Invariant Blocks Under Coprime Actions

Gunter Malle, Gabriel Navarro, Britta Späth^{1}

Received: October 15, 2014

Communicated by Thomas Peternell

Abstract. If a finite group A acts coprimely as automorphisms on a finite group G, then the A-invariant Brauer p-blocks of G are exactly those that contain A-invariant irreducible characters.

2010 Mathematics Subject Classification: 20C15, 20C20, 20C33 Keywords and Phrases: Coprime action, blocks, characters

1. Introduction

One of the most basic situations in group theory is when a group A acts by automorphisms on another group G. If we further assume that A and G are finite of coprime orders, it is well-known that most of the representation theory of G admits a version in which only the A-invariant structure is taken into account. For instance, it is true (and not trivial) that the number of irreducible complex characters of G which are fixed by A equals the number of conjugacy classes of G fixed by A. Although it is fair to say that the ordinary A-representation theory of G is mostly well developed, we cannot say the same about A-invariant modular representation theory (that is, of prime characteristic p). For instance, it is suspected that the number of Ainvariant irreducible p-Brauer characters of G is the number of A-invariant p-regular classes of G, but this conjecture continues to be open. This, together with some other problems, was proposed more than 20 years ago in [13]. The extensive research during these years on Brauer p-blocks allows us now to give a solution to Problem 6 of [13].

Theorem 1.1. Suppose that the finite group A acts by automorphisms on the finite group G with (|A|, |G|) = 1. Let p be a prime, and let B be a Brauer p-block of G. Then the following statements are equivalent:

(i) B is A-invariant, (ii) B contains some A-invariant character χ ∈ Irr(G), (iii) B contains some A-invariant Brauer character ϕ ∈ IBr(G).

1The research of the first and third author has been supported by the ERC Advanced Grant 291512

Of course, (ii) and (iii) easily imply that B is A-invariant, so all the work is concentrated in proving that (i) implies (ii) and (iii). The paper is structured in the following way. In Section 2 a version of the main theorem is proven in the case where G is quasi-simple. Afterwards in Section 3 we present a Gallagher type theorem for blocks. In connection with Dade’s ramification group from [2], revisited in Section 4, we show the existence of character triple isomorphisms having crucial properties with respect to coprime action and blocks (see Section 5). We conclude in the final section with the reductions proving how the results on quasi-simple groups imply our main statement.

Acknowledgement. We thank the referee for a careful reading of a previous version.

2. Quasi-simple groups and their central products

The aim of this section is the proof of a strengthened version of Theorem 1.1 in cases where the group G/Z(G) is the direct product of r isomorphic non-abelian simple groups. First we deal with the case where G is the universal covering group of a simple group and hence G is a quasi-simple group. In the following we use the standard notation around characters and blocks as introduced in [7] and [14]. Let p be a prime. If A acts on G, we denote by Bl_{A}(G) the set of A-invariant p-blocks of G. If Z◁ G, B ∈ Bl(G) and ν ∈ Irr(Z) we denote by Irr(B|ν) the set Irr(B) ∩ Irr(G|ν). Also, sometimes we will work in GA, the semidirect product of G with A.

Theorem 2.1. Let G be the universal covering group of a non-abelian simple group S, A a group acting on G with (|G|, |A|) = 1, B an A-invariant p-block of G, Z the Sylow p-subgroup of Z(G) and ν ∈ Irr(Z) A-invariant. Then B contains some A-invariant character χ ∈ Irr(G|ν).

This theorem is true whenever B is the principal block and Z = 1, since then χ can be chosen to be the trivial character. On the other hand when B is a block of central defect, Irr(B|ν) contains exactly one character and hence this one is A-invariant. Note that neither alternating nor sporadic simple groups possess coprime automorphisms, and that for groups of Lie type the only coprime order automorphisms are field automorphisms (up to conjugation). Thus, for the proof of the theorem, we can assume that S is simple of Lie type. We consider the following setup. Let G be a simple algebraic group of simply connected type over an algebraic closure of a finite field of characteristic r, and let F : G → G be a Steinberg endomorphism, with group of fixed points G := G^{F}. It is well known that all finite simple groups of Lie type occur as G/Z(G) with G as before, except for the Tits group ^{2}F_{4}(2)′. Since the latter does not possess coprime automorphisms, we need not consider it here. Furthermore, in all but finitely many cases, the group G is the universal covering group of S = G/Z(G). None of the exceptions, listed for example in [11, Table 24.3], has coprime automorphisms, apart from the Suzuki group ^{2}B_{2}(8). But for 2^{2}.^{2}B_{2}(8), the outer automorphism of order three permutes the three non-trivial central characters, hence our claim holds. For the proof of Theorem 2.1 we may thus assume that G and S are as above, and that A induces a (necessarily cyclic) group of coprime field automorphisms on G. We first discuss the action of such automorphisms on the Lusztig series E(G, s) ⊆ Irr(G) of irreducible characters of G, where s runs over semisimple elements of a dual group G^{∗} = G^{∗}^{F} of G. Let G ↪ G̃ be a regular embedding, that is, G̃ is connected reductive with connected center and with derived subgroup G. Corresponding to this there exists a surjection G̃^{∗} → G^{∗} of dual groups. Note that all field automorphisms of G are induced by those of G̃. Let γ be a field automorphisms of G̃. We denote the corresponding field automorphism of G̃^{∗} also by γ. Let s̃ ∈ G̃^{∗} := G̃^{∗F} be semisimple. Now by [10, Prop. 3.5], γ acts trivially on E(G̃, s̃) whenever it stabilises E(G̃, s̃). Let s ∈ G^{∗} with preimage s̃. Then E(G, s) consists of the constituents of the restrictions of characters in E(G̃, s̃) to G, see e.g. [1, Prop. 15.6(i)].

Lemma 2.2. In the above setting let γ be a coprime (field) automorphism of G. If γ stabilises E(G, s), then it fixes E(G, s) pointwise.

Proof. According to what we said before, γ can only permute the G-constituents of a fixed character χ ∈ E(G̃, s̃). We have G̃ = GZ(G̃), so |G̃| = |G| |Z(G̃)| and hence

|G̃ : GZ(G̃)| = \frac{|G̃| |G ∩ Z(G̃)|}{|G| |Z(G̃)|} = |G ∩ Z(G̃)| = |Z(G)|.

