Stabilizers of Vertices of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture. IV

Abstract

This is the fourth in a series of papers whose results imply the validity of a strong version of the Sims conjecture on finite primitive permutation groups. In this paper, the case of primitive groups with a simple socle of orthogonal Lie type and nonparabolic point stabilizer is considered. Let G be a finite group, and let M₁ and M₂ be distinct conjugate maximal subgroups of G. For any i ∈ ℕ, we define inductively subgroups (M₁, M₂)ⁱ and (M₂, M₁)ⁱ of M₁ ∩ M₂, which will be called the ith mutual cores of M₁ with respect to M₂ and of M₂ with respect to M₁, respectively. Put ${\left( {{M_1},\,{M_2}} \right)^1} = {\left( {{M_1} \cap {M_2}} \right)_{{M_1}}}$ and ${\left( {{M_2},\,{M_1}} \right)^1} = {\left( {{M_1} \cap {M_2}} \right)_{{M_2}}}$. For i ∈ ℕ, assuming that (M₁, M₂)ⁱ and (M₂, M₁)ⁱ are already defined, put ${\left( {{M_1},{M_2}} \right)^{i + 1}} = {\left( {{{\left( {{M_1},\,{M_2}} \right)}^i} \cap {{\left( {{M_2},{M_1}} \right)}^i}} \right)_{{M_1}}}$ and ${\left( {{M_2},{M_1}} \right)^{i + 1}} = {\left( {{{\left( {{M_1},\,{M_2}} \right)}^i} \cap {{\left( {{M_2},{M_1}} \right)}^i}} \right)_{M2}}$. We are interested in the case where (M₁)_G = (M₂)_G = 1 and 1 < ∣(M₁, M₂)² ∣ ≤ ∣(M₂, M₁)²∣. The set of all such triples (G, M₁, M₂) is denoted by Π. We consider triples from Π up to the following equivalence: triples (G, M₁, M₂) and (G′, $M_1^\prime$, $M_2^\prime$) from Π are equivalent if there exists an isomorphism of G onto G′ mapping M₁ onto $M_1^\prime$ and M₂ onto $M_2^\prime$. In the present paper, the following theorem is proved.

Theorem. Suppose that (G, M₁, M₂) ∈ Π, L = Soc(G) is a simple orthogonal group of dimension ≥ 7, and M₁ ∩ L is a nonparabolic subgroup of L. Then$L \cong O_8^ + \left( r \right)$, where r is an odd prime, (M₁, M₂)³ = (M₂, M₁)³ = 1, and one of the following holds

(a) r ≡ ±1 (mod 8), G is isomorphic to$O_8^ + \left( r \right)\,:\;\mathbb{Z}_3$or$O_8^ + \left( r \right)\,\;:\;\>{S_3}$, (M₁, M₂)² = Z (O₂(M₁)) and (M₂, M₁)² = Z(O₂(M₂)) are elementary abelian groups of order 2³, (M₁, M₂)¹ = O₂(M₁) and (M₂, M₁)¹ = O₂(M₂) are special groups of order 2⁹, the group M₁/O₂(M₁) is isomorphic to L₃(2) × ℤ₃or L₃(2) × S₃, respectively, and M₁ ∩ M₂is a Sylow 2-subgroup of M₁

(b) r ≤ 5, the group G/L either contains Outdiag(L) or is isomorphic to the group ℤ₄, (M₁, M₂)² = Z(O₂(M₁ ∩ L)) and (M₂, M₁)² = Z(O₂(M₂ ∩ L)) are elementary abelian groups of order 2², (M₁, M₂)¹ = [O₂(M₁ ∩ L), O₂(M₁ ∩ L)] and (M₂, M₁)¹ = [O₂ (M₂ ∩ L), O₂(M₂ ∩ L)] are elementary abelian groups of order 2⁵, O₂(M₁ ∩ L)/[O₂(M₁ ∩ L), O₂(M₁ ∩ L)] is an elementary abelian group of order 2⁶, the group (M₁ ∩ L)/O₂(M₁ ∩ L) is isomorphic to the group S₃, ∣M₁: M₁ ∩ M₂∣ = 24, ∣M₁ ∩ M₂ ∩ L∣ = 2¹¹, and an element of order 3 from M₁ ∩ M₂ (for G/L ≅ A₄or G/L ≅ S₄) induces on the group L its standard graph automorphism.

In any of cases (a) and (b), the triples (G, M₁, M₂) exist and form one equivalence class.