The lesson of this post and the next in the series is that the most interesting questions (to me, anyway) are not about the girth of the deep commuting graph but instead about the classes of groups G defined by either of the following two conditions:
- two different types of graph defined on G are equal, or are approximately equal in some way;
- one type of graph defined on G belongs to a restricted class (for example, perfect graphs, cographs, threshold graphs).
This post concerns an extension of the hierarchy of graphs on a group into a second dimension, which gives more scope to ask such questions. The results are contained in two papers on the arXiv, 2112.02395 and 2112.02613 (a revised and improved version of the second will be posted shortly).
I will restrict attention here to three types of graph defined on a group:
- the power graph: two elements joined if one is a power of the other;
- the enhanced power graph: two elements joined if both are powers of a third element;
- the commuting graph: two elements joined if they commute.
It is commonly done with these graphs to delete elements of the group joined to all others, but I will not do this, for reasons which will become clear.
The new element is to take also an equivalence relation on the group. The ones I will describe here are
- same order.
The starting point was the talk by Lavanya Selvaganesh in the research discussion at CUSAT, although very similar ideas had also been considered by other people.
So, if A is a type of graph on groups, and B is an equivalence relation, we define the B superA graph on G by the rule that x and y if there are elements u and v which are B-equivalent to x and y equivalently and are adjacent in the graph A. So, for example, we can talk about the “order superpower graph”, which was Lavanya’s subject.
We make a convention that two elements which are B-equivalent are joined in the B superA graph. This is not forced but is in a sense natural (since arguably the definitions of the graphs would give a loop at each vertex, from which this convention would follow).
This gives us nine graphs to play with, though it turns out that two of them (the order superenhanced power graph and the order supercommuting graph) are always equal (though the other eight are in general distinct). A subsidiary problem which we didn’t solve but which should not be two hard is to find a group on which all eight graphs are distinct.
As I said, I am interested in the class of groups for which two of these graphs coincide. It was already known that the power graph is equal to the enhanced power graph if and only if all elements of G have prime power order (these are the so-called EPPO groups or CP groups); and that the enhanced power graph is equal to the commuting graph if and only if all the Sylow subgroups of G are cyclic or generalized quaternion. In each case, all such groups are known, though the classifications are not trivial.
I will discuss two new cases where interesting and previously-studied classes of groups arise. In the second case, I learned something from this, which may be new to you also.
A Dedekind group is a finite group in which all subgroups are normal. Dedekind proved in 1897 that such a group is either abelian, or of the form Q×E×F, where Q is the quaternion group of order 8, E is an elementary abelian 2-group, and F is an abelian group of odd order.
Theorem: For a finite group G, the following are equivalent:
- the power graph of G is equal to the conjugacy superpower graph;
- the enhanced power graph of G is equal to the conjugacy superenhanced power graph;
- G is a Dedekind group.
A 2-Engel group is a group which satisfies the 2-Engel identity [[x,y],y] = 1, where [x,y] denotes the commutator x−1y−1xy. Thus every nilpotent group of class 2 is 2-Engel, and Hopkins and Levi independently showed that a 2-Engel group is nilpotent of class 3 (yes, that is the Levi whose name is attached to the Levi graph); both inclusions are strict.
I didn’t know prior to this work that G is a 2-Engel group if and only if every centraliser in G is a normal subgroup. This is proved in a post by Korhonen on StackExchange, using a result of Kappe.
Theorem: The commuting graph of G is equal to the conjugacy supercommuting graph if and only if G is a 2-Engel group.
There is much more to say about these graphs, for some of which I refer you to the papers. In particular, each supergraph has a “condensed” version where the vertices are the B-equivalence classes rather than elements; I used the “expanded” version in order to be able to compare the graphs. But that will do for now. The moral of this story, if it has one, is that this project is turning up interesting group theory.