MATHEMATICS

On $\mathfrak{H}_\sigma^\tau$-critical formations of finite groups

I. N. Safonova, V. V. Skrundz

Belarusian State University, Minsk

Abstract: We study $\mathfrak{H}_\sigma^\tau$-critical formations or, in other words, minimal $\tau$-closed $\sigma$-local non-$\mathfrak{H}$-formations of finite groups, where $\mathfrak{H}$ is some class of groups, $\sigma$ is a partition of the set of all prime numbers $\mathbb{P}$, and $\tau$ is a subgroup functor. A criterion for minimal $\tau$-closed $\sigma$-local formations is obtained. A description of $\sigma$-local formations of this type is given for the formations of all $\Pi$-groups, all $\sigma$-soluble and all $\sigma$-nilpotent $\Pi$-groups, where $\varnothing\ne\Pi\subseteq\sigma$. In particular, a description of minimal $\tau$-closed $\sigma$-local non-$\sigma$-soluble and non-$\sigma$-nilpotent formations of finite groups is obtained.

Keywords: finite group, formation $\sigma$-function, $\sigma$-local formation, subgroup functor, critical functor-closed $\sigma$-local formation, $\sigma$-soluble group.

UDC: 512.542

Received: 23.06.2025

DOI: 10.54341/20778708_2025_3_64_99