Some p-groups of Weilandt length three

Abstract

Finite p-groups of Wielandt length 1 are groups in which every subgroup is normal and are Dedekind groups. When the prime is odd therefore a finite p-group of Wielandt length 1 is Abelian. For an odd prime, a finite p-group of Wielandt length 2 has nilpotency class at most 3 and for such a goup to have class 3 there must be a 2-generator subgroup of this class. In this paper it is shown that for any prime p > 3 a finite p-group of Wielandt length 3 has nilpotency class at most 4, and for such a group to have class 4 there must be a 2-generator subgroup with this class. Two families of p-groups of Wielandt length 3 are described. One is a family of 3-generator groups with the property that each group modulo its Wielandt subgroup has class 2, the other is a family of 2-generator groups with the property that each group modulo its Wielandt subgroup has class 3.