极小传递置换群的渐近计数
Asymptotic enumeration of minimally transitive permutation groups
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中文总结 AI 辅助
该研究证明了极小传递置换群计数的Pyber上界在固定素数幂序列上最优，推导得出顶点传递图和有向图的数量阶，并给出Pyber上界的未发表证明，探讨对McKay–Praeger猜想的启示。
中文摘要 AI 辅助
我们证明了，对于对称群$S_n$的极小传递子群，Pyber给出的上界$2^{O(n\log(n))}$在每个固定素数的幂序列上是最优的，即使这些群按置换同构计数亦是如此。作为副产品，我们的构造表明，在每个固定素数的幂序列上，次数为$n$的极小传递置换群的最大阶为$2^{\Theta(n)}$。为完整起见，我们还给出了Pyber此前未发表的该上界的证明。我们进一步推导得出，阶为$n$的有标记顶点传递图和有向图的数量均为$2^{\Theta(n\log(n))}$，并讨论了我们的结果对McKay–Praeger猜想相关研究方法的启示。
英文摘要
We prove that Pyber's upper bound $2^{O(n\log(n))}$ for the number of minimally transitive subgroups of $S_n$ is best possible along the powers of every fixed prime, even when the groups are counted up to permutational isomorphism. As a byproduct, our construction shows that, along the powers of every fixed prime, the maximum order of a minimally transitive permutation group of degree $n$ is $2^{Θ(n)}$. For completeness, we also present Pyber's previously unpublished proof of his upper bound. We further deduce that the numbers of labelled vertex-transitive graphs and digraphs of order $n$ are both $2^{Θ(n\log(n))}$, and discuss the implications of our results for approaches to the McKay--Praeger conjecture.
