Structural Results On The Weak Commutativity Construction

TL;DR

This paper investigates the algebraic structure of the weak commutativity group, providing new formulas for its derived and lower central series, and establishing bounds on the exponents of certain sections.

Contribution

It introduces novel expressions for the derived and lower central series of the weak commutativity group and derives new bounds for the exponents of some of its sections.

Findings

New expressions for the derived series of χ(G)

New expressions for the lower central series of χ(G)

Bounds for the exponents of some sections of χ(G)

Abstract

The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. We obtain new expressions for the terms of the derived series and the lower central series of $\chi(G)$. We also present new bounds for the exponent of some sections of $\chi(G)$.