Geodesic
On two problems from “The Kourovka Notebook”
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 98-102
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We solve Problems 19.87 and 19.88 formulated by A.N. Skiba in “The Kourovka Notebook.” It is proved that if, for every Sylow subgroup $P$ of a finite group $G$ and every maximal subgroup $V$ of $P$, there is a $\sigma$-soluble ($\sigma$-nilpotent) subgroup $T$ such that $VT=G$, then $G$ is $\sigma$-soluble ($\sigma$-nilpotent, respectively).
Keywords: finite group, $\sigma$-nilpotent group, partition of the set of all prime numbers, Sylow subgroup, maximal subgroup.
Mots-clés : $\sigma$-soluble group 
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S. F. Kamornikov; V. N. Tyutyanov. On two problems from “The Kourovka Notebook”. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 98-102. http://geodesic.mathdoc.fr/item/TIMM_2021_27_1_a9/

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[2] Skiba A.N., “On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups”, J. Algebra, 436 (2015), 1–16 | DOI | MR | Zbl

[3] Guralnick R.M., “Subgroups of prime power index in a simple group”, J. Algebra, 81:2 (1983), 304–311 | DOI | MR | Zbl

[4] Doerk K., Hawkes T., Finite soluble groups, Walter de Gruyter, Berlin; N Y, 1992, 891 pp. | MR