| Popis: |
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 $$2^{\lfloor \log _2(c)\rfloor } \cdot \exp (G)^{\lfloor \log _2(c)\rfloor +1}$$ ( $$p=2$$ ). |
