论文标题
具有可溶性稳定器的原始组的基本尺寸
Base sizes for primitive groups with soluble stabilisers
论文作者
论文摘要
让$ g $是一个有限的原始置换组，在$ω$上，带有点稳定器$ h $。回想一下，如果$ g $的稳定剂微不足道，则$ω$的子集是$ g $的基础。我们定义了$ g $的基本大小，表示为$ b（g，h）$，是$ g $的最小尺寸。确定群体的基础规模是排列群体理论中的一个基本问题，历史悠久，可以追溯到19世纪。在这里，我们的主要动机之一是1996年的Seress定理，该定理指出$ b（g，h）\ leqslant 4 $如果$ g $是可溶的。在本文中，我们通过证明所有有限原始$ g $的$ b（g，h）\ leqslant 5 $，并具有可溶点稳定剂$ h $。这是最好的。我们还确定了所有几乎简单组的确切基础大小，并在此环境中研究随机基础。例如，我们证明，$ 4 $随机元素以$ω$形式形成$ 1 $ $ 1 $ as $ | g | $倾向于无穷大的概率。
Let $G$ be a finite primitive permutation group on a set $Ω$ with point stabiliser $H$. Recall that a subset of $Ω$ is a base for $G$ if its pointwise stabiliser is trivial. We define the base size of $G$, denoted $b(G,H)$, to be the minimal size of a base for $G$. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that $b(G,H) \leqslant 4$ if $G$ is soluble. In this paper we extend Seress' result by proving that $b(G,H) \leqslant 5$ for all finite primitive groups $G$ with a soluble point stabiliser $H$. This bound is best possible. We also determine the exact base size for all almost simple groups and we study random bases in this setting. For example, we prove that the probability that $4$ random elements in $Ω$ form a base tends to $1$ as $|G|$ tends to infinity.
