Conjectures on character degrees for the simple Lyons group

Masato Sawabe ^{1}

Department of Mathematics, Kumamoto University, Kumamoto, 860-8555 Japan Email: sawabe@math.sci.kumamoto-u.ac.jp

and

Katsuhiro Uno

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

Abstract

We prove a version of Isaacs-Navarro conjecture for the simple Lyons group. The original conjecture is a refinement of Alperin-McKay conjecture which deals with irreducible characters of height zero and the normalizer of a defect group. But we look at also the normalizers of radical p-chains and consider the alternating sum of the numbers of characters of any given defect.

1 Introduction

Let G be a finite group and p a prime. Recently, a new type of a conjecture, which concerns the p′-parts of degrees of complex irreducible characters of G with height zero, is proposed by Isaacs and Navarro [8]. (Conjecture 2.2 below) Though the original version considers only height zero characters of G and the normalizer of a defect group, here we consider a version of Dade’s type. Namely, we consider the alternating sum of the numbers of relevant characters of the normalizers of pchains with any fixed defect. (Conjecture 2.1) It is proposed first in [14] and implies immediately the ordinary form of Dade’s conjecture (Conjecture 2.3 or [2]).

Let Ly be the simple Lyons group. It has order 2^{8}·3^{7}·5^{6}·7·11·31·37·67, and its Schur multiplier and the outer automorphism group are both trivial. The purpose of this paper is to prove that Conjecture 2.1 for Ly is true. According to [3], the ordinary form of Dade’s conjecture is equivalent to the inductive form, if the Schur multiplier and the outer automorphism group are trivial. Thus the inductive form will also be confirmed for Ly. Moreover, for p = 7, 11, 31, 37 or 67, a Sylow p-subgroup of Ly is cyclic of prime order. For those primes, the normalizers of p-chains which we have to consider are just Ly itself and the normalizer of a Sylow p-subgroup, and furthermore, it is known that p-blocks of these normalizers with positive defect have only characters of height zero. Hence Conjectures 2.2 and 2.1 are equivalent to each other. Since Conjecture 2.2 is proved for cyclic defect group cases in Theorem 2.1 of [8], it suffices to treat the primes 2, 3 and 5 here. We also remark that for Conjecture 2.1, we count the number of characters satisfying a certain congruent relation such as (|G|/χ(1))_{p′} ≡ ±r mod p, where (|G|/χ(1))_{p′} is the p′-part of |G|/χ(1) for a irreducible character χ of G. Thus for primes 2 and 3, Conjecture 2.1 is nothing but the ordinary form of Dade’s conjecture.

1Supported by a grant from JSPS Research Fellowships for Young Scientists.

It is well known that Ly contains a maximal subgroup isomorphic to 3McL2, the automorphism group of the covering group of the McLaughlin simple group. In particular, a part of the 3-local structure of Ly is described by 3McL2, and we will use this fact in the present paper. However, it is worth mentioning that even for the 2-structure, it is quite useful. In fact, investigation of so called radical 2-chains of Ly can be reduced in part to that for 3McL2, and all the important 2-chains of Ly come from those of 3McL2. Hence, for primes 2 and 3, we will use the results proved for 3McL2. Other information we will use are as follows. The character tables of several subgroups of Ly can be found in ATLAS [1] and the GAP library. Also, maximal local subgroups of Ly are determined in [15].

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. Furthermore, S_{n} and A_{n} denote the symmetric group and the alternating group of degree n, respectively, and D_{8}, Q_{8} and SD_{16} denote the dihedral group of order 8, the quaternion group and the semidihedral group of order 16, respectively. Finally, M_{10} and M_{11} denote the Mathieu groups, while M_{i} for i ≤ 3 will be used for other groups.

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 p-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 r, we denote by Irr(H, [r]) the set of those φ in Irr(H) such that

(|H|/φ(1))_{p′} ≡ ±r mod p.

We also define Irr(H, d, [r]) by

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

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, [r]), and Irr(H, d, [r]) respectively by Irr(H, d|θ), Irr(H, [r]|θ) and Irr(H, d, [r]|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, d|θ) and so on denote the intersections of Irr(H, B) with the relevant subsets of Irr(H). 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 cardinality of Irr(∗) is in general denoted by k(∗).

For modular representation theory of finite groups, we refer to [4] and [12].

Acknowledgment. The authors are very much grateful to Professor S.Yoshiara for his many helpful suggestions, which are crucial in determining radical p-chains of Ly.

2 Conjectures of Dade, Isaacs and Navarro

We describe the conjecture considered in the present paper. 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 following conjecture first appeared in [14] and is an alternating sum version of Isaacs-Navarro conjecture (Conjecture 2.2 below).

Conjecture 2.1 Let p be a prime and G a finite group with O_{p}(G) = 1. Let B be a p-block of G. 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, [r]) = 0

for all d and r.

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 r with 0 < r < p, let m_{r}(B) be the number of irreducible characters in B with height zero and the p′-part of the degrees are congruent to ±r modulo p. Then Isaacs and Navarro propose the following conjecture in [8].

Conjecture 2.2 (Isaacs-Navarro) In the notation above, let c be the p′-part of |G : N_{G}(D)|. Then

m_{cr}(B) = m_{r}(b)

for all r with 0 < r < p.

Conjecture 2.1 immediately implies the ordinary form of Dade’s conjecture stated below.

Conjecture 2.3 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.

Now we give the following easy remarks.

Lemma 2.4 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 2.5 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, [r]|θ) to Irr(G, d, [r]|θ) sending any φ in Irr(T, d, [r]|θ) to φ^{G}.

Finally, we state some well known results on extensions of characters. We will use them in the present paper without referring. For the proof, see 5.20 of [9] and (11.22), (11.31), (8.16) and (6.17) of [7].

Lemma 2.6 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 θ can be extended to T in each of the following cases.

(i) θ(1) = 1 and T is a semidirect product T = N:T′ for some subgroup T′ of T. (ii) The Schur multiplier of T/N is trivial. In particular, every Sylow subgroup

of T/N is cyclic.

(iii) |N| and |T/N| are relatively prime.

Moreover, if one of the above holds, then fixing an extension θ̃ of θ to T, we have a bijection from Irr(T/N, d, [r]) to Irr(T, d + d(θ), [cr]|θ) sending any λ in Irr(T/N, d, [r]) to λθ̃, where c is the p′-part of |N|/θ(1).

3 The case of p = 2 for Ly

In this section, we assume that G = Ly and p = 2. Thus we can forget r.

3.1 Radical 2-chains

There is only one conjugacy class of involutions. Let z be an involution of Ly. By Proposition 7 of [16] there are 17 conjugacy classes of non-trivial radical 2subgroups of Ly. Among those, ⟨z⟩ is a radical 2-subgroup. Moreover, we have N_{G}(z) = 2.A_{11}, and A_{11} has 15 conjugacy classes of radical 2-subgroups. (See Lemma 6 of [16].) Proposition 7 of [16] also yields that for each radical subgroup R of A_{11}, the inverse image ⟨z⟩R in 2·A_{11} is a radical 2-subgroup of Ly and this gives an injection from the set of conjugacy classes of radical 2-subgroups R of A_{11} to that of conjugacy classes of radical 2-subgroups ⟨z⟩R of Ly. However, among these ⟨z⟩R, two subgroups F_{4} = ⟨z⟩U_{0} ≅ 2^{4} and V_{2} = ⟨z⟩U_{02} ≅ 2^{2}.2^{4}, in the notation of [16], have normalizers in G not contained in C_{G}(z) and the others have normalizers contained in C_{G}(z). There exist two more conjugacy classes of radical 2-subgroups of Ly. Namely, V_{2}^{(1)} ≅ 2^{2}[2^{5}] and F_{3} ≅ 2^{3}. In the proof of Proposition 7 of [16], it is shown that V_{2}^{(1)} contains V_{2} and we have Z(V_{2}) = Z(V_{2}^{(1)}) ≅ 2^{2}, and thus N_{G}(V_{2}^{(1)}) ≤ N_{G}(Z(V_{2})) = N_{G}(V_{2}). Hence Lemma 2.4 yields the following.

