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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

2 Answers 2

up vote 10 down vote accepted

In order to try getting this question off the unanswered list, here is an answer:

The result turned out not to be previously known, and has now been published in Communications in Algebra 40 (2012), no. 5, 1856–1859, "Bounding the derived length of a solvable group: an improvement on a result by Gluck" (http://ams.org/mathscinet-getitem?mr=2924487).

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).

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@JackSchmidt Thank you, I edited in the link. –  Tobias Kildetoft Feb 17 at 8:03

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 $5$.)

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I am not aware of examples for $p$-groups. I know that there is an example of a solvable group $G$ with $|cd(G)| = dl(G) = 5$, but I don't know if it has been determined whether or not there can be an example with $cd(G) = 6$ (the case with $5$ is not at all new after all). –  Tobias Kildetoft Feb 7 at 8:38
    
BTW, I find that this problem (the Taketa inequality) has some interesting and unusual properties: It is (at least half) conjectured that an even stronger result should hold, and we know it to hold in a rather large number of special cases. And yet, we still somehow seem to be far from being able to prove it (at least this is my impression). –  Tobias Kildetoft Feb 7 at 9:46

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