THE NUCLEAR LENGTH OF A CHARACTER

Wayne W. Wheeler

School of Mathematical Sciences Australian National University Canberra ACT 0200 Australia (E-mail: wheeler@pell.anu.edu.au)

10 September 1993

1. Introduction

If π is a set of primes, then the character theory of π-separable groups demonstrates a remarkable richness that is not shared by finite groups in general. For example, in [1] Gajendragadkar defined the set 𝔛_{π}(G) of π-special characters of G and showed that if α is π-special and β is π′-special, then αβ is irreducible. Because a product of characters is so rarely irreducible, this result seems quite surprising. The idea of factoring characters into a product of π-special and π′-special parts was exploited by Isaacs, who defined a set of irreducible characters B_{π}(G) containing the π-special characters [2]. When π = p′, the complement of a single prime p, the characters in B_{p′}(G) provide canonical lifts for the irreducible Brauer characters of G. Thus the set IBr(G) of irreducible Brauer characters of G is obtained simply by restricting the characters in B_{p′}(G) to the p-regular elements of G. More generally, the characters in B_{π′}(G) act as a π-analogue of Brauer characters, and Slattery has used them to develop a theory of π-blocks in π-separable groups (See [4], [5], and [6]). More recently, Isaacs and Navarro have defined weights and vertices for characters and have proven a π-analogue of Alperin’s Weight Conjecture for π-separable groups [3].

The point of this paper is to return to Isaacs’ definition of B_{π}(G) in [2] and study it more closely. If χ ∈ Irr(G), then Isaacs constructs a subgroup I and a character ψ ∈ Irr(I) with ψ^{G} = χ. The pair (I, ψ) is called a standard inducing pair, and (I, ψ) = (G, χ) if and only if χ factors as the product of a π-special and a π′-special character. One can also construct a standard inducing pair for ψ, and this process leads inductively to a pair (W, γ) such that γ^{G} = χ and γ is the product of a π-special and a π′-special character. The pair (W, γ) is called a nucleus for χ and is unique up to G-conjugacy. Isaacs defines the set B_{π}(G) to consist of those irreducible characters χ for which there is a nucleus (W, γ) such that γ is itself π-special.

But it is easy to see that this construction produces more than just the conjugacy class of pairs (W, γ). In particular, associated to the character χ ∈ Irr(G) is a

Typeset by AMS-TEX

nonnegative integer given by the number of times one must apply the inductive procedure before obtaining a nucleus for χ. We call this number the nuclear length of χ, and we write it as nl(χ). The purpose of this paper is to study the invariant given by the nuclear length. We will see that it is related to both the character degree χ(1) and the π-length ℓ_{π}(G) of G. Specifically, we will prove the following result.

Theorem A. Let G be a π-separable group, and let χ ∈ Irr(G). Then nl(χ) is bounded by both of the following numbers:

(1) 2ℓ_{π}(G).

(2) The number of prime divisors (counting multiplicities) of χ(1).

In fact, the bound given in (2) is a very easy consequence of the definition of the nucleus. When χ ∈ B_{π′}(G), however, we can get somewhat better bounds, and in this case neither bound seems to follow directly from the definitions.

Theorem B. Let G be a π-separable group, and let χ ∈ B_{π′}(G). Then nl(χ) is bounded by both of the following numbers:

(1) ℓ_{π}(G).

(2) The number of prime divisors (counting multiplicities) of χ(1)_{π}.

Here χ(1)_{π} denotes the π-part of χ(1).

Section 2 is devoted to a summary of relevant definitions and results from [1] and [2], leading to a precise definition of the nuclear length. Theorems A and B are proven in Section 3.

2. The Character Theory of π-Separable Groups

The purpose of this section is to define the nuclear length and to give a brief review of the relevant aspects of character theory for π-separable groups. The results of interest here come almost exclusively from the work of Gajendragadkar [1] and Isaacs [2]. We assume throughout the section that π is a set of primes and G is a π-separable group.

Recall that if χ is a character of G, then the determinantal order o(χ) is the order of det χ in the group of linear characters of G. This concept forms the basis for the following definition from [1].

