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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MZM_2022_112_6_a4,
     author = {V. I. Murashka},
     title = {On {Questions} {Posed} by {Shemetkov,} {Ballester-Bolinches,} and {Perez-Ramos} in {Finite} {Group} {Theory}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {839--849},
     publisher = {mathdoc},
     volume = {112},
     number = {6},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a4/}
}
TY  - JOUR
AU  - V. I. Murashka
TI  - On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory
JO  - Matematičeskie zametki
PY  - 2022
SP  - 839
EP  - 849
VL  - 112
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a4/
LA  - ru
ID  - MZM_2022_112_6_a4
ER  - 
%0 Journal Article
%A V. I. Murashka
%T On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory
%J Matematičeskie zametki
%D 2022
%P 839-849
%V 112
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a4/
%G ru
%F MZM_2022_112_6_a4
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/