论文标题
极端原始群体的分类
The classification of extremely primitive groups
论文作者
论文摘要
令$ g $为一个有限的原始排列组，在$ω$上，具有非平凡点稳定器$g_α$。我们说，如果$g_α$以$ω\ setminus \ {α\} $为原始的每个轨道，则$ g $是极其原始的。这些群体自然出现在几种不同的情况下，他们的研究可以追溯到1920年代的曼宁工作。在本文中，我们确定了几乎简单的极其原始的群体，其中Socle是一种杰出的谎言类型。通过将此结果与伯恩斯，普拉格和血清的早期工作相结合，这完成了几乎简单的极端原始群体的分类。此外，鉴于Mann，Praeger和Seress的结果，我们的主要定理对所有有限的极具原始组进行了完整的分类，以至于有限的许多仿射例外（并且猜测没有例外）。在此过程中，我们还为特殊群体的原始行动基本大小建立了几个新结果，这可能是独立的。
Let $G$ be a finite primitive permutation group on a set $Ω$ with nontrivial point stabilizer $G_α$. We say that $G$ is extremely primitive if $G_α$ acts primitively on each of its orbits in $Ω\setminus \{α\}$. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.
