A dual version of Huppert’s ρ-σ conjecture

by

Gunter Malle FB Mathematik Universität Kaiserslautern Postfach 3049 D-67653 Kaiserslautern, GERMANY E-mail: malle@mathematik.uni-kl.de

Alexander Moretó Departament d’Àlgebra Universitat de València 46100 Burjassot. València. SPAIN E-mail: Alexander.Moreto@uv.es

The first author’s research is partially supported by the Spanish Ministerio de Educación y Ciencia, grants MTM2004-06067-C02-01, MTM2004-04665, the FEDER and the Programa Ramón y Cajal.

Abstract

Huppert’s ρ-σ conjecture asserts that any finite group has some character degree that is divisible by “many” primes. In this note, we consider a dual version of this problem, and we prove that for any finite group there is some prime that divides “many” character degrees.

1 Introduction

One of the main problems on character degrees of finite groups is Huppert’s ρ-σ conjecture, which roughly asserts that any finite group G has some (complex) irreducible character whose degree is divisible by “many”, say one-third, of the primes that divide some character degree of G. More precisely, let ρ(G) be the set of primes that divide some character degree of G and let σ(G) be the maximum number of different primes that divide any given character degree of G. The weak ρ-σ conjecture asserts that there exists an integer valued function g such that for any finite group G, |ρ(G)| ≤ g(σ(G)). The strong ρ-σ conjecture asserts that |ρ(G)| ≤ 3σ(G), or perhaps even there exists some additive constant C such that for any finite group G, |ρ(G)| ≤ 2σ(G) + C and C can be taken to be 0 is G is solvable (see p. 220 and Remarks 17.9(a) of [11]). The strong ρ-σ conjecture remains open. The weak ρ-σ conjecture was finally settled in [14].

In this note, we consider a sort of converse to Huppert’s ρ-σ conjectures: is it true that for any finite group G there is some prime that divides “many” of the character degrees of G? Again, one can state two different versions of this problem. We say that a finite group G satisfies property P_{k} if for any prime p the number of different character degrees of G that are divisible by p is at most k. The weak form asserts that there exists an integer valued function f such that for any finite group that satisfies property P_{k}, the number of character degrees of G is at most f(k). The strong form could assert that the number of character degrees of a nonabelian group that satisfies property P_{k} is at most 4k or even 3k+C for some universal constant C, that can be taken to be 0 if G is solvable. In other words, given any nonabelian finite group there is one prime that divides at least one-fourth of the character degrees. A family of examples showing that the bound | cd(G)| ≤ 3k would be sharp for infinitely many values of k and solvable groups was constructed in [1]. (As usual, we write cd(G) to denote the set of degrees of the irreducible characters of G.) This bound was suggested in [1] and conjectured in [13]. It was proved in [1] for k ≤ 3 and in [13] for k = 4. As the groups L_{2}(2^{f}) show, this bound does not hold for arbitrary finite groups.

Regarding the weak form of the problem it was proved in [1] that for solvable groups there is a quadratic bound. This was later improved in McVey’s Ph.D. Thesis [12] but his bound is still quadratic. In this article, we consider the problem for arbitrary finite groups for the first time, and settle the weak form of the problem.

Theorem A. There exists an integer-valued function f such that if G is a finite group with the property that for any prime p, there are at most k members of cd(G) that are divisible by p, then | cd(G)| ≤ f(k).

Our methods would allow us to give an explicit bound. However, this would probably be far from best possible so, for the sake of simplicity, we have refrained from doing so. As usual, we will say that a quantity is kbounded if it is bounded by some real valued function that depends only on k.

Now, we explain the structure of this paper. In Section 2 we prove several elementary results that will be necessary in the proof of Theorem A. In Section 3 we prove Theorem A and several related results for simple groups and in Section 4 we complete the proof of Theorem A.

2 Preliminary results

We recall the following from the introduction:

Definition 2.1. We say that a finite group G satisfies property P_{k} if for any prime p the number of different character degrees of G that are divisible by p is at most k.