Definition 2.1. Let G be a π-separable group. A character χ ∈ Irr(G) is said to be π-special if the following conditions are satisfied:

(1) χ(1) is a π-number.

(2) If S is any subnormal subgroup of G and σ is any irreducible constituent of χ_{S}, then the determinantal order o(σ) is a π-number.

The set of π-special characters of G is denoted by 𝔛_{π}(G).

The following result is almost an immediate consequence of the above definition. It appears as Proposition 4.1 of [1].

Proposition 2.2. Let G be a π-separable group, and let χ ∈ Irr(G) be π-special. If S ◁◁ G and σ ∈ Irr(S) is a constituent of χ_{S}, then σ is π-special.

Proposition 2.3. Let G be a π-separable group, and let N be a normal subgroup of G such that G/N is a π-group. If η ∈ Irr(N) is π-special, then every irreducible constituent of η^{G} is π-special.

Proof. See Proposition 4.5 of [1]. □

Proposition 2.4. Let G be a π-separable group, and let N be a normal subgroup of G such that G/N is a π′-group.

(1) If χ ∈ Irr(G) is π-special, then χ_{N} is irreducible and π-special.

(2) If η ∈ Irr(N) is π-special and G-invariant, then η has a π-special extension η̂ ∈ Irr(G). Moreover, η̂ is the only irreducible constituent of η^{G} that is π-special.

Proof. See Proposition 4.3 of [1]. □

The next result, which appears as Proposition 7.1 of [1], shows that π-special characters have a remarkable multiplicative property.

Proposition 2.5. Let G be a π-separable group. If α ∈ 𝔛_{π}(G) and β ∈ 𝔛_{π′}(G), then αβ is irreducible. Moreover, if α′ ∈ 𝔛_{π}(G) and β′ ∈ 𝔛_{π′}(G) satisfy αβ = α′β′, then α = α′ and β = β′.

As in [2], we will say that a character χ ∈ Irr(G) is π-factorable if it can be written as a product αβ with α ∈ 𝔛_{π}(G) and β ∈ 𝔛_{π′}(G). The following result, which is an easy consequence of Propositions 2.2 and 2.5, is given as Corollary 2.6 of [2].

Corollary 2.6. Let G be a π-separable group, and let χ ∈ Irr(G) be π-factorable. If S ◁◁ G, then every irreducible constituent of χ_{S} is π-factorable.

The main point of [2] is to use the concept of π-factorability as the basis for defining a certain subset B_{π}(G) of Irr(G) containing the π-special characters. Because the definitions involved in the construction of B_{π}(G) are very important to the work presented in the next section, we recall the construction in detail.

Consider the set P of all pairs (H, θ), where H is a subgroup of G and θ ∈ Irr(H). If (K, φ) ∈ P, then we write (H, θ) ≤ (K, φ) if H ⊆ K and θ is a constituent of φ_{H}. If g ∈ G, let (H, θ)^{g} = (H^{g}, θ^{g}). Then ≤ is a partial order on P, and the conjugation action of G respects the partial order. We will write I_{G}(H, θ) for the stabilizer of (H, θ) in G.

A pair (S, σ) is called a π-factorable subnormal pair if S ◁◁ G and σ is πfactorable. The set of π-factorable subnormal pairs in G is denoted by F(G), and we write F^{∗}(G) for the set of maximal elements of F(G). The following proposition describes a fundamental property of the partial order on F(G).

Proposition 2.7. Let G be a π-separable group, and let χ ∈ Irr(G). Then there is a pair (S, σ) ∈ F^{∗}(G) with (S, σ) ≤ (G, χ). Moreover, if (U, φ) ∈ F(G) with (U, φ) ≤ (G, χ), then (U, φ)^{g} ≤ (S, σ) for some g ∈ G.

Proof. See Theorem 3.2 of [2]. □

Proposition 2.8. Let G be a π-separable group, let (S, σ) ∈ F^{∗}(G), and set I = I_{G}(S, σ). Then induction defines a bijection Irr(I|σ) → Irr(G|σ). Moreover, if S < G, then I < G.

