FC-groups with finitely many automorphism orbits

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 $G^{\prime }$ 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 $\omega (G)⩽8$, 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.

arXiv commentSubmitted to an internacional journal

References in corpus (2)

• Finite Groups with 6 or 7 Automorphism Orbits
• On finite groups with few automorphism orbits

Cited by in corpus (2)

• Soluble Groups with few orbits under automorphisms
• Virtually nilpotent groups with finitely many orbits under automorphisms