Our first result will be used many times in the remainder of this paper without further explicit mention.

Lemma 2.2. Let N be a normal subgroup of a finite group G. If G satisfies property P_{k}, then G/N satisfies property P_{k}.

Proof. This is an immediate consequence of the fact that cd(G/N) ⊆ cd(G).

The next three lemmas relate character degrees of a group and a normal subgroup.

Lemma 2.3. Let G be a finite group with normal subgroup N. Then | cd(G)| ≤ | cd(N)|Ω(|G/N|), where Ω(n) is the number of divisors of the integer n.

Proof. This follows from Corollary 11.29 of [4], for instance.

Given a normal subgroup N of a finite group G and n ∈ cd(N) we write cd(G|N, n) to denote the set of character degrees of G that lie over characters of N of degree n.

Lemma 2.4. Let G be a finite group satisfying property P_{k}. Let N be a normal subgroup of G and 1 < n ∈ cd(N). Then | cd(G|N, n)| ≤ k.

Proof. This is a straightforward consequence of the fact that all the members of cd(G|N, n) are multiples of n.

Property P_{k} is not inherited by normal subgroups, but we have the following.

Lemma 2.5. Let G be a finite group satisfying property P_{k}. If N is a normal subgroup of G and the number of divisors of |G : N| is l, then N satisfies property P_{kl}

Proof. Assume that a prime p divides kl + 1 character degrees of N. It follows from property P_{k} that the number of character degrees of G lying over these kl + 1 character degrees of N is at most k. This means that for some of these at most k degrees of G, the number of character degrees of N lying under it is bigger than l. But this is impossible.

The next lemma will allow us to “forget” the character degrees that are multiples of a given finite number of primes.

Lemma 2.6. Let G be a finite group satisfying property P_{k}. If π is a finite set of primes and G has m character degrees of π′-degree, then | cd(G)| ≤ m + |π|k.

Proof. For any prime p ∈ π, G has at most k character degrees that are divisible by p. If d ∈ cd(G) is not divisible by any prime in π, then it has π′-degree and the result follows.

Our last result in this section is well-known. Since we have not been able to find a reference in the literature, we include a short proof. If a group G acts on a set Ω, we say that X ⊆ Ω lies in a regular orbit if the size of its orbit is |G/ Ker_{G} Ω|.

Lemma 2.7. Assume that a finite abelian group G acts on a set Ω. Then G has a regular orbit on P(Ω).

Proof. We argue by induction on |Ω|. Assume first that the action is not transitive and let Ω = Ω_{1} ∪ Ω_{2} be a decomposition of Ω as the union of two proper disjoint G-invariant subsets. By the inductive hypothesis, there exist X_{1} ⊆ Ω_{1} and X_{2} ⊆ Ω_{2} in regular orbits under the action of G/ Ker_{G} Ω_{1} and G/ Ker_{G} Ω_{2}, respectively. Let X = X_{1} ∪ X_{2}. Hence X ⊆ Ω lies in a regular orbit under the action of G. Therefore we may assume that the action of G on Ω is transitive. Also, we may assume that this action is faithful.

Since G is abelian all the point stabilizers are trivial, so every singleton in P(Ω) is in a regular orbit.

3 Simple groups

In this section we prove Theorem A and some related results for simple groups. All results except for Proposition 3.7, Corollary 3.8 and Lemma 3.9 are of an asymptotic nature. Hence, in their proofs, we may (and will) ignore any finite number of simple groups, like the sporadic groups, the Tits group, groups of Lie type with exceptional Schur multiplier, and alternating groups 𝔄_{n} for all n ≤ n_{0}, for some bound n_{0}.

Proposition 3.1. For all n ≥ 3 we have:

n/2 ≤ min{k | 𝔄_{n} satisfies P_{k}} ≤ | cd(𝔄_{n})| ≤ n!
n/2 ≤ min{k | 𝔖_{n} satisfies P_{k}} ≤ | cd(𝔖_{n})| ≤ n!