Proof. See Theorem 4.4 and Lemma 4.5 of [2]. □

We are now ready to present Isaacs’ definition of the nucleus of a character. Suppose that G is π-separable, and let χ be any irreducible character of G. By Proposition 2.7 there is a pair (S, σ) ∈ F^{∗}(G) with (S, σ) ≤ (G, χ). Let I = I_{G}(S, σ). Then Proposition 2.8 implies that there is a unique character ψ ∈ Irr(I|σ) such that ψ^{G} = χ. In this way we associate a pair (I, ψ) with the original pair (G, χ), and (I, ψ) is unique up to G-conjugacy. The pair (I, ψ) is called a standard inducing pair for χ. If χ is π-factorable, then S = G so that (I, ψ) = (G, χ). But if χ is not π-factorable, then S < G and hence I < G by Proposition 2.8. We can then repeat the process by finding a standard inducing pair for ψ. Proceeding by induction, we see that there is a chain

(G, χ) = (I_{0}, ψ_{0}) > (I_{1}, ψ_{1}) > ⋯ > (I_{n}, ψ_{n}) = (W, γ)

such that (I_{j}, ψ_{j}) is a standard inducing pair for ψ_{j−1} and γ is π-factorable. Any such pair (W, γ) is called a nucleus for χ, and the set of all nuclei for χ is denoted nuc(χ). Thus nuc(χ) consists of a single G-conjugacy class of pairs, and if (W, γ) is a nucleus for χ, then γ^{G} = χ. Finally, B_{π}(G) denotes the set of all χ ∈ Irr(G) such that there is a pair (W, γ) ∈ nuc(χ) with γ ∈ 𝔛_{π}(G). Note that if (I, ψ) is a standard inducing pair for χ, then nuc(ψ) ⊆ nuc(χ). Thus χ ∈ B_{π}(G) if and only if ψ ∈ B_{π}(I).

To reach the main subject of this paper, we now need only one simple observation about Isaacs’ definition of the nucleus. Let χ ∈ Irr(G). Because the standard inducing pair (I, ψ) is unique up to G-conjugacy, so is the pair (W, γ). But we can say more: in fact, the entire chain of pairs from (G, χ) to (W, γ) is unique up to G-conjugacy. In particular, the length of the chain is uniquely determined by χ. This observation suggests the following definition.

Definition 2.9. Let G be a π-separable group, and let χ ∈ Irr(G). Let (W, γ) ∈ nuc(χ), and let

(G, χ) = (I_{0}, ψ_{0}) > (I_{1}, ψ_{1}) > ⋯ > (I_{n}, ψ_{n}) = (W, γ)

be a chain such that (I_{j}, ψ_{j}) is a standard inducing pair for ψ_{j−1}. Then we say that χ has nuclear length n. We write nl(χ) for the nuclear length of χ.

The nuclear length can be regarded as a number that measures how close χ is to being π-factorable. Indeed, it is immediate from the definition that χ is π-factorable if and only if nl(χ) = 0.

We end this section with a few more results that will be needed to study nuclear lengths in the next section. Because it seems to be convenient for our purposes to focus on B_{π′}(G) instead of B_{π}(G), we have stated the results below for this set of characters.

Proposition 2.10. Let G be a π-separable group, and let N be a normal subgroup of G. Suppose that η ∈ Irr(N).

(1) Assume that G/N is a π′-group and χ ∈ Irr(G|η). Then η ∈ B_{π′}(N) if and only if χ ∈ B_{π′}(G).

(2) Assume that G/N is a π-group. Then η ∈ B_{π′}(N) if and only if there exists a character χ ∈ B_{π′}(G) lying over η. Moreover, in this case χ is the unique irreducible constituent of η^{G} in B_{π′}(G).

Proof. See Theorems 6.2 and 7.1 of [2]. □

The following proposition is essentially proven in Theorem 6.2 and Proposition 7.3 of [2]. Because not all parts are explicitly stated there, we have sketched a proof for the convenience of the reader.

