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A strong version of the Sims conjecture for primitive parabolic permutation representations of finite simple groups Lie types \(G_2\), \(F_4\) and \(E_6\). (Russian. English summary) Zbl 1436.20022

Summary: For a finite group \(G\), subgroups \(M_1\) and \(M_2\) of \(G\) and any \(i \in \mathbb{N}\), the subgroups \((M_1, M_2)^i\) and \((M_2, M_1)^i\) of \(M_1 \cap M_2\) are defined, inductively on \(i\), as follows: \[(M_1, M_2)^1= (M_1 \cap M_2)_{M_1},(M_2, M_1)^1= (M_1 \cap M_2)_{M_2},\] \[(M_1, M_2)^{i+1}= ((M_2, M_1)^i)_{M_1},(M_2, M_1)^{i+1}= (M_1, M_2)^i_{M_2}.\] Here, for \(H \leq G\), \(H_G\) denotes \(\bigcap_{g \in G}gHg^{-1}\). Denote by \(\Pi\) the set of all triples \((G, M_1, M_2)\) such that \(G\) is a finite group, \(M_1\) and \(M_2\) are distinct conjugate maximal subgroups of \(G\), \((M_1)_G= (M_2)_G= 1\), and \(1< |(M_1, M_2)^2| \leq |(M_2, M_1)^2|\). The 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 \(M'_2\). The present paper is a continuation of the investigations by A. S. Kondrat’ev and V. I. Trofimov [Dokl. Math. 59, No. 1, 741–743 (1999; Zbl 0986.20002); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 364, No. 6, 741–743 (1999); Proc. Steklov Inst. Math. 289, S146–S155 (2015; Zbl 1325.20001); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 20, No. 2, 143–152 (2014); Proc. Steklov Inst. Math. 295, S89–S100 (2016; Zbl 1364.20002); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 22, No. 2, 177–187 (2015); Proc. Steklov Inst. Math. 299, S113–S122 (2017; Zbl 1386.20001); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 22, No. 4, 163–172 (2016); Proc. Steklov Inst. Math. 307, S64–S87 (2019; Zbl 1454.20004); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 24, No. 3, 109–132 (2018)] on a description of the set \(\Pi \). It is obtained the description up to equivalence all triples \((G, M_1, M_2)\) from \(\Pi\) in the case whe \(G\) is a finite simple group of Lie type \(G_2\), \(F_4\) or \(E_6\), and \(M_1\) is a parabolic maximal subgroup of \(G\).

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20B15 Primitive groups
20D30 Series and lattices of subgroups
20E28 Maximal subgroups
20F05 Generators, relations, and presentations of groups
20G40 Linear algebraic groups over finite fields
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References:

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