Structural Results On The Weak Commutativity Construction

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 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 $\chi (G)$.