Conway广群及其相关群
Conway's groupoid and its relatives
摘要：
在1997年，John Conway在一个大小为$13$的集合上构造了一个$6$-重传递置换子集$M_{13}$，其中固定任何给定点的子集同构于Mathieu群$M_{12}$。该构造是通过在射影平面${\rm PG}(2,3)$上的“移动计数器难题”来实现的。我们讨论的后果和概括康威的建设。特别是，我们探讨了如何使用各种设计和超图，而不是${\rm PG}(2,3)$，以获得有趣的类似物的$M_{13}$，我们把这些类似物作为康威广群。提出了一些悬而未决的问题。
英文摘要：
In 1997, John Conway constructed a $6$-fold transitive subset $M_{13}$ of permutations on a set of size $13$ for which the subset fixing any given point was isomorphic to the Mathieu group $M_{12}$. The construction was via a "moving-counter puzzle" on the projective plane ${\rm PG}(2,3)$. We discuss consequences and generalisations of Conway's construction. In particular we explore how various designs and hypergraphs can be used instead of ${\rm PG}(2,3)$ to obtain interesting analogues of $M_{13}$; we refer to these analogues as Conway groupoids. A number of open questions are presented.
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