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 (Formula presented.), whose edge set is contained in that of the commuting graph of (Formula presented.) and contains the enhanced power graph of (Formula presented.). We call this graph the deep commuting graph of (Formula presented.). Two elements of (Formula presented.) are joined in the deep commuting graph if and only if their inverse images in every central extension of (Formula presented.) 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 (Formula presented.) act as automorphisms of the deep commuting graph.