AI 中文总结
反驳多循环猜想，证明存在无半正则自同构的顶点传递图。采用Chen等人方法构造度为16464的难以捉摸群\(7^6.\mathrm{PSU}_3(3)\)，其是2 - 闭的，此例扩展出多循环猜想无穷反例及相关顶点传递图。
AI 中文摘要
我们反驳了多循环猜想，该猜想称每个传递2 - 闭置换群是非难以捉摸的，即包含一个素数阶的错排。实际上，我们证明了一个更强的结果，回答了Marušič和Jordan的一个长期问题：存在一个无半正则自同构的顶点传递图。为此，我们采用了Chen等人最近开发的通过非分裂扩张构造难以捉摸群的方法，构造了一个度为16464的难以捉摸群\(7^6.\mathrm{PSU}_3(3)\)。我们表明这个群是其七个轨道图的全自同构群，因此是2 - 闭的。我们的例子扩展到多循环猜想的无穷多个反例，以及无穷多个无半正则自同构的顶点传递图。
英文摘要
We disprove the Polycirculant Conjecture, which states that every transitive 2-closed permutation group is non-elusive, i.e. contains a derangement of prime order. In fact, we prove a stronger result, answering a long-standing question of Marušič and Jordan: there exists a vertex-transitive graph admitting no semiregular automorphism. To do so, we employ recently developed methods of Chen et al. for constructing elusive groups via non-split extensions, allowing us to construct an elusive group $7^6.\mathrm{PSU}_3(3)$ of degree 16,464. We show that this group is the full automorphism group of seven of its orbital graphs and hence is 2-closed. Our example extends to infinitely many counterexamples of the Polycirculant Conjecture, and infinitely many vertex-transitive graphs admitting no semiregular automorphism.
Comments11 pages
