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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 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 ?


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.


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|.

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Could you explain the word "primitive"? It doesn't seem to be explained in the linked article. –  François Brunault Sep 29 '12 at 13:23
    
"χ is primitive (that is, χ is not induced from any proper subgroup of G)," en.wikipedia.org/wiki/Clifford_theory –  Alexander Chervov Sep 29 '12 at 13:29
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