Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

TL;DR

This paper establishes bounds on the Fitting height of finite groups with fixed-point-free automorphisms satisfying polynomial identities, linking algebraic properties of automorphisms to the group's structural complexity.

Contribution

It proves new bounds on the Fitting height of finite groups based on polynomial identities satisfied by automorphisms, extending previous results to more general polynomials.

Findings

Fitting height is bounded by degree of polynomial for primitive polynomials.

Fitting height is bounded by number of irreducible factors for any polynomial in certain groups.

Bounds are stronger than previous bounds based on automorphism order when polynomial degree is small.

Abstract

Let $f(x)$ be a non-zero polynomial with integer coefficients. An automorphism $\varphi$ of a group $G$ is said to satisfy the elementary abelian identity $f(x)$ if the linear transformation induced by $\varphi$ on every characteristic elementary abelian section $S$ of $G$ is annihilated by $f(x)$. We prove that if a finite (soluble) group $G$ admits a fixed-point-free automorphism $\varphi$ satisfying an elementary abelian identity $f(x)$, where $f(x)$ is a primitive polynomial, then the Fitting height of $G$ is bounded in terms of $\operatorname{deg}(f(x))$. We also prove that if $f(x)$ is any non-zero polynomial and $G$ is a $\sigma'$-group for a finite set of primes $\sigma=\sigma(f(x))$ depending only on $f(x)$, then the Fitting height of $G$ is bounded in terms of the number $\operatorname{irr}(f(x))$ of irreducible factors in the decomposition of $f(x)$. These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length $\alpha (|\varphi|)$ of $\langle\varphi\rangle$ when $\operatorname{deg} (f(x))$ or $\operatorname{irr}(f(x))$ is small in comparison with $\alpha (|\varphi|)$.