Generating groups for nilpotent varieties

Abstract

Let ℜc denote the variety of all nilpotent groups of class ≦ c, that is, ℜc is the class of all groups satisfying the law, where we define, as usual, and, inductively, . Further, let Fk(ℜc) denote a free group of ℜe of rank k. In her book Hanna Neumann ([4], Problem 14) poses the following problem: Determine d(c), the least k such that Fk(ℜc) generates ℜc. Further, she suggests, incorrectly, that d(c) = [c/2] + l. However, as we shall prove here, the correct answer is d(c) = c—1, for c ≦ 3. 2 More generally, we shall prove the following result.