FC-groups with finitely many automorphism orbits

TL;DR

This paper characterizes the structure of FC-groups with finitely many automorphism orbits, showing finiteness of the derived subgroup and a specific decomposition, and classifies infinite FC-groups with up to eleven orbits.

Contribution

It provides new structural results for FC-groups with finitely many automorphism orbits, including decomposition and classification of certain infinite cases.

Findings

Derived subgroup of such FC-groups is finite.

FC-groups decompose into torsion and divisible parts.

Infinite FC-groups with ≤8 orbits are either soluble or a product involving A_5.

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 $\omega(G)$. In this paper we prove that if $G$ is an FC-group with finitely many automorphism orbits, then the derived subgroup $G'$ is finite and $G$ admits a decomposition $G = Tor(G) \times D$, where $Tor(G)$ is the torsion subgroup of $G$ and $D$ is a divisible characteristic subgroup of $Z(G)$. We also show that if $G$ is an infinite FC-group with $\omega(G) \leqslant 8$, then either $G$ is soluble or $G \cong A_5 \times H$, where $H$ is an infinite abelian group with $\omega(H)=2$. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits.