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 ?

  • $\begingroup$ There is of course also the relation $c_j | n$. Aside from that, I don't know equalities but there are also inequalities. For example, we have $n \leq N(r)$ where $N(r)$ is a bound depending only on $r$. See Appendix B of Keith Conrad's expository paper math.uconn.edu/~kconrad/blurbs/grouptheory/conjclass.pdf $\endgroup$ – François Brunault Oct 20 '11 at 17:01
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    $\begingroup$ There is also $\sum_j \s_jm_j =$ number of elements of order 2, where $s_j$ are the Frobenius-Schur indicators (always 0 or $\pm 1$, depending on whether the characters and their representations are realisable over $\mathbb{R}$). $\endgroup$ – Alex B. Oct 21 '11 at 1:19
  • $\begingroup$ mathoverflow.net/questions/77517/… Here some related congruences are described $\endgroup$ – Alexander Chervov Aug 10 '12 at 21:18

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 ${\bf Z}(G)$.


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