Lemma 3.1 In the notation above, among those in R_{2}(Ly), it suffices to consider only

1 < ⟨z⟩,
1 < V_{2},
1 < V_{2} < R < ⋯, with R ≠ V_{2}^{(1)},
1 < ⟨z⟩ < V_{2} < ⋯,
1 < F_{4} < ⋯,
1 < ⟨z⟩ < F_{4} < ⋯, and
1 < F_{3} < ⋯,

where 1 < P_{1} < ⋯ and 1 < P_{1} < P_{2} < ⋯ include the case of 1 < P_{1} and 1 < P_{1} < P_{2}, respectively in the above.

We know that N_{G}(V_{2}) ≤ N_{G}(Z(V_{2})) and thus N_{G}(V_{2}) ≤ 3McL2. (§2 of [15]) Moreover, N_{G}(V_{2}) = (V_{2} ×3)(S_{3} ×S_{3}), where the first S_{3} acts trivially on Z(V_{2}) ≅ 2^{2} and non-trivially on 3, while the second S_{3} acts faithfully on Z(V_{2}) and trivially on 3. Note also that in this notation, we have V_{2}^{(1)} = V_{2}(2 × 1), since Z(V_{2}^{(1)}) = Z(V_{2}). Moreover, the intersection of N_{G}(V_{2}) = (V_{2}×3)(S_{3}×S_{3}) and 3McL is (V_{2}×3)(3×S_{3}), since an outer automorphism of 3McL inverts each element of 3 in 3McL. See the argument preceding Proposition 7 of [16]. Since a radical subgroup of a direct product is conjugate to a direct product of radical subgroups of direct factors, radical 2-chains starting with 1 < V_{2} are conjugate to

C_{00} : 1 < V_{2},
C_{01} : 1 < V_{2} < V_{2}(2 × 1),
C_{02} : 1 < V_{2} < V_{2}(1 × 2),
C_{03} : 1 < V_{2} < V_{2}(2 × 2),
C_{04} : 1 < V_{2} < V_{2}(2 × 1) < V_{2}(2 × 2), or
C_{05} : 1 < V_{2} < V_{2}(1 × 2) < V_{2}(2 × 2).

But, in this notation, V_{2}^{(1)} = V_{2}(2 × 1). Thus we must consider only C_{00}, C_{02}, C_{03} and C_{05}. Moreover, it is clear that N_{G}(C_{03}) = N_{G}(C_{05}). Hence the following holds.

Lemma 3.2 Among radical 2-chains starting with 1 < V_{2}, it suffices to consider only the following.

\begin{matrix}Q & N_{G}(Q) \\ 1 < V_{2} & (V_{2} × 3)(S_{3} × S_{3}) \\ 1 < V_{2} < V_{2}(1 × 2) & (V_{2} × 3)(S_{3} × 2) \end{matrix}

On the other hand, since we may assume that Z(V_{2}) ≅ 2^{2} contains z, we have N_{G}(V_{2})∩C_{G}(z) ≅ (V_{2} ×3)(S_{3} ×2). Thus radical 2-chains starting with 1 < ⟨z⟩ < V_{2} are conjugate to

C_{10} : 1 < ⟨z⟩ < V_{2},
C_{11} : 1 < ⟨z⟩ < V_{2} < V_{2}(2 × 1),
C_{12} : 1 < ⟨z⟩ < V_{2} < V_{2}(1 × 2),
C_{13} : 1 < ⟨z⟩ < V_{2} < V_{2}(2 × 2),
C_{14} : 1 < ⟨z⟩ < V_{2} < V_{2}(2 × 1) < V_{2}(2 × 2), or
C_{15} : 1 < ⟨z⟩ < V_{2} < V_{2}(1 × 2) < V_{2}(2 × 2).

We clearly have N_{G}(C_{10}) = N_{G}(C_{12}) ≅ (V_{2} × 3)(S_{3} × 2) and N_{G}(C_{11}) = N_{G}(C_{13}) = N_{G}(C_{14}) = N_{G}(C_{15}) ≅ (V_{2} × 3)(2 × 2). Hence we have the following.

Lemma 3.3 Radical 2-chains starting with 1 < ⟨z⟩ < V_{2} have no contribution in the alternating sum.

We have N_{G}(F_{4}) ≅ (2^{4} × 3)·A_{7} ≤ 3McL and A_{7} has three conjugacy classes of radical 2-subgroups. Namely, two conjugacy classes of subgroups E_{1} and E_{2} isomorphic to 2^{2} and a Sylow 2-subgroup isomorphic to D_{8}, which satisfy N_{A_{7}}(E_{1}) ≅ (A_{4} × 3) : 2, N_{A_{7}}(E_{2}) ≅ S_{4} and N_{A_{7}}(D_{8}) ≅ D_{8}. On the other hand, [11] classifies radical 2-subgroups of 3McL and the inverse images of E_{i} and D_{8} in N_{G}(F_{4}) are all radical 2-subgroups of 3McL. In fact, in the notation of [11], their inverse images correspond to R_{1,2,1}, R_{2,1,1} ≅ F_{4} : 2^{2} and R_{1,1,1,1} ≅ F_{4} : D_{8}, respectively. Moreover, R_{1,2,1} = V_{2} in our notation. (See 4.4 of [11].) Since every subgroup appearing in any radical 2-chain of 3McL is a radical 2-subgroup of 3McL (4.4.7 of [11]), radical 2-chains of G starting with 1 < F_{4} or 1 < ⟨z⟩ < F_{4} are G-conjugate to

C_{20} : 1 < F_{4},
C_{21} : 1 < F_{4} < V_{2},
C_{22} : 1 < F_{4} < V_{2} < R_{1,1,1,1},
C_{23} : 1 < F_{4} < R_{2,1,1},
C_{24} : 1 < F_{4} < R_{2,1,1} < R_{1,1,1,1},
C_{25} : 1 < F_{4} < R_{1,1,1,1}
C_{30} : 1 < ⟨z⟩ < F_{4},
C_{31} : 1 < ⟨z⟩ < F_{4} < V_{2},
C_{32} : 1 < ⟨z⟩ < F_{4} < V_{2} < R_{1,1,1,1},
C_{33} : 1 < ⟨z⟩ < F_{4} < R_{2,1,1},
C_{34} : 1 < ⟨z⟩ < F_{4} < R_{2,1,1} < R_{1,1,1,1}, or
C_{35} : 1 < ⟨z⟩ < F_{4} < R_{1,1,1,1}.

However, as is claimed in 4.4.49 of [11], we have N_{3McL}(R_{2,1,1}) ≤ N_{3McL}(z) and N_{3McL}(R_{1,1,1,1}) ≤ N_{3McL}(z). Thus N_{G}(C_{2i}) = N_{G}(C_{3i}) for i with 2 ≤ i ≤ 5. The structure of their normalizers are also determined in 4.4 of [11], and we have the following.

Lemma 3.4 Among radical 2-chains starting with 1 < F_{4} or 1 < ⟨z⟩ < F_{4}, it suffices to consider only the following.

\begin{matrix}Q & N_{G}(Q) \\ C_{20} : 1 < F_{4} & (F_{4} × 3)·A_{7} \\ C_{21} : 1 < F_{4} < V_{2} & (V_{2} × 3)(3 × S_{3}) \\ C_{30} : 1 < ⟨z⟩ < F_{4} & (F_{4} × 3)·GL_{3}(2) \\ C_{31} : 1 < ⟨z⟩ < F_{4} < V_{2} & (V_{2} × 3)((1 × 3):2) ≅ (V_{2} × 3)(1 × S_{3}) \end{matrix}

Finally, we consider radical 2-chains starting with 1 < F_{3}. Since N_{G}(F_{3})/F_{3} ≅ L_{3}(2), they are G-conjugate to

C_{40} : 1 < F_{3},
C_{41} : 1 < F_{3} < F_{3}U_{p},
C_{42} : 1 < F_{3} < F_{3}U_{p} < F_{3}U_{pℓ},
C_{43} : 1 < F_{3} < F_{3}U_{ℓ},
C_{44} : 1 < F_{3} < F_{3}U_{ℓ} < F_{3}U_{pℓ}, or
C_{45} : 1 < F_{3} < F_{3}U_{pℓ},

where U_{ℓ} and U_{p} are the unipotent radical of parabolic subgroups of L_{3}(2) which are a line stabilizer H_{ℓ} and a point stabilizer H_{p}, respectively, and U_{pℓ} = U_{p}U_{ℓ} .

