Conjectures on character degrees for the simple Thompson group

Dedicated to Professor Yukio Tsushima on his sixtieth birthday

Katsuhiro Uno

Department of Mathematics, Osaka University, Toyonaka, Osaka, 560-0043 Japan Email: uno@math.wani.osaka-u.ac.jp

1 Introduction

Let T h be the simple Thompson group. It has order 2^{15}·3^{10}·5^{3}·7^{2}·13·19·31, and its Schur multiplier and the outer automorphism group are both trivial.

In this paper, we prove that Dade’s conjecture for T h is true. Here we mean by saying the conjecture the ordinary form of the conjecture. (See [2].) According to [4], the ordinary form is equivalent to the inductive form if the Schur multiplier and the outer automorphism group are trivial. Moreover, if a defect group of a block is cyclic, then the ordinary form is proved to be true for this block in [2]. Thus, it suffices to treat the primes 2, 3, 5 and 7. The character tables of several subgroups of T h are available. We use those found in ATLAS [1] and GAP library. Also, maximal local subgroups of T h are determined in [12] and [18]. These information are quite useful. In the proof of the conjecture, we use a reduction theorem, which is a very special case of the one proved in §16 of [3].

Let G be a finite group and p a prime. Quite recently, a new type of a conjecture, which concerns the p′-part of character degrees, is proposed by Isaacs and Navarro [10]. Though the original version considers only height zero characters of G and the normalizer of a defect group, here we prove a version of Dade’s type. Namely, we prove that the alternating sum of the numbers of relevant characters of the normalizers of p-chains with any fixed defect is zero. Since we count the number of characters satisfying a certain congruent relation such as χ(1)_{p′} ≡ ±κ mod p, where χ(1)_{p′} is the p′-part of χ(1), it suffices to consider the case of p ≥ 5. However, for p = 13, 19 or 31, a Sylow p-subgroup of T h is cyclic of prime order. Thus the normalizers of chains are just T h and the normalizer of a Sylow p-subgroup, and moreover, it is known that those p-blocks of the normalizers have only characters of height zero. Hence the assertion of our version of the conjecture is equivalent to that of Isaacs and Navarro, and the latter is proved for cyclic defect group cases in Theorem 2.1 of [10]. Thus when proving the alternating sum version of the conjecture, we treat only the cases of p = 5 and 7 in this paper. Our version of the conjecture is proposed in §3 below.

We denote by A : B and A · B a split and a non-split extension of A by B, respectively. We use n to denote a cyclic group of order n. Moreover, for a prime p, an elementary abelian group of order p^{n} is denoted by p^{n}. For an odd prime p, we denote by p_{+}^{1+2n} an extra special group of order p^{2n+1} and exponent p. The plus type extra special group of order 2^{2n+1} is denoted by 2_{+}^{1+2n}. Finally, S_{n} and A_{n} denote the symmetric group and the alternating group of degree n, respectively, and D_{8} denotes the dihedral group of order 8.

Throughout the paper, a character means an irreducible complex character unless otherwise noted. For a fixed prime p, we use the following notation. Let G be a finite group and H a subgroup of G. The principal block of H is denoted by B_{0}(H), and the set of characters φ of H is denoted by Irr(H). If the p-part of |H|/φ(1) is p^{d}, then we say that φ has defect d = d(φ). For an integer d, we denote by Irr(H, d) the set of those φ in Irr(H) with d(φ) = d. Also for an integer κ, we denote by Irr(H, [κ]) the set of those φ in Irr(H) such that

(|H|/φ(1))_{p′} ≡ ±κ mod p.

We also put Irr(H, d, [κ]) by

Irr(H, d, [κ]) = Irr(H, d) ∩ Irr(H, [κ]).

For a normal subgroup K of H and a character θ of K, we use Irr(H|θ) to denote the set of those φ in Irr(H) such that φ is lying over θ, that is, the restriction of φ to K has θ as its irreducible constituent. We denote the intersections of Irr(H|θ) with Irr(H, d), Irr(H, [κ]), and Irr(H, d, [κ]) respectively by Irr(H, d|θ), Irr(H, [κ]|θ) and Irr(H, d, [κ]|θ). Moreover, for a p-block B of G, we use Irr(H, B) to denote the set of characters of H belonging to some p-block b of H inducing B. The sets Irr(H, B|θ), Irr(H, B, d), Irr(H, B|θ) and so on denote the intersections of Irr(H, B) with the relevant sets. Sometimes θ is replaced by a subset Θ of Irr(K). If this is the case, for example, Irr(H|Θ) means the set of those φ in Irr(H) such that φ is lying over some θ in Θ. The cardinalities of Irr(∗) is in general denoted by k(∗).

Acknowledgment. The author is very much grateful to Professor S.Yoshiara and Dr. M.Sawabe for their many helpful suggestions, which are crucial in determining radical p-chains of T h. Also the author is grateful to Professor G. Navarro who sent him their manuscript.

2 Dade’s conjecture

We describe the ordinary form of the conjecture. Let G be a finite group and p a prime. A radical p-chain of G is a chain

C : P_{0} < P_{1} < P_{2} < ⋯ < P_{n}

of p-subgroups P_{i} of G such that

(i) P_{0} = O_{p}(G) and
(ii) P_{i} = O_{p}(∩_{j=0}^{i}N_{G}(P_{j})) for all i = 0, 1, ⋯, n.

For those C, we write by N_{G}(C) the normalizer ∩_{i=0}^{n}N_{G}(P_{i}) of C in G, and by |C|, the length n of C. Let R_{p}(G) be the set of all radical p-chains of G. The group G acts on R_{p}(G) by conjugation. We denote a set of representatives of G-conjugacy classes of R_{p}(G) by R_{p}(G)/G.

If a p-subgroup P of G satisfies P = O_{p}(N_{G}(P)), then we say that P is a radical p-subgroup of G. (Note that in this paper, O_{p}(G) is also a radical subgroup of G.) The set of radical p-subgroups of G is denoted by B_{p}(G). Thus,

C : P_{0} < P_{1} < P_{2} < ⋯ < P_{n}

is a radical p-chain if and only if P_{0} = O_{p}(G), P_{i} ≤ N_{G}(P_{j}) for i, j with j ≤ i and P_{i+1}/P_{i} lies in B_{p}(∩_{j=0}^{i}N_{G}(P_{j})/P_{i}) for all i = 0, 1, ⋯, n − 1.

The ordinary form of Dade’s conjecture is stated as follows.

Conjecture 2.1 Let p be a prime and G a finite group with O_{p}(G) = 1. Consider the above situation. If the defect d(B) of B is greater than 0, then

∑_{C∈R_{p}(G)/G}(−1)^{|C|}k(N_{G}(C), B, d) = 0

for all d.

3 Conjecture of Isaacs and Navarro

In this section, we describe the Isaacs and Navarro conjecture and an alternating sum version of it. Let G be a finite group and p a prime. Let B be a p-block of G with defect group D and b the p-block of N_{G}(D) which corresponds to B by the first main theorem of Brauer. For an integer κ with 0 < κ < p, let M_{κ}(B) be the number of irreducible characters in B with height zero and the p′-part of the degrees are congruent to ±κ modulo p. Then Isaacs and Navarro propose the following conjecture in [10].

Conjecture 3.1 (Isaacs-Navarro) In the notation above, let c be the p′-part of |G : N_{G}(D)|. Then

M_{cκ}(B) = M_{κ}(b)

for all κ with 0 < κ < p.

However, in the above, there seems to be no reason why we take only height zero characters if we deal with also the normalizers of chains. By considering resemblance to the relationship between McKay conjecture and the ordinary form of Dade’s conjecture, we propose the following.

Conjecture 3.2 Let p be a prime and G a finite group with O_{p}(G) = 1. Consider the above situation. If the defect d(B) of B is greater than 0, then

∑_{C∈R_{p}(G)/G}(−1)^{|C|}k(N_{G}(C), B, d, [κ]) = 0

for all d and κ.

Of course the other forms such as invariant, projective forms of the conjecture can be set. However, detail of them and the relationship between Conjecture 3.2 and the original version of the conjecture by Isaacs and Navarro will be considered elsewhere. Here remark that Conjecture 3.2 implies Conjecture 2.1.

Now we give the following easy remarks.

Lemma 3.3 Let G be a finite group and fix a radical p-subgroup P of G with P ≠ O_{p}(G). Suppose that a radical p-subgroup P̃ of G with P̃ > P satisfies N_{G}(P̃) ⊆ N_{G}(P). Then P̃ is a radical p-subgroup of N_{G}(P), and putting

\begin{matrix}R_{1} = \{ C : O_{p}(G) < P < P̃ < ⋯ < P_{n} ∈ R_{p}(G) | |C| ≥ 2 \} and \\ R_{2} = \{ C : O_{p}(G) < P̃ < ⋯ < P_{n} ∈ R_{p}(G) | |C| ≥ 1 \}, \end{matrix}

there is a bijection f : R_{1} → R_{2} sending any O_{p}(G) < P < P̃ < ⋯ in R_{1} into O_{p}(G) < P̃ < · · ·. Moreover, for any C in R_{1}, we have N_{G}(C) = N_{G}(f(C)) and |C| = |f(C)| + 1.

By the above, in the alternating sum appearing in the conjectures, we can take radical p-chains which do not lie in R_{1} or R_{2}. Moreover, concerning Clifford theory, the following is important.

Lemma 3.4 Let G be a finite group and N a normal subgroup of G. Let θ be a character of N, and T the stabilizer of θ in G. Then there exists a bijection from Irr(T, d, [κ]|θ) to Irr(G, d, [κ]|θ) sending any φ in Irr(T, d, [κ]|θ) to φ^{G}.

4 A reduction theorem

In this section, we prove a theorem which reduces the problem to subsections. It is a very special case of Theorem 16.4 of [3], though here it is slightly modified and takes κ into account. We can give a proof here, because it is not very complicated in our case. Consider the following situation.

Hypothesis 4.1 Let p be a prime and G a finite group of the form

G = (Q × H):⟨σ⟩,

for a p-subgroup Q, a subgroup H and a p-regular element σ of G such that σ normalizes Q and H.

5

For any C ∈ R_{p}(G), each subgroup P appearing in C has the form P = Q × \overline{P} for a p-subgroup \overline{P} of H. Furthermore, there is a bijection

f : R_{p}(G)/G → R_{p}(H:⟨σ⟩)/(H:⟨σ⟩)

sending any class of

C : O_{p}(G) = Q × O_{p}(H) < Q × \overline{P}_{1} < ⋯

to the class of

\overline{C} : O_{p}(H:⟨σ⟩) = O_{p}(H) < \overline{P}_{1} < ⋯,

and we have |C| = |\overline{C}| and N_{G}(C) = Q:N_{H:⟨σ⟩}(\overline{C}).

