Abstract
A subgroup \( A \) of a group \( G \) is \( G \)-permutable in \( G \) if for every subgroup \( B\leq G \) there exists \( x\in G \) such that \( AB^{x}=B^{x}A \). A subgroup \( A \) is hereditarily \( G \)-permutable in \( G \) if \( A \) is \( E \)-permutable in every subgroup \( E \) of \( G \) which includes \( A \). The Kourovka Notebook has Problem 17.112(b): Which finite nonabelian simple groups \( G \) possess a proper hereditarily \( G \)-permutable subgroup? We answer this question for the exceptional groups of Lie type. Moreover, for the Suzuki groups \( G\cong{{}^{2}\!\operatorname{B}_{2}}(q) \) we prove that a proper subgroup of \( G \) is \( G \)-permutable if and only if the order of the subgroup is 2. In particular, we obtain an infinite series of groups with \( G \)-permutable subgroups.
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Funding
The work was supported by a joint grant of the Belorussian Republican Foundation for Fundamental Research (Project F23RSF-237) and the Russian Science Foundation No. 23-41-10003, https://rscf.ru/en/project/23-41-10003/.
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 5, pp. 935–945. https://doi.org/10.33048/smzh.2023.64.504
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Galt, A.A., Tyutyanov, V.N. On the Existence of Hereditarily \( G \)-Permutable Subgroups in Exceptional Groups \( G \) of Lie Type. Sib Math J 64, 1110–1116 (2023). https://doi.org/10.1134/S003744662305004X
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DOI: https://doi.org/10.1134/S003744662305004X