From Isaacs et.al. 2005

Conjecture C. Let χ be a primitive irreducible character of an arbitrary finite group G. Then χ(1) divides | clG(g)| for some element g ∈ G.

Here, of course, we have written clG(g) to denote the class of g in G. We have checked that Conjecture C holds for all irreducible characters (primitive or not) of all groups in the Atlas 1.

Question 1 What is motivation for this ? Is it possible to describe what are conjugacy class(es) should correspond to irreducible representation in this way ?

Question 2 Is it still open ?

The authors write:

We now digress to explain our original motivation for considering these questions. There are numerous parallels and analogies between theorems concerning the of set irreducible character degrees of a finite group and theorems concerning the set of conjugacy class sizes of such groups. This suggests that perhaps there are some subtle arithmetic connections between these two sets of integers associated with a given group. One such connection that is easy to see is that each prime number that divides an irreducible character degree of G must also divide some class size of G. If G is solvable, then S. Dolfi showed that more is true. He proved [2] that given any two distinct primes p and q such that pq divides some irreducible character degree of a solvable group G, then pq also divides some class size of G. One might conjecture that the analogous assertion for three or more distinct primes is also true, but as far as we know, this remains open.