A note on the Wielandt series of a metabelian p-group

Research Report MRR97-023

A note on the Wielandt series of a metabelian p-group

Elizabeth A. Ormerod

Abstract: This paper provides some results about the effect on the Wielandt series of a metabelian p-group when the Wielandt subgroup is either as small as it can be or as large as it can be. For a nilpotent group the Wielandt subgroup is between the centre and second centre of the group. We show that if the Wielandt subgroup is equal to the second centre of the group, or if the second Wielandt subgroup is equal to the second centre of the group, then the remaining terms of the Wielandt series are the same as the appropriate terms of the the upper central series. We also show that if the Wielandt subgroup is equal to the centre of the group, and the Wielandt length of the group is at most p, then the nilpotency class of the group is equal to the Wielandt length. Examples are provided to show that these results are as good as can be expected.

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