The geometry of elation groups of a finite projective space

TL;DR

This paper explores the geometric structure of elation groups in finite projective spaces, revealing a correspondence with Singer group orbits and determining their count.

Contribution

It establishes a one-to-one correspondence between conjugacy classes of elation groups and orbits of Singer groups, and calculates the number of such elation groups.

Findings

Correspondence between elation group conjugacy classes and Singer group orbits

Identification of orbits on projective subspaces of specific dimensions

Quantification of the total number of elation groups

Abstract

We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a 1-1 correspondence from conjugacy classes of such groups and orbits on projective subspaces (of a suitable dimension) of Singer groups of projective spaces. Together with a recent result of Drudge we establish the number of these elation groups.