Let $G$ be a group and consider the group $\chi(G)$ obtained from the free product $G \ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. In the last 44 years, this group has been a formidable tool for obtaining finiteness conditions in Group Theory. In this talk, we want to present some important results related to the group $\chi (G)$. Moreover, we want to establish some properties of the exponent of $\chi(G)$ when $G$ has finite exponent.