Fix a character θ of Q. Let ⟨τ⟩ be the stabilizer of θ in ⟨σ⟩, where τ = σ^{m} for some divisor m of |⟨σ⟩|. The stabilizer of θ in G is (Q × H):⟨τ⟩, and the stabilizer I(C, θ) of θ in N_{G}(C) can be written as

I = I(C, θ) = Q:N_{H:⟨τ⟩}(\overline{C}).

Note that θ can be extended to (Q × H) : ⟨τ⟩, since it has an extension θ × 1_{H} to Q × H which is ⟨τ⟩-invariant. In particular, θ can be extended to I(C, θ) for any C.

Let b be a p-block of N_{G}(C) such that b contains a some character in Irr(N_{G}(C))|θ). Then there exists a p-block b′ of I inducing b. (See V.3.1 of [13].) Let \overline{b}_{1}, \overline{b}_{2}, ⋯, \overline{b}_{t} be p-blocks of N_{H:⟨τ⟩}(\overline{C}) dominated by b′. See Chapter V, §8 of [13]. Since θ can be extended to I(C, θ) and since |Q|/θ(1) is a d(θ)-th power of p, it follows from Clifford theory that

k(I, b′, d, [κ]|θ) = ∑_{j=1}^{t} k(N_{H:⟨τ⟩}(\overline{C}), \overline{b}_{j}, d − d(θ), [κ]).

Since k(N_{G}(C), b, d, [κ]|θ) is the sum of k(I, b′, d, [κ]|θ), where b′ ranges in the set of p-blocks of I inducing b, we have the following.

Theorem 4.2 Under Hypothesis 4.1, let θ be a character of Q with defect d(θ) and Θ the G-orbit of θ, and let ⟨τ⟩ be the stabilizer of θ in ⟨σ⟩. If

∑_{\overline{C}∈R_{p}(H:⟨τ⟩)/H:⟨τ⟩}(−1)^{|\overline{C}|}k(N_{H:⟨τ⟩}(\overline{C}), \overline{B}, d, [κ]) = 0

for all p-blocks \overline{B} of H:⟨τ⟩, all d with d ≠ 0, and all κ, then we have

∑_{C∈R_{p}(G)/G}(−1)^{|C|}k(N_{G}(C), B, d, [κ]|Θ) = 0

for all p-block B of G, all d with d ≠ d(θ), and all κ.

Proof. Use the notation given preceding the theorem. Since the stabilizer of θ in G is (Q × H)⟨τ⟩, its G-orbit Θ consists of m characters. Since ⟨σ⟩ is abelian, I(C, θ) = I(C, θ^{g}) = Q: N_{H:⟨τ⟩}(\overline{C}) for all g ∈ G, and thus the N_{G}(C)-orbits of θ and θ^{g} consist of the same number of characters. Since the N_{G}(C)-orbit of θ consists of |N_{G}(C) : (Q:N_{H:⟨τ⟩}(\overline{C}))| characters, the number of N_{G}(C)-orbits in Θ is

m(C, θ) = m/|N_{G}(C) : (Q:N_{H:⟨τ⟩}(\overline{C}))| = m/|N_{H:⟨σ⟩}(\overline{C}) : N_{H:⟨τ⟩}(\overline{C}))|
= |(H:⟨σ⟩)|/|(H:⟨τ⟩)| × (|N_{H:⟨σ⟩}(\overline{C})|/|N_{H:⟨σ⟩}(\overline{C}) ∩ (H:⟨τ⟩|)^{−1}
= |(H:⟨σ⟩)|/|(H:⟨τ⟩)| × (|N_{H:⟨σ⟩}(\overline{C})H:⟨τ⟩|/|H:⟨τ⟩|)^{−1}
= |(H:⟨σ⟩)|/ × (|N_{H:⟨σ⟩}(\overline{C})(H:⟨τ⟩|))^{−1}.

Let θ = θ_{1}, θ_{2}, ⋯, θ_{m(C,θ)} form a set of representatives of N_{G}(C)-orbits in the G-orbit of θ.

Now, the H : ⟨σ⟩-orbit of the radical p-chain \overline{C} of H consists of |(H : ⟨σ⟩) : N_{H:⟨σ⟩}(\overline{C})| chains. But, it is a disjoint union of H:⟨τ⟩-orbits, each of which consists of |H⟨τ⟩ : N_{H:⟨τ⟩}(\overline{C})| chains and we have

|H⟨τ⟩ : N_{H:⟨τ⟩}(\overline{C})| = |H⟨τ⟩ : N_{H:⟨σ⟩}(\overline{C}) ∩ H⟨τ⟩| = |N_{H:⟨σ⟩}(\overline{C})(H:⟨τ⟩) : N_{H:⟨σ⟩}(\overline{C})|

Thus the number of H⟨τ⟩-orbits in the disjoint union is in total m(C, θ).

On the other hand, let B be a p-block of G and consider all p-blocks b of N_{G}(C) inducing B and having some character in Irr(N_{G}(C))|θ). (If such a b exists, then B has a character lying over θ. See V.3.10 of [13].) Each of those is induced from some p-block b′ of I(C, θ). Take all such b and b′ and all p-blocks \overline{b}_{1}, \overline{b}_{2}, ⋯, \overline{b}_{s} of N_{H:⟨τ⟩}(\overline{C}) dominated by one of these b′’s. Note that each \overline{b}_{j} can be induced to a p-block of H : ⟨τ⟩. Of course, they depend on C. Let C_{0} be the trivial chain of G consisting only one group O_{p}(G). Then I(C_{0}, θ) = (Q × H):⟨τ⟩, and we have p-blocks \overline{B}_{1}, \overline{B}_{2}, ⋯, \overline{B}_{u} of N_{H:⟨τ⟩}(\overline{C_{0}}) = H: ⟨τ⟩ by the above process applied to C_{0}. Then, it is easy to see that \overline{B}_{1}, \overline{B}_{2}, ⋯, \overline{B}_{u} are exactly the p-blocks of H : ⟨τ⟩ induced form some \overline{b}_{j}. (This is proved in much more general situation in Theorem 14.3 of [3].) From the arguments above and preceding the theorem, we have

∑_{C∈R_{p}(G)/G}(−1)^{|C|}k(N_{G}(C), B, d, [κ]|Θ)
= ∑_{C∈R_{p}(G)/G} ∑_{i=1}^{m(C,θ)}(−1)^{|C|}k(N_{G}(C), B, d, [κ]|θ_{i})
= ∑_{C∈R_{p}(G)/G} ∑_{i=1}^{m(C,θ)}(−1)^{|C|}k(I(C, θ_{i}), B, d, [κ]|θ_{i})
= ∑_{C∈R_{p}(G)/G} m(C, θ)(−1)^{|C|}k(I(C, θ), B, d, [κ]|θ)
= ∑_{\overline{C}∈R_{p}(H:⟨σ⟩)/H:⟨σ⟩)} m(C, θ)(−1)^{|\overline{C}|} ∑_{j=1}^{u} k(N_{H:⟨τ⟩}(\overline{C}), \overline{B}_{j}, d − d(θ), [κ])
= ∑_{\overline{C}∈R_{p}(H)/H:⟨σ⟩)} m(C, θ)(−1)^{|\overline{C}|} ∑_{j=1}^{u} k(N_{H:⟨τ⟩}(\overline{C}), \overline{B}_{j}, d − d(θ), [κ])
= ∑_{\overline{C}∈R_{p}(H)/H:⟨τ⟩)}(−1)^{|\overline{C}|} ∑_{j=1}^{u} k(N_{H:⟨τ⟩}(\overline{C}), \overline{B}_{j}, d − d(θ), [κ])
= ∑_{\overline{C}∈R_{p}(H:⟨τ⟩)/H:⟨τ⟩)}(−1)^{|\overline{C}|} ∑_{j=1}^{u} k(N_{H:⟨τ⟩}(\overline{C}), \overline{B}_{j}, d − d(θ), [κ])
= ∑_{j=1}^{u} ∑_{\overline{C}∈R_{p}(H:⟨τ⟩)/H:⟨τ⟩)}(−1)^{|\overline{C}|}k(N_{H:⟨τ⟩}(\overline{C}), \overline{B}_{j}, d − d(θ), [κ])

for all d and κ. Now, by the assumption, the above alternating sum is zero for all d with d ≠ d(θ). This completes the proof. □

Remark. One might think that the condition d ≠ d(θ) is not necessary in the conclusion of the above. However, if O_{p}(H) = \{1\} and \overline{B} is of defect zero, then the alternating sum in the assumption of the theorem is 1 for d = 0, since k(N_{H:⟨τ⟩}(\overline{C}), \overline{B}, 0) = 0 for all non-trivial chain \overline{C} in R_{p}(H:⟨τ⟩). Thus it is reasonable to restrict the assumption either to p-blocks \overline{B} of positive defect or to d with d ≠ 0. However, if we take the former, it becomes slightly difficult to state the conclusion.

The above theorem will be applied in the following situation.

Hypothesis 4.3 Let p be a prime and G a finite group such that G has a p-subgroup Q with

N_{G}(Q) = (Q × H):⟨σ⟩ = N_{G}(Q × O_{p}(H))

for a subgroup H of G and a p-regular element σ of G which normalizes Q and H.

Then Q × O_{p}(H) is a radical p-subgroup of G, and Hypothesis 4.1 is satisfied for N_{G}(Q) in place of G there. Assume further that O_{p}(G) = {1} and let R′_{p}(G) be the set of radical p-chains of G starting with 1 < Q × O_{p}(H). Then there is a bijection

F : R′_{p}(G)/N_{G}(Q) → R_{p}(N_{G}(Q))/N_{G}(Q)

sending any class of

C : 1 < Q × O_{p}(H) < P_{1} < ⋯

to the class of

C′ : O_{p}(N_{G}(Q)) = Q × O_{p}(H) < P_{1} < ⋯,

such that |C| = |F(C)| + 1 and N_{G}(C) = N_{G}(F(C)). Thus Theorem 4.2 can be used to reduce partly the problem for G to that for H:⟨σ⟩. In particular, if Q is abelian, we have the following.

