Finite groups admitting a coprime automorphism satisfying an additional polynomial identity

TL;DR

This paper extends classical results on finite groups with coprime automorphisms by establishing bounds on the soluble radical's Fitting height and index when automorphisms satisfy polynomial identities, with explicit bounds especially for small degrees.

Contribution

It generalizes known bounds for automorphisms of coprime order to cases where automorphisms satisfy polynomial identities, providing explicit bounds based on polynomial degree.

Findings

Bound on Fitting height of the soluble radical

Bound on the index of the soluble radical

Explicit bounds for small polynomial degrees

Abstract

It is known that a finite group with an automorphism $\varphi$ of coprime order has a soluble radical of $(|\varphi|,|C_G(\varphi)|)$-bounded Fitting height and index. We extend this classic result as follows. Let $f(x) = a_0 + a_1 \cdot x + \cdots + a_d \cdot x^d \in \mathbb{Z}[x]$ be a primitive polynomial and let $G$ be a finite group with an automorphism $\varphi$ of coprime order satisfying $ g^{a_0} \cdot \varphi(g)^{a_1} \cdots \varphi^d(g)^{a_d} = 1 $, for all $g \in G$. Then the soluble radical of $G$ has $(d,|C_G(\varphi)|)$-boundex Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree $d$.