From [Isaacs et.al. 2005][1]

> 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.



  [1]: http://www.uv.es/amoquin/35.pdf