Let $c\geq 0$, $d\geq 2$ 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 $c\leq 2^d+2^{ d-1}-3$ 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.
Categories: math.GR