On the exponent of the Weak commutativity group $\chi (G)$

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 $g\in G$. The group $\chi (G)$ is an extension of $D(G)=[G,G^{\phi }]$ by $G\times G$. We prove that if $G$ is a finite solvable group of derived length $d$, then $\exp (D(G))$ divides $\exp (G{)}^d$ if $|G|$ is odd and $\exp (D(G))$ divides $2^{d-1}\cdot \exp (G{)}^d$ if $|G|$ is even. Further, if $p$ is a prime and $G$ is a $p$-group of class $p-1$, then $\exp (D(G))$ divides $\exp (G)$. Moreover, if $G$ is a finite $p$-group of class $c\ge 2$, then $\exp (D(G))$ divides $\exp (G{)}^{\lceil {\log }_{p-1}(c+1)\rceil }$ ($p\ge 3$) and $\exp (D(G))$ divides $2^{\lfloor {\log }_2(c)\rfloor }\cdot \exp (G{)}^{\lfloor {\log }_2(c)\rfloor +1}$ ($p=2$).