TITLE: Is this a well-known bound for the derived length of a finite group?
QUESTION [13 upvotes]: Let $cd(G)$ be the set of degrees of the irreducible complex characters of the finite group $G$.
It is conjectured that if $G$ is solvable then $dl(G)\leq |cd(G)|$ and it is a result by Gluck that $dl(G) \leq 2|cd(G)|$.
I have managed to show that $dl(G)\leq 2|cd(G)|-3$ and I was wondering if this is a well-known bound and whether even better bounds are known.
Edit Forgot to mention that this is for $|cd(G)|\geq 3$.

REPLY [8 votes]: Taketa proved that $dl(G) \le |cd(G)|$ for M-groups, so this is called the "Taketa inequality", and it is conjectured that it holds for all solvable groups. I think that perhaps G. Seitz first posed that conjecture, but it is sometimes attributed to me. I was the first person to prove any bound at all, and then Gluck squeezed a better bound out of my method. (My bound was something like $3|cd(G)|$.)
Although no one has succeeded in proving the "Taketa inequality" for all solvable groups, examples suggest that much better bounds might exist. Perhaps some sort of logarithmic bound is the correct one. The situation seems to be wide open even for $p$-groups, where the Taketa inequality definitely does hold because $p$-groups are M-groups. I think that it is unknown, for example, if a $p$-group $G$ with $|cd G| = 4$, can have derived length as large as $4$. (If this is known for $4$, I am sure it is not known for $5$.)