Abstract
We give an example of a series of finite \( D_{\pi} \)-groups where for each group \( G \) in the series and its \( \pi \)-Hall subgroup \( H \), the inequality \( H\cap H^{x}\cap H^{y}\neq 1 \) holds for all \( x,y\in G \). Thus a negative answer is obtained both to Problem 7.3 by Vdovin and Revin and its analog—Question 18.31 in The Kourovka Notebook. We also describe the subgroups \( \operatorname{Min}_{G}(H,H,H) \) and \( \min_{G}(H,H,H) \).
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Funding
The author was supported by the Russian Foundation for Basic Research (Grant no. 20–01–00456) and the Russian Academic Excellence Project (Agreement 02.A03.210006 of 27.08.2013).
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 866–869. https://doi.org/10.33048/smzh.2022.63.412
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Zenkov, V.I. On Intersections of \( \pi \)-Hall Subgroups in Finite \( D_{\pi} \)-Groups. Sib Math J 63, 720–722 (2022). https://doi.org/10.1134/S0037446622040127
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DOI: https://doi.org/10.1134/S0037446622040127