Lemma 3.5 Radical 2-chains starting with 1 < F_{3} have no contribution in the alternating sum.

Proof. A standard argument gives N_{G}(C_{42}) = N_{G}(C_{44}) = N_{G}(C_{45}). Thus, it suffices to consider only C_{40}, C_{41}, C_{43} and C_{44}. Their normalizers are F_{3}L_{3}(2), F_{3}H_{p}, F_{3}H_{ℓ} and F_{3}H_{pℓ}, respectively, where H_{pℓ} = H_{p} ∩ H_{ℓ}. Each of them has only the principal 2-block. Let N_{G}(F_{3}) act on F_{3} from the right, and fixing a basis {v_{1}, v_{2}, v_{3}} of F_{3}, we may assume that H_{p} is isomorphic to

\{[\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ 0 & 0 & 1 \end{matrix}] | [\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] ∈ GL_{2}(2), a_{13}, a_{23} ∈ 𝔽_{2}\}

and H_{ℓ} is isomorphic to

\{[\begin{matrix}1 & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & a_{32} & a_{33} \end{matrix}] | [\begin{matrix}a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix}] ∈ GL_{2}(2), a_{12}, a_{13} ∈ 𝔽_{2}\}.

For i = 1, 2, 3, let θ_{i} be the character of F_{3} defined by θ_{i}(v_{i}) = −1 and θ_{i}(v_{j}) = 1 if i ≠ j. Then the stabilizer of θ_{1} in F_{3}L_{3}(2) is F_{3}H_{ℓ}, which implies that its stabilizer in F_{3}H_{p} is F_{3}H_{pℓ}. Thus Clifford theory (Lemma 2.5) yields that

k(F_{3}L_{3}(2), d|θ_{1}) = k(F_{3}H_{ℓ}, d|θ_{1}) and k(F_{3}H_{p}, d|θ_{1}) = k(F_{3}H_{pℓ}, d|θ_{1})

for all d. Moreover, the stabilizer of θ_{2} in F_{3}H_{ℓ} and in F_{3}H_{pℓ} coincide. Thus we have

k(F_{3}H_{ℓ}, d|θ_{2}) = k(F_{3}H_{pℓ}, d|θ_{2}).

Furthermore, it follows from Theorem 2.7 of [13] that

\begin{matrix}k(F_{3}L_{3}(2), d|1_{F_{3}}) − k(F_{3}H_{p}, d|1_{F_{3}}) − k(F_{3}H_{ℓ}, d|1_{F_{3}}) + k(F_{3}H_{pℓ}, d|1_{F_{3}}) \\ = \{ \begin{matrix}0 & if d ≠ 3, \\ 1 & if d = 3. \end{matrix} \end{matrix}

On the other hand, the orbits of θ_{1} under the actions of F_{3}L_{3}(2), F_{3}H_{ℓ}, F_{3}H_{p} and F_{3}H_{pℓ} consist of 7, 1, 3 and 1 characters, respectively, and those of θ_{2} under the actions of F_{3}H_{ℓ} and F_{3}H_{pℓ} consist of 6 and 2 characters, respectively. Hence, {1_{F_{3}}, θ_{1}} forms a set of representatives of F_{3}L_{3}(2)-orbits in Irr(F_{3}) and {1_{F_{3}}, θ_{1}, θ_{2}} forms one for F_{3}H_{ℓ}-orbits. However, the stabilizers of θ_{3} in F_{3}H_{p} and in F_{3}H_{pℓ} are

\{[\begin{matrix}a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{matrix}] | [\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] ∈ GL_{2}(2)\} and \{[\begin{matrix}1 & a_{12} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] | a_{12} ∈ 𝔽_{2}\},

respectively, and their orbits of θ_{3} both consist of 4 characters. Thus {1_{F_{3}}, θ_{1}, θ_{3}} forms a set of representatives for F_{3}H_{p}-orbits and {1_{F_{3}}, θ_{1}, θ_{2}, θ_{3}} forms one for F_{3}H_{pℓ}-orbits. Hence now it suffices to treat Irr(F_{3}H_{p}|θ_{3}) and Irr(F_{3}H_{pℓ}|θ_{3}). By Lemma 2.6 (ii), θ_{3} can be extended to its stabilizers in F_{3}H_{p} and in F_{3}H_{pℓ}. Hence, Irr(F_{3}H_{p}|θ_{3}) consists of two characters of degree 2^{2} and one character of degree 2^{3} and Irr(F_{3}H_{pℓ}|θ_{3}) consists of two characters of degree 2^{2}. Therefore, the conclusion holds.

Summarizing the above, we have the following.

Proposition 3.6 To prove that the conjecture holds for Ly, it suffices to consider the alternating sum with respect to the following radical 2-chains.

\begin{matrix}Q & N_{G}(Q) \\ C_{0} : 1 & G \\ C_{1} : 1 < ⟨z⟩ & 2·A_{11} \\ C_{2} : 1 < V_{2} & (V_{2} × 3)(S_{3} × S_{3}) \\ C_{3} : 1 < V_{2} < V_{2}(1 × 2) & (V_{2} × 3)(S_{3} × 2) \\ C_{20} : 1 < F_{4} & (F_{4} × 3)·A_{7} \\ C_{21} : 1 < F_{4} < V_{2} & (V_{2} × 3)(3 × S_{3}) \\ C_{30} : 1 < ⟨z⟩ < F_{4} & (F_{4} × 3)·GL_{3}(2) \\ C_{31} : 1 < ⟨z⟩ < F_{4} < V_{2} & (V_{2} × 3)((1 × 3):2) ≅ (V_{2} × 3)(1 × S_{3}) \end{matrix}

Moreover, since C_{3} and C_{10} : 1 < ⟨z⟩ < V_{2} has the same normalizer and length, we can replace C_{3} above by C_{10}.

3.2 2-blocks of G and C_{G}(z)

As is shown on p.175 of ATLAS [1], G has 53 characters. With the numbering there, it follows that G has twelve 2-blocks,

\begin{matrix}B_{0} = B_{0}(G), \{χ_{7}\}, \{χ_{8}\}, \{χ_{26}\}, \{χ_{27}\}, \{χ_{28}\}, \{χ_{39}\}, \{χ_{40}\}, \{χ_{41}\}, \{χ_{42}\}, \{χ_{43}\}, \\ B_{1} = \{χ_{i} | i = 2, 3, 4, 5, 6, 12, 13, 18, 21, 22, 24, 25, 29, 30, 35, 36, 37, 46 \} \end{matrix}

where χ_{7}(1) = χ_{8}(1) = 2^{8} ·7·67, and χ_{26}(1) = χ_{27}(1) = χ_{28}(1) = 2^{8} ·7·11·31·37, and χ_{39}(1) = χ_{40}(1) = χ_{41}(1) = χ_{42}(1) = χ_{43}(1) = 2^{8}·3^{7}·7·11, and B_{0}(G) and B_{1} have defect 8 and 7, respectively. On the other hand, C_{G}(z) ≅ 2.A_{11} and it has the principal block B_{0}(2.A_{11}) and the 2-block

B′_{1} = { χ_{j} | j = 2, 5, 6, 7, 8, 9, 18, 19, 24, 26, 28, 29, 31, 32, 33, 42, 43, 44, 45, 46, 47 },

where the numbering used above come from p.76 of ATLAS [1], and B_{0}(2.A_{11}) and B′_{1} have defect 8 and 7, respectively. The value of a character at a 3A-element of Ly determines the 2-block to which it belongs. Thus the character tables give the following. (For d which does not appear below, the relevant number is zero. This convention will be used throughout.)

\begin{matrix}d & 8 & 7 & 6 & 5 & 4 & 3 \\ k(G, B_{0}, d) & 8 & 6 & 6 & 0 & 4 & 1 \\ k(C_{G}(z), B_{0}, d) & 8 & 6 & 4 & 2 & 7 & 1 \end{matrix}  \begin{matrix}d & 7 & 6 & 5 & 4 & 3 \\ k(G, B_{1}, d) & 8 & 2 & 2 & 5 & 1 \\ k(C_{G}(z), B_{1}, d) & 8 & 2 & 2 & 8 & 1 \end{matrix}

3.3 2-blocks of other normalizers

Let i be in {2, 10, 20, 21, 30, 31}. Then N_{G}(C_{i}) is contained in 3McL2, since either V_{2} or F_{4} appears in C_{i}. (See 3.1.) Now we use the results in [11]. Notice that an outer automorphism of 3McL interchanges the two non-trivial elements in Z(3McL), leaves z and V_{2} invariant, and sends F_{4} to a subgroup not 3McL-conjugate to F_{4}. Hence the following holds.

