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Dear all,

For a finite group $G$, let $m(G)$ denote the largest number of irreducible characters of the same degree of $G$. You can say that $m(G)$ is the largest multiplicity of character degrees of $G$. I wonder if the following is true: $$m(G)\leq |N|m(G/N) \text{ for every normal subgroup } N \text{ of } G.$$ For my purpose, I only need the inequality for cyclic $N$. But I strongly believe that the inequality is good in general.

Do you see a counterexample or an idea of proof?

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This appears to fail for $G$ dihedral of order 30, with $N=C3$. Here $m(G)=7$ (7 2-dimensionals) but $|N|=3$ and $m(G/N)=2$. Are you sure this is what you mean? –  Tim Dokchitser Apr 29 '13 at 7:21
    
Have no idea how to solve this problem. But do remember that M. Isaacs (author of a book on Character Theory) has written papers that have a lot to do with the set of character degrees. –  P Vanchinathan Apr 29 '13 at 8:22
    
@Tim Dokchitser: Thank you, Tim. Do you see a counterexample in the case $G/N$ is non-solvable, say $G/N\cong A_5$? It is hard to work with multiplicities since they do not have a characterization, as far as I know. –  Hung Nguyen Apr 29 '13 at 15:22

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