A brief interlude to describe another recent preprint, and as in the preceding post I will concentrate on one result in the paper.
I don’t know why it happens, but in this project one of the most interesting graph parameters turns out to be the clique number. I have talked in an earlier post about the fact that the clique number of the power graph of a group G is the largest clique number of a cyclic subgroup, and the clique number of the cyclic group of order n is at most three times φ(n) where φ is Euler’s totient (in fact the limit superior of the ratio is 2.6481017597…).
The graph of concern here is the condensed version of a conjugacy supergraph, as defined in the preceding post. Recall that the commuting graph of a group has two vertices joined if they commute (that is, generate an abelian group). Replacing “abelian” by “nilpotent” or “soluble” here gives the nilpotency or solubility graph. Now the conjugacy super-solubility graph has x and y joined if there are conjugates u and v of x and y respectively such that u and v generate a soluble group. To condense it, we shrink each conjugacy class to a single vertex, and discard the identity. So the vertices are the non-trivial conjugacy classes, two classes joined if they contain elements generating a soluble group.
The analogous graphs for “abelian” or “nilpotent” were investigated by Herzog, Longobardi and Maj and by Mohammadian and Erfanian respectively, under the names “CCC graph” and “NCC graph” respectively, so the one considered here is the SCC graph (soluble conjugacy class graph). The investigation of the soluble case was begun by Parthajit Bhowal and Rajat Kanti Nath, who invited first me and then Benjamin Sambale to join the project.
The result I want to advertise is the following.
Theorem: Given a positive integer d, there are only finitely many finite groups whose SCC graph has clique number d. In particular, the only finite groups whose SCC graph is triangle-free are the cyclic groups of orders 1, 2 and 3 and the dihedral group of order 6.
This theorem does depend on the Classification of the Finite Simple Groups. I will say just a few words about the proof.
First, if G is soluble, then its SCC graph is complete, and so the result follows from the theorem of Landau in 1903 asserting that there are only finitely many groups with a given number of conjugacy classes. (Our theorem could be regarded as a strengthening of Landau’s.)
If G contains an element g whose order is the product of two primes p and q, then g, gp and gq define a triangle in the SCC graph. So if the graph is triangle-free, then every element of G has prime power order; G is a CP group, or EPPO group, as in the preceding post. Completing the proof involves a little digging into the structure of these groups, using properties of simple groups from the ATLAS of finite groups.
The general theorem in the insoluble case involves several reductions, and some knowledge about groups of Lie type; I will simply refer to version 2 of the paper, shortly to appear on the arXiv. We don’t have a good upper bound for the order of a group whose SCC graph has given clique number; the case of clique number 3 might be an interesting problem to tackle. For clique number 3 we meet insoluble groups; for example, the SCC graph of the alternating group A5 has four vertices (one class of elements of order 2, one of order 3, and two of order 5); the involutions are joined to all the other classes, and the two classes of order 5 to one another.
In the spirit of yesterday’s post, here are two questions we can’t answer:
- For which groups do the SCC and NCC graphs coincide?
- For which groups is the expanded SCC graph equal to the solubility graph?
It is possible that the answers might be “nilpotent groups” and “soluble groups” respectively.