As irreducible characters of the central product GZ(G̃) restrict irreducibly to G, any irreducible character of G̃ has at most |Z(G)| irreducible constituents upon restriction to G. It is easily checked that all primes not larger than |Z(G)| divide |G|, so that all prime divisors of the order of γ are larger than the number of such constituents. Thus the action has to be trivial. □

There are two quite different types of behaviour now. Either p is the defining characteristic of G, then coprime field automorphisms fix all p-blocks (but certainly not all irreducible characters); or p is different from the defining characteristic, in which case all characters in an invariant block are fixed individually (but not all p-blocks are invariant):

Proposition 2.3. In the above situation, Theorem 2.1 holds when p is the defining characteristic of G.

Proof. In this case the p-blocks of positive defect of a group of Lie type G are in bijection with the characters of Z(G), by a result of Humphreys [5]. Since the claim is certainly true for the principal block, we may assume that Z(G) ≠ 1, and so in particular G is not a Suzuki or Ree group. For each type of group and each γ-stable 1 ≠ ν ∈ Z(G) we give in Table 1 a semisimple element s of the dual group G^{∗} of G with the following properties: the Lusztig series E(G, s) of irreducible characters lies in Irr(G|ν), and the class of s is γ-invariant (since s corresponds to the γ-stable central character ν). It then follows that E(G, s) is stable under all field automorphisms of G that stabilise ν, and this implies by Lemma 2.2 that the characters in E(G, s) are individually stable, hence provide characters as claimed. □

Let us now turn to the case where p is not the defining characteristic of G.

Table 1. Semisimple elements

G  C_{G^{∗}}(s) o(ν)  conditions
SL_{n}(q) GL_{n−1}(q) divides (n, q − 1) (n, q − 1) > 1
SU_{n}(q) GU_{n−1}(q) divides (n, q + 1) (n, q + 1) > 1
Spin_{2n+1}(q) C_{n−1} 2 q odd
Sp_{2n}(q) GO_{2n}^{±}(q) 2  q odd
Spin_{2n}^{±}(q)  B_{n−1} divides 4  q odd
E_{6}(q)  D_{5} 3  q ≡ 1 (mod 3)
2E_{6}(q)  ^{2}D_{5} 3  q ≡ 2 (mod 3)
E_{7}(q)  E_{6} 2  q ≡ 1 (mod 4)
E_{7}(q)  ^{2}E_{6} 2  q ≡ 3 (mod 4)

Proposition 2.4. In the above situation, when p is different from the defining characteristic of G, if B is a γ-invariant p-block of G, then all χ ∈ Irr(B) are fixed by γ. In particular Theorem 2.1 holds in this case.

Proof. Let B be a p-block of G, Z = Z(G)_{p} and ν ∈ Irr(Z). Let γ be a coprime (field) automorphism of G fixing B and ν. By a result of Broué and Michel there exists a semisimple p′-element s ∈ G^{∗} such that B ⊆ E_{p}(G, s). Let G ↪ G̃ be a regular embedding, with corresponding epimorphism G̃^{∗} → G^{∗} of dual groups. Let s̃ ∈ G̃^{∗} be a preimage of s. Since the class of s is γ-stable, the same argument as in the proof of Lemma 2.2 shows that the class of s̃ is also γ-stable. Since the centraliser of s̃ is connected, this means that we may assume without loss of generality that s̃ itself is γ-stable, and so is CG̃^{∗}(s̃). Now consider H := (G̃^{∗})^{γ}, the fixed point subgroup of G̃^{∗} under γ. This is again a group of Lie type, of the same type as G̃^{∗}. Since p divides |Z(G)| by assumption, it also divides the order of the Weyl group of G. By [10, Prop. 3.12], then C_{H}(s̃) = CG̃^{∗}(s̃)^{γ} contains a Sylow p-subgroup of CG̃^{∗}(s̃). In particular, every semisimple p-element in CG̃^{∗}(s̃) has a γ-stable conjugate t̃ ∈ G̃^{∗}. Then E(G, s̃t̃) is γ-stable, which implies that E(G, st) is γ-stable, and hence fixed pointwise by γ, by Lemma 2.2. Thus, all elements of B ∩ E_{p}(G, s) are fixed by γ, as claimed. □

We next prove an analogous result for Brauer characters:

Theorem 2.5. Let G be the universal covering group of a non-abelian simple group S, A a group acting on G with (|G|, |A|) = 1, and B ∈ Bl_{A}(G). Then there exists an A-invariant Brauer character ϕ ∈ IBr(B).

Proof. As argued in the proof of Theorem 2.1we may assume that S is of Lie type and A induces coprime field automorphisms. Moreover, G is not an exceptional covering group of S. First assume that p is the defining characteristic of G. Let B be an A-invariant block of G, corresponding to the central character ν of G. Then there is a faithful irreducible Brauer character ϕ of G/ ker(ν) corresponding to a suitable fundamental weight ω of the underlying algebraic group as given in Table 2. Note that we only need to consider cases when Z(G) ≠ 1, which explains the restrictions in the last column of the table. If G is untwisted, defined over 𝔽_{q} with q = p^{f}, then a generator γ of A has order a with f = ka. By Steinberg’s tensor product theorem (see [11, Thm. 16.12]) then ϕ′ := ⨂_{i=0}^{a−1} γ^{i}(ϕ) is an irreducible Brauer character of G corresponding to the weight ∑_{i=0}^{a−1} p^{ik}ω, which is γ-invariant. It lies over the character ν^{d} of Z(G), with d = ∑_{i=0}^{a−1} p^{ik}. Since |Z(G)| divides p^{k} −1 and a is prime to |Z(G)|, this is again a faithful character of Z(G). Thus, this construction yields an invariant Brauer character in the p-block lying above ν^{d}. Starting instead with ϕ in the p-block above ν^{c}, with cd ≡ 1 (mod o(ν)), we find an invariant character in B.

Table 2. Faithful Brauer characters

G  ϕ(1) weight  conditions
SL_{n}(q), SU_{n}(q)  \binom{n}{i} ω_{i} (1 ≤ i ≤ n − 1) (n, q ± 1) > 1
Spin_{2n+1}(q)  2^{n} ω_{n}  q odd
Sp_{2n}(q)  2n ω_{1}  q odd
Spin_{2n}^{±}(q) 2n, 2^{n−1} ω_{1}, ω_{n−1}, ω_{n}  q odd
E_{6}(q),^{2}E_{6}(q)  27 ω_{1}  3 |̸q
E_{7}(q)  56 ω_{7}  q odd

If G is twisted, we may assume that it is not of type ^{2}B_{2}, ^{2}G_{2} or ^{2}F_{4} and that the twisting has order 2 (since else there is just one p-block of positive defect). So G is defined over 𝔽_{q^{2}}, with q = p^{f} and a|f. The same argument as before applies in this case as well. Now assume that p is different from the defining characteristic of G, and let B be an A-invariant p-block. Then B is contained in E_{p}(G, s) for some semisimple element s ∈ G^{∗}, and we showed in Proposition 2.4that all elements in B are fixed by A. Since any irreducible Brauer character in B is an integral linear combination of ordinary irreducible characters in B restricted to p′-classes, it follows that IBr(B) is fixed point-wise by A as well. □

We conclude this section with the analogous result on central products of quasi-simple groups.

