FC-groups with finitely many automorphism orbits

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). 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)×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 ω(G)⩽8, then either G is soluble or G≅A5×H, where H is an infinite abelian group with ω(H)=2. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits.