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

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 $k$.