Corollary 2.6. Let G be a finite group and let A act on G with (|G|, |A|) = 1. Assume that G/Z(G) is the direct product of r isomorphic non-abelian simple groups that are transitively permuted by A. Let B ∈ Bl(G) be A-invariant.

(a) Let Z ∈ Syl_{p}(Z(G)) and ν ∈ Irr(Z). Then B contains some A-invariant χ ∈ Irr(G|ν).

(b) B contains some A-invariant ϕ ∈ IBr(G).

Proof. Both parts can be shown analogously. We give here the proof of part (a). Let S be the simple non-abelian group such that G/Z(G) is isomorphic to the r-fold direct product of groups isomorphic to S.

It suffices to prove the statement in the case where G is perfect. Indeed, we have G = [G, G]Z(G) by the given structure of G. Assume that the statement holds in the case of perfect groups. Hence the block B′ ∈ Bl([G, G]) covered by B has then an A-invariant character χ_{0} lying over ν_{Z(G)∩[G,G]}. The character χ_{0} · ν defined as in [8, Section 5] as the unique character in Irr(G|χ_{0}) ∩ Irr(G|ν) is then necessarily A-invariant. In the following we consider the case where G is perfect. Accordingly G has a universal covering group, namely

X := Ŝ × ⋯ × Ŝ = Ŝ^{r} (r factors) ,

where Ŝ is the universal covering group of S. Let ϵ : X → G be the associated canonical epimorphism. Because of [3, 5.1.4] there is a canonical action of A on X induced by the action of A on G, such that ϵ is A-equivariant. Note that the action of A on X is coprime, since the prime divisors of |G| and |X| coincide. For 1 ≤ i ≤ r let X_{i} := {(1, …, 1, x, 1, …, 1) | x ∈ Ŝ}, which is canonically isomorphic to Ŝ via ι_{i} : x ↦ (1, …, 1, x, 1, …, 1). By assumption A acts transitively on the set of groups X_{i}. So for every 1 ≤ i ≤ r there exists an element a_{i} ∈ A such that X_{1}^{a_{i}} = X_{i}. The character ν ∈ Irr(Z) can be uniquely extended to a character ν̃ ∈ Irr(Z(G)) such that Irr(B|ν̃) ≠ ∅. Via ϵ the character ν̃ corresponds to some character ν̂ ∈ Irr(Z(X)) that can be written as

ν̂ = ν̂_{1} × ⋯ × ν̂_{r}

for suitable ν̂_{i} ∈ Irr(Z(Ŝ)) The block B corresponds to a unique block B̂ ∈ Bl(X), see [14, (9.9) and (9.10)]. Accordingly B̂ is A-invariant and can be written as B̂ = B̂_{1} × ⋯ × B̂_{r} where B̂_{i} ∈ Bl(Ŝ). Let A_{i} be the stabiliser of X_{i} in A. The action of A_{i} on X_{i} induces via ι_{i} a coprime action of A_{i} on Ŝ stabilising B̂_{i}. According to Theorem 2.1 there exists an A_{1}-invariant character ψ_{1} ∈ Irr(B̂_{1}|ν̂_{1}). There exists a unique ψ_{i} ∈ Irr(Ŝ) with (ψ_{1} ∘ ι_{1}^{−1})^{a_{i}} = ψ_{i} ∘ ι_{i}^{−1}. Since B̂ and ν̂ are A-invariant the character ψ_{i} belongs to Irr(B̂_{i}|ν̂_{i}). Accordingly the character ψ := ψ_{1} × ⋯ × ψ_{r} belongs to Irr(B̂|ν̂). In the next step we prove that ψ is A-invariant. Let ϕ_{i} := ψ_{i} ∘ ι_{i}^{−1} ∈ Irr(X_{i}). Then it is sufficient to prove ϕ_{i}^{a} = ϕ_{j} for any a ∈ A and every 1 ≤ i, j ≤ r with X_{i}^{a} = X_{j}. The equality X_{i}^{a} = X_{j} implies (X_{1}^{a_{i}})^{a} = X_{1}^{a_{j}}, and hence a_{i}aa_{j}^{−1} ∈ N_{A}(X_{1}) = A_{1}. On the other hand by the definition of ψ_{i} and ϕ_{i} we have

(ϕ_{i})^{a} = (ϕ_{1})^{a_{i}a} = (ϕ_{1})^{a_{i}aa_{j}^{−1}a_{j}} = ϕ_{1}^{a_{j}},

since ϕ_{1} is A_{1}-invariant. This proves that ψ is A-invariant as required. Part (b) follows from these considerations by applying Theorem 2.5. □

3. A Gallagher type theorem for blocks

Let G be a finite group and N◁ G. P. X. Gallagher proved that if θ ∈ Irr(N) has an extension θ̃ ∈ Irr(G), then the map Irr(G/N) → Irr(G|θ) given by \overline{η} ↦ ηθ̃ is a bijection, where η ∈ Irr(G) is the lift of \overline{η}. (See Corollary (6.17) of [7].) Now we need a similar theorem for blocks, see Theorem 3.4. For a character θ of G we denote by bl(θ) the p-block of G containing θ.

Lemma 3.1. Let N◁ G, b ∈ Bl(N) and θ ∈ Irr(b). Assume there exists an extension θ̃ ∈ Irr(G) of θ. Then the map

υ : Bl(G/N) → Bl(G|b) given by bl(\overline{η}) ↦ bl(θ̃η)

is surjective, where η ∈ Irr(G) is the lift of \overline{η} ∈ Irr(G/N) .

