Between the enhanced power graph and the commuting graph

Abstract The purpose of this note is to define a graph whose vertex set is a finite group , whose edge set is contained in that of the commuting graph of and contains the enhanced power graph of . We call this graph the deep commuting graph of . Two elements of are joined in the deep commuting graph if and only if their inverse images in every central extension of commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of act as automorphisms of the deep commuting graph.