Abstract
The reduced power graph of a group G is a graph whose vertices are the non-identity elements of G, with two vertices connected by an edge if and only if one is a power of the other. Akbari and Ashrafi conjectured 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 disprove this conjecture by providing a complete description of when the reduced power graphs of PGLn(Fq) are connected for all q and all n≥3. We also provide upper bounds on their diameters and, in case of disconnection, a description of all connected components. Our results have implications for the study of power graphs of other families of groups.
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