This is the fourth in a series of papers whose results imply the validity of a strengthened version of the Sims conjecture on finite primitive permutation groups. In this paper, the case of primitive groups with simple socle of orthogonal Lie type and nonparabolic point stabilizer is considered. Let $G$ be a finite group, and let $M_1$ and $M_2$ be distinct conjugate maximal subgroups of $G$. For any $i\in \mathbb N$, we define inductively subgroups $(M_1,M_2) ^{i}$ and $(M_2,M_1)^{i}$ of $M_1\cap M_2$, which will be called the $i$th mutual cores of $M_1$ with respect to $M_2$ and of $M_2$ with respect to $M_1$, respectively. Put $(M_1,M_2)^{1}=(M_1\cap M_2)_{M_1}$ and $(M_2,M_1)^{1}=(M_1\cap M_2)_{M_2}$. For $i\in \mathbb N$, assuming that $(M_1,M_2)^{i}$ and $(M_2,M_1)^{i}$ are already defined, put $(M_1,M_2)^{i+1} = ((M_1,M_2)^{i}\cap (M_2,M_1)^{i})_{M_1}$ and $(M_2,M_1)^{i+1} = ((M_1,M_2)^{i}\cap (M_2,M_1)^{i})_{M_2}$. We are interested in the case when $(M_1)_G=(M_2)_G=1$ and $1<|(M_1,M_2)^{2}| \leq |(M_2,M_1)^{2}|$. The set of all such triples $(G, M_1, M_2)$ is denoted by $\Pi$. We consider triples from $\Pi$ up to the following equivalence: triples $(G,M_1,M_2)$ and $(G',M'_1,M'_2)$ from $\Pi$ are equivalent if there exists an isomorphism from $G$ to $G'$ mapping $M_1$ to $M'_1$ and $M_2$ to $M'_2$. In the present paper, the following theorem is proved.
Theorem.
Suppose that $(G, M_1, M_2)\in\Pi$, $L=Soc(G)$ is a simple orthogonal group of degree $\geq 7$, and $M_1\cap L$ is a nonparabolic subgroup of $L$. Then $Soc(G)\cong P\Omega^+_8(r)$, where $r$ is an odd prime, $(M_1, M_2)^3=(M_2,M_1)^3=1$, and one of the following statements holds$:$
$\mathrm (a)$ $r\equiv\pm1 (\mathrm{mod} 8)$; the group $G$ is isomorphic to $P\Omega^+_8(r):{\mathbb Z}_3$ or $P\Omega^+_8(r):S_3$; $(M_1, M_2)^2=Z(O_2(M_1))$ and $(M_2, M_1)^2=Z(O_2(M_2))$ are elementary abelian groups of order $2^3$; $(M_1, M_2)^1=O_2(M_1)$ and $(M_2,
M_1)^1=O_2(M_2)$ are special groups of order $2^9$; the group $M_1/O_2(M_1)$ is isomorphic to $L_3(2)\times {\mathbb Z}_3$ or $L_3(2)\times S_3$, respectively; and $M_1\cap M_2$ is a Sylow $2$-subgroup in $M_1$;
$\mathrm (b)$ $r\leq 5$; $G/L$ either contains $Outdiag(L)$ or is isomorphic to ${\mathbb Z}_4$; $(M_1, M_2)^2=Z(O_2(M_1\cap L))$ and $(M_2, M_1)^2=Z(O_2(M_2\cap L))$ are elementary abelian groups of order $2^2$; $(M_1, M_2)^1=(O_2(M_1\cap L))'$ and $(M_2, M_1)^1=(O_2(M_2\cap L))'$ are elementary abelian groups of order $2^5$; $O_2(M_1\cap L)/(O_2(M_1\cap L))'$ is an elementary abelian group of order $2^6$; the group $(M_1\cap L)/O_2(M_1\cap L)$ is isomorphic to the group $S_3$; $|M_1:M_1\cap M_2|=24$; $|M_1\cap M_2\cap L|=2^{11}$; and an element of order $3$ from $M_1\cap M_2$ (if it exists) induces on the group $L$ its graph automorphism.
In any of cases $\mathrm (a)$ and $\mathrm (b)$, the triples $(G,M_1,M_2)$ from $\Pi$ exist and form one class up to equivalence.