Lemma 3.7 (i) For i = 2, 10, N_{G}(C_{i}) is not contained in 3McL, and has the principal 2-block B_{0}(N_{G}(C_{i})) and the other 2-block B_{1}(N_{G}(C_{i})) of defect 7.

(ii) For i = 20, 21, 30, 31, N_{G}(C_{i}) ≤ 3McL, and N_{G}(C_{i}) has the principal 2block B_{0}(N_{G}(C_{i})) and the other two complex conjugate 2-blocks B_{1}(N_{G}(C_{i})) and B_{1}(N_{G}(C_{i}))^{∗} of defect 7.

Moreover, the characters of B_{0}(N_{G}(C_{i})) lie in Irr(N_{G}(C_{i})|1_{Z(3McL)}) and those in B_{1}(N_{G}(C_{i})) lie in Irr(N_{G}(C_{i})|ρ) ∪ Irr(N_{G}(C_{i})|ρ^{−1}) in the case (i) above, where ρ is a non-trivial character of Z(3McL), and we may assume that those in B_{1}(N_{G}(C_{i})) lie in Irr(N_{G}(C_{i})|ρ) in the case (ii). Furthermore, B_{0}(N_{G}(C_{i})) induces B_{0}, and B_{1}(N_{G}(C_{i})) and B_{1}(N_{G}(C_{i}))^{∗} induce B_{1}.

The sets of degrees of characters in B_{j}(N_{G}(C_{i})), for j = 0, 1, can be described as multisets as follows. The right most column shows the references found in [11].

Block  Degrees  References
B_{0}(N_{G}(C_{2})) {1^{4}, 2^{4}, 4^{1}, 6^{2}, 9^{4}, 12^{5}, 24^{2}}  4.7.14 and 4.9.17
B_{0}(N_{G}(C_{10})) {1^{4}, 2^{2}, 3^{4}, 4^{4}, 6^{4}, 8^{4}, 16}  4.7.6 and 4.9.6
B_{0}(N_{G}(C_{20})) {1, 6, 10^{2}, 14^{2}, 15^{2}, 21, 35, 45^{2}, 90, 105, 120} 4.6.10
B_{0}(N_{G}(C_{21})) {1^{2}, 2^{4}, 3^{4}, 6^{2}, 9^{2}, 12^{2}, 24}  4.7.14
B_{0}(N_{G}(C_{30})) {1, 3^{2}, 6, 7^{3}, 8^{4}, 14, 21^{2}, 24^{2}}  4.6.2
B_{0}(N_{G}(C_{31})) {1^{2}, 2, 3^{6}, 4^{2}, 6, 8^{4}}  4.7.6

Hence, by Lemma 3.7, we have the following, which together with the table found in the previous subsection yields that Conjecture 2.3 is true for p = 2 and B_{0}.

\begin{matrix}d & 8 & 7 & 6 & 5 & 4 & 3 & (−1)^{|C_{i}|} \\ k(N_{G}(C_{2}), B_{0}, d) & 8 & 6 & 6 & 2 & 0 & 0 & −1 \\ k(N_{G}(C_{10}), B_{0}, d) & 8 & 6 & 4 & 4 & 1 & 0 & +1 \\ k(N_{G}(C_{20}), B_{0}, d) & 0 & 8 & 6 & 0 & 1 & 0 & −1 \\ k(N_{G}(C_{21}), B_{0}, d) & 0 & 8 & 6 & 2 & 1 & 0 & +1 \\ k(N_{G}(C_{30}), B_{0}, d) & 0 & 8 & 2 & 0 & 6 & 0 & +1 \\ k(N_{G}(C_{31}), B_{0}, d) & 0 & 8 & 2 & 2 & 4 & 0 & −1 \end{matrix}

Next, consider B_{1}. For C_{20} and C_{21}, we have the following.

Degrees  Reference
Irr(N_{G}(C_{20})|ρ) {6, 15^{3}, 21^{2}, 24^{2}, 45^{2}, 90, 105, 120} 4.10.8
Irr(N_{G}(C_{21})|ρ) {3^{6}, 6^{2}, 9^{2}, 12^{2}, 24}  4.10.5

Moreover, it is claimed in 4.10.3 of [11] that for i = 30, 31, we have

k(N_{G}(C_{i}), d|1_{Z(3McL)})) = k(N_{G}(C_{i}), d|ρ)

for all d. Thus we have the following, which means that the chains C_{20}, C_{21}, C_{30} and C_{31} give no contribution to the alternating sum.

\begin{matrix}d & 7 & 6 & 5 & 4 & 3 & (−1)^{|C_{i}|} \\ k(N_{G}(C_{20}), B_{1}, d|ρ) & 8 & 2 & 0 & 3 & 0 & −1 \\ k(N_{G}(C_{21}), B_{1}, d|ρ) & 8 & 2 & 2 & 1 & 0 & +1 \\ k(N_{G}(C_{30}), B_{1}, d|ρ) & 8 & 2 & 0 & 6 & 0 & +1 \\ k(N_{G}(C_{31}), B_{1}, d|ρ) & 8 & 2 & 2 & 4 & 0 & −1 \end{matrix}

Finally, consider C_{2} and C_{10}. Let i be 2 or 10. Since their normalizers are not contained in 3McL, the inertia subgroup in N_{G}(C_{i}) of any character of N_{G}(C_{i}) ∩ 3McL lying over ρ is N_{G}(C_{i}) ∩ 3McL. Thus by Lemma 2.5, the induction gives a bijection

Irr(N_{G}(C_{i}) ∩ 3McL, d|ρ) → Irr(N_{G}(C_{i}), d|ρ)

for all d. Thus the following holds.

Degrees  References
B_{1}(N_{G}(C_{2})) {6^{6}, 12^{2}, 18^{2}, 24^{2}, 48} 4.10.5
B_{1}(N_{G}(C_{10})) {2^{2}, 4, 6^{6}, 8^{2}, 12, 16^{4}} 4.10.3 and 4.7.6

Hence, by Lemma 3.7, we have the following, which together with the table found in the previous subsection yields that Conjecture 2.3 is true for p = 2 and B_{1}.

\begin{matrix}d & 7 & 6 & 5 & 4 & 3 & (−1)^{|C_{i}|} \\ k(N_{G}(C_{2}), B_{1}, d) & 8 & 2 & 2 & 1 & 0 & −1 \\ k(N_{G}(C_{10}), B_{1}, d) & 8 & 2 & 2 & 4 & 0 & +1 \end{matrix}

4 The case of p = 3 for Ly

In this section, we assume that G = Ly and p = 3. Thus we can forget r.

