Arithmetic relations between degrees of irrchar and cardinals of conjugacy classes Perhaps this question arose already in MO. If so, then I'm ready to delete it.
If $G$ is a finite group, I denote c $=(c_1,\ldots,c_r)$ the cardinals of its congacy classes and m $=m_1,\ldots,m_s$ the degrees of its irreducible complex representations. The following arithmetical properties are well-know:
$$s=r,\qquad m_j|n=\sum_jc_j,\qquad\sum_jm_j^2=n.$$
Is there any other relation between c and m ?
 A: There are some very general, but fairly weak inequalities relating the vectors $\bf c$ and $\bf m$. For example,
$$
c_i \le {n \over u} \qquad {\rm and} \qquad m_i \le \sqrt{n \over v}\,,
$$
where $u$ is the number of $1$s in $\bf m$ and $v$ is the number of $1$s in $\bf c$.
To see why these inequalities hold, observe that $u = |G:G'|$, so $n/u = |G'|$. Every conjugacy class is contained in some coset of $G'$, so has size at most $|G'|$. Also,
$v = |{\bf Z}(G)|$, and as is well known, $|G:{\bf Z}(G)|$ is an upper bound for the squares of the irreducible character degrees of $G$.
In fact, one can make a somewhat stronger statement. The list $\bf c$ can be partitioned in such a way that the sum of the $c_i$ in each part is exactly $n/u$, and similarly, ${\bf m}$ can be partitioned in such a way that the sum of the squares of the $m_i$ in each part is exactly $n/v$. The partition of $\bf c$ corresponds to the classes contained in the various cosets of $G'$ and the partition of $\bf m$ corresponds to the characters lying over the various linear characters of ${\bf Z}(G)$. 
