论文标题
有限置换组的高度和关系复杂性
On the height and relational complexity of a finite permutation group
论文作者
论文摘要
令$ g $为$ t $的$ω$的排列组。我们说，如果$λ\subseteqΩ$是独立的集合，则如果其点稳定器不等于$λ$的任何适当子集的点稳定器。我们将$ g $的高度定义为独立集的最大尺寸，并表示此数量$ \ mathrm {h}（g）$。 在本文中，我们研究$ g $原始的情况下的$ \ mathrm {h}（g）$。我们的主要结果断言，$ \ mathrm {h}（g）<9 \ log t $，否则$ g $是在一个特定的经过精心良好的家庭中（“原始的大型基本组”）。该结果的直接推论是对具有较大“关系复杂性”的原始置换群体的表征，后者的数量是Cherlin在对置换基团模型理论的研究中引入的统计量。 我们还研究$ \ mathrm {i}（g）$，是$ g $的不剩余基数的最大长度，在这种情况下，我们证明，如果$ g $是原始的，那么$ \ mathrm {i}（g）<7 \ log t $又是$ g $，又是$ g $，$ g $在某个特定的家庭中（其中包括原始的大型基团）。
Let $G$ be a permutation group on a set $Ω$ of size $t$. We say that $Λ\subseteqΩ$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $Λ$. We define the height of $G$ to be the maximum size of an independent set, and we denote this quantity $\mathrm{H}(G)$. In this paper we study $\mathrm{H}(G)$ for the case when $G$ is primitive. Our main result asserts that either $\mathrm{H}(G)< 9\log t$, or else $G$ is in a particular well-studied family (the "primitive large--base groups"). An immediate corollary of this result is a characterization of primitive permutation groups with large "relational complexity", the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\mathrm{I}(G)$, the maximum length of an irredundant base of $G$, in which case we prove that if $G$ is primitive, then either $\mathrm{I}(G)<7\log t$ or else, again, $G$ is in a particular family (which includes the primitive large--base groups as well as some others).