4.1 Radical 3-subgroups and radical 3-chains

Maximal 3-local subgroups are classified in §3 of [15]. There are two conjugacy classes 3A and 3B of elements of order three, and a maximal 3-local subgroup is conjugate to one of

\begin{aligned}M_{1} = N_{G}(3A) & ≅ 3·McL:2, \\ M_{2} = N_{G}(3A_{2}B_{2}) & ≅ 3^{2+4} :2A_{5}·D_{8}, \\ M_{3} = N_{G}(3A_{11}B_{110}) & ≅ 3^{5} :(2 × M_{11}). \end{aligned}

Let us examine radical 3-subgroups of M_{i}’s. First of all, O_{3}(M_{i}) is a radical 3-subgroups of M_{i} and thus of G for each i. Moreover, a radical 3-subgroup R of M_{i} must contain O_{3}(M_{i}). (Prop.1.4 of [2]) In particular, R/O_{3}(M_{i}) is a radical 3-subgroup of M_{i}/O_{3}(M_{i}). Let us treat M_{i} individually. Radical 3-subgroups of M_{1}/O_{3}(M_{1}) ≅ McL: 2 are classified as follows. See also [11]. It has two conjugacy classes of maximal 3-local subgroups

\begin{aligned}L_{1} & ≅ 3_{+}^{1+4}(2A_{5}2), \\ L_{2} & ≅ 3^{4}M_{10}. \end{aligned}

See Lemma 5.6 of [5]. The groups O_{3}(L_{1}) ≅ 3_{+}^{1+4} and O_{3}(L_{2}) ≅ 3^{4} are radical 3-subgroups of McL : 2. On the other hand, it is clear that non-trivial radical 3-subgroups of L_{1}/O_{3}(L_{1}) ≅ 2A_{5}2 and L_{2}/O_{3}(L_{2}) ≅ M_{10} ≅ A_{6}2 are their Sylow 3-subgroups only, and they are just Sylow 3-subgroups T ≅ 3^{4}3^{2} of McL:2. Hence a non-trivial radical 3-subgroup of McL: 2 is conjugate to one of O_{3}(L_{1}), O_{3}(L_{2}) and T.

Let R be a 3-radical subgroup of G. Suppose that N_{G}(R) ≤ M_{1}. Then from the above paragraph, R/O_{3}(N_{G}(3A)) is trivial or conjugate to one of 3_{+}^{1+4}, 3^{4} and 3^{4}3^{2}. If R/O_{3}(N_{G}(3A)) is conjugate to 3_{+}^{1+4}, then it follows from the third paragraph of §3 of [15] that the inverse image of Z(R/O_{3}(N_{G}(3A))) in R is a 3^{2}-subgroup with two subgroups of type 3A and 3B each. Thus N_{G}(R) ≤ M_{2}, which implies that R ≅ O_{3}(M_{2}). If R/O_{3}(N_{G}(3A)) is conjugate to 3^{4}, then R is generated by subgroups of type 3A and thus R ≅ 3^{5} ≅ O_{3}(M_{3}). If R/O_{3}(N_{G}(3A)) is conjugate to T, then R is a Sylow 3-subgroup S of G. Suppose that N_{G}(R) ≤ M_{2}. Then, it is clear from the order that R is conjugate to O_{3}(M_{2}) or S. Suppose that N_{G}(R) ≤ M_{3}. Then, R is conjugate to O_{3}(M_{3}) or S, since radical 3-subgroups of M_{11} are trivial or a Sylow 3-subgroup.

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

O_{3}(M_{1}), O_{3}(M_{2}), O_{3}(M_{3}), and S.

Now look at radical 3-chains of G. Let R_{1} be the set of radical 3-chains of G starting with 1 < O_{3}(N_{G}(3A)) and R_{2} the set of radical 3-chains of 3·McL: 2. By the argument above, there exist a bijection

f : R_{1} → R_{2},

which preserves the normalizers and changes the parity of length. Moreover, two chains in R_{1} are G-conjugate if and only if their images under f are N_{G}(3A)- conjugate. Since Dade’s conjecture is proved for 3·McL: 2 in [11], we are allowed not to take the chains in R_{1} into account in the alternating sum.

On the other hand, it is seen on p.405 of [15] that C_{G}(3A_{2}B_{2}) ≅ 3^{2+4} :2A_{5}, which contains S. Thus we have Z(S) ≅ 3A_{2}B_{2}. This implies that N_{G}(S) ≤ N_{G}(Z(S)) ≤ M_{2}. Hence the two chains 1 < S and 1 < O_{3}(M_{2}) < S have the same normalizers and their lengths have different parity. Finally, the normalizer of a Sylow 3-subgroup of M_{11} is 3^{2} :SD_{16}. Summarizing the above, we have the following.

Proposition 4.1 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 < O_{3}(M_{2}),  3^{2+4} :2A_{5}·D_{8},
C_{3} : 1 < O_{3}(M_{3}),  3^{5} :(2 × M_{11}),
C_{4} : 1 < O_{3}(M_{3}) < S, 3^{5} :(2 × 3^{2} :SD_{16}),

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

4.2 3-blocks of G, M_{2} and M_{3}

The character table tells us that G has the following 3-blocks. (See [1] and [6].)

B_{0} = B_{0}(G), {χ_{20}, χ_{21}, χ_{22}}, {χ_{12}, χ_{49}, χ_{51}}, {χ_{39}}, {χ_{40}}, {χ_{41}}, {χ_{42}}, {χ_{43}}.

Here, χ_{12}(1) = 2 · 3^{6} · 31 · 67, χ_{49}(1) = 3^{6} · 5 · 7 · 31 · 67, χ_{51}(1) = 3^{6} · 11 · 37 · 67, χ_{20}(1) = χ_{21}(1) = χ_{22}(1) = 2·3^{6}·11·31·37, and χ_{39}(1) = χ_{40}(1) = χ_{41}(1) = χ_{42}(1) = χ_{43}(1) = 2^{8}·3^{7}·7·11. The block B_{0} is of defect 7, and other blocks have defect one or zero. Indeed, the two blocks of defect one have O_{3}(M_{1}) as their defect groups. Thus, it suffices to consider B_{0}. Since M_{2} and M_{3} have normal selfcentralizing subgroups, they have only the principal 3-blocks. Moreover, they appear as 6th and 7th maximal subgroups in GAP library. Their character tables give the following.

\begin{matrix}d & 7 & 6 & 5 & 4 \\ k(G, B_{0}, d) & 27 & 6 & 3 & 6 \\ k(M_{2}, B_{0}, d) & 27 & 6 & 18 & 6 \\ k(M_{3}, B_{0}, d) & 27 & 6 & 3 & 0 \end{matrix}

4.3 3-blocks of N_{G}(S)

The group N_{G}(S) ≅ 3^{5} : (2 × 3^{2} : SD_{16}) has a normal selfcentralizing subgroup E = O_{3}(M_{3}) ≅ 3^{5}. Thus it has only the principal 3-block. (V.3.11 of [4].)

Now we examine the structure of N_{G}(S). According to Proposition 2.6 and the discussion thereafter in [10], E has a subgroup Z(S) ≅ 3^{2} = 3A_{2}B_{2}, and we have N_{G}(Z(S)) = M_{2} ≅ 3^{2+4} : 2A_{5} ·D_{8}. Notice also that C_{G}(Z(S)) ≅ 3^{2+4} : 2A_{5}. Thus there exists a homomorphism α : M_{2} → GL_{2}(3) such that Kerα = C_{G}(Z(S)) and Imα ≅ D_{8}. Restrict α to N_{G}(S). We know that |M_{2} : N_{G}(S)| = 10. Since a Sylow 3-subgroup of 2A_{5} has the normalizer with index 10 in 2A_{5}, we can conclude that the image of the restriction of α is also D_{8}. In fact, it is a subsection of 2 × SD_{16} isomorphic to D_{8}. It is not difficult to see that a subgroup of GL_{2}(3) isomorphic to D_{8} is determined uniquely up to GL_{2}(3)-conjugate. It then follows that there exist two N_{G}(S)-orbits in Irr(Z(S)) \ {1_{Z(S)}}, each of which consists of 4 characters.

Now we treat characters of N_{G}(S) lying over non-trivial characters of Z(S). Let λ_{1} and λ_{2} be representatives of N_{G}(S)-orbits in Irr(Z(S)) \ {1_{Z(S)}}. On the degrees of characters of S lying over some λ_{i}, the following holds.

Lemma 4.2 For each i, Irr(S|λ_{i}) consists of three characters all of whose degrees are 3^{2}. Moreover, Irr(S) \ Irr(S/Z(S)) consists of 24 characters.

