# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 136, 2016, pp. 19–34**

**DOI: 10.4171/RSMUP/136-3**

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

Witold Tomaszewski^{[1]}(1) Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100, Gliwice, Poland

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.

*Keywords: *Group laws, abelian groups, commutation of elements

Tomaszewski Witold: On laws of the form $ab\equiv ba$ equivalent to the abelian law. *Rend. Sem. Mat. Univ. Padova* 136 (2016), 19-34. doi: 10.4171/RSMUP/136-3