# Is this a well-known bound for the derived length of a finite group?

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 $|cd(G)|\geq 3$.

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The bounds I (dimly) remember that were better all had extra hypotheses. They were "much" better, as in dl(G) ≤ C*|cd(G)| + D where C < 2, and the hypotheses were often "mild" (like "odd order" or so), but I don't recall any that were comparable to yours: better D with no extra hypothesis (beyond the obvious, cd(G) ≥ some small integer). Marty Isaacs is a good person to ask. –  Jack Schmidt Jul 29 '10 at 16:55
Yes, I know that for groups of odd order, we have C = 1 and D = 0 (by a result of Berger). If all character degrees are odd, combining this with Ito-Michler gives C = 1 and D = 1. Does Isaacs use this site, or how can I go about asking him? –  Tobias Kildetoft Jul 29 '10 at 21:04
For large values of $|cd(G)|$ however, a better bound is known, namely $dl(G)\leq |cd(G)| + 24\rm{log}_2(|cd(G)|) + 364$ which gives a better bound when $|cd(G)|\geq 588$ (this bound is due to Thomas Keller).
There are also better bounds known if one puts various assumptions on the group $G$. For example, if $|G|$ is odd it is a result of Berger that $dl(G)\leq |cd(G)|$, and as mentioned in the question, this bound is in fact conjectured to hold for all finite solvable groups (I just picked what is probably the simplest of the conditions known to be sufficient for the inequality to hold, as there are many others, even strictly weaker ones that $|G|$ being odd).