THE INDUCTIVE MCKAY CONDITION FOR SIMPLE GROUPS NOT OF LIE TYPE

GUNTER MALLE

Abstract. We show that the alternating groups, the sporadic groups and the exceptional covering groups of groups of Lie type satisfy the inductive condition required to verify the McKay conjecture for finite groups.

1. Introduction

More than thirty years ago John McKay observed that the number of complex irreducible characters of odd degree of some finite simple groups coincided with the same number for the normalizers of their Sylow 2-subgroups. This was then extended by Alperin to the conjecture that for any finite group G and for any prime p the number |Irr_{p′}(G)| of complex irreducible characters of p′-degree coincides with the number |Irr_{p′}(N_{G}(P))| of complex irreducible characters of p′-degree of the normalizer N_{G}(P) of any Sylow p-subgroup P of G.

This is now known to hold for p-solvable groups by work of Wolf [19]. In [14] we showed that the McKay conjecture on character degrees holds for a finite group G and a prime p if all non-abelian simple groups X involved in G are good for p, i.e., they satisfy a certain inductive condition on the complex irreducible characters of perfect central extensions of X. In the same paper, we proved that the simple groups L_{2}(q), the Suzuki groups and the Ree groups in characteristic 3 are good for all p.

The aim of this paper is to reduce the question to one about non-exceptional covering groups of groups of Lie type. More precisely we show the following:

Theorem 1.1. The inductive form of the McKay conjecture is true for those quasisimple groups S which are not non-exceptional covering groups of a finite group of Lie type.

In order to prove the theorem we have to handle the alternating groups, the exceptional covering groups of groups of Lie type, and the sporadic groups. This is done in Theorems 3.1, 4.1 and 5.1 in the next three sections.

2. Some reductions

In [14] we showed that the McKay-conjecture holds for a finite group G and a prime p if a certain inductive condition is satisfied for all finite simple groups involved in G. We start by recalling this condition on finite simple groups. In fact, we will only formulate a restricted version of this condition which is sufficient for the purpose of this paper.

Date: September 5, 2006.

Let X be a non-abelian simple group and p a prime. Let Z be a factor group of the Schur multiplier of X, of order prime to p, and S the perfect central extension of X with center Z(S) = Z. Let ν denote a faithful character of Z and P a Sylow p-subgroup of S. Let A = C_{Aut(S)}(Z) ∩ N_{Aut(S)}(P). Assume that there exists a subgroup N_{S}(P) ≤ T < S of S normalized by A and an A-equivariant bijection

∗ : Irr_{p′}(S|ν) ⟶ Irr_{p′}(T |ν), θ ↦ θ^{∗},

with the following properties. For θ ∈ Irr_{p′}(S|ν) with inertia group A_{θ} in A, let G be a group containing S as a normal subgroup with Z = Z(G) = C_{G}(S) and such that N_{G}(P) induces precisely A_{θ} on S. To any G-invariant character θ of S there is attached a cohomology class [θ]_{G/S} controlling the extendibility of θ to G (see [13, Chapt. 11], for example). Assume then that the cohomology classes [θ]_{G/S} and [θ^{∗}]_{N_{G}(T)/T} coincide under the natural isomorphism induced by G/S ≅ N_{G}(T)/T. If these conditions are satisfied for all choices of θ ∈ Irr_{p′}(S|ν) then we say that the pair (X, Z) is very good for p.

A simple group X such that (X, Z) is very good for p for all choices of Z is good for p in the sense of [14], but the converse is not necessarily true.

Except for one example, all the perfect groups S considered in this paper are such that A/(A ∩ S) is trivial or cyclic of prime order. In this case, the condition on cohomology elements is trivially satisfied, since the relevant cohomology groups are trivial, by [13, Cor. 11.22]. Then, it remains to prove the existence of an Aequivariant bijection as above. This can be done as follows:

Lemma 2.1. Let G be a finite group, H ≤ G a normal subgroup of prime index r with Z(G) = Z(H), and P a Sylow p-subgroup of G for some prime p. Assume that

|Irr_{p′}(G|ν)| = |Irr_{p′}(N_{G}(P)|ν)| and |Irr_{p′}(H|ν)| = |Irr_{p′}(N_{H}(P ∩ H)|ν)|

for some ν ∈ Irr(Z(G)) and either (i) p ≠ r, or (ii) N_{H}(P ∩ H) = N_{G}(P) ∩ H.