Proof. According to [16, Lemma 2.2] we have for every g ∈ G that

λ_{θ̃η}(ℭ𝔩_{G}(g)^{+}) = λθ̃_{L}(ℭ𝔩_{L}(g)^{+})λ\overline{η}(ℭ𝔩_{G/N}(\overline{g})^{+}),

where L is defined by L/N := C_{G/N}(\overline{g}) and \overline{g} = gN. This implies that the blocks bl(θ̃η) and bl(θ̃η′) coincide for every two characters \overline{η}, \overline{η}′ ∈ Irr(G/N) with bl(\overline{η}) = bl(\overline{η}′). Hence υ is well-defined. On the other hand, every block of Bl(G|b) has a character in Irr(G|θ) (by [14, (9.2)]) and such a character can be written as θ̃η for some \overline{η} ∈ Irr(G/N), by Gallagher’s theorem. This proves that υ is surjective. □

In general, the map in Lemma 3.1 is not a bijection. (For instance, if b has a defect group D such that C_{G}(D) ⊆ N, then it is well-known that there is a unique block of G covering b, see [17, Lemma 3.1]. On the other hand, G/N might have many p-blocks. Take, for instance, G = SL_{2}(3), N = Q_{8}, p = 2, and θ ∈ Irr(N) the irreducible character of degree 2.) Our aim in this section is to find general conditions which guarantee that the map in Lemma 3.1 is a bijection. The following statement follows also from Theorem 3.3(d) of [4], where a Morita equivalence between the involved blocks is proven. For completeness we nevertheless give here an alternative character theoretic proof.

Lemma 3.2. Let N◁ G and b ∈ Bl(N) with trivial defect group. Let θ ∈ Irr(b). Assume there exists an extension θ̃ ∈ Irr(G) of θ. Then the map

υ : Bl(G/N) → Bl(G|b) given by bl(\overline{η}) ↦ bl(θ̃η)

is a bijection.

Proof. By Lemma 3.1, we know that υ is surjective. Let \overline{η}, \overline{η}′ ∈ Irr(G/N) with bl(η) ≠ bl(η′). By [14, Exercise (3.3)] there exists some \overline{g} ∈ (G/N)^{0} with λ\overline{η}(ℭ𝔩_{G/N}(\overline{g})^{+}) ≠ λ\overline{η}′(ℭ𝔩_{G/N}(\overline{g})^{+}). Assume there exists some c ∈ G with cN = \overline{g} with

λθ̃_{L}(ℭ𝔩_{L}(c)^{+})^{∗} ≠ 0,

where L is defined by L/N := C_{G/N}(\overline{g}). Then bl(θ̃η) ≠ bl(θ̃η′), since

λ_{θ̃η}(ℭ𝔩_{G}(c)^{+}) = λθ̃_{L}(ℭ𝔩_{L}(c)^{+})λ\overline{η}(ℭ𝔩_{G/N}(\overline{g})^{+}).

Thus, let g ∈ G with gN = \overline{g}. Note that gN is closed under L-conjugation, and let 𝕊 be a representative set of the L-conjugacy classes contained in gN, i.e.

⋃_{s∈𝕊}^{.} ℭ𝔩_{L}(s) = gN.

By [7, Lemma (8.14)] we have

∑_{c∈gN} θ̃(c)θ̃(c^{−1}) = |N|.

This implies

∑_{s∈𝕊} |ℭ𝔩_{L}(s)|θ̃(s)θ̃(s^{−1}) = |N|.

Dividing by θ(1) we obtain

∑_{s∈𝕊} λθ̃_{L}(ℭ𝔩_{L}(s)^{+})θ̃(s^{−1})^{∗} = ( ∑_{s∈𝕊} \frac{|ℭ𝔩_{L}(s)|θ̃(s)}{θ(1)}θ̃(s^{−1}))^{∗} = (\frac{|N|}{θ(1)})^{∗} ≠ 0

since θ is of defect 0. This implies that for some c ∈ 𝕊 we have λθ̃_{L}(ℭ𝔩_{L}(c)^{+})^{∗} ≠ 0 as required. □

Next, we generalise Lemma 3.2. Recall Theorem 2.1 of [15]: for a finite group X, a p-subgroup Y ◁ X and an X-invariant character ν ∈ Irr(Y) there is a natural bijection dz(X/Y) → rdz(X|ν), χ ↦ χ_{ν}, with \frac{χ(1)_{p}}{ν(1)_{p}} = \frac{|X|_{p}}{|Y |_{p}}, where dz(X) denotes the characters lying in a defect zero block and rdz(X|ν) the set of characters χ ∈ Irr(X|ν), see [15] for the definition of this bijection.

Lemma 3.3. Let N◁ G, and suppose that D◁ G is contained in N. Let μ ∈ Irr(D) be G-invariant. Let θ ∈ dz(N/D). Then θ extends to G/D if θ_{μ} extends to G.

Proof. Note that since θ_{μ} extends to G, the character μ is G-invariant. The bijection dz(N/D) → rdz(N|μ), χ ↦ χ_{μ} is G-equivariant. Hence θ is G/D-invariant. Suppose that θ_{μ} extends to G. In order to show that θ extends to G it is enough to prove that θ extends to Q, whenever Q/N is a Sylow q-subgroup of G/N for some prime q. If Q/N ∈ Syl_{p}(G/N), then θ considered as a character of N/D has defect zero, and therefore it extends to Q/D in this case (see, for instance, Problem (3.10) of [14]). So θ as a character of N extends to Q. Now suppose that Q/N is a q-group for some q ≠ p. We know that θ_{μ} extends to some η ∈ Irr(Q). According to its degree, η ∈ rdz(Q|μ). Then η = γ_{μ} for some γ ∈ dz(Q/D). By the values of the functions γ_{μ} and using that μ̂(g) ≠ 0 whenever g_{p} ∈ D we check that γ_{N} = θ. □

Theorem 3.4. Let N◁ G and b ∈ Bl(N) with a defect group D with DC_{G}(D) = G. Let θ ∈ Irr(b). Assume there exists an extension θ̃ ∈ Irr(G) of θ. Then the map

υ : Bl(G/N) → Bl(G|b) given by bl(\overline{η}) ↦ bl(θ̃η)

is a bijection.

Proof. By Lemma 3.1 it is sufficient to prove that | Bl(G/N)| = | Bl(G|b)|. Because of DC_{G}(D) = G the defect group D is normal in N. By [14, Thm. (9.12)], there exists a unique character θ_{1} ∈ Irr(N) of b with D ⊆ ker θ_{1} that is a lift of some \overline{θ}_{1} ∈ dz(N/D). By Lemma 3.3 the character \overline{θ}_{1} extends to G/D. Write \overline{N} = N/D and \overline{G} = G/D. Then Lemma 3.2 applies to the character \overline{θ}_{1} ∈ Irr(N/D) associated to θ_{1}. Then | Bl(\overline{G}/\overline{N})| = | Bl(\overline{G}|\overline{b})|, where \overline{b} = bl(\overline{θ}_{1}). Now by [14, Thm. (9.10)] there is a canonical bijection between the blocks of \overline{G} and the blocks of G, given by covering. Using [14, Thm. (9.2)] with Brauer characters, we easily check that under this bijection, a block \overline{B} of \overline{G} covers \overline{b} if and only if B covers b. This proves the statement. □

