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Let G be a group,and let α be a regular automorphism of order p2 of G,where p is a prime.If G is polycyclic-by-finite and the map ψ:G → G defined by gψ =[g,α] is surjective,then G is soluble.If G is polycyclic,then CG(αp) and G/[G,αp] are both nilpotent-by-finite.