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Finite groups in which the normalizers of pairwise intersections of Sylow 2-subgroups have odd indices. (English. Russian original) Zbl 1238.20017

Proc. Steklov Inst. Math. 259, Suppl. 2, S163-S177 (2007); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 13, No. 2, 90-103 (2007).
From the introduction: V. V. Kabanov, A. A. Makhnev, and A. I. Starostin [Algebra Logika 15, 655-659 (1976; Zbl 0369.20010)] proved that if in a finite group the intersection of two Sylow 2-subgroups is normal in at least one of them, then it is normal in both subgroups; from this they derived a description of the structure of finite groups with this property. In the Kourovka Notebook [2002; Zbl 0999.20001], they posed Problem 5.14(v) of describing finite groups in which the normalizer of the intersection of any two Sylow 2-subgroups has odd index. For the sake of brevity, let us call such groups IN-groups.
In the present paper, the authors describe all semisimple (i.e., with a trivial solvable radical) IN-groups and thereby solve Problem 5.14(v). The following theorem is proved. Theorem. A nontrivial group \(G\) is a semisimple IN-group if and only if \(K_1'\times\cdots\times K_n'\leq O^{2'}(G)\leq K_1\times\cdots\times K_n\), where each of the groups \(K_1,\dots,K_n\) is isomorphic to one of the following groups: \(L_2(q)\) for \(3<q\not\equiv\pm 1\pmod{16}\), \(L_3(2^m)\), \(U_3(2^m)\) with \(m>1\), \(\text{PSp}_4(2^m)\) with \(m>1\), \(^2B_2(q)\) with \(q>2\), \(^2G_2(q)\) with \(q>3\), \(U_3(3)\), \(G_2(3)\), \(A_7\), \(J_1\), \(S_7\), and the extension of the group \(L_2(q^2)\) with \(q\equiv\pm 3\pmod 8\) by the group of field automorphisms of order 2.

MSC:

20D05 Finite simple groups and their classification
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D06 Simple groups: alternating groups and groups of Lie type
20D25 Special subgroups (Frattini, Fitting, etc.)
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