4. Dade’s Ramification Group

In order to further generalise Theorem 3.4, we need to go deeper and use a subgroup with remarkable properties introduced by E. C. Dade in 1973, see [2]. This subgroup is key in the remainder of this paper. We shall use M. Murai’s version of it (see [12]). Suppose that M◁ T and that θ ∈ Irr(M) is T -invariant. If x, y ∈ T are such that [x, y] ∈ M, then Dade and Isaacs defined a complex number ⟨⟨x, y⟩⟩_{θ}, in the following way: since M⟨y⟩/M is cyclic, it follows that θ extends to some ψ ∈ Irr(M⟨y⟩). Now, ψ^{x} is some other extension and by Gallagher’s theorem, there exists some λ ∈ Irr(M⟨y⟩) such that ψ^{x} = λψ. Now ⟨⟨x, y⟩⟩_{θ} = λ(y). The properties of this (well-defined) number are listed in Lemma (2.1) and Theorem (2.3) of [6] which essentially assert that ⟨⟨, ⟩⟩_{θ} is multiplicative (in both arguments). Thus, if H, K are subgroups of T with [H, K] ⊆ M, then we have uniquely defined a subgroup

H^{⊥} ∩ K := {k ∈ K | ⟨⟨k, h⟩⟩_{θ} = 1 for all h ∈ H} .

Note that by the definition of ⟨⟨, ⟩⟩_{θ} this group always contains M ∩ K.

Lemma 4.1. Suppose that M◁ T and that θ ∈ Irr(M) is T -invariant. Suppose that H, K are subgroups of T with [H, K] ⊆ M, and M ⊆ K. If ρ ∈ Irr(K) extends θ, then ρ_{H^{⊥}∩K} is H-invariant.

Proof. Notice that M ⊆ H^{⊥} ∩ K. Let ν = ρ_{H^{⊥}∩K}. We claim that ν is H-invariant. (Since [H, K] ⊆ M, notice that H normalises every subgroup between M and K.) Now, let h ∈ H and x ∈ H^{⊥} ∩ K. We want to show that ν^{h}(x) = ν(x). Let J = M⟨m⟩, and write (ν_{J})^{h} = λν_{J}, where λ ∈ Irr(J/M). Then λ(x) = ⟨⟨h, x⟩⟩_{θ} = 1, and ν^{h}(x) = ν(x). □

For any block b ∈ Bl(N) with defect group D, we call a character η ∈ Irr(DC_{N}(D)) canonical character of b if D ⊆ ker η and bl(η)^{N} = b, see [14, p. 204].

Theorem 4.2. Let N◁ G, and let b ∈ Bl(N) be G-invariant. Then there exists a subgroup N ⊆ G[b]◁ G, uniquely determined by G and b, satisfying the following properties.

(a) Suppose that D is any defect group of b, and let η ∈ Irr(DC_{N}(D)) be a canonical character of b. Let K be the stabiliser of η in DC_{G}(D), and let H be the stabiliser of η in N_{N}(D). Then G[b] = N(H^{⊥} ∩ K).

(b) If B ∈ Bl(G) covers B′ ∈ Bl(G[b]) and B′ covers b, then B is the only block of G that covers B′.

Proof. We have that N_{N}(D) and DC_{G}(D) are normal subgroups of N_{G}(D) whose intersection is DC_{N}(D). In particular, [H, K] ⊆ DC_{N}(D) for every pair of subgroups H ⊆ N_{N}(D) and K ⊆ DC_{G}(D). Now, part (a) follows from [12, Thm. 3.13], while part (b) follows from [12, Thm. 3.5]. □

The subgroup G[b] has many further properties, but we restrict ourselves to those that shall be used in this paper. For our later considerations we mention the following.

Corollary 4.3. Let N◁ G, and let b ∈ Bl(N) be G-invariant. For Z◁ G with N ∩ Z = 1 let \overline{b} ∈ Bl(NZ/Z) be the induced block. Then G[b]/Z = \overline{G}[\overline{b}] for \overline{G} := G/Z.

Proof. ¿From the assumptions we see Z ⊆ C_{G}(N) and so Z ⊆ G[b] by Theorem 4.2(a). Let D be a defect group of b, η a canonical character of b, K the stabiliser of η in DC_{G}(D) and H the stabiliser of η in N_{N}(D). By Theorem 4.2(a) we have G[b] = N(H^{⊥} ∩ K). Now DZ/Z is a defect group of b, \overline{η} is the canonical character of b induced by η, \overline{K} := KZ/Z the stabiliser of \overline{η} in \overline{D}C\overline{G}(\overline{D}) and \overline{H} := HZ/Z the stabiliser of \overline{η} in N_{NZ/Z}(\overline{D}). For k ∈ K and h ∈ H we have

⟨⟨k, h⟩⟩_{η} = ⟨⟨kZ, hZ⟩⟩\overline{η}.

This leads to \overline{H}^{⊥} ∩ \overline{K} = (H^{⊥} ∩ K)Z/Z. Together with Theorem 4.2(a) this implies the statement. □

The properties of G[b] allow us to generalise Theorem 3.4 to the following situation.

Theorem 4.4. Let N◁ G and b ∈ Bl(N). Let θ ∈ Irr(b). Assume there exists an extension θ̃ ∈ Irr(G) of θ. If G[b] = G, then the map

υ : Bl(G/N) → Bl(G|b) given by bl(\overline{η}) ↦ bl(θ̃η)

is a bijection.

Proof. By Lemma 3.1 it is enough to show that | Bl(G|b)| = | Bl(G/N)|. By Theorem 4.2(a), we have that G = NC_{G}(D), where D is any defect group of b. Let b′ be any block of DC_{N}(D) with defect group D inducing b. By [9, Thm. C(a.2)], there exists some θ′ ∈ Irr(b′) that extends to K = DC_{G}(D). By Lemma 3.3 the unique (canonical) character η ∈ Irr(b′) that has D in its kernel extends to some η̃ ∈ Irr(K). Let T be the stabiliser of η in N_{G}(D), hence DC_{G}(D) = K ⊆ T ⊆ N_{G}(D). Let H = T ∩ N. Accordingly [H, K] ⊆ DC_{N}(D) and T = KH. By Theorem 3.4, we have that

| Bl(G/N)| = | Bl(DC_{G}(D)/DC_{N}(D))| = | Bl(DC_{G}(D)|b′)|.

