On Tamura’s identity yx=f(x,y) in groups

Abstract

The purpose of this article is to study the groups satisfying the law $yx=x^{a_1}{y}^{b_1}{x}^{a_2}{y}^{b_2}$, where a_i and b_i are positive integers. Tamura posed a problem as to when such a law implies that a group is abelian. We show that, apart from obvious reasons why such a variety should contain non-abelian groups, every elementary amenable or residually finite group satisfying such a law has to be abelian.

Keywords:

• Commutativity
• group identity
• periodic groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E10
• 20F50

View correction statement:

Erratum to: On Tamura’s identity yx = f(x, y) in groups

Additional information

Funding

The author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0222, and projects No. J1-8132, J1-7256 and N1-0061).