Certain locally nilpotent varieties of groups

Abstract

Let $c\ge 0$, $d\ge 2$ be integers and ${\mathcal{N}}_c^{(d)}$ be the variety of groups in which every $d$-generator subgroup is nilpotent of class at most $c$. N.D. Gupta posed this question that for what values of $c$ and $d$ it is true that ${\mathcal{N}}_c^{(d)}$ is locally nilpotent? We prove that if $c\le 2^d+2^{d-1}-3$ then the variety ${\mathcal{N}}_c^{(d)}$ is locally nilpotent and we reduce the question of Gupta about the periodic groups in ${\mathcal{N}}_c^{(d)}$ to the prime power finite exponent groups in this variety.

4 pages, to appear in Bull. Austral. Math. Soc