2} = |P : R| ∑_{χ∈S} χ(i)^{2}.

Since |S : P | divides χ(1) for all χ ∈ Irr(B), it follows from the above that in order to prove the equivalence of (i) and (ii), it suffices to prove that the integer ∑_{χ∈S} χ(i)^{2} belongs to J(O) if and only if the integer ∑_{χ∈S}(\frac{χ(1)}{|S:P |})^{2} belongs to J(O).

Let ω_{B} : Z(B) → Z(B)/J(Z(B)) = k be the canonical surjection. By [15] or [19, 9.3], Tr_{P}^{G}(i) is an invertible element of Z(B), hence ω_{B}(Tr_{P}^{G}(i)) ≠ 0. Recall that for any z ∈ Z(B), and any χ ∈ Irr(B), we have \frac{χ(z)}{χ(1)} ∈ O and ω_{B}(z) = \overline{\frac{χ(z)}{χ(1)}}, where for x ∈ O we denote by x̄ the image in k of x under the canonical surjection O/J(O). Thus, for any χ ∈ Irr(B), we have

0 ≠ ω_{B}(Tr_{P}^{G}(i)) = \overline{\frac{χ(Tr_{P}^{G}(i))}{χ(1)}} = \overline{|G : P |\frac{χ(i)}{χ(1)}}.

Choose λ ∈ O^{×} with λ̄ = ω_{B}(Tr_{P}^{G}(i)). The above shows that for any χ ∈ Irr(B), there exists an element t_{χ} ∈ J(O) such that

|G : P |\frac{χ(i)}{χ(1)} = λ + t_{χ}.

Since, |G : S| is invertible in O, by suitably replacing λ and t_{χ}, we obtain that there exists λ ∈ O^{×} such that for all χ ∈ Irr(B), there exists t_{χ} ∈ J(O) such that

|S : P |\frac{χ(i)}{χ(1)} = λ + t_{χ} .

Since t_{χ} ∈ J(O), we have that

∑_{χ∈S} χ(i)^{2} = ∑_{χ∈S}(λ + t_{χ})^{2}(\frac{χ(1)}{|S : P |})^{2} ≡ λ^{2} ∑_{χ∈S}(\frac{χ(1)}{|S : P |})^{2} mod (J(O)).

The equivalence of (i) and (ii) follows. If (iii) holds, then by the above, the height zero characters in B form a unique regular P/R-orbit with respect to the ∗-construction; in particular, all height zero characters of B have the same degree with p-part |S : P |, and hence (iii) implies (i). We prove next that (ii) implies (iv). The strategy is the same as used for [9, Lemma 4]. Let V denote the subset of Irr_{0}(D) consisting of P -stable elements. Note that χ ∈ Irr_{0}(B) is of height zero if and only if χ(i) is prime to p. Moreover, χ(i) is the dimension of the simple Â-module corresponding to χ under the canonical Morita equivalence between B and A. By Proposition 2.3 (i)-(iv) and Proposition 2.4 it follows that

∑_{χ∈Irr_{0}(B)} χ(i)^{2} = |P : R| ∑_{W ∈V} dim_{K}(W)^{2}.

In particular, |P : R| divides ∑_{χ∈Irr_{0}(B)} χ(i)^{2} and we have

\frac{1}{|P : R|} ∑_{χ∈Irr_{0}(B)} χ(i)^{2} = ∑_{W ∈V} dim_{K}(W)^{2}.

If W′ ∈ Irr(D)\V, then either p divides the dimension of W′ or the size of the P -orbit of W′ is divisible by p. Hence,

∑_{W ∈V} dim_{K}(W)^{2} ≡ ∑_{W ∈Irr(D)} dim_{K}(W)^{2} (mod p).

Assume that \frac{1}{|P:R|} ∑_{χ∈Irr_{0}(B)} χ(i)^{2} is a p′-integer. Then by the above, dim_{K}(D̂) = ∑_{W ∈Irr(D)} dim_{K}(W)^{2} is a p′-integer. The hyperfocal subalgebra D is a direct summand of A as a left D-module, hence projective as a left OQ-module. In particular, |Q| divides the O-rank of D. Thus Q = 1, which means that F and hence B are nilpotent. This proves that (ii) implies (iv). As mentioned earlier, the fact that (iv) implies any of (i), (ii), (iii) is immediate from the structure theorem [16, Theorem 1.6] for source algebras of nilpotent blocks. □

4. Around Conjecture 1.2.

As before, G is a finite group, B a block algebra of OG with corresponding block idempotent b = 1_{B}, and P a defect group of B. Let i ∈ B^{P} be a source idempotent and A = iBi the corresponding source algebra.

Proposition 4.1. Suppose P is abelian. Then Irr_{0}(D) = Irr(D).