Then there is an N_{G}(P)-equivariant bijection

∗ : Irr_{p′}(H|ν) → Irr_{p′}(N_{H}(P ∩ H)|ν).

Proof. By assumption H is normal in G, so G acts on Irr_{p′}(H) and on Irr_{p′}(H|ν). We denote by n_{1} the number of G-invariant p′-degree characters of H lying over ν, and by n_{2} the number of orbits of non-invariant p′-degree characters of H lying over ν. Thus |Irr_{p′}(H|ν)| = n_{1} + rn_{2}. Since G/H is cyclic, all G-invariant characters extend to G, while the others induce irreducibly. Write X for the set of characters of G lying above some member of Irr_{p′}(H|ν). Then clearly |X| = rn_{1} + n_{2} and Irr_{p′}(G|ν) ⊆ X, by Clifford theory.

Note that in both cases (i) and (ii) we have that N_{H}(P ∩ H) is normal of index r in N_{G}(P). Thus the same considerations as for (H, G) apply to the pair (N_{H}(P ∩ H), N_{G}(P)). We write |Irr_{p′}(N_{H}(P ∩ H)|ν)| = m_{1} + rm_{2} and |Y | = rm_{1} + m_{2} for the set of characters Y of N_{G}(P) lying above some member of Irr_{p′}(N_{H}(P ∩ H)|ν). In particular we have n_{1} + rn_{2} = m_{1} + rm_{2}.

Now first assume that r ≠ p. Then X = Irr_{p′}(G|ν), Y = Irr_{p′}(N_{G}(P)|ν), and hence rn_{1} + n_{2} = rm_{1} + m_{2}, which means that n_{1} = m_{1}, n_{2} = m_{2}. Thus the number of invariant and of non-invariant characters is the same for both H and N_{H}(P), and any bijection respecting G-invariance will do.

On the other hand, if r = p but N_{H}(P ∩ H) = N_{G}(P) ∩ H then Irr_{p′}(G|ν) is just equal to the set of extensions of invariant characters, hence |Irr_{p′}(G|ν)| = rn_{1}, and similarly |Irr_{p′}(N_{G}(P)|ν)| = rm_{1}. Again we obtain that n_{1} = m_{1} and n_{2} = m_{2}, and we can conclude as in the first case. □

Note that the assumption |Irr_{p′}(G)| = |Irr_{p′}(N_{G}(P))| is just the ordinary McKay conjecture for the group G. Hence we have:

Corollary 2.2. Let S be perfect with (C_{Aut(S)}(Z) : S/Z) of prime order r, where Z := Z(S) is cyclic. Assume the following:

(1) the ordinary McKay-conjecture for the prime p holds for all central factor groups of S,

(2) the ordinary McKay-conjecture for the prime p holds for all central factor groups of a group G containing S as a normal subgroup of index r and inducing C_{Aut(S)}(Z) on S,

(3) either p ≠ r, or N_{S}(P ∩ S) = N_{G}(P) ∩ S, where P is a Sylow p-subgroup of

G. Then (S/Z, Z) is very good for p.

Proof. Since Irr(S) = ∪_{ν}Irr(S|ν), the validity of the McKay-conjecture for all central factor groups implies that |Irr_{p′}(S|ν)| = |Irr_{p′}(N_{S}(P_{1})|ν)| for all ν ∈ Irr(Z), where P_{1} := P ∩S, and similarly for G. Thus Lemma 2.1 with H := S provides an N_{G}(P)- equivariant bijection ^{∗} : Irr_{p′}(S|ν) ⟶ Irr_{p′}(T |ν) with T := N_{S}(P_{1}) as required. □

The application of this criterion is sometimes facilitated by the fact that the ordinary McKay conjecture is known in the case of groups with a cyclic Sylow psubgroup [7]:

Theorem 2.3 (Dade). Let G be a finite group with cyclic Sylow p-subgroup. Then the ordinary McKay conjecture holds for G at the prime p.

