Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

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关于与 G 共轭类相关的正则图

给定一个有限群G具有正态子群ñ, 简单图$\Gamma _{\textit {G}}( \textit {N} )$是一个图，其顶点的形式为$|x^G|$， 在哪里$x\in {N\setminus {Z(G)}}$和$x^G$是个G- 共轭类ñ包含元素X. 两个顶点$|x^G|$和$|y^G|$如果它们不是互质的，则它们是相邻的。我们证明，如果$\伽玛_G(N)$是连通不完全正则图，则$N= P \times {A}$在哪里磷是一个p-group，对于一些素数p,$A\leq {Z(G)}$和$\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$.