Proof. This follows from the forward direction of the height zero conjecture (proved by Kessar and Malle in [10]), and Proposition 2.3. □

Proposition 4.2. Conjecture 1.2 holds for blocks with a normal defect group.

Proof. Suppose that the defect group P of B is normal in G. By Külshammer’s structure theorem in [12], the source algebra A is isomorphic to a twisted group algebra of P ⋊ E, where E is the inertial quotient of B. From this it follows that a hyperfocal source algebra is isomorphic to a twisted group algebra of Q ⋊ E. The result follows. □

The block B is said to be of principal type if for any subgroup Q of P, the idempotent Br_{Q}(b) is a block of kC_{G}(Q), or equivalently, if the algebra B(Q) = kC_{G}(Q)Br_{Q}(b) is a block algebra of kC_{G}(Q). In that case, we have e_{P} = Br_{P} (b), there is a unique local point γ of P on B, and the fusion system F of B on P is equal to the fusion system F_{P} (G) induced by conjugation of elements in G on subgroups of P. The principal block of OG is of principal type by Brauer’s third main theorem. The following two observations are well-known and included for convenience; the proofs are routine block theory. Recall that the multiplicity m_{γ} of a local point of P on B is the number |γ ∩ I| of elements of γ which appear in a primitive orthogonal idempotent decomposition I of b in B^{P}.

Lemma 4.3. Suppose that the defect group P of B is a Sylow-p-subgroup of G. Let γ be a local point of P on OG^{P}. Then the multiplicity m_{γ} of γ on B is prime to p and divides the order of C_{G}(P).

Proof. Let e be the unique block of kC_{G}(P) such that Br_{P} (γ) ⊆ kC_{G}(P)e. Then Br_{P} (γ) is the unique point of kC_{G}(P)e. Thus m_{γ} is the dimension of the unique (up to isomorphism) simple kC_{G}(P)e-module V. Since P is a Sylow-p-subgroup of G, it follows that C_{G}(P) = Z(P) × C′ for some p′-subgroup C′ of C_{G}(P), and hence e = e_{η}, where η is an irreducible character of C′ with coefficients in k and e_{η} is the corresponding central primitive idempotent in kC′ (which makes sense as |C′| is prime to p). Thus dim_{k}(V) = η(1) divides |C′|, whence the result. □

Proposition 4.4. Suppose that B is of principal type and that P is a Sylow-p-subgroup of G. Set N = O^{p}(G). Then b is a block idempotent of ON. Setting C = ONb, the following hold.

(i) The block C of ON is of principal type, with a defect group Q = P ∩ N = 𝔥𝔶𝔭(F) and fusion system O^{p}(F). (ii) The algebra C^{P} contains a (B^{P})^{×}-conjugate of i. (iii) If i ∈ C^{P}, then D = iCi is a hyperfocal subalgebra of the source algebra A = iBi of B. (iv) If i ∈ C^{P}, then there is a source idempotent j ∈ (iCi)^{Q} of the block C belonging to the unique local point δ of Q on C. The multiplicity m_{δ}^{γ} of δ on iCi is prime to p, and for any χ ∈ Irr(C) we have χ(i) ≡ m_{δ}^{γ}χ(j) mod p.

(v) All irreducible characters of iCi have degree prime to p if and only of all irreducible characters of jCj have degree prime to p.

Proof. Since a block idempotent is supported on p′-elements, we have b ∈ (ON)^{G}. Thus b is a G-conjugacy class sum of blocks of ON. As G = NP, the idempotent b is in fact a P -conjugacy class sum of blocks of ON. Since Br_{P} (b) ≠ 0, this implies that b is a block of ON. Then Q = P ∩ N is a Sylow-p-subgroup of N, hence a defect group of the block algebra C = ONb of ON. By Puig’s hyperfocal subgroup theorem, we have Q = 𝔥𝔶𝔭(F). Let R be a subgroup of Q. We need to show that f = Br_{R}(b) is a block of kC_{N}(R). After replacing R by a conjugate, if necessary, we may assume that C_{P} (R) is a Sylow-p-subgroup of C_{G}(R). Since C_{N}(R) is normal of ppower index in C_{G}(R), it follows that C_{G}(R) = C_{N}(R)C_{P} (R). We have Br_{C_{P} (R)}(f) ≠ 0 because f is a block of kC_{G}(R) with defect group C_{P} (R). Arguing as above for b it follows that f is a C_{P} (R)-conjugacy class of block idempotents of kC_{N}(R), and hence the condition Br_{C_{P} (R)}(f) ≠ 0 implies that f remains a block idempotent of kC_{N}(R). Thus the block C is of principal type, and hence C has the fusion system F_{Q}(N), which is equal to O^{p}(F). This proves (i). Statement (ii) follows as the inclusion C → B is a semi-covering (cf. [13, 3.9, 3.16]) and therefore any primitive (local) idempotent in C^{P} remains primitive (local) in B^{P}. Thus the unique local point γ of P on B contains an element in C. Choosing such an i ∈ γ ∩ C, it follows that iCi is a unitary P -stable subalgebra of iBi such that iCi ∩ P = i(P ∩ N)i = Qi. Statement (iii) follows by the uniqueness of hyperfocal subalgebras. By [17, 4.2], the group Q has a unique local point δ on D = iCi. This is then necessarily a defect pointed group of C, and hence any j ∈ δ is a source idempotent of C. If j′ belongs to a nonlocal point of a subgroup R of P on OG, then j′ can be written as a trace from a proper subgroup of R, and hence χ(j′) ∈ J(O) for any χ ∈ Irr(G). Thus, if χ ∈ Irr(B), then χ(1) ≡ m_{γ}χ(i) mod J(O), since γ is the unique local point of P on B. Similarly, we have χ(i) ≡ m_{δ}^{γ}χ(j) mod J(O). Since P is a Sylow-p-subgroup of G, it follows that the height zero characters in B are the characters of degree prime to p in B. Thus the previous lemma and the above congruence applied to a height zero character yield together that m_{δ}^{γ} is prime to p. This proves (iv). Multiplication by j induces a Morita equivalence between the hyperfocal subalgebra iCi of B and the source algebra jCj of C. In particular, multiplication by j induces a bijection between irreducible characters of iCi and jCj. Thus (v) follows from (iv). □

