Polycyclic group admitting an almost regular automorphism of prime order

A well-known result due to Thompson states that if a finite group G has a fixed-point-free automorphism of prime order, then G is nilpotent. In this note, giving a counterpart of Thompson's result in the context of polycyclic groups, we prove: if a polycyclic group G has an automorphism of prime order with finitely many fixed points, then G is nilpotent-by-finite.