On the Fitting height and insoluble length of a finite group

Abstract: A classical result of Baer states that an element x of a finite group $G$ is contained in the Fitting subgroup $F(G)$ of $G$ if and only if $x$ is a left Engel element of $G$. That is, $x$ is in $F(G)$ if and only if there exists a positive integer $k$ such that $[g, x, ..., x]$ (with $x$ appearing $k$ times, and using the convention $[x_1, x_2, x_3, \dots, x_k] := [[\dots [[x_1, x_2], x_3], ...], x_k])$ is trivial for all $g$ in $G$. The result was generalised by E. Khukhro and P. Shumyatsky in a 2013 paper via an analysis of the sets $E(G(k))= \{[g, x, ..., x]: g \in G\}$.

In this talk, we will continue to study the properties of these sets, concluding with the proof of two conjectures made in said paper. As a by-product of our methods, we also prove a generalisation of a result of Flavell, which itself generalises Wielandt's Zipper Lemma and provides a characterisation 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 nite group inverted by an involutory automorphism. Joint work with R.M. Guralnick.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

Series comments: Please send a message to andrea.caranti@unitn.it to receive a link to the Zoom room where the conference (a series of talks, actually) takes place.