The Fitting height of finite groups with a fixed-point-free automorphism satisfying an identity

TL;DR

This paper explores the relationship between fixed-point-free automorphisms satisfying polynomial identities and the Fitting height of finite groups, extending classical theorems and proposing a conjecture with partial confirmations.

Contribution

It formulates a conjecture linking polynomial identities of automorphisms to the Fitting height and confirms it for a broad class of polynomials with explicit constants.

Findings

Confirmed the conjecture for a large family of polynomials.

Established bounds on Fitting height based on polynomial factors.

Extended classical results of Thompson and Berger.

Abstract

Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial $f(x) = a_0 + a_1 x + \cdots + a_d x^d \in \mathbb{Z}[x] $, there is an integer $k > 0$ with the following property. Let $G$ be a finite (solvable) group with a fixed-point-free automorphism $\alpha$ satisfying $\gcd(|G|,k)= 1$ and $$\{ g^{a_0} \cdot \alpha(g)^{a_1} \cdot \alpha^2(g)^{a_2} \cdots \alpha^d(g)^{a_d} | g \in G \} = \{1\}.$$ Then the Fitting height of $G$ is at most the number of irreducible factors of $f(x)$. We confirm the conjecture for a large family of polynomials with explicit constants $k$.