Proof. We use the character table of M_{2} = N_{G}(3A_{2}B_{2}) found in GAP library. Let L = O_{3}(M_{2}) ≅ 3^{2+4}. The numbering of characters is also taken from it. Using its labels of conjugacy classes, we can see that 3a, 3b and 3c elements generate L and 3a and 3b elements do Z(S). By looking at the restrictions of characters of M_{2} to Z(S), we can see that there are two M_{2}-orbits of non-trivial linear characters of Z(S). In fact, χ_{34} and χ_{36} lie over characters of Z(S) contained in distinct M_{2}-orbits. Moreover, we have [(χ_{34})_{L}, (χ_{34})_{L}] = [(χ_{36})_{L}, (χ_{36})_{L}] = 4 and [(χ_{34})_{Z(S)}, (χ_{34})_{Z(S)}] = [(χ_{36})_{Z(S)}, (χ_{36})_{Z(S)}] = 4·3^{4}. Since λ_{i}’s are characters of Z(S), we can conclude that both (χ_{34})_{L} and (χ_{36})_{L} are the sum of four different characters, respectively, all of whose degrees are 3^{2}. Write (χ_{34})_{L} = η_{1} +η_{2} +η_{3} +η_{4} and (χ_{36})_{L} = η_{5} +η_{6} +η_{7} +η_{8}. Then (η_{i})_{Z(S)} is 3^{2} times a character of Z(S), and we may assume that (η_{1})_{Z(S)} = 3^{2}λ_{1} and (η_{5})_{Z(S)} = 3^{2}λ_{2}. In particular, C_{G}(Z(S)) ≅ 3^{2+4} : 2A_{5} stabilizes each η_{i}. Since S ≤ C_{G}(Z(S)), each η_{i} can be extended to S. Now the proof is completed.

Using the notation induced in the proof of the above lemma, we can consider Irr(N_{G}(S)|η_{i}). We know that N_{G}(S) ≤ M_{2} and N_{G}(S) is the inverse image of the normalizer of a Sylow 3-subgroup of M_{2}/L ≅ 2A_{5} ·D_{8}. Recall that η_{i} can be extended to S. Since the normalizer of a Sylow 3-subgroup of M_{2}/L ≅ 2A_{5}·D_{8} has an element which acts non-trivially on the Sylow 3-subgroup, N_{G}(S)/S ≅ 2 × SD_{16} has an element which interchanges two extensions of η_{i} to S. Recalling also the action of N_{G}(S) on Irr(Z(S)), we can conclude that there exist four N_{G}(S)-orbits in Irr(S) \ Irr(S/Z(S)) consisting of 4, 8, 4, 8 characters, respectively. Since the first factor 2 in N_{G}(S)/S ≅ 2 × SD_{16} inverts all non-trivial characters of Z(S) (Proposition 2.7 of [10]), the stabilizer of a character in Irr(S) \ Irr(S/Z(S)) within N_{G}(S)/S is isomorphic to D_{8} or Q_{8}, if the length of the orbit is 4, and to a group of order 4, if the length of the orbit is 8. Note that each character in Irr(S)\Irr(S/Z(S)) can be extended to its stabilizer, since S is a Sylow 3-subgroup. (Lemma 2.6 (iii)) Hence Clifford theory (Lemma 2.5) gives the following.

\begin{matrix}d & 7 & 6 & 5 & 4 \\ k(N_{G}(S), B_{0}, d|Irr(Z(S)) \ \{1_{Z(S)}\}) & 0 & 0 & 18 & 0 \end{matrix}

On the other hand, the action of N_{G}(S) on E/Z(S) ≅ 3^{3} can be seen as follows. Let

β : 2 × 3^{2} :SD_{16} → GL_{3}(3)

be the natural homomorphism induced from the action of N_{G}(S)/E ≅ 2 × 3^{2} :SD_{16} on E/Z(S). We claim the following.

Lemma 4.3 In the notation above, Kerβ is trivial and we have

Imβ ≅ \{[\begin{matrix}a & b & y_{1} \\ c & d & y_{2} \\ 0 & 0 & y_{3} \end{matrix}] ∈ GL_{3}(3) | [\begin{matrix}a & b \\ c & d \end{matrix}] ∈ U, y_{1}, y_{2}, y_{3} ∈ 𝔽_{3}, y_{3} ≠ 0 \},

where U is a Sylow 2-subgroup of GL_{2}(3).

Proof. Let L be the set of subgroups of S generated by a 3A-element of Ly. By Proposition 2.7 of [10], L consists of 11 subgroups and all of them are contained in E. Moreover, the first factor 2 of M_{3}/E ≅ 2 × M_{11} inverts all the elements of E and the second factor M_{11} acts naturally on L. Write L = {H_{1}, H_{2}, ⋯, H_{11}}. Since Z(S) ≅ 3A_{2}B_{2} is generated by two commuting 3A-elements, we may assume that Z(S) = ⟨H_{1}, H_{2}⟩. Now, we claim that, if i_{1}, i_{2}, i_{3} and i_{4} are distinct elements of {1, 2, ⋯, 11}, then ⟨H_{i_{1}}, H_{i_{2}}⟩ ∩ ⟨H_{i_{3}}, H_{i_{4}}⟩ = {1}. Since M_{11} acts 4-fold transitively on L, we may assume that i_{j} = j for j with 1 ≤ j ≤ 4. Let x_{j} be a generator of H_{j}. Consider ⟨x_{1}, x_{2}, x_{3}⟩. Since both ⟨x_{1}, x_{2}⟩ and ⟨x_{1}, x_{3}⟩ are 3A_{2}B_{2} and they are distinct, ⟨x_{1}, x_{2}, x_{3}⟩ has at least 3 subgroups generated by a 3B-element of Ly. Thus it has at most 10 subgroups generated by a 3A-element of Ly. Since there are 11 subgroups in L, we may assume that x_{4} is not contained in ⟨x_{1}, x_{2}, x_{3}⟩. Hence we obtain ⟨x_{1}, x_{2}⟩ ∩ ⟨x_{3}, x_{4}⟩ = {1} as desired.

Now, let \overline{g} be an element of Kerβ and take g ∈ N_{G}(S) with gE = \overline{g}. If g sends some H_{i}, 3 ≤ i ≤ 11, to H_{j}, then g^{−1}x_{i}^{−1}gx_{i} lies in Z(S)∩⟨x_{i}, x_{j}⟩ = ⟨x_{1}, x_{2}⟩∩⟨x_{i}, x_{j}⟩.

Thus we get g^{−1}x_{i}^{−1}g = x_{i}^{−1} ∈ H_{i} ∩ H_{j}, a contradiction. Hence \overline{g} acts trivially on L \ \{H_{1}, H_{2}\}. However, since M_{11} ≤ A_{11}, this \overline{g} must be an even permutation on L, and therefore it acts trivially on L. This means that \overline{g} must lie in the first factor 2. But, since the nontrivial element of this factor 2 inverts all the elements of E, we can conclude that \overline{g} is trivial. Once we know that Kerβ is trivial, then the rest is clear, since N_{G}(S)/E ≅ 2 × 3^{2} :SD_{16} has a normal subgroup 3^{2}. This completes the proof.

From the above consideration, we can see the action of N_{G}(S) on Irr(E). But, before doing it, we claim the following.

Lemma 4.4 In the notation above, every character in Irr(E) can be extended to its stabilizer in N_{G}(E), hence to that in N_{G}(S).

Proof. We use the character table of M_{3} = N_{G}(E) found in GAP library. The numbering of characters is also taken from it. Moreover, using its labels of conjugacy classes, we can see that 3a elements and 3b elements generate E. By looking at the restrictions of characters of N_{G}(E) to E, we can see that there are two N_{G}(E)-orbits of non-trivial linear characters of E. In fact, χ_{21} and χ_{25} lie over characters of E contained in distinct N_{G}(E)-orbits. Moreover, we have [(χ_{21})_{E}, (χ_{21})_{E}] = 110 and [(χ_{25})_{E}, (χ_{25})_{E}] = 132. Since E has 3^{5} − 1 = 242 non-trivial linear characters, it follows that (χ_{21})_{E} and (χ_{25})_{E} are the sums of 110 and 132 N_{G}(E)-conjugate linear characters, respectively. Hence the conclusion holds.