3. Alternating groups

We can now prove that previous results of Michler and Olsson imply the validity of the inductive condition for the alternating groups 𝔄_{n}, n ≥ 5.

Theorem 3.1. Let X = 𝔄_{n} for n ≥ 5. Then X is good for all divisors p of its order in the sense of [14].

Proof. This was shown for n ≤ 6 in [14, Thm. 15.3], so assume that n ≥ 7. In particular, Out(X) is cyclic of order 2. We have to check that for all covering groups S of X with |Z(S)| prime to p and all faithful ν ∈ Z(S) there exist bijections

∗ : Irr_{p′}(S|ν) → Irr_{p′}(N_{S}(P)|ν)

satisfying suitable compatibility properties with respect to A := N_{Aut(S)}(P), where P is a Sylow p-subgroup of S. For Out(X) cyclic, this boils down to ^{∗} being Aequivariant, that is, ^{∗} respects inertia groups in A. First assume that S = X. Then Michler and Olsson [16] show that

|Irr_{p′}(𝔖_{n})| = |Irr_{p′}(N_{𝔖_{n}}(P))| and |Irr_{p′}(𝔄_{n})| = |Irr_{p′}(N_{𝔄_{n}}(P ∩ 𝔄_{n}))|

where P denotes a Sylow p-subgroup of 𝔖_{n}. (Despite its title, this reference proves what we need also for p = 2.) The claim follows from Lemma 2.1. (For p = 2 note that the Sylow 2-subgroup of 𝔄_{n} is self-normalizing, so condition (ii) of Lemma 2.1 is satisfied.) Next assume that S = 2.𝔄_{n}, the 2-fold covering group, so p ≠ 2. Again it is shown in [16] that the McKay conjecture for both 2.𝔄_{n} and 2.𝔖_{n} is satisfied, and we may conclude with Lemma 2.1.

It only remains to treat the exceptional covering groups 3.𝔄_{7} and 6.𝔄_{7} to complete the proof. Neither 3.𝔄_{7} nor 6.𝔄_{7} admits outer automorphisms centralizing the center. Thus we just need to check that |Irr_{p′}(S|ν)| = |Irr_{p′}(N_{S}(P)|ν)| for all primes p coprime to |Z(S)| and ν a faithful character of Z(S), where S ∈ {3.𝔄_{7}, 6.𝔄_{7}}. By Theorem 2.3 the result is clear when the Sylow p-subgroups are cyclic, that is, for p = 5, 7. We are left with the case p = 2, and thus with S = 3.𝔄_{7}. According to the Atlas [6] we have |Irr_{2′}(3.𝔄_{7}|ν)| = 4, and it is straightforward to check the required equality. □

4. Exceptional multipliers of groups of Lie type

We now turn to the exceptional multipliers of groups of Lie type.

Theorem 4.1. Let X be a simple group of Lie type and S a perfect central extension such that Z = Z(S) is an exceptional factor group of the Schur multiplier of X. Then (X, Z) is very good for all primes p.

Table 1. Exceptional covering groups

\begin{matrix}S & \overline{A} & p \\ 2.L_{3}(4) & 2^{2} & 3, 5, 7 \\ 4_{1}.L_{3}(4) & 2_{3} & 3 \\ 4_{2}.L_{3}(4) & 2_{2} & 3 \\ 3_{1}.U_{4}(3) & 2_{2} & 2 \\ 3_{2}.U_{4}(3) & 2_{3} & 2 \\ 2.U_{6}(2) & 2 & 3 \\ 2.S_{6}(2) & 1 & 3 \end{matrix}  \begin{matrix}S & \overline{A} & p \\ 3.O_{7}(3) & 1 & 2 \\ 2.O_{8}^{+}(2) & 2 & 3, 5 \\ 3.G_{2}(3) & 1 & 2 \\ 2.G_{2}(4) & 2 & 3, 5 \\ 2.F_{4}(2) & 2 & 3, 5, 7 \\ 2.^{2}E_{6}(2) & 2 & 3, 5, 7 \\ 6.^{2}E_{6}(2) & 1 & 5, 7 \end{matrix}