Proposition 4.5. Conjecture 1.2 holds for blocks of finite p-solvable groups.

Proof. Suppose that G is p-solvable. Then B is source algebra equivalent to a block of principal type of a subgroup H of G such that P is a Sylow-p-subgroup of H and such that the block idempotent belongs to OO_{p′}H; this is a version of Fong-Reynolds reduction, as presented, for instance, in [8, Theorem 5.1]. Thus we may assume that B is of principal type and that P is a Sylow-p-subgroup of G. Set N = O^{p}(G). By 4.4, the block idempotent b of B remains a block idempotent in ON, and the group Q = N ∩ P = 𝔥𝔶𝔭(F) is a defect group of C = ONb. It follows from 4.4 (v) that all irreducible characters of a hyperfocal subalgebra D of B have degree prime to p if and only if all irreducible characters in C have degree prime to p. Since by a result of Gluck and Wolf [7] the height zero conjecture holds for blocks of p-solvable groups, it follows that this is equivalent to Q being abelian. □

Since [10] and [7] invoke the classification of finite simple groups, so do the above proofs of Conjecture 1.2 for blocks with an abelian defect group or blocks of p-solvable finite groups.

Proposition 4.6. For blocks of principal type and maximal defect, Brauer’s height zero conjecture implies Conjecture 1.2.

Proof. Suppose that Brauer’s height zero conjecture holds for blocks of principal type and maximal defect. Suppose that B is of principal type and that its defect group is a Sylow-p-subgroup of G. Let N, C, i, j be as in Proposition 4.4. By Proposition 4.4 (v), all irreducible characters of the hyperfocal algebra D = iCi have degree prime to p if and only if all irreducible characters of the source algebra jCj of C have degree prime to p, which by Brauer’s height zero conjecture (assumed to be true) applied to the block C is true if and only if Q is abelian. □

Remark 4.7. The converse implication in 4.6 holds for the ‘forward’ direction of the two conjectures. More precisely, suppose that Conjecture 1.2 holds for all blocks of principal type and maximal defect. Suppose that B is of principal type and that P is a Sylow-p-subgroup of G. Let N, C, i, j be as in Proposition 4.4. It follows from 2.3 that all irreducible characters of iBi have p′-degree if an only if all irreducible character of iCi are P -stable, have p′-degree, P/Q is abelian, and all 2-cocycles α_{W} as in 2.3 are trivial. Since we assume 1.2 to be true, this is equivalent to requiring that all irreducible characters of iCi are P -stable, the groups Q, P/Q are both abelian, and all α_{W} as before are trivial. We do not know whether these conditions force P to be abelian. We can show that these conditions are satisfied if P is abelian. Indeed, if P is abelian, then Q and P/Q are trivially abelian. By a theorem of Knörr in [11], every O-free B-module affording an irreducible character χ has vertex P. But then any character of C covered by χ must be P -stable (because otherwise one could write χ as being induced from a proper subgroup or G containing N, hence not containing P, yielding a lattice with vertex smaller than P). All characters of iCi have p′-degree, so all α_{W} are trivial by 2.4.

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School of Engineering and Mathematical Sciences, City University London EC1V 0HB, Great Britain

E-mail address: radha.kessar.1@city.ac.uk

School of Engineering and Mathematical Sciences, City University London EC1V 0HB, Great Britain

E-mail address: markus.linckelmann.1@city.ac.uk

Departament d’Àlgebra, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Spain.

E-mail address: gabriel.navarro@uv.es