Formations of Finite Groups in Polynomial Time: the $\mathfrak{F}$-Radical

TL;DR

This paper presents polynomial time algorithms for computing the $rak{F}$-radical of permutation groups, especially for primitive saturated formations of soluble groups, advancing computational group theory.

Contribution

It introduces efficient algorithms for calculating the $rak{F}$-radical in permutation groups for specific formations, expanding computational methods in group theory.

Findings

Polynomial time algorithm for $rak{F}$-radical computation

Algorithms for various group length measures

Application to primitive saturated formations of soluble groups

Abstract

For a Baer-local (composition) Fitting formation $\mathfrak{F}$ the polynomial time algorithm for the computation of the $\mathfrak{F}$-radical of a permutation group is suggested. In particular it is showed how one can compute the $\mathfrak{F}$-radical in case when $\mathfrak{F}$ is a primitive saturated formation of soluble groups. Moreover, the polynomial time algorithms for the computation of different lengthes associated with a group are presented.