Reduced Power Graphs of $\mathrm{PGL}_3(\mathbb{F}_q)$

TL;DR

This paper investigates the structure of reduced power graphs of the projective general linear group PGL_3 over finite fields, determining their number of connected components and diameters for all prime powers q.

Contribution

It provides exact counts of connected components and diameters of reduced power graphs for PGL_3(F_q), a comprehensive analysis across all prime powers q.

Findings

Exact number of connected components for each q

Precise diameters of each component

Complete characterization for all prime powers q

Abstract

Given a group $G$, let us connect two non-identity elements by an edge if and only if one is a power of another. This gives a graph structure on $G$ minus identity, called the reduced power graph. In this paper, we shall find the exact number of connected components and the exact diameter of each component for the reduced power graphs of $\mathrm{PGL}_3(\mathbb{F}_q)$ for all prime power $q$.