Soluble Groups with few orbits under automorphisms

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 $G=K⋊H$, where $H$ is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that $\omega (G)=3$.

Submitted to an international journal Geometriae Dedicata (2020)