Virtually nilpotent groups with finitely many orbits under automorphisms

Abstract

Let G be a group. The orbits of the natural action of \({{\,\mathrm{Aut}\,}}(G)\) on G are called automorphism orbits of G, and the number of automorphism orbits of G is denoted by \(\omega (G)\). Let G be a virtually nilpotent group such that \(\omega (G)< \infty \). 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 \(G^{'}= D \times {{\,\mathrm{Tor}\,}}(G^{'})\) where D is a torsion-free nilpotent radicable characteristic subgroup. In particular, if the maximum normal torsion subgroup \(\tau (G)\) of G is trivial, then \(G^{'}\) is nilpotent.