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

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. 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.