Proof. The simple groups of Lie type with an exceptional multiplier are the following: L_{2}(4), L_{2}(9), L_{3}(2), L_{4}(2), L_{3}(4), U_{4}(2), U_{4}(3), U_{6}(2),

S_{6}(2), O_{7}(3), O_{8}^{+}(2), Sz(8), G_{2}(3), G_{2}(4), F_{4}(2), ^{2}E_{6}(2).

Among these, L_{2}(4), L_{2}(9) and L_{4}(2) are isomorphic to alternating groups which were treated in the previous section. The groups L_{3}(2) ≅ L_{2}(7) and U_{4}(2) = S_{4}(3) are isomorphic to groups of Lie type for which the multiplier is not exceptional. The covering groups of Sz(8) were handled in [14]. We are left with the exceptional covering groups of the ten groups

L_{3}(4), U_{4}(3), U_{6}(2), S_{6}(2), O_{7}(3), O_{8}^{+}(2), G_{2}(3), G_{2}(4), F_{4}(2), ^{2}E_{6}(2).

Note that we only have to consider primes p not dividing |Z(S)|. Moreover, if Ā := C_{Aut(S)}(Z)/(S/Z) is cyclic of prime order r ≠ p, only primes p with non-cyclic Sylow p-subgroup need to be considered, by Corollary 2.2 and Theorem 2.3. The Atlas [6] shows that thus only the fourteen groups S in Table 1 with the given primes p and the given groups Ā remain. (For example, the exceptional 6-fold cover 6.L_{3}(4) with Ā = C_{2} has order divisible by p = 2, 3, 5, 7, but 2, 3 divide |S|, while the Sylow p-subgroups for p = 5, 7 are cyclic, so it does not appear in Table 1. Also recall that the 3-fold cover SL_{3}(4) of L_{3}(4) is non-exceptional.)

The only case with Ā non-cyclic is S = 2.L_{3}(4) where we have to treat all primes not dividing |Z|, that is 3,5,7. Here Ā is the Klein four group, with Schur multiplier of order 2. Thus there is just one non-trivial cohomology class, and a character with inertia factor group Ā does not extend to A if and only if the corresponding cohomology class is the non-trivial one. Hence in order to check equality of cohomology classes, we just have to compare extendibility. Now the character tables of all Sylow normalizers involved are easily calculated in GAP [9] and it follows that S = 2.L_{3}(4) is very good for all p.

We claim that in all other cases, we are in the situation of Corollary 2.2. Indeed, except for S = 3.U_{4}(3) we always have r ≠ p. But the Sylow 2-subgroups of U_{4}(3) are self-normalizing, so the alternative condition in Corollary 2.2(3) is satisfied in this case. The numbers Irr_{p′}(S|ν) and Irr_{p′}(G|ν) can now be extracted from the Atlas. So it remains to look at the Sylow normalizers. First consider the three-fold covers

S ∈ {3_{1}.U_{4}(3), 3_{2}.U_{4}(3), 3.O_{7}(3), 3.G_{2}(3)},

so p = 2. In all four cases, the Sylow 2-subgroup of X = S/Z is self-normalizing, so the normalizer of a Sylow 2-subgroup P of S is the direct product of P with a cyclic group of order 3. Thus, Irr_{p′}(N_{S}(P)) can be computed from data inside the simple group S/Z. For 2.U_{6}(2), the Sylow 3-normalizer is contained inside the subgroup 2.U_{4}(3).2_{2}, so again computations are reduced to this smaller group. For 2.G_{2}(4) the Sylow 3-normalizer is contained in the subgroup 3.L_{3}(4).2_{3}, the Sylow 5-normalizer in 2.J_{2}. With these reduction, the character tables of the Sylow normalizers can be computed in all but the last three groups using GAP.

