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.

中文翻译：

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具有原始自同构群和Sims猜想的强版本的图的顶点的稳定器。IV

这是一系列论文中的第四篇，其结果暗示了有限元原始置换组上Sims猜想的强版本的有效性。在本文中，考虑了具有正交Lie型简单底端和非抛物线点稳定器的原始群的情况。令G为有限群，令M ₁和M ₂为G的不同共轭最大子群。对于任何我∈ℕ，我们感应定义子组（中号₁，中号₂）^我和（中号₂，中号₁）^我的中号₁ ∩中号₂，这将被称为我的个相互芯中号₁对于中号₂和的中号₂相对于中号₁分别。将\（{\ left（{{M_1}，\，{M_2}} \ right）^ 1} = {\ left（{{M_1} \ cap {M_2}} \ right）_ {{M_1}}} \\ ）和\（{\ left（{{M_2}，\，{M_1}} \ right）^ 1} = {\ left（{{M_1} \ cap {M_2}} \ right）_ {{M_2}}} \）。对于我∈ℕ，假设（中号₁，中号₂）^我和（中号₂，中号₁）^我已经定义好了，把\（{\ 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}}} \）和\（{ \ 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}} \）。我们对（ M ₁） _G =（ M ₂） _G = 1且1 <∣（ M ₁， M ₂） ² ∣≤∣（ M ₂， M ₁） ²的情况感兴趣∣。所有这些三元组（G，M ₁，M ₂）的集合由Π表示。我们考虑从up到以下等价的三元组：如果存在，则从triple的三元组（G，M ₁，M ₂）和（G '，\（M_1 ^ \ prime \），\（M_2 ^ \ prime \））等价在G '上存在G的同构，将M ₁映射到\（M_1 ^ \ prime \）和M ₂映射到\（M_2 ^ \ prime \）。本文证明了以下定理。定理。假设（g ^，中号₁，中号₂）∈Π，大号= SOC（ģ）是尺寸的简单正交基≥7，并且M ₁ ∩ L是L的亚组nonparabolic然后\（L \琮O_8 ^ + \ left（r \ right）\），其中r是一个奇数素数，（M ₁，M ₂）³ =（M ₂，M ₁）³ = 1，并且下列条件之一保持（a）r≡±1（mod 8），G同构为\（O_8 ^ + \ left（r \ right）\，：\; \ mathbb {Z} _3 \）或\（O_8 ^ + \ left（r \ right） \，\;：\; \> {S_3} \），（M ₁，M ₂）² = Z（O ₂（M ₁））和（M ₂，M ₁）² = Z（O ₂（M ₂））是2 ³，（M ₁，M ₂）¹ = Ö ₂（中号₁）和（中号₂，中号₁）¹ = Ö ₂（中号₂）是为了特殊群体2 ⁹，M组₁ / Ò ₂（中号₁）是同构为L ₃（ 2）×ℤ ₃或L ₃（2）× s ^ ₃，分别和M ₁ ∩中号₂是西洛2- M的子组₁（b）中ř ≤5，基团G / L或者包含Outdiag（大号）或同构于组ℤ ₄，（中号₁，中号₂）² = Ž（Ô ₂（中号₁ ∩大号））和（中号₂，中号₁）² = ž（Ô ₂（中号₂ ∩大号））是为了初等阿贝尔群2 ²，（中号₁，中号₂）¹ = [ Ò ₂（中号₁ ∩大号），Ô ₂（中号₁ ∩大号）]和（中号₂，中号₁）¹ = [ Ò ₂（中号₂ ∩大号），Ô ₂（中号₂ ∩大号）]是为了初等阿贝尔群2 ⁵，Ô ₂（中号₁ ∩大号）/ [ Ô ₂（中号₁ ∩大号），Ô ₂（中号₁ ∩大号）]是顺序的一个初等阿贝尔群2 ⁶，基团（中号₁ ∩大号）/ Ô ₂（中号₁ ∩大号）是同构的基团S ₃，|中号₁：中号₁ ∩中号₂ | = 24，|中号₁ ∩中号₂ ∩大号| = 2 ¹¹，以及一个的顺序元件3的M ₁ ∩中号₂（对于G /大号≅甲₄或G /大号≅小号₄）在L组上诱导其标准图自同构。在任何情况（a）和（b）中，三元组（G，M ₁，M ₂）存在并形成一个等价类。