Proof. The irreducible characters of the symmetric group 𝔖_{n} are parameterized by partitions λ of n, and their degrees are given by the so-called hook formula. In particular, for the hook partitions λ_{k} = (n − k, 1^{k}) of n, where 0 ≤ k < n/2, the degree of χ_{λ} is equal to \binom{n}{k}. It is obvious that \binom{n}{k} are all different for 0 ≤ k < n/2. An irreducible character χ_{λ} of 𝔖_{n} remains irreducible upon restriction to the alternating group 𝔄_{n} if λ is not self-dual. Again, it is obvious that the hook partitions are not self-dual, for 0 ≤ k < n/2. Thus, min{k | 𝔄_{n} satisfies P_{k}} ≥ n/2 as claimed. The remaining inequalities are obvious.

The bounds obtained in this proof are very weak, and much better bounds could be established.

We next prove a similar statement for the character degrees of groups of Lie type. The last inequality has been obtained in Theorem 1.5 of [7], but for completeness we include a proof.

Proposition 3.2. There exists a monotonous function f_{2} : ℕ ⟶ ℕ such that whenever G is a finite group of Lie type of rank n then

f_{1}(n) := n/2 − 5 ≤ min{k | G satisfies P_{k}} ≤ | cd(G)| ≤ f_{2}(n).

Proof. We consider the following setup. Let G be a simple algebraic group defined over a finite field 𝔽_{q} and F : G → G the corresponding Frobenius endomorphism. Then the group of fixed points G := G^{F} is, by definition, a finite group of Lie type.

The fundamental results of Lusztig [8] on the classification of the complex irreducible characters of finite groups of Lie type show the following. The set Irr(G) is partitioned into so-called Lusztig series E(G, s), where s runs over a system of representatives of semisimple elements in the dual group G^{∗} of G. Furthermore, the degrees of characters χ ∈ E(G, s) only depend on the structure of C_{G^{∗}}(s). In particular, cd(G) is controlled by the various C_{G^{∗}}(s). Fix s ∈ G^{∗} and let C := C_{G^{∗}}(s). The connected component C^{∘} of C is a connected reductive subgroup of G^{∗}, hence completely determined by its root system. Now C/C^{∘} is a subgroup of the relative Weyl group in G^{∗} of the root system of C^{∘}, which is a section of the Weyl group of G^{∗}. In particular, for a given C^{∘} there is only a finite number of possible C’s.

Now, for a given root system of G (and hence of G^{∗}) there only exist a finite number of sub-root systems which can give rise to a connected centralizer C^{∘}. Hence, the number of different connected centralizers is bounded by a function in the root system of G. Our previous argument then shows that the number of different centralizers C_{G^{∗}}(s), up to conjugation, is bounded by a function in the root system of G. Finally, the number of characters in E(G, s) only depends on the structure of C_{G^{∗}}(s), and thus | cd(G)| is bounded in terms of a function in the root system of G, independent of the order of the underlying field. In turn, for a fixed rank n there only exist finitely many different root systems of algebraic groups of rank n, hence we have established the existence of the upper bound f_{2}.

The relation between min{k | G satisfies P_{k}} and | cd(G)| is trivial. For the lower bound we consider the elements of E(G, 1), the so-called unipotent characters. By Lusztig’s description of E(G, 1), the cardinality |E(G, 1)| only depends on the root system of G, and it tends to infinity with the rank of G. Indeed, there is a subset E of E(G, 1), the so-called principal series unipotent characters, which is in bijection with Irr(W), such that that the degree polynomials of characters in E specialize to the character degrees in Irr(W). In particular, there are at least as many different degrees of unipotent characters as there are different degrees in Irr(W). For the finitely many exceptional types, the lower bound is now easily proved by inspection.

So now assume that G is of classical type. Then, if G has rank n, Irr(W) has the symmetric group 𝔖_{n} as a quotient, so indeed we find at least n/2 different character degrees by Proposition 3.1. Moreover, all except possibly five of the degrees of unipotent characters are divisible by the defining characteristic of G (see [10, Lemma 2.12], for example). So we have n/2 − 5 ≤ min{k | G satisfies P_{k}}. This proves the validity of the lower bound.

