# Soluble Groups with few orbits under automorphisms

## TL;DR

This paper investigates soluble groups with finitely many automorphism orbits, showing they decompose into a torsion-free nilpotent part and a finite group, and classifies certain groups with exactly three automorphism orbits.

## Contribution

It establishes a structural decomposition for soluble groups with finite automorphism orbits and classifies those with three orbits, advancing understanding of automorphism actions.

## Key findings

- Existence of a torsion-free characteristic nilpotent subgroup in such groups
- Decomposition of the group as a semidirect product with a finite group
- Classification of soluble groups with exactly three automorphism orbits

## Abstract

Let $G$ be a group. The orbits of the natural action of Aut$(G)$ on $G$ are called ``automorphism orbits'' of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. We prove that if $G$ is a soluble group with finite rank such that $\omega(G)< \infty$, then $G$ contains a torsion-free characteristic nilpotent subgroup $K$ such that $G = K \rtimes H$, where $H$ is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that $\omega(G)=3$.