Proposition 2.11. Let G be a π-separable group and N ◁ G such that G/N is a π-group. Let η ∈ Irr(N), and let χ ∈ Irr(G|η). Suppose that (U, φ) ∈ F^{∗}(N) with (U, φ) ≤ (N, η) and (S, σ) ∈ F^{∗}(G) with (U, φ) ≤ (S, σ) ≤ (G, χ). Let (J, ξ) and (I, ψ) be the corresponding standard inducing pairs for η and for χ respectively. Then

(1) U = S ∩ N.

(2) I normalizes J.

(3) I ∩ J ◁ IJ.

(4) I/(I ∩ J) and J/(I ∩ J) are π-groups. Moreover, if χ ∈ B_{π′}(G), then

(5) I = I_{G}(U, φ) and J = I ∩ N.

(6) ψ lies over ξ.

Proof. The fact that U = S ∩ N is Lemma 3.3 of [2]. Moreover, Lemma 6.1 and Proposition 7.3 of [2] imply that I normalizes J, I ∩ J ◁ IJ, |I : I ∩ J| divides |G : N|^{2}, and |J : I ∩J| divides |G : N|. Hence I/(I ∩J) and J/(I ∩J) are π-groups.

Now assume that χ ∈ B_{π′}(G). Then Proposition 2.10 implies that η ∈ B_{π′}(N) and hence σ and φ are π′-special by Lemma 5.2 of [2]. Now S/U is a π-group because it is isomorphic to a subgroup of G/N. Thus Proposition 2.4 implies that σ_{U} ∈ Irr(U) and hence σ_{U} = φ. Moreover, U = S ∩ N ◁ I and I stabilizes σ_{U} = φ, so I ⊆ I_{G}(U, φ). By Lemma 6.1 of [2] we know that I_{G}(U, φ) normalizes S. But I_{G}(U, φ) must also stabilize σ because σ is the unique π′-special extension of φ by Proposition 2.4. Hence I = I_{G}(U, φ), and J = I ∩ N.

Now let ξ_{0} be an irreducible constituent of ψ_{J} lying over φ, and set η_{0} = ξ_{0}^{N}. Then Proposition 2.8 implies that η_{0} is an irreducible constituent of χ_{N}, so η_{0}^{x} = η for some x ∈ G. Then (U, φ)^{x} ≤ (N, η) and (U, φ)^{x} ∈ F^{∗}(N), so Proposition 2.7 shows that there is an n ∈ N with (U, φ)^{xn} = (U, φ). It follows that xn ∈ I, so we may assume without loss of generality that x ∈ I. Then ξ_{0}^{x} ∈ Irr(J|φ) and (ξ_{0}^{x})^{N} = η = ξ^{N}, so Proposition 2.8 implies that ξ_{0}^{x} = ξ. Hence ξ is a constituent of ψ_{J}, and (6) follows. This completes the proof. □

3. Bounding the Nuclear Length

In this section we study the nuclear length of a character χ ∈ Irr(G) and obtain the bounds given in Theorems A and B. Because the strategy is to proceed by induction on the order of the group G, we begin by considering a normal subgroup N of G, and we try to relate the nuclear length of χ to that of an irreducible constituent of χ_{N}. The first step in this direction is a technical lemma.

Lemma 3.1. Let G be a π-separable group, and let N be a normal subgroup such that G/N is either a π-group or a π′-group. Let η ∈ Irr(N), and let χ ∈ Irr(G|η). Let (U, φ) ∈ F^{∗}(N) with (U, φ) ≤ (N, η), and let (J, ξ) be the corresponding standard inducing pair for η. Then there is a pair (S, σ) ∈ F^{∗}(G) with (U, φ) ≤ (S, σ) ≤ (G, χ) such that if (I, ψ) is the corresponding standard inducing pair for χ, then (ψ_{I∩J}, ξ_{I∩J}) ≠ 0.

Proof. It suffices to consider the case in which G/N is a π-group. We prove the result by induction on nl(χ). If nl(χ) = 0, then χ and η are both π-factorable by Corollary 2.6. Hence (J, ξ) = (N, η) and (I, ψ) = (G, χ), so the conclusion of the lemma is satisfied.