Theorem 4.4 Under Hypothesis 4.3, use the above notation. Suppose that O_{p}(G) = {1} and that Q is abelian with |Q| = p^{s}. If

∑_{\overline{C}∈R_{p}(H_{1})/H_{1}}(−1)^{|\overline{C}|}k(N_{H_{1}}(\overline{C}), \overline{B}, d, [κ]) = 0

for all p-blocks \overline{B} of all H_{1} with H ≤ H_{1} ≤ H:⟨σ⟩, all d with d ≠ 0 and all κ, then we have

∑_{C∈R′_{p}(G)/G}(−1)^{|C|}k(N_{G}(C), B, d, [κ]) = 0

for all p-blocks B of G, all d with d ≠ s and all κ.

Proof. Taking all N_{G}(Q)-orbits Θ in Irr(Q), and applying Theorem 4.2, we have the desired result, since all the characters of Q have defect s. □

In the group T h, the above situation really occurs and to use Theorem 4.2 for it, we prove that the conjecture 3.2 holds for certain groups.

Proposition 4.5 The conjecture 3.2 holds for the following groups G and primes p.

(1a) Let q = p^{e}. The group G is p^{2e}: GL_{2}(q) = (F_{q})^{2} : GL_{2}(q) with the natural action of GL_{2}(q) on (F_{q})^{2}.

(1b) Let q = p^{e}. The group G is p^{2e}: SL_{2}(q) = (F_{q})^{2} : SL_{2}(q) with the natural action of SL_{2}(q) on (F_{q})^{2}.

(1c) Let q = p^{e} with p ≠ 2. Let σ be the Galois automorphism of F_{q^{2}} with σ(x) = x^{q}. The group G is (p^{4e} : SL_{2}(q^{2})) : ⟨σ⟩ = ((F_{q^{2}})^{2} : SL_{2}(q^{2})) : ⟨σ⟩ with the natural actions of SL_{2}(q) and σ.

(2) Let p = 3. Denote by σ the graph automorphism of G_{2}(3) of order two. The group G is G_{2}(3) or G_{2}(3):⟨σ⟩.

Proof. (1a), (1b). We omit the proof of (1a) and give a proof of (1b), since the argument for (1a) is similar to and simpler than that for (1b). Let H(q) be the subgroup of upper triangular matrices in SL_{2}(q).

H(q) = \{( \begin{matrix}a & b \\ 0 & a^{−1} \end{matrix} ) |a, b ∈ F_{q}, a ≠ 0\}.

Then the trivial chain (F_{q})^{2} = O_{p}(G) and (F_{q})^{2} < (F_{q})^{2} : O_{p}(H(q)) form a set of representatives of G-conjugacy classes of radical p-chains of G. The normalizer of the latter is (F_{q})^{2} : H(q). It follows from V.3.11 of [7] that G and (F_{q})^{2} : H(q) have only the principal p-blocks. We use the argument in §6 of [14]. Consider the action of SL_{2}(q) on Irr((F_{q})^{2}). Define ν : (F_{q})^{2} → C^{∗} by ν((α, β)) = (\sqrt[q]{1})^{α} for (α, β) ∈ (F_{q})^{2}. Then ν is a character of (F_{q})^{2}. The stabilizer of ν in SL_{2}(q) is

H_{1}(q) = \{( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) |b ∈ F_{q}\}.

Clearly, the trivial character 1_{(F_{q})^{2}} and ν form a set of representatives of G-conjugacy classes of Irr((F_{q})^{2}). However, there are three H(q)-orbits, one consisting of 1_{(F_{q})^{2}}, the orbit of ν and the one consisting of all the others. Let ν′ be the character of (F_{q})^{2} defined by ν′((α, β)) = (\sqrt[q]{1})^{β} for (α, β) ∈ (F_{q})^{2}. Then ν and ν′ are not H(q)- conjugate and the stabilizer of ν′ in H(q) is trivial. It follows from Clifford theory that

k((F_{q})^{2} :SL_{2}(q), d, [κ]) = k(SL_{2}(q), d − 2e, [κ]) + k(H_{1}(q), d − 2e, [κ]) and
k((F_{q})^{2} :H(q), d, [κ]) = k(H(q), d − 2e, [κ]) + k(H_{1}(q), d − 2e, [κ])
+k({1}, d − 2e, [κ]),

for all d and κ, since 1_{F_{q}} and ν can be extended to their stabilizers. (See also 5.20 of [11].) If p ≥ 5, then we have

\begin{matrix}(d, κ) & (e, 1) & (e, 2) & (0, 1) \\ k(SL_{2}(q), d, [κ]) & q − 1 & 4 & 1 \\ k(H(q), d, [κ]) & q − 1 & 4 & 0 \end{matrix}

For other d’s and κ’s which do not appear above, the numbers of relevant characters are zero. This convention is used throughout the paper. For p = 2 and 3, we can ignore κ and have the following.

\begin{matrix}p = 3 \\ d & e & 0 \\ k(SL_{2}(q), d) & q + 3 & 1 \\ k(H(q), d) & q + 3 & 0 \end{matrix} \begin{matrix}p = 2 \\ d & e & 0 \\ k(SL_{2}(q), d) & q & 1 \\ k(H(q), d) & q & 0 \end{matrix}

Thus the result holds. See also [16].

(1c). Use the argument in (1b). Our G has two conjugacy classes of radical p-chains. However, we have ((F_{q^{2}})^{2} : H(q^{2})) : ⟨σ⟩ as the normalizer of a non-trivial radical p-chain. Define θ : (F_{q^{2}})^{2} → C^{∗} by θ((α, β)) = ν(T (α), T (β)) for (α, β) ∈ (F_{q^{2}})^{2}, where T : F_{q^{2}} → F_{q} is the usual trace map. Define θ′ similarly by using ν′ instead of ν. Then θ and θ′ are ⟨σ⟩-invariant. Moreover, 1_{(F_{q})^{2}} and θ form a set of representatives of SL_{2}(q^{2}):⟨σ⟩-orbits in Irr((F_{q^{2}})^{2}), and 1_{(F_{q})^{2}}, θ and θ′ form that of H(q^{2}):⟨σ⟩-orbits. Since 1_{F_{q^{2}}}, θ and θ′ can be extended to their stabilizers, we have

k((F_{q^{2}} :SL_{2}(q^{2})):⟨σ⟩, d, [κ]) = k(SL_{2}(q^{2}):⟨σ⟩, d − 4e, [κ])
+k(H_{1}(q^{2}):⟨σ⟩, d − 4e, [κ]) and
k((F_{q^{2}} :H(q^{2})):⟨σ⟩, d, [κ]) = k(H(q^{2}):⟨σ⟩, d − 4e, [κ])

+k(H_{1}(q^{2}):⟨σ⟩, d − 4e, [κ]) + k(⟨σ⟩, d − 4e, [κ]).

Now look at the σ action on the characters of SL_{2}(q^{2}) and H(q^{2}). For characters of SL_{2}(q^{2}), see for example, [5]. The group SL_{2}(q^{2}) has (q^{2} − 3)/2 characters of degree q^{2} + 1. It is easily seen that among those, q − 2 are σ-invariant and (q − 1)^{2}/2 are not. It has also (q^{2} − 1)/2 characters of degree q^{2} − 1. None of them are σ-invariant. The characters of degree 1, q^{2}, (q^{2} − 1)/2 and (q^{2} + 1)/2 are all σ-invariant. On the other hand, H(q^{2}) has q^{2} − 1 linear characters, and among those, exactly q − 1 are σ-invariant. It also has four characters of degree (q^{2}−1)/2. They are all σ-invariant. Thus, for p ≥ 5, we have

\begin{matrix}(d, κ) & (4e, 2) & (4e, 1) & (4e, 2) & (0, 1) \\ k(SL_{2}(q^{2}):⟨σ⟩, d, [κ]) & 2(q − 1) & q(q − 1)/2 & 8 & 2 \\ k(H(q^{2}):⟨σ⟩, d, [κ]) & 2(q − 1) & q(q − 1)/2 & 8 & 0 \end{matrix}

and for p = 3, we have

\begin{matrix}d & 4e & 0 \\ k(SL_{2}(q^{2}):⟨σ⟩, d) & (q^{2} + 3q + 12)/2 & 2 \\ k(H(q^{2}):⟨σ⟩, d) & (q^{2} + 3q + 12)/2 & 0 \end{matrix}

Hence the result holds.

(2). Since p = 3, we can ignore κ in the argument. The group G_{2}(3) has 3 non-conjugate non-trivial parabolic subgroups U, U^{σ} and U_{0}, where U is a maximal parabolic subgroup and U_{0} is a Borel subgroup. Hence the following form a set of representatives of radical 3-chains of G_{2}(3). Note that two maximal parabolic subgroups are σ-conjugate and U_{0} is σ-invariant.

radical 3 − chain  normalizer parity
1,  G_{2}(3),  +
1 < O_{3}(U),  U,  −
1 < O_{3}(U)^{σ},  U^{σ},  −
1 < O_{3}(U) < O_{3}(U_{0}),  U_{0},  +
1 < O_{3}(U)^{σ} < O_{3}(U_{0}),  U_{0},  +
1 < O_{3}(U_{0}),  U_{0},  −

The group G_{2}(3) has one block of defect zero and the principal block B_{0}(G_{2}(3)), and U and U_{0} have only the principal blocks. From [6] we have the following. Note that U_{0} is denoted by B in [6].

\begin{matrix}d & 6 & 5 & 4 \\ k(G_{2}(3), B_{0}(G_{2}(3)), d) & 15 & 6 & 1 \\ k(U, B_{0}(G_{2}(3)), d) & 15 & 15 & 1 \\ k(U_{0}, B_{0}(G_{2}(3)), d) & 15 & 24 & 1 \end{matrix}

Thus the result follows for G_{2}(3). The group G_{2}(3) : ⟨σ⟩ has the following set of representatives of conjugacy classes of radical 3-chains.

radical 3 − chain  normalizer parity
1,  G_{2}(3):⟨σ⟩, +
1 < O_{3}(U), U, −
1 < O_{3}(U) < O_{3}(U_{0}), U_{0},  +
1 < O_{3}(U_{0}),  U_{0} :⟨σ⟩,  −

The group G_{2}(3) : ⟨σ⟩ has two blocks of defect zero and the principal block B_{0}(G_{2}(3) : ⟨σ⟩), and the subgroup U_{0} : ⟨σ⟩ has only the principal block. In the notation of [6], eleven characters, θ_{i} for i = 0, 1, 2, 5, 6, 7, 10, 11, θ_{12}(±1) and χ_{14}(1), of G_{2}(3) are ⟨σ⟩-invariant, and so are the ten characters χ_{1}(0, 0), χ_{1}((q−1)/2, 0), θ_{5}(0), θ_{5}(1), θ_{6}(0), θ_{6}(1), θ_{7}, θ_{8}, θ_{9}(±1) of U_{0}. The character θ_{5} of G_{2}(3) has degree 3^{6} and thus lies in a block of defect zero. All the other characters lie in the corresponding principal blocks. Thus we have the following.

\begin{matrix}d & 6 & 5 & 4 \\ k(G_{2}(3):⟨σ⟩, B_{0}(G_{2}(3):⟨σ), d) & 18 & 6 & 2 \\ k(U_{0} :⟨σ⟩, B_{0}(G_{2}(3):⟨σ), d) & 18 & 15 & 2 \end{matrix}

Thus the result follows for G_{2}(3):⟨σ⟩. □

5 The case of p = 2 for T h

In this section, we assume that G = T h and p = 2. Thus we can ignore κ.

