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 commutator structures are isomorphic, in a certain sense. Commutation, the map taking (x,y) to x⁻¹y⁻¹xy, is a map from G×G to G. But multiplying its arguments by central elements doesn’t change the value, and it maps into the derived subgroup G‘ generated by all commutators. So it can be thought of as a map from (G/Z(G))² to G‘. Then an isoclinism from G to H is a pair of isomorphisms from G/Z(G) to H/Z(H) and from G‘ to H‘ which “intertwine” the commutation maps in the obvious way.

For example, the dihedral and quaternion groups of order 8 are isoclinic. The central quotients are the Klein group of order 4, and the derived groups the cyclic group of order 2.

As an exercise, show that isoclinic groups of the same order have isomorphic commuting graphs. (The commuting graph is a sort of “kernel” of the commutation map.)

Bojan and I found that many rather esoteric parts of group theory get involved in the study: the Schur and Bogomolov multipliers, Schur covers, and so on.

Anyway, to the business in hand: Bahram Arvin, Behrouz Edalatzadeh and Ali Reza Salemkar introduce a new concept, which they call “deep isoclinism”. Roughly speaking, it bears the same relation to the deep commuting graph as isoclinism does to the commuting graph.

Two groups are deeply isoclinic if they have Schur covers which are isoclinic. Facts about Schur covers demonstrate that this notion is independent of the choice of Schur covers in the definition. (Similarly, the definition of the deep commuting graph can be simplified: it is the projection of the commuting graph in a Schur cover, and again the definition is independent of the choice of Schur cover.)

The notion differs curiously from ordinary isoclinism in being much more stringent. Two abelian groups of the same order are deep isoclinic if and only if they are isomorphic. Also, a group is deep isoclinic to the trivial group if and only if it is cyclic. Isoclinism and deep isoclinism coincide for a pair of groups if and only if the Bogomolov multiplier of each is equal to its Schur multiplier.

Just as isoclinic groups of the same order have isomorphic commuting graphs, so deep isoclinic groups of the same order have isomorphic deep commuting graphs. The converse of the first assertion is false (examples of order 64 were found by Eamonn O’Brien), but for the second assertion it is not known if it holds or not.

It also gave me great pleasure that the International Journal of Group Theory is back on-line, though of course we don’t know whether this will continue, given the state of the war. Check it out at ijgt.ui.ac.ir. Check it out: it is diamond open access, you can read everything!