Virtually nilpotent groups with finitely many orbits under automorphisms

TL;DR

This paper characterizes virtually nilpotent groups with finitely many automorphism orbits, showing their structure involves torsion and torsion-free nilpotent components, and provides conditions for their derived subgroup to be nilpotent.

Contribution

It proves a structural decomposition for virtually nilpotent groups with finitely many automorphism orbits, detailing the nature of their torsion and torsion-free parts.

Findings

G = K ⋉ H with H torsion and K torsion-free nilpotent radicable

G' decomposes as D × Tor(G') with D torsion-free nilpotent radicable

If τ(G) is trivial, then G' is nilpotent

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 $\omega(G)$. Let $G$ be a virtually nilpotent group such that $\omega(G)< \infty$. We prove that $G = K \rtimes H$ where $H$ is a torsion subgroup and $K$ is a torsion-free nilpotent radicable characteristic subgroup of $G$. Moreover, we prove that $G^{'}= D \times \Tor(G^{'})$ where $D$ is a torsion-free nilpotent radicable characteristic subgroup. In particular, if the maximum normal torsion subgroup $\tau(G)$ of $G$ is trivial, then $G^{'}$ is nilpotent.