Abstract
Let G be a solvable group and let Δ(G) be the character degree graph of G. The vertices of Δ(G) are the primes dividing character degrees of G and there is an edge between two primes if they divide a common character degree of G. In this paper, we show that the Taketa inequality dl(G) ≤ | cd(G)| holds when G is a solvable group whose degree graph Δ(G) has diameter 3.
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Aziziheris, K., Lewis, M.L.: Taketa’s theorem for some character degree sets. Arch. Math. (Basel) 100(3), 215–220 (2013)
Dugan, C.: Solvable Groups Whose Character Degree Graphs Have Diameter Three, Ph.D. Thesis: Kent State University (2007)
Garrison, S.: On Groups with a Small Number of Character Degrees, Ph.D. Thesis: University of Wisconsin, Madison (1973)
Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1976)
Isaacs, I.M., Knutson, G.: Irreducible character degrees and normal subgroups. J. Algebra 199, 302–326 (1998)
Keller, T.M.: Orbits in finite group actions. In: Proceedings of Groups St. Andrews 2001 in Oxford, London Mathematical Society Lecture Notes Series, vol. 305, pp. 306–331 (2003)
Kildetoft, T.: Bounding the derived length of a solvable group: An improvement on a result by Gluck. Comm. Algebra 40(5), 1856–1859 (2012)
Lewis, M.L.: An overview of graphs associated with character degrees and conjugacy class sizes in finite groups. Rocky Mountain J. Math. 38(1), 175–210 (2008)
Lewis, M.L.: Solvable groups with degree graphs having 5 vertices and diameter 3. Comm. Algebra 30(11), 5485–5503 (2002)
Lewis, M.L.: Derived lengths of solvable groups having five irreducible character degrees. I. Algebr. Represent. Theory 4(5), 469–489 (2001)
Lewis, M.L.: Derived lengths and character degrees. J. Algebra 126(7), 1915–1921 (1998)
Manz, O., Wolf, T.R.: Representations of Solvable Groups, London Mathematical Society Lecture Note Series, vol. 185. Cambridge University Press, Cambridge (1993)
Pálfy, P.P.: On the character degree graph of solvable groups, I: Three primes. Period. Math. Hungar. 36(1), 61–65 (1998)
Sass, C.B.: Character Degree Graphs of Solvable Groups with Diameter Three, To appear in J. Group Theory
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Presented by Radha Kessar.
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Lewis, M.L., Sass, C.B. The Taketa Problem and Character Degree Graphs with Diameter Three. Algebr Represent Theor 18, 1395–1399 (2015). https://doi.org/10.1007/s10468-015-9546-7
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DOI: https://doi.org/10.1007/s10468-015-9546-7