Now assume X = F_{4}(2). The Sylow 3-normalizer of X is contained in the subgroup L_{4}(3).2_{2}, and the one of X.2 inside L_{4}(3).2^{2}. Let H denote the full preimage of L_{4}(3).2^{2} in the double cover of F_{4}(2).2. We claim that H = 2 × L_{4}(3).2^{2}. Indeed, H is some central extension of L_{4}(3).2^{2}, with |Z(H)| = 2. Thus it’s second derived subgroup is a central extension of L_{4}(3), with centre of order at most 2. The only non-split such extension has the property that some involutions of L_{4}(3) lift to elements of order 4. But according to the Atlas all involutions of F_{4}(2).2 lift to involutions in the double cover. Thus H′′ = L_{4}(3). Now H/H′′ is a central extension of 2^{2}, all of whose elements have order 2, so H/H′′ is elementary abelian. Thus the structure of H is as claimed. This allows to check the assertion for p = 3.

The Sylow 5-normalizer of X is a (split) extension of C_{5}^{2} with the complex reflection group G_{8}, the Sylow 5-normalizer of F_{4}(2).2 is an extension of C_{5}^{2} with a group isoclinic to the complex reflection group G_{9}. The first group has trivial Schur multiplier by [17], the second has Schur multiplier C_{2}. With GAP it is possible to compute a corresponding covering group of the normalizer, thus giving the result for p = 5. The Sylow 7-normalizer of X is an extension of C_{7}^{2} with the complex reflection group G_{5}, which has Schur multiplier C_{3} by [17]. The Sylow 7-normalizer of F_{4}(2).2 is an extension of C_{7}^{2} with a group isoclinic to the complex reflection group G_{14} and has trivial Schur multiplier by GAP. Thus the Sylow 7-normalizer in 2.X is the direct product of the one in X with C_{2}, and similarly in 2.F_{4}(2).2. see Table 2 for the character degrees. This completes the proof for X = F_{4}(2).

Finally, let X = ^{2}E_{6}(2) and first p = 3. The Sylow 3-normalizer of X is contained in a subgroup O_{7}(3), the one of X.2 in a subgroup O_{7}(3).2. Since all involutions of ^{2}E_{6}(2).2 lift to involutions in the double cover, O_{7}(3) lifts to the direct product 2 × O_{7}(3), and O_{7}(3).2 lifts to (2 × O_{7}(3)).2. The number of 3′-degree characters of the latter are known from the Atlas. For p = 5 the Sylow normalizer in X.2 is contained in the maximal subgroup 𝔖_{3} × O_{8}^{+}(2).𝔖_{3}, hence isomorphic to 𝔖_{3} × 5^{2}.G_{8} (since O_{8}^{+}(2).𝔖_{3} contains the Sylow 5-normalizer of F_{4}(2)). The Sylow 5-normalizer in X is the normal subgroup (3 × 5^{2}.4𝔄_{4}).2 of index 2, with the two on top acting non-trivially on both factors. Both groups can be explicitly constructed in GAP. One finds that 𝔖_{3} × 5^{2}.G_{8} has trivial Schur multiplier, while the Schur multiplier of (3 × 5^{2}.4𝔄_{4}).2 has order 3. Thus, both lift to direct products with the center in the twofold cover. From this, the assertion can already be checked for S = 2.X. In the sixfold cover S = 6.X, the preimage of (3 × 5^{2}.4𝔄_{4}).2 has 40 faithful characters, see Table 2. The claim follows. Finally, the Sylow 7-normalizer in ^{2}E_{6}(2) is contained in F_{4}(2). The latter lifts to 2.F_{4}(2) in the exceptional covering group 2.^{2}E_{6}(2), to 2.F_{4}(2) × 2 in 2.^{2}E_{6}(2).2, and to 3 × 2.F_{4}(2) in 6.^{2}E_{6}(2). For all these groups, the number of 7′-character degrees are known by our previous computations for F_{4}(2), allowing to conclude. □

5. Sporadic groups

Finally, we treat the sporadic simple groups. Most cases were already considered before by other authors, but we still have to deal with decorations of two of the three biggest groups.

