| Abstrakt: |
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 and . For i ∈ ℕ, assuming that (M1, M2)i and (M2, M1)i are already defined, put and . 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′, , ) from Π are equivalent if there exists an isomorphism of G onto G′ mapping M1 onto and M2 onto . 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 , 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 or , (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. [ABSTRACT FROM AUTHOR] |
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