AI 中文总结
本文围绕有限群作用在有限集上的关系复杂度和高度展开，先定义概念并证明基本结果，重点研究\(PSL_{2}(q)\)和\(PGL_{2}(q)\)的基本作用，计算其高度与关系复杂度。
AI 中文摘要
对于作用在有限集上的有限群，可以计算该作用的一个称为关系复杂度的统计量。这个概念由Gregory Cherlin定义，受模型论考量的推动。另一个相关统计量是作用的高度，它为关系复杂度提供一个上界。本文定义了这两个概念并证明了一些基本结果。之后主要关注研究\(PSL_{2}(q)\)和\(PGL_{2}(q)\)的基本作用，并计算它们各自的高度和关系复杂度。
英文摘要
For a finite group acting on a finite set, a statistic called relational complexity can be calculated for the action. This notion was defined by Gregory Cherlin and motivated by considerations in model theory. Another related statistic is the height of the action, which provides an upper bound for relational complexity. In this paper, both concepts are defined and some basic results proved. The main focus later on is examining the primitive actions of $ PSL_{2} (q) $ and $ PGL_{2} (q) $ and computing both the height and relational complexity for each one.
