论文标题
非简单基团的非交易，非生成图
The non-commuting, non-generating graph of a non-simple group
论文作者
论文摘要
令$ g $为（有限或无限）组，这样$ g/z（g）$并不简单。 $ g $的非交易，非生成图$ g $的$ g \ setminus z（g）$，带有顶点$ x $和$ y $，只要$ [x，y] \ ne 1 $和$ \ langle x，y \ rangle x，y \ rangle \ ne g $。我们研究了$ g $的结构与$ξ（g）$的连接性与直径之间的关系。特别是，我们证明了该图：（i）最多可与直径连接，最多为$ 4 $； （ii）由孤立的顶点和直径的连接部分组成，最多为$ 4 $；或（iii）是两个直径$ 2 $的连接组件的结合。我们还详细描述了具有（III）类图的有限组。在同伴论文ARXIV：2212.01616中，我们考虑了$ g/z（g）$有限且简单的情况。
Let $G$ be a (finite or infinite) group such that $G/Z(G)$ is not simple. The non-commuting, non-generating graph $Ξ(G)$ of $G$ has vertex set $G \setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y] \ne 1$ and $\langle x, y \rangle \ne G$. We investigate the relationship between the structure of $G$ and the connectedness and diameter of $Ξ(G)$. In particular, we prove that the graph either: (i) is connected with diameter at most $4$; (ii) consists of isolated vertices and a connected component of diameter at most $4$; or (iii) is the union of two connected components of diameter $2$. We also describe in detail the finite groups with graphs of type (iii). In the companion paper arXiv:2212.01616, we consider the case where $G/Z(G)$ is finite and simple.