Now assume that nl(χ) > 0 so that χ is not π-factorable. Choose (S, σ) ∈ F^{∗}(G) with (U, φ) ≤ (S, σ) ≤ (G, χ), and let (I, ψ) be the standard inducing pair for χ determined by (S, σ). Then Proposition 2.11 implies that I normalizes J, I ∩ J ◁ IJ, and I/(I ∩ J) and J/(I ∩ J) are π-groups.

Let θ = ψ^{IJ}, and let ξ_{0} ∈ Irr(J|φ) be a constituent of θ_{J}. Then ξ_{0}^{N} ∈ Irr(N) by Proposition 2.8, and χ lies over ξ_{0}^{N} because it lies over ξ_{0}. Hence ξ_{0}^{N} = η^{x} for some x ∈ G. It follows that (U, φ) ≤ (N, η^{x}), so (U, φ)^{x^{−1}} ≤ (N, η) and (U, φ)^{x^{−1}} ∈ F^{∗}(N). Thus there is an n ∈ N such that (U, φ)^{x^{−1}} = (U, φ)^{n}, and nx ∈ I_{G}(U, φ). Hence we may assume that x ∈ I_{G}(U, φ) so that x normalizes J. Moreover, ξ^{x} lies over φ^{x} = φ and (ξ^{x})^{N} = (ξ^{N})^{x} = η^{x} = ξ_{0}^{N}, so ξ^{x} = ξ_{0} by Proposition 2.8. Thus (J, ξ^{x}) ≤ (IJ, θ), and (J, ξ) ≤ (IJ, θ)^{x^{−1}}.

If we now replace (S, σ) by (S, σ)^{x^{−1}}, then (I, ψ) is replaced by (I, ψ)^{x^{−1}} and (IJ, θ) by (IJ, θ)^{x^{−1}}. Hence we may assume that (J, ξ) ≤ (IJ, θ). Then ξ is a constituent of θ_{J} = (ψ^{IJ})_{J} = (ψ_{I∩J})^{J}, so there is a character ζ ∈ Irr(I ∩ J) that is a constituent of both ψ_{I∩J} and ξ_{I∩J}. This completes the proof. □

Lemma 3.2. Let G be a π-separable group, and let N be a normal subgroup of G such that G/N is either a π-group or a π′-group. Let η ∈ Irr(N), and suppose that there is a subgroup U ◁◁ N such that (U, φ) ∈ nuc(η). If χ ∈ Irr(G|η), then nl(χ) ≤ 1.

Proof. It suffices to consider the case in which G/N is a π-group. Let (S, σ) ∈ F^{∗}(G) with (U, φ) ≤ (S, σ) ≤ (G, χ), and let (I, ψ) be the standard inducing pair for χ determined by (S, σ). Since U ◁◁ N and (U, φ) ∈ nuc(η), it follows that (U, φ) ∈ F^{∗}(N) and I_{N}(U, φ) = U. Then Proposition 2.11 shows that U ◁ I and I/U is a π-group, so I/S is a π-group.

Write σ = αβ, where α ∈ 𝔛_{π}(S) and β ∈ 𝔛_{π′}(S). Then β is invariant in I and |I : S| is a π-number, so β extends to β̂ ∈ 𝔛_{π′}(I) by Proposition 2.4. Hence σ^{I} = (αβ)^{I} = α^{I} β̂. All irreducible constituents of α^{I} are π-special by Proposition 2.3, so all irreducible constituents of σ^{I} are π-factorable. In particular, ψ is π-factorable, and nl(χ) ≤ 1. □

Proposition 3.3. Let G be a π-separable group, and let N, H_{1}, and H_{2} be subgroups of G such that N ◁ H_{1} and N ◁ H_{2}. Suppose that H_{1}/N and H_{2}/N are π-groups. Let η ∈ Irr(N), and let χ_{1} ∈ Irr(H_{1}|η) and χ_{2} ∈ Irr(H_{2}|η). Then |nl(χ_{1}) − nl(χ_{2})| ≤ 1.