Lemma 3.3. If G is a quasisimple group of Lie type, then | cd(G)| is bounded by some function of | cd(G/Z(G))|.

Proof. We keep the setup from the proof of Proposition 3.2. If G is chosen to be simply-connected, then the group of fixed points G := G^{F} is the universal covering group of the simple group S := G/Z(G), except for a finite number of cases, and all finite simple groups of Lie type occur among as such an S, except for the Tits group. Clearly, Irr(S) ⊂ Irr(G) in a natural way, and in order to prove the assertion, it is enough to establish a function bounding | cd(G)| in terms of | cd(S)|. Now the unipotent characters E(G, 1) have Z(G) in their kernel, so descend to characters of S. Thus, if n denotes the rank of G, then by Proposition 3.2 we have | cd(S)| ≥ n/2 − 5, which tends to infinity with n, while | cd(G)| ≤ f_{2}(n). So our claim follows.

Lemma 3.4. There exists an integer valued function f_{3} such that if an almost simple group G with socle S satisfies property P_{k}, then S satisfies property P_{f_{3}(k)}.

Proof. First consider groups of Lie type. By Proposition 3.2 the minimal k such that S has property P_{k} is bounded above by a function only depending on the rank of G. Hence, we may assume that the rank is larger than 8, so that G is of classical type. Now the unipotent characters of S are invariant under diagonal and field automorphisms of S by Lusztig’s results [8]. Furthermore, the subgroup of Aut(S) generated by inner, diagonal and field automorphisms has index at most 6 in Aut(S). (The factor group consists of the graph automorphisms, of which there are at most six.)

Since all but at most five unipotent characters have degree divisible by p, the defining characteristic of S (see [10, Lemma 2.12]), and the number of different unipotent character degrees goes to infinity with the rank, this proves the claim for groups of Lie type.

For alternating groups, the claim follows directly from Proposition 3.1

Lemma 3.5. There exists an integer valued function f_{4} such that whenever S ≤ G ≤ Aut(S), where S is a nonabelian simple group, and G satisfies property P_{k}, then | cd(G)| ≤ f_{4}(k).

Proof. This is just Proposition 3.1 for alternating groups. We may hence assume that S is of Lie type. Assume that G is almost simple with socle S and satisfies P_{k}. Then S satisfies P_{f_{3}(k)} by Lemma 3.4, so f_{3}(k) ≥ n/2 − 5 by Proposition 3.2, where n denotes the rank of S. Thus n ≤ 2(f_{3}(k) + 5). By Lemma 2.4 we also have that

| cd(G)| ≤ k(| cd(S)| − 1) + | cd(G/S)|.

Furthermore, | cd(S)| ≤ f_{2}(n), where n denotes the Lie-rank of S, again by Proposition 3.2. Since G/S is solvable and satisfies property P_{k}, [1, Th. A] implies that the quantity | cd(G/S)| is bounded by a function g(k). (Alternatively, this follows directly from the explicit knowledge of the structure of Out(S).) Hence

\begin{matrix}| cd(G)| ≤ & k(| cd(S)| − 1) + | cd(G/S)| \\ ≤ & k f_{2}(n) + g(k) ≤ k f_{2}(2(f_{3}(k) + 5)) + g(k) \end{matrix}

is k-bounded.

Lemma 3.6. There exists an integer valued function f_{5} such that if S is a nonabelian simple group that satisfies property P_{k}, then there exists a prime divisor p of |S| such that the number of indices of the maximal subgroups of S of p′-index is at most f_{5}(k).

