A Graph Associated with $|\cd(G)|-1$ Degrees of a Solvable Group

DOI：10.3770/j.issn:1000-341X.2011.01.021

 作者 单位 梁登峰 北京工商大学计算机与信息工程学院, 北京 100048 施武杰 苏州大学数学系, 江苏 苏州 215006

令$G$是有限群.文中考虑集合$\cd(G)\backslash\{m\}$,其中 $m\in \cd(G)$.我们定义了图$\Delta(G-m)$,其顶点集合是$\rho(G-m)$,即由$\cd(G)\backslash\{m\}$中元素的素因子组成的集合.令$p,q\in\rho(G-m)$,如果$pq$整除$a\in \cd(G)\backslash\{m\}$中的某一个元素,我们就说$p$和$q$之间有一条边.文中证明了以下结果:若$G$是可解群,则$\Delta(G-m)$至多有两个连通分支.

Let $G$ be a group. We consider the set $\cd(G)\backslash\{m\}$, where $m\in \cd(G)$. We define the graph $\Delta(G-m)$ whose vertex set is $\rho(G-m)$, the set of primes dividing degrees in $\cd(G)\backslash\{m\}$. There is an edge between $p$ and $q$ in $\rho(G-m)$ if $pq$ divides a degree $a\in \cd(G)\backslash\{m\}$. We show that if $G$ is solvable, then $\Delta(G-m)$ has at most two connected components.