A test for a local formation of finite groups to be a formation of soluble groups with the Shemetkov property

TL;DR

This paper investigates conditions under which local formations of finite groups are soluble with the Shemetkov property, providing a polynomial-time check for such formations based on minimal non-formation groups.

Contribution

It introduces a polynomial-time method to verify if a local formation of finite groups with bounded prime set is soluble with the Shemetkov property, extending previous soluble-only assumptions.

Findings

Provides a polynomial-time check for local formations with bounded prime sets.

Extends known solutions to include non-soluble minimal non-formation groups.

Offers criteria to identify formations with the Shemetkov property.

Abstract

L.A. Shemetkov posed a Problem 9.74 in Kourovka Notebook to find all local formations $\mathfrak{F}$ of finite groups such that every finite minimal non-$\mathfrak{F}$-group is either a Schmidt group or a group of prime order. All known solutions to this problem are obtained under the assumption that every minimal non-$\mathfrak{F}$-group is soluble. Using the above mentioned solutions we present a polynomial in $n$ time check for a local formation $\mathfrak{F}$ with bounded $\pi(\mathfrak{F})$ to be a formation of soluble groups with the Shemtkov property where $n=\max \pi(\mathfrak{F})$.