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

TL;DR

This paper investigates the connectivity of reduced power graphs of projective general linear groups over finite fields, disproving a conjecture that only alternating groups have connected such graphs, and provides detailed structural results.

Contribution

It completely characterizes when the reduced power graphs of PGL_n(F_q) are connected, disproving a prior conjecture, and describes their diameters and components.

Findings

Reduced power graphs of PGL_n(F_q) are connected in specific cases.

The conjecture that only alternating groups have connected reduced power graphs is false.

Provides bounds on diameters and descriptions of components when disconnected.

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. It is conjectured by Akbari and Ashrafi that if a non-abelian finite simple group has a connected reduced power graph, then it must be an alternating group. In this paper, we shall give a complete description of when the reduced power graphs of $\mathrm{PGL}_n(\mathbb{F}_q)$ are connected for all $q$ and all $n\geq 3$. In particular, the conjectured by Akbari and Ashrafi is false. We shall also provide an upper bound in their diameters, and in case of disconnection, provide a description of all connected components.