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This article is cited in 1 scientific paper (total in 1 paper)
On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory
V. I. Murashka Gomel State University named after Francisk Skorina
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.
Keywords:
finite group, saturated formation, solvably saturated formation, centrally
saturated formation,
$\mathfrak{F}$-hypercenter, distributive lattice.
Received: 18.04.2022 Revised: 20.07.2022
Citation:
V. I. Murashka, “On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory”, Mat. Zametki, 112:6 (2022), 839–849; Math. Notes, 112:6 (2022), 932–939
Linking options:
https://www.mathnet.ru/eng/mzm13553https://doi.org/10.4213/mzm13553 https://www.mathnet.ru/eng/mzm/v112/i6/p839
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Abstract page: | 196 | Full-text PDF : | 31 | References: | 54 | First page: | 6 |
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