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

TL;DR

This paper presents polynomial-time algorithms for computing the $rak{F}$-hypercenter in finite groups for a broad class of formations, including several well-known group classes, advancing computational group theory.

Contribution

It introduces new polynomial-time algorithms for the $rak{F}$-hypercenter across various formations, expanding computational methods in group theory.

Findings

Algorithms for $rak{F}$-hypercenter computation are effective for multiple group classes.

Polynomial time complexity is achieved for broad family of formations.

New methods for intersection of maximal $rak{F}$-subgroups are proposed.

Abstract

For a wide family of formations $\mathfrak{F}$ (which includes Baer-local formations) it is proved that the $ \mathfrak{F}$-hypercenter of a permutation finite group can be computed in polynomial time. In particular, the algorithms for computing the $\mathfrak{F}$-hypercenter for the following classes of groups are suggested: hereditary local formations with the Shemetkov property, rank formations, formations of all quasinilpotent, Sylow tower, $p$-nilpotent, supersoluble, $w$-supersoluble and $SC$-groups. For some of these formations algorithms for the computation of the intersection of all maximal $\mathfrak{F}$-subgroups are suggested.