Geodesic
On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory
Matematičeskie zametki, Tome 112 (2022) no. 6, pp. 839-849.

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A chief factor $H/K$ of a group $G$ is said to be $\mathfrak{F}$-central if $(H/K)\rtimes (G/C_G(H/K))\in\mathfrak{F}$. In 1997, Shemetkov posed the problem of describing finite group formations $\mathfrak{F}$ such that $\mathfrak{F}$ coincides with the class of groups for which all chief factors are $\mathfrak{F}$-central. We refer to such formations as centrally saturated. We prove that the centrally saturated formations form a complete distributive lattice. As an answer to a question posed by Ballester-Bolinches and Perez-Ramos, conditions for a centrally saturated formation to be saturated and solvably saturated in the class of all groups are found. As a consequence, a criterion for hereditary Fitting formations to be solvably saturated is obtained.
Keywords: finite group, saturated formation, solvably saturated formation, centrally saturated formation, $\mathfrak{F}$-hypercenter, distributive lattice. 
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V. I. Murashka. On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory. Matematičeskie zametki, Tome 112 (2022) no. 6, pp. 839-849. http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a4/

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