The nilpotency of finite groups with a fix-point-free automorphism satisfying an identity

TL;DR

This paper extends the Frobenius conjecture by analyzing finite groups with fix-point-free automorphisms satisfying polynomial identities, showing such groups are solvable with a specific structural decomposition.

Contribution

It provides a new structural classification of finite groups with automorphisms satisfying polynomial identities, generalizing previous results and explicitly describing their solvable structure.

Findings

Groups are solvable and have a specific decomposed form

Existence of explicit invariants related to polynomial identities

Groups are constrained by the polynomial's properties and automorphism conditions

Abstract

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial $r(t) = a_0 + a_1 \cdot t + \cdots + a_d \cdot t^d \in \mathbb{Z}[t]$ that is irreducible over $\mathbb{Q}$, there exist (explicit) invariants $a,b,c \in \mathbb{N}$ with the following property. Consider a finite group with a fix-point-free automorphism ${\alpha}:{G}\longrightarrow{G}$ and suppose that for all $x \in G$ we have the equality $x^{a_0} \cdot \alpha(x^{a_1}) \cdot \alpha^2(x^{a_2})\cdots \alpha^d(x^{a_d}) = 1_G.$ Then $G$ is solvable and of the form $A \cdot (B \rtimes (C \times D))$, where $A$ is an $a$-group, $B$ is a $b$-group, $C$ is a nilpotent $c$-group, and $D$ is a nilpotent group of class at most $d^{2^d}$. Here, a group $H$ is said to be an $a$-group (resp. $b$-group or $c$-group) if the order of every $h \in H$ divides some natural power of $a$ (resp. $b$ or $c$).