5.1 Radical 2-chains

By Theorem 20 of [20], there are 19 G-conjugacy classes of non-trivial radical 2subgroups. Among them, there are radical 2-subgroups E ≅ 2_{+}^{1+8} and A ≅ 2^{5}, and we have N_{G}(E) ≅ E ·A_{9} and N_{G}(A) ≅ A·L_{5}(2) which are only maximal 2-local subgroups of G up to G-conjugacy. (Theorem 2.2 of [18].) Moreover, C_{G}(z) = N_{G}(E) ≅ E·A_{9} for the central involution z of E. Furthermore, a radical 2-subgroup of N_{G}(E) is N_{G}(E)-conjugate to the inverse image ER in N_{G}(E) of a radical 2subgroup R of A_{9} under the surjective map N_{G}(E) → A_{9}, and each ER is a radical 2-subgroup of G. Note that A_{9} has eleven classes of radical 2-subgroups, and those ER give representatives of radical 2-subgroups of G having G-conjugates whose normalizers are contained in N_{G}(E). Thus, it follows from Lemma 3.3 that radical 2-chains of G starting with 1 < ER or 1 < E < ER for each non-trivial radical 2-subgroup R of A_{9} can be eliminated in the alternating sum. In particular, it suffices to consider only 1 < E among those starting with 1 < E. To examine the situation for A, let us introduce the following notation. Fix a Borel subgroup of L_{5}(2) and let an ordered set Π = {r_{1}, r_{2}, r_{3}, r_{4}} be the set of fundamental positive roots of L_{5}(2). Then L_{5}(2)-conjugacy classes of parabolic subgroups of L_{5}(2) are parameterized by subsets of Π. For a subset J of Π, we denote the corresponding parabolic subgroup by P_{J}, and the unipotent radical of P_{J} by U_{J}. A radical 2subgroup of L_{5}(2) is L_{5}(2)-conjugate to U_{J} for some J. Thus a radical 2-subgroup of N_{G}(A) is N_{G}(A)-conjugate to AU_{J} for some J. However, as is shown in the proof of Theorem 20 of [20], if r_{1} ∉ J, then N_{G}(AU_{J}) ≤ N_{G}(E) up to conjugacy. Thus AU_{J} is G-conjugate to a radical 2-subgroup of N_{G}(E). Hence to examine radical 2-subgroups of G contained in N_{G}(A), it suffices to look at AU_{J} such that r_{1} ∈ J. Those eight AU_{J}’s with r_{1} ∈ J are representatives of remaining radical 2-subgroups of G. (Note that AU_{Π} = A.) Thus, again by Lemma 3.3, radical 2-chains of G starting with 1 < AU_{J} or 1 < A < AU_{J} for a subset J of Π with r_{1} ∈ J and J ≠ Π can be eliminated in the alternating sum. In particular, among those starting with 1 < A, it suffices to consider only 1 < A and

C : 1 < A < AU_{J_{1}} < AU_{J_{2}} < ⋯ < AU_{J_{t}} with Π ⊃ J_{1} ⊃ J_{2} ⊃ ⋯ ⊃ J_{t}

such that r_{1} ∉ J_{1}. Note that N_{G}(C) = AP_{J_{t}}. Let us fix a subset J of Π with r_{1} ∉ J, and let R′_{2}(G, A, AU_{J}) be the subset of R_{2}(G) consisting of those C above such that the last subgroup in C is AU_{J}. (So J_{t} = J in the above.) If J ≠ {r_{2}, r_{3}, r_{4}}, then we can define a map f : R′_{2}(G, A, AU_{J}) → R′_{2}(G, A, AU_{J}) by

f(C) = \{ \begin{matrix}1 < A < AU_{J_{2}} < ⋯ < AU_{J_{t}}, & if J_{1} = \{r_{2}, r_{3}, r_{4}\}, \\ 1 < A < AU_{\{r_{2},r_{3},r_{4}\}} < AU_{J_{1}} < AU_{J_{2}} < ⋯ < AU_{J_{t}}, & otherwise, \end{matrix}

in the above notation. Then f gives an involutive bijection on R′_{2}(G, A, AU_{J}) and we have N_{G}(C) = N_{G}(f(C)) and |C| = |f(C)| ± 1. Thus we can eliminate the chains in R′_{2}(G, A, AU_{J}) in the alternating sum. Now the remaining radical 2chains are 1 < A and 1 < A < AU_{J_{0}}, where J_{0} = {r_{2}, r_{3}, r_{4}}. Note that U_{J_{0}} ≅ 2^{4} and P_{J_{0}} ≅ 2^{4} : L_{4}(2). Thus the normalizer of 1 < A < AU_{J_{0}} has the structure A · (2^{4} : L_{4}(2)). Moreover, since we may assume N_{G}(AU_{J_{0}}) ≤ N_{G}(E), we have AU_{J_{0}} = E and A·(2^{4} : L_{4}(2)) = E·L_{4}(2) = E·A_{8}. Hence, we have the following.

Proposition 5.1 To prove that the conjecture holds for G and p = 2, it suffices to consider the alternating sum with respect to the following radical 2-chains.

\begin{matrix}C_{0} : 1, & N_{G}(C_{0}) = G, \\ C_{1} : 1 < E, & N_{G}(C_{1}) = N_{G}(E) ≅ E·A_{9}, \\ C_{2} : 1 < A, & N_{G}(C_{2}) = N_{G}(A) ≅ A·L_{5}(2), \\ C_{3} : 1 < A < A·2^{4}, & N_{G}(C_{3}) = N_{G}(E) ∩ N_{G}(A) ≅ A·(2^{4} : L_{4}(2)) = E·A_{8}. \end{matrix}

5.2 2-blocks of G, N_{G}(E) and N_{G}(A)

As is shown in ATLAS, G has 48 irreducible characters. From the character table of G, it follows that G has four 2-blocks,

B_{0} = B_{0}(G), {χ_{22}}, {χ_{23}}, {χ_{38}},

where χ_{22}(1) = χ_{23}(1) = 2^{15}·5^{3} and χ_{38}(1) = 2^{15}·7^{2}·19.

Since N_{G}(E) ≅ 2_{+}^{1+8} · A_{9} and N_{G}(A) ≅ A · L_{5}(2) have normal selfcentralizing 2-subgroups, they have only the principal 2-blocks, which of course induce B_{0}. (See V.3.11 of [7].) They are maximal subgroups of G and appear in the GAP library.

From their character tables, we have the following.

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 \\ k(G, B_{0}, d) & 16 & 4 & 2 & 14 & 2 & 0 & 3 & 0 & 2 & 2 & 0 \\ k(N_{G}(E), B_{0}, d) & 16 & 4 & 2 & 14 & 1 & 0 & 3 & 7 & 3 & 2 & 0 \\ k(N_{G}(A), B_{0}, d) & 16 & 4 & 2 & 14 & 2 & 0 & 2 & 0 & 0 & 0 & 1 \end{matrix}

5.3 2-blocks of N_{G}(E) ∩ N_{G}(A)

By 5.1, N_{G}(E) ∩ N_{G}(A) has a normal subgroup E ≅ 2^{5}.2^{4} ≅ 2_{+}^{1+8}, which is selfcentralizing. Thus N_{G}(E) ∩ N_{G}(A) has only the principal 2-block, which induces B_{0}. Moreover, we have (N_{G}(E) ∩ N_{G}(A))/E ≅ L_{4}(2) ≅ A_{8}.

The set Irr(E) consists of one character η of degree 2^{4} and 2^{8} characters of degree 1. From the character table of N_{G}(E), we can compute the N_{G}(E)-orbits in Irr(E). Note first that in the notation of GAP library, 1a, 2a, 2b and 4a elements of N_{G}(E) form E. Of course, the trivial character 1_{E} of E and η form their own orbits of length one. We use the same numbering of characters of N_{G}(E) as appeared in GAP library. First of all, by looking at the values at 2b and 4a elements of N_{G}(E), we can see that (χ_{21})_{E} and (χ_{32})_{E} lie over different orbits. Moreover, it is easy to see that a character of N_{G}(E) lying over a non-trivial linear character of E has a restriction which is a multiple of either (χ_{21})_{E} or (χ_{32})_{E}. Thus, Clifford theory implies that there are exactly two N_{G}(E)-orbits of non-trivial linear characters of E. Moreover, we have [(χ_{21})_{E}, (χ_{21})_{E}] = 120 and [(χ_{32})_{E}, (χ_{32})_{E}] = 135. Since E has 2^{8} − 1 = 255 non-trivial linear character, we can conclude that the two orbits consist of 120 and 135 characters, respectively. Let θ_{1} and θ_{2} be representatives of these orbits of length 120 and 135, respectively. Then 1_{E}, η, θ_{1} and θ_{2} form a set of representatives of N_{G}(E)-orbits in Irr(E). For i = 1, 2, let T_{i} be the stabilizer of θ_{i} in N_{G}(E). Note also that (χ_{21})_{E} and (χ_{32})_{E} are exactly the sum of characters in the orbits of θ_{1} and θ_{2}, respectively. This implies that θ_{i} can be extended to T_{i}, for i = 1, 2.

