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 
 conjugacy class(es) should correspond to irreducible representation in this way ?
(at least for some standard groups S_n, A_n, GL_n(F_q),...) What are representative examples?  
 
**Question 2** Is it still open ? 

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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.

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**Partial result:**

In the following, we use the notation np to denote the p-part of a positive
integer n, where p is a prime number.

Corollary D. Let χ be a primitive irreducible character of a solvable group
G, and let p be a prime divisor of |G|. Then χ(1)p divides (| clG(g)|p)
3 for some element g ∈ G.

---------

Not related results, for complteness:

Denote CV(g) fixed point subspace for g in V.

Our main result is the following.

Theorem A. Let V be a nonzero finite dimensional completely reducible
F G-module, where F is any field and G is any finite group. Assume that
CV (G) = 0 and let p be the smallest prime divisor of |G|. Then there exists
some element g ∈ G such that

$ dim CV (g) ≤ (1/p) ~ dim V $.

The fraction 1/p cannot, in general, be replaced by any smaller quantity.
In particular, this shows that Neumann’s conjecture is valid for odd-order
groups, at least...

Corollary B. Let V be a nonzero finite dimensional completely reducible
F G-module, where F is an arbitrary field and G is any finite group, and
assume that CV (G) = 0. Then

$1/ |G| \sum_{g∈G} dim CV (g) ≤ ((p + 1)/ 2p)~~ dim V$ ,

where p is the smallest prime divisor of |G|.



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