Proof. We prove the result by induction on nl(χ_{1}). If nl(χ_{1}) = 0, then χ_{1} and η are π-factorable by Corollary 2.6. Hence (N, η) ∈ nuc(η), and Lemma 3.2 shows that nl(χ_{2}) ≤ 1. Thus |nl(χ_{1}) − nl(χ_{2})| ≤ 1 in this case. Similarly, we get the desired result if nl(χ_{2}) = 0.

Thus we may assume that nl(χ_{1}) > 0 and nl(χ_{2}) > 0. Let (U, φ) ∈ F^{∗}(N) with (U, φ) ≤ (N, η), and let (J, ξ) be the corresponding standard inducing pair for η. For i = 1, 2 Lemma 3.1 implies that there is a pair (S_{i}, σ_{i}) ∈ F^{∗}(H_{i}) with (U, φ) ≤ (S_{i}, σ_{i}) ≤ (H_{i}, χ_{i}) such that if (I_{i}, ψ_{i}) is the corresponding standard inducing pair for χ_{i}, then ((ψ_{i})_{I_{i}∩J}, ξ_{I_{i}∩J}) ≠ 0. Moreover, Proposition 2.11 shows that I_{i} ∩ J ◁ I_{i}J, and I_{i}/(I_{i} ∩ J) and J/(I_{i} ∩ J) are π-groups. In particular, it follows that O^{π}(I_{i}) = O^{π}(I_{i} ∩ J) = O^{π}(J) for i = 1, 2. Set M = O^{π}(J). Let θ_{1} ∈ Irr(M) be a common constituent of (ψ_{1})_{M} and ξ_{M}, and let θ_{2} ∈ Irr(M) be a common constituent of (ψ_{2})_{M} and ξ_{M}. Then there is an x ∈ J such that θ_{1} = θ_{2}^{x}. If we replace (H_{2}, χ_{2}) by (H_{2}, χ_{2})^{x} and (S_{2}, σ_{2}) by (S_{2}, σ_{2})^{x}, then we may assume that θ_{1} = θ_{2}.

Since M = O^{π}(I_{1}) = O^{π}(I_{2}), we know that M ◁ I_{1} and M ◁ I_{2}, and I_{1}/M and I_{2}/M are π-groups. Moreover, ψ_{1} ∈ Irr(I_{1}|θ_{1}) and ψ_{2} ∈ Irr(I_{2}|θ_{1}). Thus it follows by induction that

|nl(χ_{1}) − nl(χ_{2})| = |nl(ψ_{1}) − nl(ψ_{2})| ≤ 1,

and this completes the proof. □

We will use Proposition 3.3 in the following form.

Corollary 3.4. Let G be a π-separable group, and let N be a normal subgroup of G such that G/N is either a π-group or a π′-group. If η ∈ Irr(N) and χ ∈ Irr(G|η), then nl(χ) ≤ 1 + nl(η).

Proof. It suffices to consider the case in which G/N is a π-group. The result then follows by applying Proposition 3.3 with H_{1} = G, H_{2} = N, χ_{1} = χ, and χ_{2} = η. □

Theorem 3.5. Let G ≠ 1 be a π-separable group, and let χ ∈ Irr(G). Let 1 = G_{0} ◁ G_{1} ◁ ⋯ ◁ G_{n} = G be any chain of subgroups such that G_{i}/G_{i−1} is either a π-group or a π′-group for 1 ≤ i ≤ n. Then nl(χ) ≤ n − 1.

Proof. We prove the result by induction on n. If n = 1, then G is either a π-group or a π′-group, so nl(χ) = 0 = n − 1.

Now assume that n > 1, and let η ∈ Irr(G_{n−1}) be a constituent of χ_{G_{n−1}}. Then Corollary 3.4 shows that nl(χ) ≤ 1 + nl(η). Moreover, we know by induction that nl(η) ≤ n − 2, so it follows that nl(χ) ≤ n − 1. □

We now obtain Theorem A as an easy consequence of this result.

Proof of Theorem A. To prove (1), consider the chain of subgroups

1 ◁ O_{π′}(G) ◁ O_{π′π}(G) ◁ O_{π′ππ′}(G) ◁ ⋯ ◁ G.