Proof. First assume that S is of Lie type, with Weyl group W of rank n. If S satisfies property P_{k} then k ≥ f_{1}(n) by Proposition 3.2. If M is a maximal subgroup of S, then either |S : M| is divisible by the defining characteristic p of S, or M is a maximal parabolic subgroup (see for example [3, Thm. 3.1.3]). Conjugacy classes of parabolic subgroups correspond to subsets of the set of nodes of the Dynkin diagram for W, hence the number of classes of maximal parabolic subgroups of S is at most equal to n if W has rank n. Since f_{1}(n) tends to infinity, this proves the claim for groups of Lie type.

It remains to consider alternating groups. Here, by [6, Th.] maximal subgroups of odd index of 𝔄_{n}, n ≥ 9, are either intransitive (𝔖_{k}×𝔖_{n−k})∩𝔄_{n} or imprimitive 𝔖_{k} ≀ 𝔖_{n/k} ∩ 𝔄_{n}. Clearly, the number of such subgroups is bounded in n, and n is bounded in k by Proposition 3.1.

Proposition 3.7. Let S be a finite nonabelian simple group, S ≅ L̸_{2}(3^{f}). Then there exist nonlinear characters φ, ψ ∈ Irr(S) of different degrees that extend to Aut(S).

Proof. For sporadic groups and the Tits group, this can be checked from the known character tables. For the alternating groups 𝔄_{n}, n ≥ 7, take the characters parameterized by the partitions (n − 1, 1) and (n − 2, 2). (Note that 𝔄_{6} = L_{2}(9), and 𝔄_{5} can be checked easily.) For S of Lie type it is shown in [9, Thm. 2.4] that all unipotent characters extend to their inertia groups in Aut(S), and it is known that there are at least two nonlinear unipotent characters of different degree which are invariant in Aut(S) if S is not of type A_{1}.

So finally assume that S = L_{2}(q) for q ≥ 7, q ≢ 0 (mod 3). Let χ_{1} denote the Steinberg character of S, of degree q. This extends to Aut(S) by [2]. Let χ_{2} be a character corresponding to a semisimple element of order 3 in the dual group, of degree q + ϵ if q ≡ ϵ (mod 3). It is shown in [5, Lemma 15.2] that χ_{2} extends to Aut(S).

It can be checked easily that the groups L_{2}(3^{f}) constitute counterexamples to the conclusion of the proposition. Still, we have the following:

Corollary 3.8. Let S be a nonabelian finite simple group. Then there exist non-linear φ, ψ ∈ Irr(S) of different degree such that φ extends to Aut(S) and ψ extends to its inertia subgroup in Aut(S).

Finally, we will need the following well-known fact.

Lemma 3.9. Let S be a nonabelian simple group. Write n to denote the rank of S if S is of Lie type and put n = 3 otherwise. Let π be the set of primes smaller than max{4, n + 2}. Then Out(S) has a cyclic and normal Hall π′-subgroup.

Proof. If S is not of Lie type, then Out(S) is cyclic, so we may assume that S is of Lie type.

The group of diagonal automorphisms is either the Klein four group or it is cyclic of order at most n + 1. Thus its automorphism group is a π-group. The group of graph automorphisms has order divisible by 6, so again it is a πgroup. Since the group of outer automorphisms is a semidirect product of the group of diagonal automorphisms with the field and graph automorphisms, a Hall π′-subgroup of Out(S) consists of some field automorphisms that form a normal subgroup in Out(S).

4 Proof of Theorem A

Now, we are ready to prove Theorem A. We split the proof in two parts. The argument of the first part is essentially contained in [1]

Theorem 4.1. There exists an integer-valued function f_{6} such that for any finite group G satisfying property P_{k} and which has some nonabelian solvable quotient, we have | cd(G)| ≤ f_{6}(k).

Proof. Let L be a maximal normal subgroup of G such that G/L is a nonabelian solvable group. By Lemma 12.3 of [4], G/L is a p-group for some prime p, or a Frobenius group with elementary abelian kernel and cyclic complement. (Note that an abelian Frobenius complement is cyclic.)