By applying Lemma 4.1 (with DC_{N}(D) as the normal subgroup M of T = HK and the T -invariant character η), we have that η̃ is H-invariant. Hence, we conclude that T is the stabiliser of every block of H covering b′ (using Gallagher’s theorem). Now, let b̃ = (b′)^{N_{N} (D)}. By the Harris-Knörr correspondence [14, Thm. (9.28)] we have | Bl(G|b)| = | Bl(N_{G}(D)|b̃)|. If {e_{1}, …, e_{s}} are the blocks of DC_{G}(D) covering b′, then {e_{1}^{N_{G}(D)}, …, e_{s}^{N_{G}(D)}} are all the blocks of N_{G}(D) covering b̃. Now, if (e_{i})^{N_{G}(D)} = (e_{j})^{N_{G}(D)}, then it follows that (e_{i})^{x} = e_{j} for some x ∈ N_{G}(D). Since e_{i} and e_{j} only cover the block b′, it follows that (b′)^{x} = b′, and x ∈ T. However e_{i} is T -invariant, and therefore e_{i} = e_{j}. □

5. Character triple isomorphisms under coprime actions and blocks

In considerations using character triples the existence of an isomorphic character triple whose character is linear and faithful plays an important role. We give here an Aversion of this statement that will be used later. Furthermore we analyse how blocks behave under the bijections of characters. If A acts on G, then Irr_{A}(G) denotes the set of the irreducible complex characters of G which are invariant under A. Let IBr_{A}(G) be defined analogously.

Proposition 5.1. Let A act on G with (|G|, |A|) = 1 and N◁ G be A-stable.

(a) Let θ ∈ Irr_{GA}(N). Then there exists a character triple (G^{∗}, N^{∗}, θ^{∗}), an action of A on G^{∗} stabilising N^{∗} and θ^{∗} and an isomorphism

(ι, σ) : (G, N, θ) ⟶ (G^{∗}, N^{∗}, θ^{∗}),

such that N^{∗} ⊆ Z(G^{∗}) and both ι and σ_{G} are A-equivariant.

(b) Let θ ∈ IBr_{GA}(N). Then there exists a modular character triple (G^{∗}, N^{∗}, θ^{∗}), an action of A on G^{∗} stabilising N^{∗} and θ^{∗} and an isomorphism

(ι, σ) : (G, N, θ) ⟶ (G^{∗}, N^{∗}, θ^{∗}),

such that N^{∗} ⊆ Z(G^{∗}) is a p′-group and both ι and σ_{G} are A-equivariant.

Proof. Since the proof of both statements is based on the same ideas, we give here only the proof of (a). First note that θ extends to NA because of [7, Cor. (8.16)]. By the assumptions it is clear that (GA, N, θ) is a character triple. Let P be a projective representation of GA with the following properties:

(i) P_{NA} affords an extension of θ to NA, (ii) the values of the associated factor set α : GA × GA → ℂ are roots of unity and (iii) α is constant on N × N-cosets. Let X := GA and E be the finite cyclic group generated by the values of α. For the construction of (G^{∗}, N^{∗}, θ^{∗}) and the A-action on G^{∗} we follow the proof of Theorem (8.28) in [14]. Let X̃ be constructed as there using P: the group X̃ consists of pairs (x, ϵ) with x ∈ X and ϵ ∈ E and multiplication in X̃ is given by

(x_{1}, ϵ_{1})(x_{2}, ϵ_{2}) = (x_{1}x_{2}, α(x_{1}, x_{2})ϵ_{1}ϵ_{2}).

The projective representation P lifts to a representation of X̃. Let τ be the character afforded by that representation. The groups Ñ := {(n, ϵ) | n ∈ N, ϵ ∈ E} and G̃ := {(g, ϵ) | g ∈ G, ϵ ∈ E} are normal in X̃. Since P_{NA} is a representation the set {(x, 1) | x ∈ NA} forms a group isomorphic to NA. Via this isomorphism A can be identified with a group of X̃ and acts on G̃ and Ñ. Let E_{0} := 1×E and N_{0} := N ×1. Identifying N and N_{0} we set θ̃ = θ × 1_{E}. This character is A-invariant. Via the epimorphism ι_{1} : X̃ → X, (x, ϵ) ↦ x, the character triples (X, N, θ) and (X̃, Ñ, θ̃) are isomorphic. The map λ̃ ∈ Irr(Ñ) with λ̃(n, ϵ) = ϵ^{−1} is a linear character with kernel N_{0}. By the construction of X̃ the character λ̃ is X̃-invariant. We see that τÑ = λ̃^{−1}θ̃. Now one can argue that (X̃, Ñ, λ̃) and (X̃, Ñ, θ̃) are isomorphic character triples. Analogously (X̃, Ñ, λ̃) and (X̃/N_{0}, Ñ/N_{0}, λ) are isomorphic character triples, where λ ∈ Irr(Ñ/N_{0}) is the character induced by λ̃. With G^{∗} := G̃/N_{0}, N^{∗} := Ñ/N_{0} and θ^{∗} := λ we obtain the required isomorphism

(ι, σ) : (G, N, θ) ⟶ (G^{∗}, N^{∗}, θ^{∗}).

Let ι_{2} : X̃ → X̃/N_{0} be the canonical epimorphism. Because of ker ι_{1} = E and ker ι_{2} = N_{0} the isomorphism ι is given by gN ↦ ι_{2} ∘ ι_{1}^{−1}(gN). For χ ∈ Irr(G|θ) the character σ_{G}(χ) is obtained in the following way: the character χ lifts to some τG̃μ for some μ ∈ Irr(G̃) with ker μ ≥ N and μ ∈ Irr(G̃|λ̃). Hence μ ∘ ι_{2}^{−1} is a character of Irr(G̃/N_{0}|λ). By its construction θ^{∗} is A-invariant and the maps ι and σ_{G} are A-equivariant, since τG̃ is A-invariant. □

In order to include blocks in the above result, additional assumptions are required.

Proposition 5.2. Let N◁ G and b ∈ Bl(N). Suppose that A acts on G with (|G|, |A|) = 1, such that N and b are A-stable. Assume G[b] = G.

(a) Let θ ∈ Irr_{GA}(N) ∩ Irr(b). Let (G^{∗}, N^{∗}, θ^{∗}) and (ι, σ) : (G, N, θ) ⟶ (G^{∗}, N^{∗}, θ^{∗}) be as in Proposition 5.1(a). Then two characters χ_{1}, χ_{2} ∈ Irr(G|θ) satisfy bl(χ_{1}) = bl(χ_{2}) if and only if bl(σ_{G}(χ_{1})) = bl(σ_{G}(χ_{2}))

(b) Let θ ∈ IBr_{GA}(N) ∩ IBr(b). Let (G^{∗}, N^{∗}, θ^{∗}) and (ι, σ) : (G, N, θ) ⟶ (G^{∗}, N^{∗}, θ^{∗}) be as in Proposition 5.1(b). Then two characters ϕ_{1}, ϕ_{2} ∈ IBr(G|θ) satisfy bl(ϕ_{1}) = bl(ϕ_{2}) if and only if bl(σ_{G}(ϕ_{1})) = bl(σ_{G}(ϕ_{2})).

