Fitting like subgroups of finite groups

TL;DR

This paper introduces the concept of $\\mathbb{F}$-functorial for finite groups, exploring its properties, lattice structure, and bounds on the generalized Fitting height in subgroup products.

Contribution

It defines $\\mathbb{F}$-functorials, analyzes their lattice structure, and establishes bounds on the generalized Fitting height for subgroup products.

Findings

The set of all $\\mathbb{F}$-functorials forms a complete distributive lattice.

The cardinality of this lattice is continuum.

Sharp bounds on the generalized Fitting height of mutually permutable subgroup products.

Abstract

In this paper the concept of $\mathbb{F}$-functorial of a finite group was introduced. These functorials have many properties of the Fitting subgroup of a soluble group and the generalized Fitting subgroup of a finite group. It was shown that the set of all $\mathbb{F}$-functorials is a complete distributive lattice and the cardinality of this lattice is continuum. The sharp bounds on the generalized Fitting height of mutually permutable product of two subgroups were obtained.