Assume first that G/L is a p-group. We know that G has at most k character degrees that are divisible by p, so it suffices to bound the character degrees of G that are p′-numbers. If χ ∈ Irr(G) has p′-degree, then χ_{L} is also irreducible (by Corollary 11.29 of [4], for instance). Hence, by Gallagher’s Theorem (Corollary 6.17 of [4]), χψ ∈ Irr(G) for any ψ ∈ Irr(G/L). If we take a nonlinear ψ, it follows that for any d ∈ cd(G) that is a p′-number, dψ(1) also belongs to cd(G). This implies that the number of character degrees of G that are p′-numbers bigger than 1 is at most k. Therefore, we have proved that | cd(G)| ≤ 2k + 1.

Now, we assume that G/L is a Frobenius group with elementary abelian kernel K/L of order q^{a} and cyclic complement of order f. Then cd(G/L) = {1, f}. We know that G has at most k character degrees that are divisible by q, so it suffices to bound the number of irreducible characters of G of q′- degree. Assume that φ ∈ Irr(L) is nonlinear of q′-degree and extends to K (otherwise, the irreducible characters of G lying over φ would have degrees divisible by q). Since the Frobenius complement is cyclic, by Corollaries 11.22 and 11.31 of [4], we deduce that φ extends to its inertia subgroup in G. Write T = I_{G}(φ) and let φ̃ ∈ Irr(T) be such an extension. Let μ ∈ Irr(T/L) with μ(1) = |T/K|. By Gallagher’s Theorem μφ̃ ∈ Irr(T) and by Clifford’s Correspondence χ = (μφ̃)^{G} ∈ Irr(G). It follows that

χ(1) = |G : T ||T : K|φ(1) = fφ(1) ∈ cd(G).

Hence the number of degrees of the nonlinear irreducible characters of L of q′-degree that extend to K is at most k. By Lemma 2.4, we obtain that the number of character degrees of G lying over these irreducible characters is at most k^{2}. It remains to consider the number of character degrees of G lying over linear irreducible characters of L. In other words, it remains to bound | cd(G/L′)|. But G/L′ is solvable and satisfies property P_{k}, so the result follows from Theorem A of [1].

Given a normal subgroup N of a finite group G, we write Irr(G|N) to denote the set of irreducible characters of G which do not contain N in their kernel. The notation cd(G|N) stands for the degrees of the characters in Irr(G|N).

Theorem 4.2. There exists an integer-valued function f_{7} such that for any finite group G satisfying property P_{k} and which has no nonabelian solvable quotients, we have | cd(G)| ≤ f_{7}(k).

Proof. Let L be a maximal normal subgroup of G such that G/L is not abelian. Write K = G′L. Observe that K/L is a nonabelian chief factor of G and C/L = C_{G/L}(K/L) = L/L. We have that K/L is a direct product of, say l, copies of a nonabelian simple group S. Since C = L, we have that G/L is isomorphic to a subgroup of Γ = Aut(S) ≀ 𝔖_{l}. Put U = Aut(S)^{l} ∩ G ⊴ G, so that G/U is isomorphic to a subgroup of 𝔖_{l}.

Now, we split the proof in several steps.

Step 1. We claim that l is k-bounded.

Since G/K is abelian, we have that G/U is an abelian permutation group on the set Ω = {S_{1}, …, S_{l}} consisting by l copies of S. By Lemma 2.7, G/U has a regular orbit on P(Ω), say the orbit of Δ ⊆ Ω. Since Ω − Δ also lies in a regular orbit we may assume, without loss of generality, that |Δ| ≥ |Ω|/2. Thus, it suffices to show that |Δ| is k-bounded. We may assume, without loss of generality, that Δ = {S_{1}, …, S_{u}} for some u ≥ l/2.

By Corollary 3.8, S_{i} has two nonlinear irreducible characters φ_{i} and ψ_{i} of different degrees such that φ_{i} extends to Aut(S_{i}) and ψ_{i} extends to its inertia subgroup in Aut(S_{i}). (We identify the characters φ_{i} and φ_{j} for every i, j and the same with the characters ψ_{i} and ψ_{j}.) For each 1 ≤ k ≤ u, we consider

μ_{k} = φ_{1} × ⋯ × φ_{k} × 1_{S_{k+1}} × ⋯ × 1_{S_{u}} × ψ_{u+1} × ⋯ × ψ_{l} ∈ Irr(K/L).

