Conjugacy classes of groups of prime order in $\mathrm{PGL}_{k+1}(\mathbb{C})$

TL;DR

This paper counts conjugacy classes of certain cyclic subgroups of prime order in the projective general linear group over complex numbers, linking group classifications with geometric configurations of points.

Contribution

It provides a count of conjugacy classes of admissible cyclic subgroups of prime order in PGL and explores their relation to point set configurations in projective space.

Findings

Count of conjugacy classes of prime order cyclic subgroups

Description of association between conjugacy classes and point sets

Connection between group classification and geometric configurations

Abstract

Let $\mathbb{C}$ be the field of complex numbers. Let $k$ be natural number with $k \geq 2$ and let $p$ be a rational prime. In this paper we count the number of conjugacy classes of admissible cyclic subgroups of $\mathrm{PGL}_{k+1}(\mathbb{C})$ of order $p$, where with admissible we intend those finite subgroups that can be contained in the automorphism group of a set of points in $\mathbb{P}^k(\mathbb{C})$ in general position and of cardinality $n\geq k+3$. We also describe a kind of association between the conjugacy classes of these groups and show a beautiful relation connecting this type of association and the association between point sets.