论文标题
虚拟组的虚拟生成图
The virtually generating graph of a profinite group
论文作者
论文摘要
我们考虑图形$γ_ {\ rm {firt}}（g）$，其顶点是有限生成的profinite $ g $的元素，其中两个vertices $ x $和$ y $在且仅当它们拓扑上生成$ g $的开放子群时才相邻。我们通过删除其孤立的顶点，研究了图$δ_ {\ rm {virt}}（g）$从$γ_ {\ rm {firt}}（g）$获得的连接性。特别是我们证明，对于每个正整数$ t $，都存在有限生成的Prosoluble $ g $，其属性$δ_ {\ rm {firt}}（g）$具有精确的$ t $连接的组件。此外，我们研究图$ \tildeγ_{\ rm {firt}}}（g）$，其顶点再次是$ g $的元素，并且仅在存在最小的$ g $的$ g $时，在其中两个顶点相邻。在这种情况下，我们证明了获得的子图$ \tildeδ_ {\ rm {virt}}（g）获得的$删除隔离的顶点是连接的，并且最多具有直径3。
We consider the graph $Γ_{\rm{virt}}(G)$ whose vertices are the elements of a finitely generated profinite group $G$ and where two vertices $x$ and $y$ are adjacent if and only if they topologically generate an open subgroup of $G$. We investigate the connectivity of the graph $Δ_{\rm{virt}}(G)$ obtained from $Γ_{\rm{virt}}(G)$ by removing its isolated vertices. In particular we prove that for every positive integer $t$, there exists a finitely generated prosoluble group $G$ with the property that $Δ_{\rm{virt}}(G)$ has precisely $t$ connected components. Moreover we study the graph $\tilde Γ_{\rm{virt}}(G)$, whose vertices are again the elements of $G$ and where two vertices are adjacent if and only if there exists a minimal generating set of $G$ containing them. In this case we prove that the subgraph $\tilde Δ_{\rm{virt}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.
