Regular saturated formations of finite soluble groups
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Viachaslau I. Murashka
Abstract
For a formation 𝔉 of finite groups, consider the graph whose vertices are elements of a finite group, with two vertices connected by an edge if and only if they generate a non-𝔉-group as elements of the group. A hereditary formation 𝔉 is said to be regular if the set of all isolated vertices of the described graph coincides with the intersection of all 𝔉-maximal subgroups in every group. In this paper, a constructive description of saturated regular formations of soluble groups is given, which improves the results of Lucchini and Nemmi. In particular, it is shown that saturated regular non-empty formations of soluble groups are just hereditary formations 𝔉 of soluble groups that contain every group whose cyclic primary subgroups are all 𝐾-𝔉-subnormal. Also, we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz lattice.
Funding source: Belarusian Republican Foundation for Fundamental Research
Award Identifier / Grant number: F23RNFM-63
Funding statement: This work was financially supported by the Belarussian Republican Foundation for Fundamental Research (BRFFR-RSF M, project F23RNFM-63).
Acknowledgements
The author is very grateful to the reviewer for the helpful suggestions.
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Communicated by: Andrea Lucchini
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