A Paper on Calculating the Height and Relational Complexity of the Primitive Actions of $ PSL_{2} (q) $ and $ PGL_{2} (q) $

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.