Abstract
Earlier, the author described up to conjugacy all pairs \((A,B)\) of nilpotent subgroups of a finite group \(G\) with socle \(L_{2}(q)\) for which \(A\cap B^{g}\neq 1\) for any element of \(G\). A similar description was obtained by the author later for primary subgroups \(A\) and \(B\) of a finite group \(G\) with socle \(L_{n}(2^{m})\). In this paper, we describe up to conjugacy all pairs \((A,B)\) of nilpotent subgroups of a finite group \(G\) with simple socle from the “Atlas of Finite Groups” for which \(A\cap B^{g}\neq 1\) for any element \(g\) of \(G\). The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group \(G\) for any nilpotent subgroups \(N\) contains an element \(g\) such that \(N\cap N^{g}=1\).
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This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00456).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 29, No. 2, pp. 54 - 66, 2023 https://doi.org/10.21538/0134-4889-2023-29-2-54-66.
Translated by E. Vasil’eva
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Zenkov, V.I. On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups”. Proc. Steklov Inst. Math. 323 (Suppl 1), S321–S332 (2023). https://doi.org/10.1134/S0081543823060251
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DOI: https://doi.org/10.1134/S0081543823060251