The number of different degrees among these characters is u. By the way we have chosen them, their inertia subgroup is contained in U. In fact, the inertia subgroup I := I_{G}(μ_{k}) of μ_{k} is

I = Aut(S_{1}) × ⋯ × Aut(S_{u}) × I_{Aut(S_{u+1})}(ψ_{u+1}) × ⋯ × I_{Aut(S_{l})}(ψ_{l}) ∩ U,

which does not depend on k.

Since φ_{i}, ψ_{i} and 1_{S_{i}} extend to their inertia subgroup in Aut(S_{i}), we conclude that μ_{k} extends to its inertia subgroup in G for every k. Write μ̂_{k} ∈ Irr(I) to denote the extension of μ_{k}. By Clifford’s correspondence, μ̂_{k}^{G} ∈ Irr(G). These u induced irreducible characters provide u different character degrees of G all of them being multiples of φ(1). It follows that |Δ| = u ≤ k, as desired.

Step 2. We may assume that G has a nonabelian almost simple quotient with socle S.

Since G/U is a subgroup of 𝔖_{l} and l is k-bounded, the number of divisors of the order of this group is k-bounded, and by Lemmas 2.5 and 2.3 we may assume that G = U. This implies that G has a normal subgroup M such that G/M is almost simple with socle S.

Write R/M = Soc(G/M). We know from Lemma 3.5 that | cd(G/M)| ≤ f_{4}(k). It remains to prove that | cd(G|M)| is bounded by some integer valued function of k.

We split the set cd(G|M) in three subsets. Write cd_{1}(G|M) to denote the set of degrees of the irreducible characters of G lying over irreducible characters of M that are not R-invariant, cd_{2}(G|M) to denote the set of degrees of the irreducible characters of G lying over irreducible characters of M that are R-invariant but do not extend to R and cd_{3}(G|M) to denote the set of degrees of the irreducible characters of G lying over nonprincipal irreducible characters of M that extend to R. Then

cd(G|M) = cd_{1}(G|M) ∪ cd_{2}(G|M) ∪ cd_{3}(G|M),

and it suffices to bound | cd_{i}(G|M)| for i = 1, 2, 3. This is what we will do in the next three steps.

Step 3. | cd_{1}(G|M)| is k-bounded.

Since G/M satisfies property P_{k}, Lemma 3.4 implies that R/M satisfies property P_{f_{3}(k)}. Since the characters of M we are looking at are not Rinvariant, their inertia subgroup in R is contained in some maximal subgroup of R that contains M. In particular, by Clifford’s Correspondence, the degrees of the irreducible characters of R lying over the characters of M we are considering are multiples of the index of some maximal subgroup of R/M. Since R is normal in G, the same happens with the degrees of the irreducible characters of G lying over these characters of M.

Now, Lemma 2.4 implies that for any maximal subgroup U/M of R/M, the number of character degrees of G that are multiples of |R/U| is at most k. By Lemma 3.6, there is a prime p such that the number of indices of maximal subgroups of S of p′-index is at most f_{5}(f_{3}(k)). Hence, the number of character degrees of G lying over these characters of M is at most f_{5}(f_{3}(k))k + k (the latter k corresponds to the at most k character degrees that are multiples of p).

Step 4. | cd_{2}(G|M)| is k-bounded.

Now, we consider the characters of M that are R-invariant but do not extend to R. Recall that R satisfies property P_{f_{3}(k)}. Using a character triple isomorphism (Theorem 11.28 of [4]), Lemma 3.3 and Proposition 3.1, it follows that there is a set of primes of k-bounded size such that any character degree of R arising from any of these characters of M is a multiple of some of them. The same happens with the degrees of the irreducible characters of G lying over these characters of M. As before, we obtain that the number of character degrees of G that is obtained this way is k-bounded.