Proof. We continue using the notation introduced in the proof of Proposition 5.1. Let N_{0} := N × 1 ⊆ X̃ and θ_{0} := θ × 1_{E} ∈ Irr(Ñ). Let b_{0} := bl(θ_{0}). From Corollary 4.3 we see that G[b] = G̃[b_{0}]/E, where b_{0} := bl(θ_{0}). For the proof of (a) let μ_{1} and μ_{2} ∈ Irr(G̃) with N_{0} ⊆ ker μ_{1} and N_{0} ⊆ ker μ_{2} such that τG̃μ_{1} is a lift of χ_{1} and τG̃μ_{2} is a lift of χ_{2}. Note that E ⊆ Z(G̃). According to [14, (9.9) and (9.10)] bl(χ_{1}) = bl(χ_{2}) if and only if τG̃μ_{1} and τG̃μ_{2} belong to the same block. According to Theorem 4.4 we see that the characters of G̃/N_{0} induced by μ_{1} and μ_{2} are in the same block if and only if bl(τG̃μ_{1}) = bl(τG̃μ_{2}). This proves the statement of (a). Part (b) follows from similar considerations. □

As a corollary that might be of independent interest we conclude the following.

Corollary 5.3. Let N◁ G and b ∈ Bl(N). Assume there exists some G-invariant θ ∈ Irr_{A}(N) ∩ Irr(b) or θ ∈ IBr_{A}(N) ∩ IBr(b). Let (G^{∗}, N^{∗}, θ^{∗}) and (ι, σ) : (G, N, θ) ⟶ (G^{∗}, N^{∗}, θ^{∗}) be as in Proposition 5.1. Let H := G[b] and H^{∗} the group with ι(H/N) = H^{∗}/N^{∗}.

(a) Then Bl(G|b) is in bijection with Bl(H^{∗}|b^{∗}), where b^{∗} := bl(θ^{∗}).

(b) Two characters χ_{1}, χ_{2} ∈ Irr(G|θ) satisfy bl(χ_{1}) = bl(χ_{2}) if and only if bl(σ_{G}(χ_{1})) and bl(σ_{G}(χ_{2})) cover the same block of H^{∗}.

(c) Two characters ϕ_{1}, ϕ_{2} ∈ IBr(G|θ) satisfy bl(ϕ_{1}) = bl(ϕ_{2}) if and only if bl(σ_{G}(ϕ_{1})) and bl(σ_{G}(ϕ_{2})) cover the same block of H^{∗}.

Proof. Part (a) follows directly from Proposition 5.2. Parts (b) and (c) are applications of Proposition 5.2 together with Theorem 4.2(b). □

6. Reduction

In this section we show how Theorem 1.1 is implied by the analogous statement for the central product of quasi-simple groups given in Corollary 2.6. In fact, we will work with the following slightly more general statement.

Theorem 6.1. Let the group A act on the group G with (|A|, |G|) = 1.

(a) Let Z be an A-invariant central p-subgroup of G, let ν ∈ Irr_{A}(Z) and let B ∈ Bl_{A}(G). Then there exists an A-invariant character χ ∈ Irr(B|ν).

(b) Let B ∈ Bl_{A}(G). Then there exists an A-invariant character ϕ ∈ IBr(B).

The following well-known result will be used for part (a).

Theorem 6.2. Let the group A act on the group G with (|A|, |G|) = 1 and let N◁ G be A-stable. Let θ ∈ Irr_{A}(N). Then there exists an A-invariant character in Irr(G|θ).

Proof. This is Theorem (13.28) and Corollary (13.30) of [7]. □

For the proof of Theorem 6.1(b) we need the following analogue for Brauer characters.

Theorem 6.3. Let the group A act on the group G with (|A|, |G|) = 1 and let N◁ G be A-stable. Let ϕ ∈ IBr_{A}(N). Then there exists an A-invariant character in IBr(G|ϕ).

Proof. We prove this statement by induction on |G : N| and then on |G : Z(G)|. We can assume that ϕ is G-invariant, since otherwise some ϕ′ ∈ IBr_{A}(G_{ϕ}|ϕ) exists by induction and hence ϕ^{′G} ∈ IBr_{A}(G|ϕ). Assume there exists an A-stable subgroup K◁ G with N ⪇ K ⪇ G. Then by induction there exists some A-invariant character ϕ′ ∈ IBr(K|ϕ) and one in IBr(G|ϕ′). Accordingly we can assume that G/N is a chief factor of GA and hence G/N is the direct product of isomorphic simple groups, such that A acts transitively on the factors of G/N. Then (G, N, ϕ) forms a modular character triple that is isomorphic to some (G^{∗}, N^{∗}, ϕ^{∗}) according to Proposition 5.1, such that N^{∗} ⊆ Z(G^{∗}) and p ∤ |N^{∗}|. Since the isomorphism of the character triples is A-equivariant it is sufficient to prove the statement for (G^{∗}, N^{∗}, ϕ^{∗}). For this it suffices to prove that there exists some A-invariant character in IBr(G^{∗}|θ^{∗}). If G/N is a q-group for p ≠ q, the group G^{∗} is a p′-group and the statement follows immediately from Theorem 6.2. If G/N is a p-group, the set IBr(G^{∗}|ϕ^{∗}) is a singleton. Let ν ∈ Irr(N^{∗}) be the character with ν^{0} = ϕ^{∗}. (Note that ν is unique since N^{∗} is a p′-group.) Since ν is A-invariant there exists some A-invariant χ ∈ Irr(G^{∗}|ν) by Theorem 6.2. This character hence belongs to an A-invariant block B ∈ Bl(G^{∗}). Further G^{∗}/N^{∗} is the direct product of isomorphic non-abelian simple groups that are permuted transitively by A. According to Corollary 2.6 there exists an A-invariant Brauer character in IBr(B). □

We start proving Theorem 6.1 in a series of intermediate results, working by induction first on |G/Z(G)| and second on |G|. It is clear that we may assume that Z ∈ Syl_{p}(Z(G)).

