The Wielandt series of metabelian groups

The Wielandt subgroup of a group is the intersection of the normalisers of its subnormal subgroups. It is non-trivial in any finite group and thus gives rise to a series whose length provides a measure of the complexity of the group's subnormal structure. In this paper results of Ormerod concerning the interplay between the Wielandt series and upper central series of metabelian p-groups, p odd, are extended to the class of all odd order metabelian groups. These extensions are formulated in terms of a natural generalisation of the upper central series which arises from Casolo's strong Wielandt subgroup, the intersection of the centralisers of a group's nilpotent subnormal sections.