Virtually nilpotent groups with finitely many 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 $\omega (G)$. Let $G$ be a virtually nilpotent group such that $\omega (G)<\infty$. We prove that $G=K⋊H$ 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 $G^{{}^{\prime }}$ is nilpotent.

References in corpus (1)

• Soluble Groups with few orbits under automorphisms