Abstract
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.
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Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 839–849 https://doi.org/10.4213/mzm13553.
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Murashka, V.I. On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory. Math Notes 112, 932–939 (2022). https://doi.org/10.1134/S000143462211027X
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DOI: https://doi.org/10.1134/S000143462211027X