Elizabeth A. Ormerod

The Wielandt subgroup of metacyclic p-groups

The Wielandt subgroup is the intersection of the normalisers of all the subnormal subgroups of a group. For a finite group it is a non-trivial characteristic subgroup, and this makes it possible to define an ascending normal series terminating at the group. This series is called the Wielandt series and its length determines the Wielandt length of the group. In this paper the Wielandt subgroup of a metacyclic p –group is identified, and using this information it is shown that if a metacyclic p –group has Wielandt length n , its nilpotency class is n or n + 1.

A note on p -groups with power automorphisms

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 ω 1 ( G ) is defined by: ω 1 ( G ) = ω( G ) and for i ≥ 1, ω i+1 ( G )/ ω( G ) = ω( G /ω i , ( G )). The subgroups of the upper central series we denote by ζ i ( G ).