论文标题
周期组中两个要素的产品的顺序
The order of the product of two elements in the periodic groups
论文作者
论文摘要
令$ g $为一个定期组，让$ lcm（g）$为G $ in G $的所有$ x \ in的集合，这样$ o（x^nz）$将$ o（x^n）$的最不常见的倍数划分为$ g $ in $ g $的所有$ o（x^n）$和$ o（z）$。在本文中，我们证明$ lcm（g）$生成的子组是$ g $的本地nilpotent特性子组，每当$ g $是本地有限的组。对于$ x，y \ in g $，每当$ o（xy）$将$ o（x）$和$ o（y）$的最小常见的倍数分配为$ o（xy）$时，顶点$ x $连接到顶点$ y $。令$ deg（g）$为$ g $超过$ g $的所有$ veg（g）$的总和。我们证明，对于任何有限的组$ g $，带有$ h（g）$ fosacy类，$ deg（g）= | g |（h（g）+1）$，仅当$ g $是一个阿贝里安集团。
Let $G$ be a periodic group, and let $LCM(G)$ be the set of all $x\in G$ such that $o(x^nz)$ divides the least common multiple of $o(x^n)$ and $o(z)$ for all $z$ in $G$ and all integers $n$. In this paper, we prove that the subgroup generated by $LCM(G)$ is a locally nilpotent characteristic subgroup of $G$ whenever $G$ is a locally finite group. For $x,y\in G$ the vertex $x$ is connected to vertex $y$ whenever $o(xy)$ divides the least common multiple of $o(x)$ and $o(y)$. Let $Deg(G)$ be the sum of all $deg(g)$ where $g$ runs over $G$. We prove that for any finite group $G$ with $h(G)$ conjugacy classes, $Deg(G)=|G|(h(G)+1)$ if and only if $G$ is an abelian group.
