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 \(g \in G\). The group \(\chi (G)\) is an extension of \(D(G) = [G,G^{\varphi }]\) 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 \(2^{\lfloor \log _2(c)\rfloor } \cdot \exp (G)^{\lfloor \log _2(c)\rfloor +1}\) (\(p=2\)).

弱交换群\(\chi (G)\)由两个同构群G和\(G^{\varphi }\) 产生，服从关系\([g,g^{\varphi }]=1\ )对于所有\(g \in G\)。的组\（\志（G）\）是一个扩展\（d（G）= [G，G ^ {\ varphi}] \）由\（G \乘以G \） 。我们证明如果G是导出长度为d的有限可解群，则\(\exp (D(G))\)除以\(\exp (G)^{d}\)如果 | G | 是奇数且\(\exp (D(G))\)除以\(2^{d-1}\cdot \exp (G)^{d}\)如果 |G | 是均匀的。此外，如果p是素数和G ^是一个p类-基\（P-1 \） ，然后\（\ EXP（d（G））\）划分\（\ EXP（G）\） 。此外，如果G是类\(c\ge 2\)的有限p -群，则\(\exp (D(G))\)除以\(\exp (G)^{\lceil \log _{ p-1}(c+1)\rceil }\) ( \(p\ge 3\) ) 和\(\exp (D(G))\)除以\(2^{\lfloor \log _2(c )\rfloor } \cdot \exp (G)^{\lfloor \log _2(c)\rfloor +1}\) ( \(p=2\) )。