Formations of Finite Groups in Polynomial Time: $\mathfrak{F}$-residuals and $\mathfrak{F}$-subnormality

TL;DR

This paper demonstrates that for many formations of finite groups, key subgroup properties like the $rak{F}$-residual and $rak{F}$-subnormality can be efficiently computed or checked in polynomial time, advancing computational group theory.

Contribution

It establishes polynomial-time algorithms for computing $rak{F}$-residuals and checking $rak{F}$-subnormality in finite groups for a broad class of formations.

Findings

$rak{F}$-residuals can be computed in polynomial time for many formations.

$rak{F}$-subnormality can be checked in polynomial time when $rak{F}$ is hereditary.

The results extend the computational understanding of subgroup properties in finite groups.

Abstract

For a wide family of formations $\mathfrak{F}$ it is proved that the $ \mathfrak{F}$-residual of a permutation finite group can be computed in a polynomial time. Moreover, if in the previous case $\mathfrak{F}$ is hereditary, then an $\mathfrak{F}$-subnormality of a subgroup can be checked in a polynomial time.