On laws of the form $ab\equiv ba$ equivalent to the abelian law

Abstract

N.D. Gupta has proved that groups which satisfy the laws $[x,y]\equiv [x{,}_{n}y]$ for $n=2,3$ are abelian. Every law $[x,y]\equiv [x{,}_{n}y]$ can be written in the form $ab\equiv ba$ where $a,b$ belong to a free group $F_2$ of rank two, and the normal closure of $⟨a,b⟩$ coincides with $F_2$. In this work we investigate laws of this form. In particular, we discuss certain classes of laws and show that the metabelian groups which satisfy them are abelian.