On the Wielandt Subgroup in a p-Group of Maximal Class

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 suflcient 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) ∩ 1(G).