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

Tomaszewski, Witold

doi:10.4171/RSMUP/136-3

On laws of the form

a b \equiv b a

equivalent to the abelian law

Tomaszewski, Witold ¹

¹ Silesian University of Technology, GLIWICE, POLAND

Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 19-34

Résumé

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 $a b \equiv b a$ 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.

DOI : 10.4171/RSMUP/136-3

Classification : 20
Mots-clés : Group laws, abelian groups, commutation of elements

Affiliations des auteurs :

Tomaszewski, Witold ¹

¹ Silesian University of Technology, GLIWICE, POLAND

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     author = {Tomaszewski, Witold},
     title = {On laws of the form $ab\equiv ba$ equivalent to the abelian law},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {19--34},
     year = {2016},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {136},
     doi = {10.4171/RSMUP/136-3},
     url = {https://www.numdam.org/articles/10.4171/RSMUP/136-3/}
}

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Tomaszewski, Witold. On laws of the form $ab\equiv ba$ equivalent to the abelian law. Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 19-34. doi: 10.4171/RSMUP/136-3

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