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Proceedings of the Institute of Mathematics of the NAS of Belarus

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On critical σ-local formations of finite groups

Abstract

In this article we study minimal $\sigma$-local non-$\mathfrak H$-formations of finite groups (or, in other words, $\mathfrak H_\sigma$-critical formations), i. e. such $\sigma$-local formations not included in the class of groups $\mathfrak H$, all of whose proper $\sigma$-local subformations are contained in $\mathfrak H$. A description of minimal $\sigma$-local non$\mathfrak H$-formations for an arbitrary $\sigma$-local formation $\mathfrak H$ of classical type is obtained (а $\sigma$-local formation is called a $\sigma$-local formation of classical type if it has a $\sigma$-local definition such that all its non-Abelian values are $\sigma$-local). The main result of the work in the class of $\sigma$-local formations solves the problem of L. A. Shemetkov (1980) on the description of critical formations for given classes of finite groups. As corollaries, descriptions of $\mathfrak H_\sigma$-critical formations are given for a number of specific classes of finite groups, such as the classes of all $\sigma$-nilpotent, meta-$\sigma$-nilpotent groups, as well as the class all groups with $\sigma$-nilpotent commutator subgroup.

For citations:


Safonova I.N. On critical σ-local formations of finite groups. Trudy Instituta matematiki. 2023;31(2):63-80. (In Russ.)

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ISSN 1812-5093 (Print)