Next, consider characters of E whose kernels contain Z(S). They can be regarded as those of E/Z(S). It follows from the standard argument (§4 and §6 of [13]) that there are three N_{G}(S)-orbits in Irr(E/Z(S)). Namely, the orbit of 1_{E}, that of θ_{1} whose stabilizer T_{1} in N_{G}(S) satisfies

T_{1}/E ≅ \{[\begin{matrix}a & b & y_{1} \\ c & d & y_{2} \\ 0 & 0 & 1 \end{matrix}] ∈ GL_{3}(3) | [\begin{matrix}a & b \\ c & d \end{matrix}] ∈ U, y_{i} ∈ 𝔽_{3}\} ≅ 3^{2} :U,

and that of θ_{2} whose whose stabilizer T_{2} in N_{G}(S) satisfies

T_{2}/E ≅ \{[\begin{matrix}a & b & y_{1} \\ 0 & 1 & 0 \\ 0 & 0 & ±1 \end{matrix}] ∈ GL_{3}(3) | [\begin{matrix}a & b \\ 0 & 1 \end{matrix}] ∈ U, y_{1} ∈ 𝔽_{3}\} ≅ 2 × (3:2).

Notice that in T_{2}/E, we have b = 0, since the 2×2 matrix there is a 2-element. Thus the orbits of θ_{1} and θ_{2} consist of 2 and 24 characters, respectively, and they give all 27 characters of E/Z(S). Lemma 4.4 and the character tables of (2 × 3^{2} :SD_{16}) and T_{i}/E give the following. (See also Lemma 2.6.)

\begin{matrix}d & 7 & 6 & 5 & 4 \\ k(N_{G}(S), B_{0}, d|1_{E}) & 18 & 0 & 0 & 0 \\ k(N_{G}(S), B_{0}, d|θ_{1}) & 9 & 0 & 0 & 0 \\ k(N_{G}(S), B_{0}, d|θ_{2}) & 0 & 6 & 0 & 0 \end{matrix}

Summarizing the above, we have the following.

\begin{matrix}d & 7 & 6 & 5 & 4 \\ k(N_{G}(S), B_{0}, d) & 27 & 6 & 18 & 0 \end{matrix}

Hence, by Proposition 4.1, Conjecture 2.3 holds for G in the case of p = 3.

5 The case of p = 5 for Ly

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

5.1 Radical 5-subgroups and radical 5-chains

In [15] it is shown that there are two conjugacy classes of maximal 5-local subgroups. One is the normalizer N_{G}(C) of a 5A-subgroup C ≅ 5 and the other is the normalizer N_{G}(A) of a 5A^{3}-subgroup A ≅ 5^{3}. Their structures are

N_{G}(C) ≅ 5_{+}^{1+4} : 2A_{6} · 4, and N_{G}(A) ≅ 5^{3} · L_{3}(5).

Let E = O_{5}(N_{G}(C)) ≅ 5_{+}^{1+4}. Let Q be a radical 5-subgroup of G. If N_{G}(Q) ≤ N_{G}(C), then we have O_{5}(N_{G}(C)) ≤ Q and thus Q = E or Q ≅ S, where S is a Sylow 5-subgroup of G. If N_{G}(Q) ≤ N_{G}(A), then we have Q = A·U, where U is the unipotent radical of a parabolic subgroup of L_{3}(5). Note that we have N_{G}(A·U) ≤ N_{G}(Z(A·U)). On the other hand, since C_{G}(A) = A, we have Z(A·U) = C_{A}(U), which has order 5, if U contains a point stabilizer and has order 5^{2} otherwise. Since A consists of 5A-elements, we have N_{G}(A·U) ≤ N_{G}(C) in the former case, and since N_{G}(5A^{2}) ≤ N_{G}(A) by §4 of [15], we have N_{G}(A·U) ≤ N_{G}(A) in the latter case. Note that one of A·U in the former case is a Sylow 5-subgroup S of G. Let U_{ℓ} and U_{p} be the unipotent radical of parabolic subgroups of L_{3}(5) which are a line stabilizer H_{ℓ} and a point stabilizer H_{p}, respectively. Then, we have E = AU_{p}, and comparing the orders of the centers, we can see that E ≅ 5_{+}^{1+4} and A·U_{ℓ} are not G-conjugate. The above argument implies that a non-trivial radical 5-subgroup of G is conjugate to one of

E = AU_{p} = O_{5}(N_{G}(C)) ≅ 5_{+}^{1+4}, O_{5}(N_{G}(A)) ≅ 5^{3}, A·U_{ℓ} and S.

Now look at radical 5-chains of G. Since N_{G}(A·U_{ℓ}) ≤ N_{G}(A), Lemma 2.4 yields that radical 5-chains starting with 1 < A·U_{ℓ} and those starting with 1 < A < A·U_{ℓ} can be eliminated in the alternating sum.

Thus, it suffices to consider the following representatives of G-conjugacy classes of radical 5-chains of G.

C_{0} : 1,
C_{1} : 1 < E = O_{5}(N_{G}(C)),
C_{2} : 1 < E < S,
C_{3} : 1 < S,
C_{4} : 1 < A,
C_{5} : 1 < A < A·U_{p},
C_{6} : 1 < A < A·U_{p} < S,
C_{7} : 1 < A < S.

The previous argument gives N_{G}(S) ≤ N_{G}(C), which implies that N_{G}(C_{2}) = N_{G}(C_{3}). On the other hand, we have N_{G}(C_{6}) = N_{G}(C_{7}) by a standard argument. Hence we have the following.

Proposition 5.1 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_{1} : 1 < E, & N_{G}(C_{1}) = N_{G}(C) ≅ 5_{+}^{1+4} :2A_{6}·4 \\ C_{4} : 1 < A, & N_{G}(C_{4}) = N_{G}(A) ≅ 5^{3}·L_{3}(5), \\ C_{5} : 1 < A < A·U_{p}, & N_{G}(C_{5}) ≅ 5^{3}·H_{p} ≅ 5^{3}·(5^{2} :GL_{2}(5)). \end{matrix}

5.2 5-blocks of G, N_{G}(C) and N_{G}(A)

The character table tells us that G has the following 5-blocks. We write the set of characters in B_{0}(G) as a multiset by their degrees and the multiplicity is written exponentially. Notice but the exponent of 5 is not the multiplicity, but the genuine exponent.

B_{0} = B_{0}(G) = {1, (5·496)^{2}, 45694, 48174^{2}, 120064^{2}, 381766, 5·230547, 5^{3}·12276, 3028266, 5·614792, 5·845339, 4997664, 5·1075866, 5·2151772,

11834746, 5·3381356, 18395586^{3}, 5^{3}·153698, 22609664^{3},
(5·5450544)^{4}, 5·5757444, 5·5917573, 5^{2}·1229584, 5·6762712,
5·7377504, 43110144^{5}, 5^{3}·353276, 45648306^{2}, (5^{3}·365552)^{2},
5·10598931, 56022921, 71008476},

{χ_{24}}, {χ_{25}}, {χ_{38}}, {χ_{50}}, {χ_{52}}.

Here, χ_{24}(1) = χ_{25}(1) = 2^{2} · 5^{6} · 11 · 31, χ_{38}(1) = 5^{6} · 37 · 67, χ_{50}(1) = 3 · 5^{6} · 31 · 37 and χ_{52}(1) = 2·5^{6}·31·67. The block B_{0} is of defect 6. Thus, it suffices to consider B_{0}. On the other hand, by V.3.11 of [4], N_{G}(C) and N_{G}(A) have only the principal blocks. They appear as the 3rd and 5th maximal subgroups of G in the GAP library. The characters in their principal 5-blocks are as follows.