Keywords: finite primitive permutation group, stabilizer of a point, Sims conjecture, almost simple group, group of orthogonal Lie type
The paper was received by the Editorial Office on Deсember 25, 2017.
Funding Agency: This work was supported by the Russian Science Foundation (project no. 14-11-00061-П).
Anatolii Semenovich Kondrat’ev, Dr. Phys.-Math. Sci., N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: A.S.Kondratiev@imm.uran.ru
Vladimir Ivanovich Trofimov, Dr. Phys.-Math. Sci., N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, trofimov@imm.uran.ru
REFERENCES
1. Bourbaki N. Groupes et algebres de Lie (Chapt. IV–VI). Paris: Hermann, 1968, 282 p. doi: 10.1007/978-3-540-34491-9 . Translated to Russian under the title Gruppy i algebry Li (glavy IV – VI). Moscow: Mir Publ., 1972, 334 p.
2. Kondrat’ev A.S. Normalizers of the Sylow 2-subgroups in finite simple groups. Math. Notes, 2005, vol. 78, no. 3-4, pp. 338–346. doi: 10.1007/s11006-005-0133-9 .
3. Kondrat’ev A.S., Trofimov V.I. Stabilizers of graph’s vertices and a strengthened version of the Sims conjecture. Dokl. Math., 1999, vol. 59, no. 1, pp. 113–115.
4. Kondrat’ev A.S., Trofimov V.I. Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture, I,II, III. Proc. Inst. Steklov Math., 2015, vol. 289, suppl. 1, pp. 146–155, doi: 10.1134/S0081543815050132 ; 2016, vol. 295, suppl. 1, pp. 89–100, doi: 10.1134/S0081543816090108 ; 2017, vol. 299, suppl. 1, pp. 113–122, doi: 10.1134/S0081543817090140 .
5. Aschbacher M. On the maximal subgroups of the finite classical groups. Invent. Math., 1984, vol. 76, no. 3, pp. 469–514. doi: 10.1007/BF01388470 .
6. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A. Atlas of finite groups. Oxford: Clarendon Press, 1985, 252 p. ISBN: 0198531990 .
7. Barry M.J.J. Large abelian subgroups of Chevalley groups. J. Austral. Math. Soc., Ser. A, 1979, vol. 27, no. 1, pp. 59–87. doi: 10.1017/S1446788700016645 .
8. Bray J.N., Holt D.F., Roney-Dougal C.M. The maximal subgroups of the low-dimensional finite classical groups. Cambridge: Cambridge Univ. Press, 2013, London Math. Soc. Lect. Note Ser., vol. 407, 438 p. doi: 10.1017/CBO9781139192576 .
9. Carter R.W. Simple groups of Lie type. London: Wiley, 1972, 331 p. ISBN: 0471137359 .
10. The GAP Group. GAP – Groups, Algorithms, and Programming, Ver. 4.9.1., 2018. Available on: https://www.gap-system.org .
11. Gerono G.C. Note sur la r$\acute{\mathrm{e}}$solution en nombres entiers et positifs de l’$\acute{\mathrm{e}}$quation $x^m = y^n + 1$. Nouv. Ann. Math. (2), 1870, vol. 9, pp. 469–471.
12. Gorenstein D. Finite groups. N Y: Harper and Row, 1968, 528 p. ISBN (2nd ed.): 978-0821843420 .
13. Gorenstein D., Lyons R., Solomon R. The classification of the finite simple groups. Providence: Amer. Math. Soc., 1998, Number 3, Part I, Ser. Math. Surveys Monogr., vol. 40, no. 3, 420 p. ISBN: 978-0-8218-0391-2 .
14. Kleidman P.B. The maximal subgroups of the finite 8-dimensional orthogonal groups $P\Omega^+_8 q)$ and of their automorphism groups. J. Algebra, 1987, vol. 110, no. 1, pp. 173–242. doi: 10.1016/0021-8693(87)90042-1 .
15. Kleidman P.B., Liebeck M.W. The subgroup structure of the finite classical groups. Сambridge: Cambridge Univ. Press, 1990, London Math. Soc. Lect. Note Ser., vol. 129, 304 p. ISBN: 0-521-35949-X .
16. Timmesfeld F.G. Amalgams with rank 2 groups of Lie-type in characteristic 2. Preprint. Giessen: University Giessen, 1984, 126 p.
