On the generalized Fitting height and insoluble length of finite groups

TL;DR

This paper proves two conjectures related to the Fitting height and insoluble length of finite groups, generalizes a subgroup characterization, and explores implications for groups with involutory automorphisms.

Contribution

It confirms conjectures of Khukhro and Shumyatsky and generalizes Flavell's result, extending understanding of subgroup structures and group automorphisms.

Findings

Proved conjectures on Fitting height and insoluble length

Generalized a subgroup characterization related to maximal subgroups

Derived applications to groups with involutory automorphisms

Abstract

We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by-product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes Wielandt's Zipper Lemma and provides a characterization of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a finite group inverted by an involutory automorphism.