The number of successive quotients in this series that are π-groups is ℓ_{π}(G), and the number that are π′-groups is at most ℓ_{π}(G)+1. Thus the total number of quotients in the series is at most 2ℓ_{π}(G) + 1, and Theorem 3.5 implies that nl(χ) ≤ 2ℓ_{π}(G).

To prove (2), we merely use the definition of the nucleus. If (W, γ) ∈ nuc(χ), then there is a chain

(W, γ) = (I_{n}, ψ_{n}) < (I_{n−1}, ψ_{n−1}) < ⋯ < (I_{0}, ψ_{0}) = (G, χ)

in which (I_{j}, ψ_{j}) is a standard inducing pair for ψ_{j−1}. Since W = I_{n} < I_{n−1} < ⋯ < I_{0} = G, we see that n = nl(χ) is at most the number of prime divisors (counting multiplicities) of |G : W |. Because |G : W | divides γ^{G}(1) = χ(1), the result follows. □

We now turn our attention to characters in B_{π′}(G). The assumption that χ ∈ B_{π′}(G) allows us to obtain better bounds on nl(χ) with rather little additional effort. Indeed, the following proposition already improves upon the conclusion of Corollary 3.4.

Proposition 3.6. Let G be a π-separable group, and let N be a normal subgroup such that G/N is a π′-group. Let χ ∈ B_{π′}(G), and let η be an irreducible constituent of χ_{N}. Then nl(χ) = nl(η).

Proof. We prove the result by induction on |G|. If nl(χ) = 0, then χ ∈ B_{π′}(G) is π-factorable and hence χ is π′-special. Then η is also π′-special by Proposition 2.2, so nl(η) = 0. Conversely, suppose that nl(η) = 0. Because η ∈ B_{π′}(N) by Proposition 2.10, it follows that η is π′-special. Then Proposition 2.3 implies that χ is π′-special, so nl(χ) = 0.

Now suppose that nl(χ) > 0 so that nl(η) > 0 as well. Let (U, φ) ∈ F^{∗}(N) with (U, φ) ≤ (N, η), and let (J, ξ) be the corresponding standard inducing pair for η. Then by Lemma 3.1 there is a pair (S, σ) ∈ F^{∗}(G) with (U, φ) ≤ (S, σ) ≤ (G, χ) such that if (I, ψ) is the corresponding standard inducing pair for χ, then (ψ_{I∩J}, ξ_{I∩J}) ≠ 0. Let ζ ∈ Irr(I ∩ J) be a constituent of both ψ_{I∩J} and ξ_{I∩J}. Because nl(χ) > 0 and nl(η) > 0, we have I < G and J < N ⊆ G. Moreover, Proposition 2.11 shows that I normalizes J, I ∩J ◁ IJ, and I/(I ∩J) and J/(I ∩J) are both π′-groups. Thus by induction it follows that nl(ψ) = nl(ζ) and nl(ξ) = nl(ζ). Hence nl(χ) = 1+nl(ψ) = 1+nl(ξ) = nl(η), and this completes the proof. □

Proposition 3.7. Let G be a π-separable group and N a normal subgroup of G such that G/N is a π-group. Let χ ∈ B_{π′}(G), and let η be an irreducible constituent of χ_{N}. If η is G-invariant, then nl(χ) = nl(η).

Proof. We prove the result by induction on nl(η). If nl(η) = 0, then η is π′-special because η ∈ B_{π′}(G) by Proposition 2.10. Since η is G-invariant by assumption, Proposition 2.4 shows that η has a unique π′-special extension η̂ ∈ Irr(G). In particular, η̂ ∈ B_{π′}(G). Thus χ = η̂ by Proposition 2.10, and nl(χ) = 0.

