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19.87 (2018)

Solved

Suppose that for every Sylow subgroup $P$ of a finite group $G$ and every maximal subgroup $V$ of $P$ there is a $\sigma$-soluble subgroup $T$ such that $VT = G$. Is it true that then $G$ is $\sigma$-soluble?

Contributor: A. N. Skiba

Progress

Yes, it is true (A.-M. Liu, W. Guo, I. N. Safonova, A. N. Skiba, J. Algebra, 585 (2021), 280–293); another solution using CFSG is in (S. F. Kamornikov, V. N. Tyutyanov, Trudy Inst. Mat. Mekh. UrO RAN, 27, no. 1 (2021), 98–102 (Russian); X. Yi, S. F. Kamornikov, V. N. Tyutyanov, Probl. Physics, Math. Techn., 46, no. 1 (2021), 50–53).

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