On the centralizer and the commutator subgroup of an automorphism

Let φ be an automorphism of a group G. In this paper, we study the influence of its centralizer ({C_G(varphi)}) on its commutator subgroup ({[G,varphi]}) when G is polycyclic or metabelian. For instance, when G is metabelian and φ fixed-point-free of prime order p, we prove that ({[G,varphi]}) is nilpotent of class ≤ p. Also, when G is polycyclic and φ of order 2, we show that if ({C_G(varphi)}) is finite, then so are ({G/[G,varphi]}) and ({[G,varphi]'}) .