An Exploration of the Symmetry Groups of Certain Configurations of Points

TL;DR

This paper investigates the symmetry groups of specific geometric point-line configurations, classifies their automorphisms, and explores generalized cyclic configurations to derive new geometric insights.

Contribution

It introduces a classification approach for symmetry groups of certain configurations and applies it to generalized cyclic configurations, revealing new geometric properties.

Findings

Classification of symmetry groups for (9_3) and (10_3) configurations

Connection between graph automorphisms and configuration symmetries

Geometric results on generalized cyclic configurations

Abstract

We start by introducing the basics of configurations of points and lines, and then move into discussing symmetry groups of these configurations. Specifically, we explore how we might classify the symmetries of $(9_3)$ and $(10_3)$ geometric configurations, given the graph automorphisms of their underlying set-configurations. Finally, we show how a specific class of combinatorial configurations called generalized cyclic configurations can be explored using this terminology, and give several interesting geometric results.