The Wielandt subgroup of a group G,denoted by w(G),is the intersection of the normalizers of all subnormal subgroups of G. In this paper,the authors show that for a p-group of maximal class G,either wi(G) =ζi(G) for all integer i or wi(G) =ζi+1(G) for every integer i,and w(G/K) =ζ(G/K) for every normal subgroup K in G with K ≠ 1.Meanwhile,a necessary and sufficient condition for a regular p-group of maximal class satisfying w(G) =ζ2(G) is given.Finally,the authors prove that the power automorphism group PAut(G) is an elementary abelian p-group if G is a non-abelian pgroup with elementary ζ(G) ∩ (υ)1(G).