Let us examine the individual orbits. From the character table of L_{4}(2) ≅ A_{8}, we have

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(E) ∩ N_{G}(A), d | 1_{E}) & 8 & 2 & 2 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{matrix}  (1)

The character η is N_{G}(E)-invariant. Recall that N_{G}(E)/E ≅ A_{9}. If η has an extension to N_{G}(E), then, since A_{9} has eight characters if odd degree by ATLAS, N_{G}(E) has at least eight characters of defect 11. However, as is seen in the previous subsection, this is not the case. Hence, η can not be extended to N_{G}(E). Consider Irr(N_{G}(E) | η). Let ζ be the non-trivial character of Z(2.A_{9}). Since η does not have an extension to N_{G}(E) and since (N_{G}(E) ∩ N_{G}(A))/E corresponds to L_{4}(2) ≅ A_{8} in N_{G}(E)/E ≅ A_{9}, the theory of character triple (Chap.11 of [9]) tells us that there is a bijection σ from Irr(N_{G}(E) ∩ N_{G}(A) | η) to Irr(2.A_{8} | ζ) such that φ(1)/η(1) = σ(φ)(1)/ζ(1) for all φ in Irr(N_{G}(E) | η). (See (11.28), (11.23) and (11.24) of [9].) Thus the character table of 2.A_{8} gives the following.

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(E) ∩ N_{G}(A), d |η) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 & 0 & 1 \end{matrix}  (2)

The group T_{1}/E has index 120 in N_{G}(E)/E ≅ A_{9}. Form p.37 of ATLAS, it must follow that T_{1}/E ≅ L_{2}(8) : 3. To obtain N_{G}(E) ∩ N_{G}(A)-orbit of θ_{1}, we consider A_{8}∩(L_{2}(8):3). Since |A_{8}| = 2^{6}·3^{2}·5·7 and |L_{2}(8):3| = 2^{3}·3^{3}·5·7, the group A_{8}∩(L_{2}(8):3) is a proper subgroup of L_{2}(8) : 3. On the other hand, |A_{8} : (A_{8} ∩ T_{1})| ≤ 120. Thus it follows from p. 6 of ATLAS that A_{8} ∩ (L_{2}(8):3) is isomorphic to either L_{2}(8) or 2^{3} :7:3. However, if the former was true, then it would follow from p.22 of ATLAS that L_{2}(8) ≤ A_{7}, which would contradict p.10 of ATLAS, since A_{7} does not have a subgroup isomorphic to L_{2}(8). Thus we can conclude that A_{8} ∩ (L_{2}(8):3) ≅ 2^{3} :7:3. In particular, the N_{G}(E) ∩ N_{G}(A)-orbit of θ_{1} consists of 120 characters. Since θ_{1} can be extended to N_{G}(A) ∩ T_{1} and since the 2-part of |A_{8} : A_{8} ∩ (L_{2}(8) : 3)| is 2^{3}, the character table of 2^{3} :7:3 gives the following.

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(E) ∩ N_{G}(A), d |θ_{1}) & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}  (3)

The group T_{2}/E has index 135 in A_{9}. From p.37 of ATLAS, we may assume that T_{2}/E ≤ A_{8} and p. 22 of ATLAS implies that T_{2}/E ≅ 2^{3} :L_{3}(2). Thus we may assume that T_{2} ≤ N_{G}(E)∩N_{G}(A), and then it follows that the N_{G}(E)∩N_{G}(A)-orbit of θ_{2} consists of 15 characters. Since θ_{2} can be extended to T_{2}, the character table of 2^{3} :L_{3}(2) gives the following.

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(E) ∩ N_{G}(A), d | θ_{2}) & 8 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}  (4)

Finally, let θ_{3} be a character in the N_{G}(E)-orbit of θ_{2} such that θ_{2} and θ_{3} are not N_{G}(E) ∩ N_{G}(A)-conjugate. Let T_{3} be the stabilizer of θ_{3} in N_{G}(E). Since we assume that T_{2}/E ≤ A_{8}, it follows that T_{3}/E is contained in an A_{9}-conjugate of A_{8} which is different from the original A_{8}. This means that (N_{G}(E) ∩ N_{G}(A) ∩ T_{3})/E is isomorphic to a subgroup of A_{7}. Recall that |T_{3}/E| = |T_{2}/E| = 2^{6} ·3·7, |A_{7}| = 2^{3}·3^{2}·5·7 and |N_{G}(E) ∩ N_{G}(A) : (N_{G}(E) ∩ N_{G}(A) ∩ T_{3})| ≤ 135 − 15 = 120. From these, it must follow that |N_{G}(E) ∩ N_{G}(A) : (N_{G}(E) ∩ N_{G}(A) ∩ T_{3})| = 120. In particular, the N_{G}(E) ∩ N_{G}(A)-orbit of θ_{3} consists of 120 characters. Moreover, since |(N_{G}(E) ∩ N_{G}(A) ∩ T_{3})/E| = 2^{3}·3·7, p.10 of ATLAS implies that (N_{G}(E) ∩ N_{G}(A) ∩ T_{3})/E ≅ L_{2}(7). Since θ_{3} can be extended to T_{3}, the character table of L_{2}(7) gives the following.

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(E) ∩ N_{G}(A), d | θ_{3}) & 0 & 0 & 0 & 4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \end{matrix}  (5)

Since 1_{E}, η, θ_{i}, i = 1, 2, 3 form a set of representatives of N_{G}(E) ∩ N_{G}(A)-orbits in Irr(E), the above tables (1) – (5) give the following.

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(E) ∩ N_{G}(A), d) & 16 & 4 & 2 & 14 & 1 & 0 & 2 & 7 & 1 & 0 & 1 \end{matrix}

5.4 Conjecture

Summarizing the above, we have the following.

\begin{matrix}d & 15 & 14 & 13 & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 & parity \\ k(G, B_{0}, d) & 16 & 4 & 2 & 14 & 2 & 0 & 3 & 0 & 2 & 2 & 0 & + \\ k(N_{G}(E), B_{0}, d) & 16 & 4 & 2 & 14 & 1 & 0 & 3 & 7 & 3 & 2 & 0 & − \\ k(N_{G}(A), B_{0}, d) & 16 & 4 & 2 & 14 & 2 & 0 & 2 & 0 & 0 & 0 & 1 & − \\ k(N_{G}(E) ∩ N_{G}(A), B_{0}, d) & 16 & 4 & 2 & 14 & 1 & 0 & 2 & 7 & 1 & 0 & 1 & + \end{matrix}

Hence, by Proposition 5.1, Conjecture 2.1 holds for G in the case of p = 2.

6 The case of p = 3 for T h

In this section, we assume that G = T h and p = 3. Thus we can ignore κ.

6.1 Radical 3-subgroups and radical 3-chains

Maximal 3-local subgroups are classified in §4 of [18]. There are three conjugacy classes 3A, 3B and 3C of elements of order three, and we have

\begin{aligned}L_{1} = N_{G}(3A) & ≅ (3 × G_{2}(3)) : 2, \\ L_{2} = N_{G}(3B) & ≅ (3^{3} × 3_{+}^{1+2})·3_{+}^{1+2} : 2S_{4}, \\ L_{3} = N_{G}(3C) & ≅ (3 × 3^{4} : 2A_{6}):2 ≅ 3^{5} :2S_{6}. \end{aligned}

Let T denote the normal subgroup 3^{3} × 3_{+}^{1+2} of N_{G}(3B) and let C = O_{3}(N_{G}(3A)). There is a central 3B^{2} group in T, and we have

L_{4} = N_{G}(3B^{2}) ≅ (3^{2} · [3^{7}])·2S_{4}.

Any 3-local subgroups of G is contained in one of L_{1}, L_{2}, L_{3} and L_{4}. (See Theorem 4.1 of [18].) Thus a radical 3-subgroup of G is that of L_{i} for some i.

Let us examine radical 3-subgroups of L_{i}’s. First of all, O_{3}(L_{i}) is a radical 3subgroups of L_{i} and thus of G for each i. Moreover, a radical 3-subgroup R of L_{i} must contain O_{3}(L_{i}). (Prop.1.4 of [2]) In particular, R/O_{3}(L_{i}) is a radical 3subgroup of L_{i}/O_{3}(L_{i}). Let us treat L_{i} individually. The group G_{2}(3) : 2 is the finite Chevalley group G_{2}(3) with the non-trivial graph automorphism of order 2. Thus, up to G_{2}(3) : 2-conjugate, there are two non-trivial radical 3-subgroups of G_{2}(3):2. Namely, U ≅ 3^{2} × 3_{+}^{1+2} with normalizer U:GL_{2}(3) and U_{0} ≅ 3^{2}·(3 × 3_{+}^{1+2}) with normalizer U_{0} : D_{8} in G_{2}(3) : 2, the unipotent radical of a maximal parabolic subgroup and the unipotent radical of a Borel subgroup, respectively. On the other hand, the groups 2S_{4} and 2A_{6} :2 have only one non-trivial radical 3-subgroup up to conjugate. Namely, their Sylow 3-subgroups. Let S and P be Sylow 3-subgroups of L_{2} and L_{3}, respectively. Note that S is a Sylow 3-subgroup of L_{2}, L_{4} and of G.

Let R be a 3-radical subgroup of G. Suppose that N_{G}(R) ≤ L_{1}. Then R ∩ (3 × G_{2}(3)) is a radical subgroup of 3 × G_{2}(3) and thus R is conjugate to C = O_{3}(L_{1}), C×U or C×U_{0}. From the information in ATLAS [1] (pp.60–61), it follows that [U, U] is of order 3 generated by 3A or 3B element of G_{2}(3) and the second commutator subgroup of U_{0} is also of order 3 generated by 3A or 3B element of G_{2}(3). Hence, if R is conjugate to C × U or C × U_{0}, then by p.20 of [18], N_{G}(R) is contained in L_{2} = N_{G}(3B), and thus, R is a radical 3-subgroup of L_{2}. But, by comparing order of R and radical subgroups of L_{2}, we can see that it is not the case. Hence we can conclude that R = O_{3}(L_{1}). Now suppose that N_{G}(R) ≤ L_{3}. Then R is conjugate to O_{3}(L_{3}) or P. Let z be a 3C element of G. Then C_{G}(z) = 3×3^{4} :2A_{6} ≅ 3×3^{4} :SL_{2}(9), in which the group SL_{2}(9) acts naturally on 3^{4} ≅ (F_{9})^{2} (p.21 of [18]). Thus P is a Sylow 3-subgroup of C_{G}(z), and [P, P] is a one-dimensional subgroup of (F_{9})^{2}, in which all the nonzero vectors are transitive under the action of an element of order 8 of SL_{2}(9). Consider conjugacy classes of non-trivial elements of [P, P] in G. Since there is no elementary abelian subgroup of order 9 whose non-trivial elements are all 3C-elements (the 5th paragraph of p.21 of [18]), every non-trivial element of [P, P] is either 3A or 3B-element. Moreover, since O_{3}(L_{3}) ≅ 3^{5} has type 3C_{1}B_{40}C_{80} (2nd paragraph of p.21 of [18]), we can conclude that all the non-trivial elements of [P, P] are 3B-elements. On the other hand, by the argument in the 2nd paragraph of p.22 of [18], O_{3}(L_{3}) ≅ 3^{5} is conjugate to a subgroup of T = 3^{3} × 3^{1+2} ⊴ N_{G}(3B). Hence [P, P] is a 3B^{2}-group in T. As is stated in the 3rd paragraph of p.23 of [18], N_{G}([P, P]) is either L_{4} or is contained in L_{2}. Thus we have N_{G}(P) ≤ N_{G}([P, P]) ≤ L_{i} for i = 2 or i = 4, and P is a radical 3-subgroup of L_{2} or L_{4}. But, by comparing order of P and radical subgroups of L_{2} and L_{4}, we can see that it is not the case. Hence we can conclude that R = O_{3}(L_{3}). Now, suppose that N_{G}(R) ≤ L_{2} or or N_{G}(R) ≤ L_{4}. Then R is conjugate to O_{3}(L_{2}), O_{3}(L_{4}) or S.