Let N◁ G such that G/N is chief factor of GA. Then by Glauberman’s Lemma, [7, Lemma (13.8)], there exists b ∈ Bl_{A}(N) such that B ∈ Bl(G|b). (Recall that by [14, Cor. (9.3)], G acts transitively on the set of blocks of N covered by b.) Another application of Glauberman’s Lemma shows that b has an A-invariant defect group D. Next, notice that Z ⊆ N in part (a) of Theorem 6.1. Otherwise, we have that NZ = G, and therefore G/N is a p-group. In particular B is the only block covering b ([14, Cor. (9.6)]). Let ν ∈ Irr_{A}(Z). Since b has an A-invariant character χ_{1} ∈ Irr(N|ν_{Z∩N}) by induction, the character χ_{1} · ν defined as in [8, Section 5] has the required properties and we are done. Analogously one can argue that Z(G) ⊆ N since there exists a unique μ ∈ Irr(Z(G)|ν) in a block of Z(G) covered by B.

Lemma 6.4. We can assume that G_{b} = G.

Proof. By the Fong-Reynolds theorem [14, Thm. (9.14)], there exists a unique B̃ ∈ Bl(G_{b}|b) with B̃^{G} = B. Since b is A-invariant, so is G_{b}. Also, B̃ is A-invariant by uniqueness. Notice that Z ⊆ Z(G) ⊆ G_{b}. If G_{b} < G, then by induction G_{b} has an Ainvariant character χ_{0} ∈ Irr(B̃) ∩ Irr_{A}(G_{b}|ν) and ϕ_{0} ∈ IBr(B̃). Now, the characters χ_{0}^{G} and ϕ_{0}^{G} are irreducible, A-invariant and belong to B. □

Lemma 6.5. We can assume that G[b] = G.

Proof. By Theorem 4.2(b) there exists a unique B′ ∈ Bl(G[b]|b) with (B′)^{G} = B. Since G[b] is uniquely determined by b, we have that G[b] is A-stable and by uniqueness that B′ is A-invariant. Note that Z(G) ⊆ G[b] by Theorem 4.2(a). If G[b] ≠ G then we can conclude by induction that there exist some A-fixed χ_{0} ∈ Irr(G[b]|ν) ∩ Irr(B′) and ϕ_{0} ∈ IBr(B′). By Theorem 6.2 and 6.3 there exist some χ ∈ Irr(G|χ_{0}) and ϕ ∈ IBr(G|ϕ_{0}), respectively that is A-fixed. Those characters belong to Bl(G|B′) = {B}. □

Lemma 6.6. We can assume that N = Z(G).

Proof. By induction on |G : Z(G)| we see that b contains an A-invariant character θ ∈ Irr(b|ν), respectively θ ∈ IBr(b). By the above we can assume G[b] = G. Let (G^{∗}, N^{∗}, θ^{∗}) be the character triple associated to (G, N, θ) and σ_{G} the Aequivariant bijection Irr(G|θ) → Irr(G^{∗}|θ^{∗}) from Proposition 5.1(a), respectively the A-equivariant bijection IBr(G|θ) → IBr(G^{∗}|θ^{∗}) from Proposition 5.1(b). According to Proposition 5.2(b) there is some block C ∈ Bl(G^{∗}) such that σ_{G}(Irr(B|θ)) = Irr(C|θ^{∗}) or σ_{G}(IBr(B|θ)) = IBr(C|θ^{∗}), respectively.

If N ≠ Z(G) then Irr(C|θ^{∗}) and IBr(C|θ^{∗}) both contain A-invariant characters because |G^{∗} : Z(G^{∗})| = |G : N| < |G : Z(G)|. Since σ_{G} is A-equivariant this proves the statement. □

Proof of Theorem 6.1. Now it remains to consider the case where Z(G) = N. Since N was chosen such that G/N is a chief factor of GA, the quotient G/N is the direct product of isomorphic simple groups that are transitively permuted by A. If G/N is non-abelian, Corollary 2.6 applies and proves the statement. Otherwise G/N is an elementary abelian p-group or a p′-group. In the first case Bl(G|b) is a singleton and the statement follows from Theorems 6.2 and 6.3. In the latter case B has a central defect group and the sets Irr(B|ν) and IBr(B) are singletons. □

References

[1] M. Cabanes, M. Enguehard, Representation theory of finite reductive groups. Cambridge University Press, Cambridge, 2004.

[2] E.C. Dade. Block extensions. Illinois J. Math. 17 (1973), 198–272.

[3] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups. Number 3. Part I. Chapter A. American Mathematical Society, Providence, 1998.

[4] M. E. Harris. Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. Algebra 41 (2013), 3509–3540.

[5] J.E. Humphreys. Defect groups for finite groups of Lie type. Math. Z. 119 (1971), 149–152.

[6] I. M. Isaacs. Characters of solvable and symplectic groups. Amer. J. Math. 95 (1973), 594–635.

[7] I. M. Isaacs. Character Theory of Finite Groups. Academic Press, New York, 1976.

[8] I. M. Isaacs, G. Malle and G. Navarro. A reduction theorem for the McKay conjecture. Invent. Math. 170(1) (2007), 33–101.

[9] S. Koshitani and B. Späth. Clifford theory of characters in induced blocks. To appear in Proc. Amer. Math. Soc, 2013.

[10] G. Malle and G. Navarro. Extending characters from Hall subgroups. Doc. Math. 16 (2011), 901–919.

[11] G. Malle and D. Testerman. Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011.

[12] M. Murai. On blocks of normal subgroups of finite groups. Osaka J. Math. 50 (2013), 1007–1020.

[13] G. Navarro. Some open problems on coprime action and character correspondences. Bull. London Math. Soc. 26 (1994), 513–522.

[14] G. Navarro. Characters and Blocks of Finite Groups. London Math. Soc. Lecture Note Series. Cambridge University Press, Cambridge, 1998.

[15] G. Navarro. Actions and characters in blocks. J. Algebra 275 (2004), 471–480.

[16] G. Navarro and B. Späth. On Brauer’s height zero conjecture. J. Eur. Math. Soc. 16 (2014), 695–747.

[17] G. Navarro and P. H. Tiep. Brauer’s height zero conjecture for the 2-blocks of maximal defect. J. Reine Angew. Math. 669 (2012), 225–247.

Gunter Malle FB Mathematik TU Kaiserslautern Postfach 3049 67653 Kaiserslautern Germany. malle@mathematik.uni-kl.de Gabriel Navarro Departament d’Àlgebra Universitat de València Dr. Moliner 50 46100 Burjassot Spain. gabriel.navarro@uv.es

Britta Späth FB Mathematik TU Kaiserslautern Postfach 3049 67653 Kaiserslautern Germany. spaeth@mathematik.uni-kl.de