Theorem 5.1. Let X be a sporadic simple group. Then X is good for all primes p.

Proof. The ordinary McKay conjecture was proved for all sporadic simple groups by Wilson [18]. Thus it remains to check the sporadic groups with non-trivial multiplier and/or outer automorphism group. Note that Out(X) is either trivial or cyclic of order 2 for sporadic groups, so we are in the situation of Corollary 2.2. In the papers [1, 2, 3, 4, 5, 8, 10, 11, 12, 15], the invariant projective version of Dade’s conjecture is checked for the groups

He, O′N, Ru, HN, Co_{1}, F i_{22}, McL, HS, Suz, M_{22}, J_{3}.

This implies the validity of the ordinary McKay conjecture for all bicyclic extensions, hence our assertion follows by Corollary 2.2 for the named groups. The only other sporadic groups with non-trivial multiplier or outer automorphism group are

M_{12}, J_{2}, F i′_{24}, B.

For S = 2.M_{12} and S = 2.J_{2} direct computation of Sylow normalizers in lowdimensional representations allows to conclude.

For X = F i′_{24} we have to check X.2 and 3.X. For X.2 the Sylow p-subgroups are non-abelian for p = 2, 3, 5, 7, for 3.X we have to check the primes p = 2, 5, 7. The Sylow 2-subgroup of X is self-normalizing, so the same is true for X.2. But the Sylow 2-subgroup of X.2 is contained in the maximal subgroup (2 × 2.F i_{22}).2, for which the character degrees can be found in the Atlas. It follows moreover that the Sylow 2-normalizer in 3.X is the direct product with C_{3}, reducing this situation to the case of X. This completes the verification for p = 2.

The Sylow 3-normalizer of X.2 is contained in the maximal subgroup of type 3^{2+4+8}.(𝔖_{5} × GL_{2}(3)), and the action of (𝔖_{5} × GL_{2}(3)) on the 3^{8} is described in [18, p.302]. This information allows to compute the character table of the Sylow 3-normalizer explicitly, see Table 2, finishing off the prime p = 3. The Sylow 5normalizer of X.2 is contained in the maximal subgroup 𝔖_{4} × O_{8}^{+}(3).𝔖_{3}. The latter has 5 · 20 characters of degree prime to 5, as X.2. According to [18, p.299], the structure of the Sylow 5-normalizer in X is (𝔄_{4} × 5^{2}.4𝔄_{4}).2, hence an extension of the Klein four group with the Sylow 5-normalizer in ^{2}E_{6}(2). Here, the Schur multiplier is cyclic of order 6, and the corresponding 3-fold covering has 80 faithful characters (of degree prime to 5), as desired. The Sylow 7-normalizer in X.2 may be constructed explicitly from the information in [18, Sect. 2]. The Sylow 7-normalizer in X is contained in the maximal subgroup He.2, which lifts to 3 × He.2 in 3.X, allowing to complete the case p = 7.

For S = 2.B we have to check the primes p = 3, 5, 7. By [18, Sect. 4] the Sylow 3-normalizer in B is contained in the maximal subgroup F i_{23}, which lifts to 2×F i_{23} in S. From the Atlas it can be checked that the number of faithful characters of 3′-degrees of S and of F i_{23} agree. By [18, Table III] the Sylow 5-normalizer in B is contained in the maximal subgroup HN.2, which lifts to (2 × HN).2 in S. The result now follows from the Atlas. The Sylow 7-normalizer in B has the structure (2^{2} × 7^{2}.G_{5}).2 with the primitive complex reflection group G_{5} and the top 2 acting simultaneously on the two factors. The Schur multiplier of this group is C_{2}, and the corresponding covering group has 27 characters faithful on the center, the same as the number of faithful characters of 2.B of degree prime to 7, see Table 2. This completes the proof. □

References

[1] J. An, The Alperin and Dade conjectures for the simple Held group. J. Algebra 189 (1997), 34–57.

[2] J. An, E. A. O’Brien, The Alperin and Dade conjectures for the O’Nan and Rudvalis simple groups. Comm. Algebra 30 (2002), 1305–1348.

