Between the enhanced power graph and the commuting graph. Issue 2 (8th August 2022) - Cameron, Peter J.; Kuzma, Bojan
Abstract: The purpose of this note is to define a graph whose vertex set is a finite group G G, whose edge set is contained in that of the commuting graph of G G and contains the enhanced power graph of G G . We call this graph the deep commuting graph of G G . Two elements of G G are joined in the deep commuting graph if and only if their inverse images in every central extension of G G 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 G G act as automorphisms of the deep commuting graph.