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Classifying character degree graphs with six vertices

Abstract

We investigate prime character degree graphs of solvable groups that have six vertices. There are 112 non-isomorphic connected graphs with six vertices, of which all except nine are classified in this paper. We also completely classify the disconnected graphs with six vertices.

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References

  • Akhlaghi, Z., Casolo, C., Dolfi, S., Khedri, K., Pacifici, E.: On the character degree graph of solvable groups. Proc. Am. Math. Soc. 146(4), 1505–1513 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  • Bissler, M.W.: Character degree graphs of solvable groups. OhioLINK Electronic Theses and Dissertations Center, Columbus, Ohio. Dissertation (Ph.D.)–Kent State University (2017)

  • Bissler, M.W., Laubacher, J.: Classifying families of character degree graphs of solvable groups. Int. J. Group Theory 8(4), 37–46 (2019)

    MathSciNet  Google Scholar 

  • Bissler, M.W., Laubacher, J., Lyons, C.F.: On the absence of a normal nonabelian Sylow subgroup. Comm. Algebra (2018). https://doi.org/10.1080/00927872.2018.1498871

  • Bissler, M.W., Lewis, M.L.: A family of graphs that cannot occur as character degree graphs of solvable groups (2017). arXiv:1707.03020

  • Cvetković, D., Petrić, M.: A table of connected graphs on six vertices. Discrete Math. 50, 37–49 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  • Huppert, B.: Endliche gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer, Berlin, New York (1967)

    Google Scholar 

  • Huppert, B.: Research in representation theory at Mainz (1984–1990). In: Michler, G.O., Ringel, C.M. (eds.) Representation theory of finite groups and finite-dimensional algebras (Bielefeld, May 15–17, 1991), Progr. Math. vol. 95, pp. 17–36. Basel (1991)

  • Isaacs, I.M.: Character theory of finite groups. Dover Publications, Inc., New York (1994) (Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)])

  • Lewis, M.L.: Solvable groups whose degree graphs have two connected components. J. Group Theory 4(3), 255–275 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  • Lewis, M.L.: A solvable group whose character degree graph has diameter 3. Proc. Am. Math. Soc. 130(3), 625–630 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  • Lewis, M.L.: Solvable groups with character degree graphs having 5 vertices and diameter 3. Commun. Algebra 30(11), 5485–5503 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  • Lewis, M.L.: Classifying character degree graphs with 5 vertices. In: Finite groups 2003, pp. 247–265. Walter de Gruyter, Berlin (2004)

  • Lewis, M.L.: An overview of graphs associated with character degrees and conjugacy class sizes in finite groups. Rocky Mt. J. Math. 38(1), 175–211 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  • Pálfy, P.P.: On the character degree graph of solvable groups. I. Three primes. Period. Math. Hung. 36(1), 61–65 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  • Pálfy, P.P.: On the character degree graph of solvable groups. II. Disconnected graphs. Stud. Sci. Math. Hung. 38, 339–355 (2001)

    MathSciNet  MATH  Google Scholar 

  • Sass, C.B.: Prime character degree graphs of solvable groups having diameter three. OhioLINK Electronic Theses and Dissertations Center, Columbus, Ohio. Dissertation (Ph.D.)–Kent State University (2014)

  • Sass, C.B.: Character degree graphs of solvable groups with diameter three. J. Group Theory 19(6), 1097–1127 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  • Zhang, J.: On a problem by Huppert. Bejing Daxue Xuebao Ziran Kexue Ban 34(2–3), 143–150 (1998)

    MathSciNet  MATH  Google Scholar 

  • Zuccari, C.P.M.: Regular character degree graphs. J. Algebra 411, 215–224 (2014)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Mark W. Bissler.

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Bissler, M.W., Laubacher, J. & Lewis, M.L. Classifying character degree graphs with six vertices. Beitr Algebra Geom 60, 499–511 (2019). https://doi.org/10.1007/s13366-019-00437-y

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  • DOI: https://doi.org/10.1007/s13366-019-00437-y

Keywords

  • Character degree graphs
  • Solvable groups

Mathematics Subject Classification

  • 20C15
  • 05C25
  • 20D10