A note on p-groups with power automorphisms

Abstract

Let G be a group. The norm, or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G. In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω(G). The Wielandt series of subgroups ω1(G) is defined by: ω1(G) = ω(G) and for i ≥ 1, ωi+1(G)/ ω(G) = ω(G/ωi, (G)). The subgroups of the upper central series we denote by ζi(G).