Now assume that nl(η) ≥ 1. Choose (U, φ) ∈ F^{∗}(N) with (U, φ) ≤ (N, η) and (S, σ) ∈ F^{∗}(G) with (U, φ) ≤ (S, σ) ≤ (G, χ). Let (J, ξ) and (I, ψ) be the corresponding standard inducing pairs for η and for χ respectively. Then Proposition 2.11 shows that U = S ∩ N, J = I ∩ N, I = I_{G}(U, φ), and ψ lies over ξ. Hence I/J is a π-group. Moreover, if x ∈ I, then x stabilizes (U, φ) and (N, η), and x normalizes J. Hence x must stabilize ξ, and we see that ξ is I-invariant. Thus it follows by induction that nl(χ) = nl(ψ) + 1 = nl(ξ) + 1 = nl(η), as desired. □

Theorem 3.8. Let G be a π-separable group and χ ∈ B_{π′}(G). Let 1 = G_{0} ◁ G_{1} ◁ ⋯ ◁ G_{n} = G be a chain of subgroups such that G_{i}/G_{i−1} is either a π-group or a π′-group for 1 ≤ i ≤ n. Choose a corresponding chain (G_{0}, χ_{0}) ≤ (G_{1}, χ_{1}) ≤ ⋯ ≤ (G_{n}, χ_{n}) = (G, χ). Then nl(χ) is at most the number of indices i with 1 ≤ i ≤ n for which the following two conditions are satisfied:

(1) G_{i}/G_{i−1} is a π-group.

(2) χ_{i−1} is not G_{i}-invariant.

Proof. We prove the result by induction on n. If n ≤ 1, then G is either a π-group or a π′-group, so nl(χ) = 0.

Now suppose that n > 1. For 1 ≤ m ≤ n let f(m) be the number of indices i with 1 ≤ i ≤ m such that (1) and (2) hold. Then we know by induction that nl(χ_{n−1}) ≤ f(n − 1). If G_{n}/G_{n−1} is a π′-group, then f(n − 1) = f(n) and nl(χ) = nl(χ_{n−1}) by Proposition 3.6. Similarly, when G_{n}/G_{n−1} is a π-group and χ_{n−1} is G-invariant, it follows that f(n − 1) = f(n) and nl(χ) = nl(χ_{n−1}) by Proposition 3.7. Thus we may assume that G_{n}/G_{n−1} is a π-group and χ_{n−1} is not G-invariant. In this case f(n) = f(n − 1) + 1 and nl(χ) ≤ nl(χ_{n−1}) + 1 by Corollary 3.4, so we again conclude that nl(χ) ≤ f(n). □

Finally, we can now prove Theorem B.

Proof of Theorem B. The bound given in (1) follows by applying Theorem 3.8 to the chain of subgroups

1 ◁ O_{π′}(G) ◁ O_{π′π}(G) ◁ O_{π′ππ′}(G) ◁ ⋯ ◁ G.

We prove by induction on |G| that the number described in (2) also bounds nl(χ). Let N be a maximal normal subgroup of G, and let η ∈ Irr(N) be a constituent of χ_{N}. Then there is an integer e ≥ 1 such that χ(1) = e|G : I_{G}(η)|η(1), and e divides |G : N|. If G/N is a π′-group, then nl(χ) = nl(η) by Proposition 3.6. But χ(1)_{π} = η(1)_{π}, so the result follows by induction.

Now suppose that G/N is a π-group. Then nl(χ) ≤ nl(η) + 1 by Corollary 3.4, and η(1)_{π} divides χ(1)_{π}, so it suffices by induction to show that if nl(χ) = nl(η)+1, then χ(1)_{π} ≠ η(1)_{π}. But we know by Proposition 3.7 that if nl(χ) = nl(η)+1, then η is not G-invariant. Hence χ(1)_{π} = e|G : I_{G}(η)|η(1)_{π} ≠ η(1)_{π}, and this completes the proof. □

It would be interesting to know whether the bounds on nl(χ) given in Theorems A and B are the best possible in any reasonable sense. In the case of Theorem B, for instance, it is easy to find individual examples of a π-separable group G with a character χ ∈ B_{π′}(G) such that nl(χ) = ℓ_{π}(G). But for every nonnegative integer n we can ask whether there is a π-separable group G of π-length n and a character χ ∈ B_{π′}(G) of nuclear length n. This question, along with the analogous questions for the other bounds on nl(χ), seems to be difficult to answer.

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