Step 5. | cd_{3}(G|M)| is k-bounded.

Finally, we consider the irreducible characters of M that extend irreducibly to R. Let ψ be such a character. If ψ is linear, then the character degrees of G lying over ψ belong to cd(G/G′′). By Benjamin’s result, the cardinality of this set is k-bounded. Hence, we may assume that ψ is not linear. Let K/R be a Hall π′-subgroup of G/R, where π is the same as in Lemma 3.9. By Proposition 3.2, |π| is bounded in terms of k and by Lemma 2.6, it suffices to bound the number of character degrees of G that are π′-numbers. Hence, we may assume that G = K.

Since R/M is perfect, it follows from Gallagher’s Theorem that ψ has a unique extension ψ̂ ∈ Irr(R). Since G/R is cyclic, ψ̂ extends to its inertia subgroup in G. Using Clifford’s Theorem, we have shown that for any ψ ∈ Irr(M) that extends to R, ψ(1)|G : I_{G}(ψ̂)| ∈ cd(G), where ψ̂ is the unique extension of ψ to R. But by Gallagher’s theorem, ψ(1)|G : I_{G}(ψ̂)|d ∈ cd(G), where 1 < d is the degree of some irreducible character of R/M that extends to Aut(R/M). Therefore, the number of different possibilities for ψ(1)|G : I_{G}(ψ̂)| as ψ runs over the nonlinear irreducible characters of M that extend to R is at most k.

Write d_{1}, …, d_{w} for the w ≤ k different possibilities for ψ(1)|G : I_{G}(ψ̂)| as ψ runs over the nonlinear irreducible characters of M that extend to R. We deduce from Clifford’s Correspondence and Gallagher’s Theorem that the character degrees of G lying over irreducible characters of M that extend to R have the form d_{i}t, where t is a character degree of some almost simple group with socle S. It follows from our hypothesis that for any i, the number of degrees of the form d_{i}t is at most k. We conclude that the number of character degrees of G lying over nonlinear irreducible characters of M that extend to R is at most k^{2}. This completes the proof.

Proof of Theorem A. It suffices to take f = max{f_{6}, f_{7}}.

References

[1] D. Benjamin, Coprimeness among irreducible character degrees of finite solvable groups. Proc. Amer. Math. Soc. 125 (1997) 2831–2837.

[2] W. Feit, Extending Steinberg characters. Pp. 1–9 in: Linear algebraic groups and their representations, Contemp. Math. 153, Amer. Math. Soc., Providence, 1993.

[3] D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups, Number 3. Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, 1994.

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

[5] I. M. Isaacs, G. Malle, G. Navarro, A reduction theorem for the McKay conjecture. Submitted, 2005.

[6] M. W. Liebeck, J. Saxl, The primitive permutation groups of odd degree. J. London Math. Soc. 31 (1985), 250–264.

[7] M. W. Liebeck, A. Shalev , Character degrees and random walks in finite groups of Lie type. Proc. London Math. Soc. 90 (2005), 61–86.

[8] G. Lusztig, Characters of reductive groups over a finite field. Annals of Math. Studies, 107. Princeton Univ. Press, Princeton, NJ, 1984.

[9] G. Malle, Extensions of unipotent characters and the inductive McKay condition. In preparation, 2007.

[10] G. Malle, A. E. Zalesskiĭ, Prime power degree representations of quasi-simple groups. Archiv Math. 77 (2001), 461–468.

[11] O. Manz, T. R. Wolf, Representations of Solvable Groups. Cambridge University Press, Cambridge, 1993.

[12] J. K. McVey, Bounding the number of character degrees using generalized relative primeness conditions. Ph.D. Thesis, Kent State University, 2001.

[13] J. K. McVey, Prime divisibility among character degrees. Comm. Algebra 32 (2004), 3391–3402.

[14] A. Moretó, A proof of Huppert’s ρ-σ conjectures for nonsolvable groups. Int. Math. Research Notices 54 (2005), 3375–3383.