Table 2. Character degrees for some Sylow normalizers

\begin{matrix}G & p & Irr_{p′}(N_{G}(P)|ν) \\ 2.F_{4}(2) & 7 & 1^{9}, 2^{9}, 3^{3}, 24^{6} \\ 2.F_{4}(2).2 & 7 & 1^{6}, 2^{9}, 3^{6}, 4^{3}, 48^{3} \\ 2.^{2}E_{6}(2) & 5 & 1^{4}, 2^{12}, 3^{4}, 4^{8}, 6^{2}, 24^{4}, 48^{2} \\ 2.^{2}E_{6}(2).2 & 5 & 1^{8}, 2^{16}, 3^{8}, 4^{10}, 6^{4}, 8^{2}, 24^{8}, 48^{4} \\ 6.^{2}E_{6}(2) & 5 & 3^{8}, 6^{6}, 24^{4}, 48^{2} \\ F i′_{24}.2 & 3 & 1^{16}, 2^{32}, 4^{24}, 8^{8}, 16 \\ F i′_{24}.2 & 7 & 1^{12}, 2^{15}, 12^{6}, 36^{2} \\ 3.F i′_{24} & 5 & 3^{12}, 6^{12}, 9^{4}, 12^{2}, 24^{4}, 48^{2}, 72^{4} \\ 2.B & 7 & 2^{6}, 4^{9}, 6^{6}, 8^{3}, 48^{3} \end{matrix}

[3] J. An, E. A. O’Brien, Conjectures on the character degrees of the Harada-Norton simple group HN. Israel J. Math. 137 (2003), 157–181.

[4] J. An, E. A. O’Brien, The Alperin and Dade conjectures for the Conway simple group Co_{1}. Algebr. Represent. Theory 7 (2004), 139–158.

[5] J. An, E. A. O’Brien, The Alperin and Uno conjectures for the Fischer simple group Fi_{22}. Comm. Algebra 33 (2005), 1529–1557.

[6] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of Finite Groups. Clarendon Press, Oxford, 1985.

[7] E. C. Dade, Blocks with cyclic defect groups. Ann. of Math. 84 (1966), 20–48.

[8] G. Entz, H. Pahlings, The Dade conjecture for the McLaughlin group. pp. 253–266 in: Groups St. Andrews 1997 in Bath, I, Cambridge Univ. Press, Cambridge, 1999.

[9] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4; 2004,

http://www.gap-system.org.

[10] N. M. Hassan, E. Horváth, Dade’s conjecture for the simple Higman-Sims group. pp. 329– 345 in: Groups St. Andrews 1997 in Bath, I, Cambridge Univ. Press, Cambridge, 1999

[11] F. Himstedt, Die Dade-Vermutungen für die sporadische Suzuki-Gruppe. Diplomarbeit, RWTH Aachen, 1999.

[12] J. Huang, Counting characters in blocks of M_{22}. J. Algebra 191 (1997), 1–75.

[13] M. Isaacs, Character Theory of Finite Groups, Dover, New York, 1994.

[14] M. Isaacs, G. Malle, G. Navarro, A reduction theorem for the McKay conjecture, submitted, 2005.

[15] S. Kotlica, Verification of Dade’s conjecture for Janko group J_{3}. J. Algebra 187 (1997), 579–619.

[16] G. Michler, J. B. Olsson, The Alperin-McKay conjecture holds in the covering groups of symmetric and alternating groups, p ≠ 2. J. Reine Angew. Math. 405 (1990), 78–111.

[17] R. van der Hout, The Schur multipliers of the finite primitive complex reflection groups. Indag. Math. 39 (1977), 101–113.

[18] R. A. Wilson, The McKay conjecture is true for the sporadic simple groups. J. Algebra 207 (1998), 294–305.

[19] T. R. Wolf, Variations on McKay’s character degree conjecture. J. Algebra 135 (1990), 123–138.

Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D–67653 Kaiserslautern, Germany.

E-mail address: malle@mathematik.uni-kl.de