Deep isoclinism

Graphs and groups, in my view, are two subjects engaged in a wide-ranging dialogue at present. Graphs can be used to describe interesting classes of groups, and groups to construct interesting graphs. But I am delighted that recently, in a paper published last year in the International Journal of Group Theory, a new concept in group theory has come up, based on a paper on a particular graph defined on groups, the so-called “deep commuting graph”. The deep commuting graph saw the light in my paper with Bojan Kuzma in the Journal of Graph Theory. Two vertices are joined if their inverse images in any central extension of the group commute. (A central extension of G is a group H with a central subgroup Z such that H/Z is isomorphic to G.) The deep commuting graph is contained in the commuting graph (as a spanning subgraph) and contains the enhanced power graph. Central to its study is the notion of isoclinism of groups, invented by Philip Hall. Two groups are isoclinic if their…