Finiteness conditions for the weak commutativity group and related constructions

Abstract

The weak commutativity group $\chi (G)$ is generated by two isomorphic groups G and $G^{\phi }$ subject to the relations $[g,g^{\phi }]=1$ for all $g\in G$. We establish a finiteness criterion for the subgroups of $\chi (G)$ in terms of the set $T_{\chi }(G)=\{[g_1,g_2^{\phi }]|g_i\in G\}$. We also obtain sufficient conditions under which some specific (local) classes of groups are invariant under the operator χ. Moreover, we prove that if G is a locally finite group with $\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}(G)=n$, then $\chi (G)$ is locally finite and has finite n-bounded exponent.

Communicated by Mark Lewis

KEYWORDS:

• Finiteness conditions
• local properties
• non-abelian tensor square
• weak commutativity

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E25
• 20E30
• 20E34
• 20J99

Acknowledgments

We are thankful to the referee for his/her helpful comments, which improved the final version of this paper.

Additional information

Funding

This work was partially supported by FAPDF/Brazil and “Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq”.