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 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 ≅ A 5 × 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. [ABSTRACT FROM AUTHOR]