The above arguments imply that a non-trivial radical 3-subgroup of G is conjugate to one of the following.

O_{3}(L_{1}), O_{3}(L_{2}), O_{3}(L_{3}), O_{3}(L_{4}), S

Now look at radical 3-chains of G. By the argument above, radical 3-chains starting with 1 < C are conjugate to

1 < C,
1 < C < C × U,
1 < C < C × U_{0} or
1 < C < C × U < C × U_{0}.

Radical 3-chains starting with 1 < O_{3}(L_{2}) or 1 < O_{3}(L_{3}) are conjugate to 1 < O_{3}(L_{2}), 1 < O_{3}(L_{2}) < S, 1 < O_{3}(L_{3}) or 1 < O_{3}(L_{3}) < P. Finally, radical 3-chains starting with 1 < O_{3}(L_{4}) are conjugate to 1 < O_{3}(L_{4}) or 1 < O_{3}(L_{4}) < S, and 1 < S is the unique radical 3-chain starting with 1 < S. Since 1 < O_{3}(L_{4}) < S and 1 < S have the same normalizers and their length have opposite parity, we will not take them into account in the alternating sum. Summarizing the above, we have the following.

Proposition 6.1 To prove that the conjecture holds for G and p = 3, it suffices to consider the alternating sum with respect to the following radical 3-chains.

radical 3 − chain  normalizer
C_{0} : 1,  G,
C_{1} : 1 < C = O_{3}(N_{G}(3A)),  (3 × G_{2}(3)) : 2,
C_{2} : 1 < C < C × U,  3 × U:GL_{2}(3)(< 3 × G_{2}(3)),
C_{3} : 1 < C < C × U_{0}  (3 × U_{0}):D_{8}
C_{4} : 1 < C < C × U < C × U_{0} 3 × U_{0} :2^{2} < (3 × G_{2}(3))
C_{5} : 1 < O_{3}(N_{G}(3B)),  (3^{3} × 3_{+}^{1+2})·3_{+}^{1+2} : 2S_{4},
C_{6} : 1 < O_{3}(N_{G}(3B)) < S,  (3^{3} × 3_{+}^{1+2})·3_{+}^{1+2} : (3:2^{2}),
C_{7} : 1 < O_{3}(N_{G}(3C)),  (3 × 3^{4} : 2A_{6}):2 ≅ 3^{5} :2S_{6},
C_{8} : 1 < O_{3}(N_{G}(3C)) < P,  (3 × (3^{4} : (3^{2} :8))):2,
C_{9} : 1 < O_{3}(N_{G}(3B^{2})),  (3^{2}·[3^{7}])·2S_{4}.

The notation used in this subsection will be used throughout this section, though sometimes other letters have different meanings when appearing in different subsections.

6.2 3-blocks of G and a reduction

The character table tells us that G has the following 3-blocks. (See also [8].)

B_{0} = B_{0}(G), B′ = {χ_{25}, χ_{43}, χ_{45}}, {χ_{12}}, {χ_{13}}, {χ_{42}}, {χ_{48}}.

Here χ_{25}(1) = 2^{3}·3^{9}·31, χ_{43}(1) = 3^{9}·5^{3}·31, χ_{45}(1) = 3^{9}·7·19·31, χ_{12}(1) = χ_{13}(1) = 3^{10}·13, χ_{42}(1) = 3^{10}·5·13·19, χ_{48}(1) = 2^{3}·3^{10}·13·31. The block B_{0} and B′ are of defect 10 and 1, respectively. Indeed, B′ has C as a defect group. Thus, it suffices to consider B_{0}. The character table of G tells us that we have the following.

\begin{matrix}d & 10 & 9 & 8 & 7 & 6 & 5 \\ k(G, B_{0}, d) & 15 & 6 & 1 & 18 & 0 & 1 \end{matrix}

Note that C_{1}, C_{2}, C_{3} and C_{4} are representatives of radical 3-chains of G starting with 1 < O_{3}(N_{G}(3A)) and C_{7} and C_{8} are representatives of radical 3-chains of G starting with 1 < O_{3}(N_{G}(3C)). Furthermore, we have

\begin{matrix}C_{G}(3A) ≅ 3 × G_{2}(3), & N_{G}(3A) = N_{G}(C) ≅ (3 × G_{2}(3)):2, \\ C_{G}(3C) ≅ 3 × 3^{4} :2A_{6} ≅ 3 × 3^{4} :SL_{2}(9), & N_{G}(3C) ≅ (3^{5} :SL_{2}(9)):⟨σ⟩, \end{matrix}

where in C_{G}(3C) the group SL_{2}(9) acts naturally on 3^{4} ≅ (F_{9})^{2}, and in N_{G}(3C) the element σ inverts the 3C-element and acts as the Galois automorphism of order two on (F_{9})^{2} : SL_{2}(9). (For the structure of N_{G}(3A), see the previous subsection.) Hence Theorem 4.4 and Proposition 4.5 imply that

∑_{i=1}^{4}(−1)^{|C_{i}|}k(N_{G}(C_{i}), B_{0}, d) = ∑_{i=7}^{8}(−1)^{|C_{i}|}k(N_{G}(C_{i}), B_{0}, d) = 0

for all d with d ≠ 1. (Here a cyclic group generated by a 3A- or a 3C-element is the group Q in Theorem 4.4.) Consider the case of d = 1. Note that the factor groups N_{G}(C_{i})/O_{3}(N_{G}(3A)), for i = 2, 3, 4, and N_{G}(C_{i})/O_{3}(N_{G}(3C)) for i = 7, 8 do not have characters of defect zero. Thus k(N_{G}(C_{i}), B_{0}, 1) = 0 for i = 2, 3, 4, 7 and 8. (See also the proof of Theorems 4.2 and 4.4.) The group N_{G}(3A) appears as the 5th maximal subgroup of G in the GAP library. It has the principal block and one non-principal 3-block consisting of two characters of degree 3^{6} and one character of degree 3^{6}·2. (In the notation given in GAP library they are χ_{23}, χ_{24} and χ_{49}. This block is of defect 1 and induces B′.) The principal block of N_{G}(3A) does not have a character of defect 1. Thus k(N_{G}(C_{1}), B_{0}, 1) = 0. Hence we have the following.

Proposition 6.2 To prove that the conjecture holds for G and p = 3, it suffices to consider the alternating sum with respect to the radical 3-chains C_{0}, C_{5}, C_{6} and C_{9}.

6.3 3-blocks of N_{G}(3B) and N_{G}(3B) ∩ N_{G}(S)

The group N_{G}(3B) ≅ (3^{3} × 3_{+}^{1+2})·3_{+}^{1+2} : 2S_{4} appears as the 6th maximal subgroup of G in the GAP library. It has 62 characters and only the principal block. Since the local principal block induces B_{0}, we have the following.

\begin{matrix}d & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(3B), B_{0}, d) & 15 & 15 & 7 & 18 & 6 & 1 \end{matrix}

Let us turn to N_{G}(3B) ∩ N_{G}(S). Let N be O_{3}(N_{G}(3B)) ≅ (3^{3} × 3_{+}^{1+2})·3_{+}^{1+2}. From the character table of N_{G}(3B), we can compute the character degrees of N. Our strategy is as follows. Let χ be a character of N_{G}(3B). By Clifford theory, the restriction χ_{N} of χ to N has a form

χ_{N} = e ∑_{i=1}^{m} θ_{i},

where e is an integer and θ_{i}’s are N_{G}(3B)-conjugate characters of N. The inner product [χ_{N}, χ_{N}] is then e^{2}m. Moreover, θ_{1}(1) = χ(1)/em is a power of 3 and \frac{1}{e}χ_{N} is a (not necessarily irreducible) character of N. Using the above, we can determine N_{G}(3B)-orbits of Irr(N) only by computing [χ_{N}, χ_{N}]. The result is as follows. We use the same numbering of characters of N_{G}(3B) as in appeared in GAP library. Notice also that, if e = 1 for some χ, then characters appearing in χ_{N} can be extended to its stabilizer, and that the order of the stabilizer is |N_{G}(3B)|/m = 48|N|/m.

First of all, χ_{1}, χ_{2}, ⋯, χ_{8} have N in their kernels, and thus they lie over 1_{N}. Characters χ_{9}, χ_{10}, ⋯, χ_{18} have T = 3^{3} × 3_{+}^{1+2} in their kernels, and thus they can be considered as characters lying over those of O_{3}(N_{G}(3B))/T ≅ 3_{+}^{1+2}. In fact, we have the following.

i = 9, 10,  χ_{i}(1) = 8, (χ_{i})_{N} = ∑_{j=1}^{8} λ′_{j}, λ′_{j}(1) = 1
i = 11,  χ_{11}(1) = 16, (χ_{11})_{N} = 2 ∑_{i=j}^{8} λ′_{j},
i = 12, 13, 14, χ_{i}(1) = 6, (χ_{i})_{N} = ∑_{i=j}^{2} μ′_{j}, μ′_{j}(1) = 3
i = 15, 16, 17, χ_{i}(1) = 12, (χ_{i})_{N} = 2 ∑_{i=j}^{2} μ′_{j},
i = 18,  χ_{18}(1) = 18, (χ_{18})_{N} = 3 ∑_{i=j}^{2} μ′_{j}.

For the other characters, we have the following. Here for characters of N, different letters signify different characters.

