论文标题
在弱通勤组$χ（g）$的指数上
On the exponent of the Weak commutativity group $χ(G)$
论文作者
论文摘要
弱的通勤组$χ（g）$由两个同构组$ g $和$ g^{φ} $生成。组$χ（g）$是$ d（g）= [g，g^φ] $ by $ g \ times g $的扩展名。我们证明，如果$ g $是有限解决长度$ d $的有限溶液组，则$ \ exp（d（g））$ divides $ \ exp（g）^{d} $如果$ | g | $是奇数，并且$ \ \ exp（d（g）此外，如果$ p $是素数，而$ g $是$ p $ - 类$ p-1 $，则$ \ exp（d（g））$ divides $ \ exp（g）$。此外，如果$ g $是有限的$ p $ - 类$ c \ geq 2 $，则$ \ exp（d（g））$ divides $ \ exp（g）^{\ lceil \ log_ {p-1}（p-1}（c+1） \ log_2（c）\ rfloor} \ cdot \ exp（g）^{\ lfloor \ log_2（c）\ rfloor+1} $（$ p = 2 $）。
The weak commutativity group $χ(G)$ is generated by two isomorphic groups $G$ and $G^{φ}$ subject to the relations $[g,g^φ]=1$ for all $g \in G$. The group $χ(G)$ is an extension of $D(G) = [G,G^φ]$ 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 $c\geq 2$, then $\exp(D(G))$ divides $\exp(G)^{\lceil \log_{p-1}(c+1)\rceil}$ ($p\geq 3$) and $\exp(D(G))$ divides $2^{\lfloor \log_2(c)\rfloor} \cdot \exp(G)^{\lfloor \log_2(c)\rfloor+1}$ ($p=2$).