B_{0}(N_{G}(C)) = {1^{4}, (5·1)^{8}, 16^{4}, 9^{4}, (5·2)^{4}, 4^{8}, (5·4)^{2}, 144^{4}, (5·96)^{2}, 576, 5·192, 5^{2}·4, (5^{2}·16)^{2}, (5^{3}·4)^{2}, (5^{2}·32)^{4}, 5^{2}·36, (5^{3}·8)^{3}},

B_{0}(N_{G}(A)) = {1, 5·6, 31^{3}, 96^{10}, 124^{10}, 5^{3}·1, (5·31)^{3}, 186, 5·124, (5·372)^{2}, (5·496)^{2}, 5^{2}·124, (5·248)^{2}, 5·744},

Since |G : N_{G}(C)| ≡ 1 and |G : N_{G}(A)| ≡ −1 mod 5, we have the following.

\begin{matrix}(d, r) & (6, 1) & (5, 1) & (5, 2) & (4, 1) & (4, 2) & (3, 1) & (3, 2) \\ k(G, B_{0}, d, [r]) & 25 & 12 & 5 & 1 & 0 & 2 & 3 \\ k(N_{G}(C), B_{0}, d, [r]) & 25 & 12 & 5 & 4 & 4 & 2 & 3 \\ k(N_{G}(A), B_{0}, d, [r]) & 25 & 8 & 4 & 1 & 0 & 1 & 0 \end{matrix}

5.3 5-blocks of 5^{3}·H_{p}

The group 5^{3} ·H_{p} is contained in N_{G}(C) ∩ N_{G}(A). See 5.1. It has a normal selfcentralizing 5-subgroup E ≅ 5^{3}.5^{2} ≅ 5_{+}^{1+4} and hence has only the principal 5-block, which induces B_{0}. (V.3.11 of [4].) Note also that |G : 5^{3}·H_{p}| ≡ 1 mod 5. Let us fix the notation as follows. There exists a homomorphism α : N_{G}(A) → L_{3}(5) such that Kerα = A. Let N_{G}(A) act on A from the right, and fixing a basis {v_{1}, v_{2}, v_{3}} of A, we can identify Imα with L_{3}(5) = SL_{3}(5). Then, we may assume that H_{p} is isomorphic to

\{[\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{matrix}] | [\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] ∈ GL_{2}(5), a_{33}det [\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] = 1, a_{13}, a_{23} ∈ 𝔽_{5}\}

and H_{ℓ} is isomorphic to

\{[\begin{matrix}a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & a_{32} & a_{33} \end{matrix}] | [\begin{matrix}a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix}] ∈ GL_{2}(5), a_{11}det [\begin{matrix}a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix}] = 1, a_{12}, a_{13} ∈ 𝔽_{5}\}.

Then we have

O_{5}(H_{p}) ≅ \{[\begin{matrix}1 & 0 & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \end{matrix}] | a_{13}, a_{23} ∈ 𝔽_{5}\}

and E = AO_{5}(H_{p}) ≅ 5_{+}^{1+4} whose center is generated by v_{3}. First of all, counting the number of characters of H_{p} ≅ 5^{2} :GL_{2}(5) by a standard argument (§4 and §6 in [13]), we have the following.

\begin{matrix}(d, r) & (6, 1) & (5, 1) & (5, 2) & (4, 1) & (4, 2) & (3, 1) & (3, 2) \\ k(5^{3}·H_{p}, B_{0}, d, [r]|1_{A}) & 25 & 4 & 0 & 0 & 0 & 0 & 0 \end{matrix}  (1)

Let θ_{1} be the character of A defined by θ_{1}(v_{1}) = θ_{1}(v_{2}) = 1 and θ_{1}(v_{3}) = ε, where ε is a primitive 5th root of unity in the complex number field. Then the stabilizer T_{1} of θ_{1} in 5^{3}·H_{p} satisfies

T_{1}/A = \{[\begin{matrix}a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{matrix}] | [\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] ∈ SL_{2}(5)\}.

On the other hand, since Z(E) = ⟨v_{3}⟩, we have Irr(E|θ_{1}) = {η}, where η is a faithful character of E with degree 5^{2}. The stabilizer of η in N_{G}(E) = N_{G}(C) is equal to C_{G}(5A), which is isomorphic to E : 2A_{6} by [15]. This implies that the stabilizer of η in 5^{3} ·H_{p} is E : SL_{2}(5). In particular, all characters of E with degree 5^{2} are 5^{3}·H_{p}-conjugate. Since the Schur multiplier of SL_{2}(5) is trivial, η has an extension to E:SL_{2}(5). (Lemma 2.6 (ii)) Hence the character table of SL_{2}(5) gives the following.

\begin{matrix}(d, r) & (6, 1) & (5, 1) & (5, 2) & (4, 1) & (4, 2) & (3, 1) & (3, 2) \\ k(5^{3}·H_{p}, B_{0}, d, [r]|θ_{1}) & 0 & 0 & 0 & 4 & 4 & 1 & 0 \end{matrix}  (2)

(The fact that η can be extended to its stabilizer in N_{G}(E) can be seen also from the character table of N_{G}(E) found in the GAP library. We can find a character of N_{G}(E) whose restriction to E has η as its constituent with multiplicity one.) Moreover, the 5^{3}·H_{p}-orbit of θ_{1} consists of |H_{p}|/|SL_{2}(5)| = 100 characters.

Let θ_{2} be the character of A defined by θ_{2}(v_{1}) = ε and θ_{2}(v_{2}) = θ_{2}(v_{3}) = 1. Then the stabilizer T_{2} of θ_{2} in 5^{3}·H_{p} satisfies

T_{2}/A = \{[\begin{matrix}1 & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{22}^{−1} \end{matrix}] | a_{12}, a_{13}, a_{22}, a_{23} ∈ 𝔽_{5}, a_{22} ≠ 0\} ≤ H_{p} ∩ H_{ℓ}.

Thus, the 5^{3} ·H_{p}-orbit of θ_{2} consists of 24 characters. Now we look at Irr(T′_{2}|θ_{2}), where T′_{2} is the stabilizer of θ_{2} in N_{G}(A). Note that T_{2} ≤ T′_{2}. Clearly, N_{G}(A) acts on the set of non-trivial characters of A transitively, and we can see that T′_{2} ≤ AH_{ℓ}. However, θ_{2} can not be extended to T′_{2}. To see it, first notice that T′_{2} has a normal subgroup Kerθ_{2} ≅ 5^{2}, and T′_{2}/Kerθ_{2} ≅ 5.(5^{2} :SL_{2}(5)). It follows from the character table of N_{G}(A) found in the GAP library that we have Irr(O_{5}(5.(5^{2} :SL_{2}(5)))|θ_{2}) = {μ}, where μ is a character of degree 5, and μ can be extended to 5.(5^{2} : SL_{2}(5)). Alternatively, this fact can be seen from Proposition 2.17 (b) of [10], since the second subgroup in the upper central series of a Sylow 5-subgroup of G is precisely Z(AU_{ℓ}) = Kerθ_{2} and its normalizer in G is AH_{ℓ}. Since T_{2} contains O_{5}(T′_{2}), we have Irr(T_{2}|θ_{2}) = Irr(T_{2}|μ) and the stabilizer of μ in T_{2} modulo AU_{ℓ} is isomorphic to

\{[\begin{matrix}a_{22} & a_{23} \\ 0 & a_{22}^{−1} \end{matrix}] | a_{22}, a_{23} ∈ 𝔽_{5}, a_{22} ≠ 0\}.

The set of degrees of characters of the above group is {1^{4}, 2^{4}}. Since μ can be extended to T_{2}, we have the following.

\begin{matrix}(d, r) & (6, 1) & (5, 1) & (5, 2) & (4, 1) & (4, 2) & (3, 1) & (3, 2) \\ k(5^{3}·H_{p}, B_{0}, d, [r]|θ_{2}) & 0 & 4 & 4 & 0 & 0 & 0 & 0 \end{matrix}  (3)

Since 1_{A}, θ_{1} and θ_{2} form a set of representatives of the 5^{3} ·H_{p}-orbits in Irr(A),

(1) - (3) give the following.

\begin{matrix}(d, r) & (6, 1) & (5, 1) & (5, 2) & (4, 1) & (4, 2) & (3, 1) & (3, 2) \\ k(5^{3}·H_{p}, B_{0}, d, [r]) & 25 & 8 & 4 & 4 & 4 & 1 & 0 \end{matrix}

Hence, by Proposition 5.1, Conjecture 2.1 holds for G in the case of p = 5.

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