χ ∈ Irr(N_{G}(3B)) Character degree [χ_{N}, χ_{N}]  χ_{N}  Character degree
χ_{19}, χ_{20}  χ_{19}(1) = 8  8  ∑_{i=1}^{8} λ_{i}  λ_{1}(1) = 1
χ_{21}  χ_{21}(1) = 16  32  2 ∑_{i=1}^{8} λ_{i}  λ_{1}(1) = 1
χ_{22}, χ_{23}, χ_{24}  χ_{22}(1) = 16  16  ∑_{i=1}^{16} ν_{i}  ν_{1}(1) = 1
χ_{25}, χ_{28}  χ_{25}(1) = 24  24  ∑_{i=1}^{24} τ_{i}  τ_{1}(1) = 1
χ_{26}, χ_{27}  χ_{26}(1) = 24  24  ∑_{i=1}^{24} τ′_{i}  τ′_{1}(1) = 1
χ_{29}, χ_{30}, χ_{31}  χ_{29}(1) = 48  16  ∑_{i=1}^{16} ξ_{i}  ξ_{1}(1) = 3
χ_{32}, χ_{33}, χ_{34}, χ_{35}  χ_{32}(1) = 27  3  ∑_{i=1}^{3} θ_{i}  θ_{1}(1) = 9
χ_{36}, χ_{37}, χ_{38}, χ_{39}  χ_{36}(1) = 27  3  ∑_{i=1}^{3} θ′_{i}  θ′_{1}(1) = 9
χ_{40}, χ_{41}, χ_{44}  χ_{40}(1) = 54  12  2 ∑_{i=1}^{3} θ_{i}  θ_{1}(1) = 9
χ_{42}, χ_{43}, χ_{45}  χ_{42}(1) = 54  12  2 ∑_{i=1}^{3} θ′_{i}  θ′_{1}(1) = 9
χ_{46}, χ_{47}  χ_{46}(1) = 72  8  ∑_{i=1}^{8} ϕ_{i}  ϕ_{1}(1) = 9
χ_{49}  χ_{46}(1) = 144  32  2 ∑_{i=1}^{8} ϕ_{i}  ϕ_{1}(1) = 9
χ_{48}, χ_{50}, χ_{51}  χ_{48}(1) = 144  16  ∑_{i=1}^{16} η_{i}  η_{1}(1) = 9
χ_{52}, χ_{53}  χ_{52}(1) = 216  24  ∑_{i=1}^{24} ζ_{i}  ζ_{1}(1) = 9
χ_{54}, χ_{55}  χ_{54}(1) = 216  24  ∑_{i=1}^{24} ζ′_{i}  ζ′_{1}(1) = 9
χ_{56}, χ_{57}, χ_{58}  χ_{56}(1) = 162  2  ∑_{i=1}^{2} ψ_{i}  ψ_{1}(1) = 81
χ_{59}, χ_{60}, χ_{61}  χ_{59}(1) = 324  8  2 ∑_{i=1}^{2} ψ_{i}  ψ_{1}(1) = 81
χ_{62}  χ_{62}(1) = 486  18  3 ∑_{i=1}^{2} ψ_{i}  ψ_{1}(1) = 81

Notice that all characters of N have extensions to their stabilizers. Let I and I′ be the stabilizers in N_{G}(3B) and in N_{G}(3B)∩N_{G}(S), respectively, of representatives of N_{G}(3B)-orbits in Irr(N). The structure of I/N can easily be obtained from the length of orbits. There are two isomorphism classes of subgroups of N_{G}(3B)/N ≅ 2S_{4} of order 6. However, from the degrees of characters of N_{G}(3B) lying over the character of N, we can see I/N ≅ S_{3}. The structures of I/N are as follows. Also I′ for typical characters of N are given. Here D and D′ are Sylow 2-subgroups of N_{G}(3B) and N_{G}(3B) ∩ N_{G}(S), respectively.

1_{N} λ′_{1} μ′_{1} λ_{1} ν_{1} τ_{1} τ′_{1} ξ_{1} θ_{1} θ′_{1} ϕ_{1} η_{1} ζ_{1} ζ′_{1} ψ_{1}
I/N 2.S_{4} S_{3} SL_{2}(3) S_{3} 3 2 2 3 D D S_{3} 3 2 2 SL_{2}(3)
I′/N 2S_{3} S_{3}  6  S_{3} 3 2 2 3 D′ D′ S_{3} 3 2 2  6

If |I : I′| = 4, then the N_{G}(3B)-orbit is also an N_{G}(3B)∩N_{G}(S)-orbit. There are two conjugacy classes of involutions in N_{G}(3B). Let z and z′ be representatives of them, and we assume that z corresponds to the central involution of 2S_{4} (indicated as 2a in GAP library) and z′ does to a non-central one. Note that z lies in N_{G}(3B)∩ N_{G}(S) and we may suppose that z′ lies in N_{G}(3B) ∩ N_{G}(S). If I/N ≅ S_{3}, then z does not lie in I. We may suppose that z′ lies in the stabilizer of λ′_{1}. Then the N_{G}(3B) ∩ N_{G}(S)-orbit of λ′_{1} consists of λ′_{1} and (λ′_{1})^{z}, since I = I′. Say (λ′_{1})^{z} = λ′_{2}. Note that the stabilizers of λ′_{1} and λ′_{2} in N_{G}(3B) are the same. Since the image of z′ in 2S_{4} is contained in exactly two conjugates of S_{3}, we may assume that z′ lies in the stabilizer of λ′_{3} in N_{G}(3B) ∩ N_{G}(S), which is ⟨z′⟩N, since N_{G}(3B) ∩ N_{G}(S)/N has only one S_{3} as a subgroup. Then the N_{G}(3B)∩N_{G}(S)-orbit of λ′_{3} is {λ′_{3}, λ′_{4}, ⋯, λ′_{8}}. This argument applies also for λ_{i}’s and ϕ_{i}’s. On the other hand, if z stabilizes some character of N, then it stabilizes all the characters in its N_{G}(3B)-orbit. Thus, it is easy to show that, if χ(z) = 0, then z does not stabilize any constituents of χ_{N}. Hence z is not in the stabilizer of τ_{i}, τ′_{i}, ζ_{i} and ζ′_{i}. The N_{G}(3B)/N-conjugacy class of z′N consists of 12 elements and among them 6 elements lie in N_{G}(3B) ∩ N_{G}(S)/N, while the N_{G}(3B) ∩ N_{G}(S)/N-conjugacy class of z′N consists of 3 elements. Thus, we may assume that each of the stabilizers of τ_{2i−1} in N_{G}(3B)∩N_{G}(S) for 1 ≤ i ≤ 3 consists of some N_{G}(3B) ∩ N_{G}(S)-conjugate of z′ and each of the stabilizers of τ_{2i−1} in N_{G}(3B) ∩ N_{G}(S) for 4 ≤ i ≤ 6 consists of some N_{G}(3B)-conjugate but not an N_{G}(3B) ∩ N_{G}(S)-conjugate of z′. Then we can conclude that N_{G}(3B) ∩ N_{G}(S)- orbits of {τ_{i}} are {τ_{1}, τ_{2}, ⋯, τ_{6}}, {τ_{7}, τ_{8}, ⋯, τ_{12}} and {τ_{13}, τ_{14}, ⋯, τ_{24}}, where τ_{2i} = (τ_{2i−1})^{z} for 1 ≤ i ≤ 6. In particular, the stabilizer of τ_{13} in N_{G}(3B) ∩ N_{G}(S) is N. This argument applies also for τ′_{i}’s, ζ_{i}’s and ζ′_{i}’s. Finally, suppose that the stabilizer of ν_{1} in N_{G}(3B)∩N_{G}(S) has order 3 and its N_{G}(3B)∩N_{G}(S)-orbit is {ν_{1}, ν_{2}, ν_{3}, ν_{4}}. Then the stabilizer of ν_{5} in N_{G}(3B)∩N_{G}(S) must be N and its N_{G}(3B)∩N_{G}(S)-orbit is {ν_{5}, ν_{6}, ⋯, ν_{16}}. This argument applies also for ξ_{i}’s and η_{i}’s.

From the above observation we can compute k(N_{G}(3B) ∩ N_{G}(S), d|γ) for a character γ of N and d, 5 ≤ d ≤ 10.

\begin{matrix}γ\d & 10 & 9 & 8 & 7 & 6 & 5 \\ 1_{N} & 6 & 0 & 0 & 0 & 0 & 0 \\ λ_{1}, λ′_{1} & 3 & 0 & 0 & 0 & 0 & 0 \\ λ_{3}, λ′_{3} & 0 & 2 & 0 & 0 & 0 & 0 \\ μ′_{1} & 0 & 6 & 0 & 0 & 0 & 0 \\ ν_{1} & 3 & 0 & 0 & 0 & 0 & 0 \\ ν_{5} & 0 & 1 & 0 & 0 & 0 & 0 \\ τ_{1}, τ_{7}, τ′_{1}, τ′_{7} & 0 & 2 & 0 & 0 & 0 & 0 \\ τ_{13}, τ′_{13} & 0 & 1 & 0 & 0 & 0 & 0 \\ ξ_{1} & 0 & 3 & 0 & 0 & 0 & 0 \\ ξ_{5} & 0 & 0 & 1 & 0 & 0 & 0 \\ θ_{1}, θ′_{1} & 0 & 0 & 0 & 4 & 0 & 0 \\ ϕ_{1} & 0 & 0 & 3 & 0 & 0 & 0 \\ ϕ_{3} & 0 & 0 & 0 & 2 & 0 & 0 \\ η_{1} & 0 & 0 & 3 & 0 & 0 & 0 \\ η_{5} & 0 & 0 & 0 & 1 & 0 & 0 \\ ζ_{1}, ζ_{7}, ζ′_{1}, ζ′_{7} & 0 & 0 & 0 & 2 & 0 & 0 \\ ζ_{13}, ζ′_{13} & 0 & 0 & 0 & 1 & 0 & 0 \\ ψ_{1} & 0 & 0 & 0 & 0 & 6 & 0 \end{matrix}

Consequently, we obtain the following.

\begin{matrix}d & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(3B) ∩ N_{G}(P), B_{0}, d) & 15 & 24 & 7 & 21 & 6 & 0 \end{matrix}

6.4 3-blocks of N_{G}(3B^{2})

The group N_{G}(3B^{2}) ≅ (3^{2}·[3^{7}])·2S_{4} appears as the 7th maximal subgroup of G in the GAP library. It has 52 characters and only the principal block. Since the local principal block induces B_{0}, we have the following.

\begin{matrix}d & 10 & 9 & 8 & 7 & 6 & 5 \\ k(N_{G}(3B^{2}), B_{0}, d) & 15 & 15 & 1 & 21 & 0 & 0 \end{matrix}

6.5 Conjecture

From the results in 6.2, 6.3 and 6.4, we have the following.

\begin{matrix}d & 10 & 9 & 8 & 7 & 6 & 5 & parity \\ k(G, B_{0}, d) & 15 & 6 & 1 & 18 & 0 & 1 & + \\ k(N_{G}(3B), B_{0}, d) & 15 & 15 & 7 & 18 & 6 & 1 & − \\ k(N_{G}(3B) ∩ N_{G}(S), B_{0}, d) & 15 & 24 & 7 & 21 & 6 & 0 & + \\ k(N_{G}(3B^{2}), B_{0}, d) & 15 & 15 & 1 & 21 & 0 & 0 & − \end{matrix}

Hence, by Proposition 6.2, Conjecture 2.1 holds for G in the case of p = 3.

