无限本原群的粗糙端
Rough ends of infinite primitive groups
摘要：
如果$G$是集合$Ω$的一组排列，则$G$的子轨道是作用在$Ω$上的点稳定子$G_α$的轨道。这些次轨道的基数是$G$的次。每一个有限次的无限本原置换群$G$忠实地充当一个具有点集$Ω$的局部有限连通点本原有向图$Γ$的自同构群，从而存在$G$在$Γ$的端点上的自然作用.证明了如果$G$在逐点收敛的置换拓扑中是闭的，则$G$的结构由作用在$Γ$端点上的$G$的任意轨道的长度决定.研究一个有限生成群的凯莱图的端点以确定群的结构常常是富有成效的。B. Kr{<$}n和R. G. M{<$}ller最近将Cayley图推广到他们称之为{ \it粗糙Cayley图}，他们称这个图的端点为群的{ \it粗糙端点}。由此可知，$Γ$的端点是$G$的粗端点，因此我们的结果等价于说，一个子度都是有限的闭本原群$G$的结构由$G$在其粗端点上的任何轨道的长度决定。
英文摘要：
If $G$ is a group of permutations of a set $Ω$, then the suborbits of $G$ are the orbits of point-stabilisers $G_α$ acting on $Ω$. The cardinalities of these suborbits are the subdegrees of $G$. Every infinite primitive permutation group $G$ with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph $Γ$ with vertex set $Ω$, and there is consequently a natural action of $G$ on the ends of $Γ$. We show that if $G$ is closed in the permutation topology of pointwise convergence, then the structure of $G$ is determined by the length of any orbit of $G$ acting on the ends of $Γ$. Examining the ends of a Cayley graph of a finitely generated group to determine the structure of the group is often fruitful. B. Kr{ö}n and R. G. M{ö}ller have recently generalised the Cayley graph to what they call a {\it rough Cayley graph}, and they call the ends of this graph the {\it rough ends} of the group. It transpires that the ends of $Γ$ are the rough ends of $G$, and so our result is equivalent to saying that the structure of a closed primitive group $G$ whose subdegrees are all finite is determined by the length of any orbit of $G$ on its rough ends.
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