Generating groups for nilpotent varieties
- 1 February 1970
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of the Australian Mathematical Society
- Vol. 11 (1) , 28-32
- https://doi.org/10.1017/s1446788700005929
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.Keywords
This publication has 3 references indexed in Scilit:
- Generating groups of nilpotent varietiesBulletin of the American Mathematical Society, 1968
- Varieties of GroupsPublished by Springer Nature ,1967
- On varieties generated by a finitely generated groupMathematische Zeitschrift, 1964