A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). So far I have been unable to find a proof of this theorem anywhere. The only references I have seen are to Isaacs' book on character theory (where he only mentions that it has been proven by S. Garrison), and to the Ph.d thesis of S. Garrison (which has not been published, so not much help there). oes anyone know where one might find the proof?
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asked Apr 13, 2010 at 13:05
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$\begingroup$ What are $cd$ and $dl$? I read your first sentenced and wondered why is he taking absolute value of the cohomological dimension? :) $\endgroup$– Mariano Suárez-ÁlvarezCommented Apr 13, 2010 at 14:06
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2$\begingroup$ cd(G) = { χ(1) : χ in Irr(G) } is the set of character degrees of G, and dl(G) is the derived length of G. The set of character degrees, even just its size, exerts quite a bit of control over the structure of a group. This is the focus of chapter 12 of Isaacs's book, and still has lots of interesting open problems. For many groups it is quite difficult / infeasible to get the character table, but often the degrees are known, and often the degrees are all that are needed. $\endgroup$– Jack SchmidtCommented Apr 13, 2010 at 15:21
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2 Answers
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A new proof was published in:
Isaacs, I. M.; Knutson, Greg. "Irreducible character degrees and normal subgroups." J. Algebra 199 (1998), no. 1, 302–326. MR1489366 DOI:10.1006/jabr.1997.7191
This was extended to cd(G)=5 in:
Lewis, Mark L. "Derived lengths of solvable groups having five irreducible character degrees. I." Algebr. Represent. Theory 4 (2001), no. 5, 469–489. MR1870501 DOI: 10.1023/A:1012706718244
It mentions that "Because of the length and complexity of his argument, Garrison never published this result." and has some other useful comments.
answered Apr 13, 2010 at 15:41
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$\begingroup$ I looked through the first article, but none of the theorems in that is the one I mentioned. $\endgroup$ Commented Apr 13, 2010 at 16:49
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1$\begingroup$ It is Theorem C, page 303: dl(G) = dl(G')+1, cd(G) = cd(G|G')+1, and theorem C says that if cd(G|G')=3, then dl(G')≤3. This is stated twice on that page, but not in italics. $\endgroup$ Commented Apr 13, 2010 at 17:34
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Sidney Garrison wrote a 1973 dissertation directed by Marty Isaacs at Wisconsin: On Groups with a Small Number of Character Degrees. There is a related paper MR0407120 (53 #10903) 20C15, Garrison, Sidney C., Bounding the structure constants of a group in terms of its number of irreducible character degrees. J. Algebra 32 (1974), no. 3, 623–628. For a solvable group, Fitting length is shown to be bounded by the number of irreducible character degrees. Then four unrelated papers through 1986, the last with S. Gagola at Kent State (by then Garrison was apparently unaffiliated). This much I get from MathSciNet, but Marty Isaacs could fill in more details.
answered Apr 13, 2010 at 14:43
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$\begingroup$ You might also check out chapter 14, section 1, page 1ff of volume 2 of Berkovich and Zhmud's Characters of Finite Groups. It is summarizing that paper, and has some related results, but not the exact one being asked about. $\endgroup$ Commented Apr 13, 2010 at 15:37