AI 中文总结
研究双轨道置换群\(G\)，证明其要么含错排，要么含唯一不动点的素数幂阶元素，还给出特定条件下含错排的推论，通过归结单群情况及利用相关分类证明结果。
AI 中文摘要
设\(G\)是\(n\gt2\)个点上的双轨道置换群。我们证明\(G\)要么包含一个错排，要么包含一个具有唯一不动点的素数幂阶元素。作为推论，如果\(G\)的轨道长度为\(n_1\)和\(n_2\)，且\(\gcd(n_1,n_2 - 1)=\gcd(n_1 - 1,n_2)=1\)，那么\(G\)包含一个错排。\(n_1 = n_2\)的特殊情况最近由埃利斯和哈珀猜想，并在各种限制假设下得到证明。我们通过归结为单群的情况并利用布博洛尼、斯皮加和魏格尔关于单群的正规\(2 -\)覆盖的分类来证明我们的结果。
英文摘要
Let $G$ be a two-orbit permutation group on $n > 2$ points. We show that $G$ contains either a derangement or an element of prime-power order with a unique fixed point. As a corollary, if the orbits of $G$ have length $n_1$ and $n_2$ and $\gcd(n_1, n_2-1) = \gcd(n_1-1, n_2) = 1$, then $G$ contains a derangement. The special case $n_1 = n_2$ was recently conjectured by Ellis and Harper and proved under various restrictive hypotheses. We prove our result by reducing to the case of simple groups and leveraging the classification of normal $2$-coverings of simple groups due to Bubboloni, Spiga, and Weigel.
Comments9 pages
