On the wielandt subgroup in a p-group of maximal class

Abstract

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 w i (G) = ζ i (G) for all integer i or w i (G) = ζ i+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) = ζ 2(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 p-group with elementary $$\zeta (G) \cap \mho _1 (G)$$ .