7 The case of p = 5 for T h

In this section, we assume that G = T h and p = 5.

7.1 Radical 5-chains

ATLAS shows that there is the unique conjugacy class of elements of order 5, and a cyclic subgroup C of order 5 satisfies |N_{G}(C)| = 2^{5}·3· 5^{3}. On the other hand, a Sylow 5-subgroup of G is an extra special group S ≅ 5_{+}^{1+2} and we have |N_{G}(S)| = 2^{5} ·3·5^{3}. (See p.299 of [19].) Since N_{G}(S) ≤ N_{G}(Z(S)), we must have N_{G}(S) = N_{G}(Z(S)). Thus, a subgroup of order 5 is not a radical 5-subgroup of G. There exists a subgroups E ≅ 5^{2} of G with N_{G}(E) ≅ E : GL_{2}(5). (See [1].) This E is a radical 5-subgroup of G and any subgroup of order 5^{2} is conjugate to E. Hence the trivial subgroup, E and S form a set of representatives of G-conjugacy classes of radical 5-subgroups of G. Thus representatives of G-conjugacy classes of radical 5-chains of G are as follows. Here P is a Borel subgroup of GL_{2}(5).

\begin{matrix}C_{0} : 1, & N_{G}(C_{0}) = G, \\ C_{1} : 1 < E, & N_{G}(C_{1}) = N_{G}(E) ≅ E:GL_{2}(5), \\ C_{2} : 1 < E < S, & N_{G}(C_{2}) ≅ E:P, \\ C_{3} : 1 < S, & N_{G}(C_{3}) = N_{G}(S) ≅ 5_{+}^{1+2} :4S_{4}. \end{matrix}

Now Proposition 4.5 implies that N_{G}(E) and E : P have only the principal 5blocks and k(N_{G}(E), B_{0}(G), d, [κ]) = k(E:P, B_{0}(G), d, [κ]) for all d and κ. (See also Proposition 2.3 of [17] or Theorem 6.2 of [14].) Hence we have the following.

Proposition 7.1 To prove that the conjecture holds for G and p = 5, it suffices to consider the alternating sum with respect to the following radical 5-chains.

\begin{matrix}C_{0} : 1, & N_{G}(C_{0}) = G, \\ C_{3} : 1 < S, & N_{G}(C_{3}) = N_{G}(S) ≅ 5_{+}^{1+2} :4S_{4}. \end{matrix}

7.2 5-blocks of G and N_{G}(S)

The group G has no block of defect one ([8]), and by V.3.11 of [7], N_{G}(E) has only the principal block. Thus G does not have a block of defect two by Brauer’s first main theorem. The character table shows that the characters whose degrees are not divisible by 5^{3} form B_{0}(G) which has 27 characters. Each of the other 21 forms of course a block of defect zero. Thus G has the following twenty two 5-blocks. We write characters in B_{0}(G) by their degrees and the multiplicity is written exponentially.

B_{0} = B_{0}(G)

= {1, 248, 4123, 30628, 61256, (5·17199)^{2}, 767637^{2}, 779247^{2}, 1707264^{2}, 5·490048, 2572752, 3376737, 4881384, 11577384, 5·3307824, (5·4265352)^{2}, 30507008, 44330496, 5·14585103, 81153009, 91171899, 190373976}, {χ_{4}}, {χ_{5}}, {χ_{7}}, {χ_{11}}, {χ_{16}}, {χ_{22}}, {χ_{23}}, {χ_{24}}, {χ_{26}}, {χ_{27}}, {χ_{28}}, {χ_{29}}, {χ_{30}}, {χ_{31}}, {χ_{34}}, {χ_{37}}, {χ_{39}}, {χ_{41}}, {χ_{43}}, {χ_{44}}, {χ_{47}}.

The group N_{G}(S) ≅ 5_{+}^{1+2} :4S_{4} appears as the 9th maximal subgroup of G in the GAP library. Since it has only the principal 5-block, we have the following.

B_{0}(N_{G}(S)) = {1^{4}, 2^{6}, 3^{4}, 4^{2}, (5·4)^{3}, 24^{4}, (5·8)^{3}, 5·12}.

Since |G|_{5′} ≡ |N_{G}(S)|_{5′} ≡ 1 mod 5, we obtain the following, and Conjecture 3.2 holds for G in the case of p = 5 by Proposition 7.1.

\begin{matrix}(d, κ) & (3, 1) & (3, 2) & (2, 1) & (2, 2) \\ k(G, B_{0}, d, [κ]) & 10 & 10 & 3 & 4 \\ k(N_{G}(S), B_{0}, d, [κ]) & 10 & 10 & 3 & 4 \end{matrix}

8 The case of p = 7 for T h

In this section, we assume that G = T h and p = 7.

8.1 Radical 7-chains

ATLAS shows that there is the unique conjugacy class of elements of order 7, and a cyclic subgroup C of order 7 has the normalizer N_{G}(C) ≅ (7 : 3 × L_{2}(7)) : 2 (p.20 of [18]). Thus C is a radical 7-subgroup of G. Hence, the trivial subgroup, C and a Sylow 7-subgroup E ≅ 7^{2} of G form a set of representatives of G-conjugacy classes of 7-radical subgroups of G. Hence, representatives of G-conjugacy classes of radical 7-chains of G are as follows.

\begin{matrix}C_{0} : 1, & N_{G}(C_{0}) = G, \\ C_{1} : 1 < C, & N_{G}(C_{1}) ≅ (7:3 × L_{2}(7)):2 \\ C_{2} : 1 < E, & N_{G}(C_{2}) ≅ 7^{2} :(3 × 2S_{4}) \\ C_{3} : 1 < C < E, & N_{G}(C_{3}) ≅ (7:3 × P):2. \end{matrix}

8.2 7-blocks of G and a reduction

The character table tells us that G has the principal 7-block B_{0} = B_{0}(G), one 7-block B′ of defect 1, and fourteen 7-blocks of defect 0. We have Irr(G, B_{0}) = Irr(G, B_{0}, 2) and Irr(G, B′) = Irr(G, B′, 1), and since |G|_{7′} ≡ 4 mod 7, the characters are distributed as follows.

Irr(G, B_{0}, 2, [1]) = {248, 30628, 767637^{2}, 4881384, 6696000^{2}, 16539120, 76271625}, Irr(G, B_{0}, 2, [2]) = {30875, 147250, 2450240, 6669000^{2}, 51684750, 72925515,

77376000, 190373976},

Irr(G, B_{0}, 2, [3]) = {1, 27000^{2}, 61256, 957125, 1707264^{2}, 4096000^{2}}, Irr(G, B′, 1) = Irr(G, B′, 1, [3]) = {χ_{3}, χ_{20}, χ_{24}, χ_{32}, χ_{39}, χ_{45}, χ_{47}}, {χ_{9}}, {χ_{10}}, {χ_{14}}, {χ_{15}}, {χ_{21}}, {χ_{26}}, {χ_{31}}, {χ_{34}}, {χ_{35}}, {χ_{36}}, {χ_{37}}, {χ_{38}}, {χ_{40}}, {χ_{46}}.

See also 3.5.2 of [15] or [19] for B_{0} and [8] for B′. We have

\begin{matrix}(d, κ) & (2, 1) & (2, 2) & (2, 3) \\ k(G, B_{0}, d, [κ]) & 9 & 9 & 9 \end{matrix}

Note that C_{1} and C_{3} are representatives of radical 7-chains of G starting with 1 < C, and

C_{G}(C) ≅ 7 × L_{2}(7), N_{G}(C) ≅ (7:3 × L_{2}(7)):2 = (7 × L_{2}(7)):6.

Since Conjecture 3.2 holds for the principal blocks of 3 × L_{2}(7) and 3 × L_{2}(7) : 2, Theorem 4.4 implies that

k(N_{G}(C_{1}), B_{0}, d, [κ]) = k(N_{G}(C_{3}), B_{0}, d, [κ])

for all d with d ≠ 1 and all κ. Moreover, it is easy to see that k(N_{G}(C_{1}), 1) = 7 and k(N_{G}(C_{3}), 1) = 0. However, by Brauer’s first main theorem, the seven characters of N_{G}(C) with defect 1 must form a unique 7-block inducing B′, and this implies that k(N_{G}(C_{1}), B_{0}, 1) = 0. Moreover, since |N_{G}(C)|_{7′} ≡ 4 mod 7, we have Irr(N_{G}(C), B′, 1) = Irr(N_{G}(C), B′, 1, [3]). Consequently, we have the following.

Proposition 8.1 To prove that the conjecture holds for G and p = 7, it suffices to consider B_{0} and the alternating sum with respect to the radical 3-chains C_{0} and C_{2}.

8.3 7-blocks of N_{G}(E)

By V.3.11 of [7], N_{G}(E) ≅ 7^{2} :(3 × 2S_{4}) has only the principal 7-block B_{0}(N_{G}(E)). It appears as the 11th maximal subgroup of G in the GAP library. Its character table gives (See also p.298 of [19].)

B_{0}(N_{G}(E)) = {1^{6}, 2^{9}, 3^{6}, 4^{3}, 48^{3}},

which implies the following, since |N_{G}(E)|_{7′} ≡ 4 mod 7.

\begin{matrix}(d, κ) & (2, 1) & (2, 2) & (2, 3) \\ k(N_{G}(E), B_{0}, d, [κ]) & 9 & 9 & 9 \end{matrix}

From the results in 8.2, 8.3 and Proposition 8.1 imply that Conjecture 3.2 holds for G in the case of p = 7.

Note that Rouquier [15] proved that there exists a perfect isometry between B_{0} and its Brauer correspondent, the principal 7-block of a Sylow normalizer N_{G}(E).

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