### Witold Tomaszewski

Silesian University of Technology, Gliwice, Poland

## Abstract

N.D. Gupta has proved that groups which satisfy the laws $[x,y]\equiv [x,_ny]$ for $n=2,3$ are abelian. Every law $[x,y]\equiv [x,_ny]$ 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 $\langle a,b \rangle$ 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.

## Cite this article

Witold Tomaszewski, On laws of the form $ab\equiv ba$ equivalent to the abelian law. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 19–34

DOI 10.4171/RSMUP/136-3