Virtually nilpotent groups with finitely many orbits under automorphisms

TL;DR

This work investigates finite automorphism-orbit numbers $\omega(G)$ for virtually nilpotent groups, showing that $\omega(G)<\infty$ forces a rigid semidirect-product structure $G=K\rtimes H$ with $K$ torsion-free nilpotent radicable and $H$ torsion, and that the derived subgroup splits as $G^{\prime}=D\times\Tor(G^{\prime})$ with $D$ torsion-free nilpotent radicable. The authors develop a framework linking $\omega$-behavior under automorphisms to module-theoretic decompositions, prove a criterion for finite $\omega$ in semidirect products, and derive a main theorem describing the precise decomposition and the nilpotence of $G^{\prime}$ when $\tau(G)=1$, with broader implications for virtually soluble cases and corollaries for finite-rank groups. The results advance the understanding of how automorphism orbits constrain the internal structure of virtually nilpotent groups and provide tools for classifying groups by their automorphism-action signatures.

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 $ω(G)$. Let $G$ be a virtually nilpotent group such that $ω(G)< \infty$. 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 $τ(G)$ of $G$ is trivial, then $G^{'}$ is nilpotent.

Theorems & Definitions (10)

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• proof : Proof of Theorem \ref{['pro1']}
• proof : Proof of Corollary \ref{['pro2']}
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• proof : Proof of Theorem \ref{['thm:linear']}