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Simple Groups Whose Gruenberg–Kegel Graph or Solvable Graph is Split

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Abstract

A graph is split if there is a partition of its vertex set into a clique and an independent set. In this paper, we determine when the Gruenberg–Kegel graph, the solvable graph, and the compact forms of these graphs associated with finite nonabelian simple groups are split. In particular, it is proved that the compact form of the Gruenberg–Kegel graph of any finite simple group is split.

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Notes

  1. In fact, Bang [6] proved in 1886 that \(n^i-1\) has a primitive prime divisor for all \(n\geqslant 2\) and \(i>2\) except for \(n=2\) and \(i=6\). Then, Zsigmondy [37] proved in 1892 that for coprime integers \(a> b\geqslant 1\) and \(i> 2\), there exists a prime r dividing \(a^i-b^i\) but not \(a^k-b^k\) for \(1\leqslant k<i\), except when \(a=2\), \(b=1\), and \(i=6\).

References

  1. Abe, S., Iiyori, N.: A generalization of prime graphs of finite groups. Hokkaido Math. J. 29(2), 391–407 (2000)

    Article  MathSciNet  Google Scholar 

  2. Aleeva, M.R.: On the composition factors of finite groups having the same set of element orders as the group \(U_3(q)\). Sib. Math. J. 43, 195–211 (2002)

    Article  MathSciNet  Google Scholar 

  3. Amberg, B., Kazarin, L.: On the soluble graph of a finite simple group. Comm. Algebra 41(6), 2297–2309 (2013)

    Article  MathSciNet  Google Scholar 

  4. Artin, E.: The orders of the linear groups. Comm. Pure Appl. Math. 8, 355–365 (1955)

    Article  MathSciNet  Google Scholar 

  5. Artin, E.: Collected Papers, pp. 8–10. Addison-Wesley, Boston (1965)

    Book  Google Scholar 

  6. Bang, A.S.: Taltheoretiske Unders\(\phi \)gelser. Tidsskrift Math., 4(5), 70–80 and 130–137 (1886)

  7. Brandl, R., Shi, W.J.: A characterization of finite simple groups with abelian Sylow \(2\)-subgroups. Ricerche Mat. 42(1), 193–198 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Cao, H.P., Chen, G., Grechkoseeva, M.A., Mazurov, V.D., Shi, W.J., Vasil’ev, A.V.: Recognition of the finite simple groups \(F_4(2^m)\) by the spectrum. Sib. Math. J. 45(6), 1031–1035 (2004)

    Article  Google Scholar 

  9. Carter, R.W.: Simple Groups of Lie Type, Pure and Applied Mathematics, vol. 28. Wiley, London (1972)

    Google Scholar 

  10. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, oxford (1985)

    MATH  Google Scholar 

  11. Darafsheh, M.R., Moghaddamfar, A.R., Zokayi, A.R.: A recognition of simple groups \({\rm PSL}(3, q)\) by their element orders. Acta Math. Sci. Ser. B (Engl. Ed.) 24(1), 45–51 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Deng, H., Shi, W.J.: The characterization of Ree groups \(^2F_4(q)\) by their element orders. J. Algebra 217(1), 180–187 (1999)

    Article  MathSciNet  Google Scholar 

  13. Földes, S., Hammer, P.L.: Split graphs. In: Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 311–315. Louisiana State University, Baton Rouge (1977)

  14. Gruber, A., Keller, T., Lewis, M.L., Naughton, K., Strasser, B.: A characterization of the prime graphs of solvable groups. J. Algebra 442, 397–422 (2013)

    Article  MathSciNet  Google Scholar 

  15. Gupta, R., Ram Murty, M.: A remark on Artin’s conjecture. Invent. Math. 78, 127–130 (1984)

    Article  MathSciNet  Google Scholar 

  16. Hagie, M.: The prime graph of a sporadic simple group. Comm. Algebra 31(9), 4405–4424 (2003)

    Article  MathSciNet  Google Scholar 

  17. Hammer, P.L., Simeone, B.: The splittance of a graph. Combinatorica 1(3), 275–284 (1981)

    Article  MathSciNet  Google Scholar 

  18. Heath-Brown, D.R.: Artin’s conjecture for primitive roots. The Quarterly Journal of Mathematics 37(1), 27–38 (1986)

    Article  MathSciNet  Google Scholar 

  19. Huppert, B., Blackburn, N.: Finite Groups, vol. 3. Springer, Berlin (1982)

    Book  Google Scholar 

  20. Isaacs, I.M.: Finite Group Theory, Graduate Studies in Mathematics, vol. 92. American Mathematical Society, Providence (2008)

    Google Scholar 

  21. Kleidman, P.B., Liebeck, M.W.: The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, vol. 129. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  22. Kondrat’ev, A.S.: On prime graph components of finite simple groups. USSR-Sb. 67(1), 235–247 (1990)

    Article  MathSciNet  Google Scholar 

  23. Kondrat’ev, A.S., Mazurov, V.D.: Recognition of alternating groups of prime degree from their element orders. Sib. Math. J. 41(2), 294–302 (2000)

    Article  Google Scholar 

  24. Lucido, M.S., Moghaddamfar, A.R.: Groups with complete prime graph connected components. J. Group Theory 7(3), 373–384 (2004)

    Article  MathSciNet  Google Scholar 

  25. Mazurov, V.D.: Recognition of finite simple groups \(S_4(q)\) by their element orders. Algebra Logic 41(2), 93–110 (2002)

    Article  MathSciNet  Google Scholar 

  26. Staroletov, A.M.: On the recognizability of the simple groups \(B_3(q)\), \(C_3(q)\) and \(D_4(q)\) by the spectrum. Sib. Math. J. 53(3), 532–538 (2012)

    Article  MathSciNet  Google Scholar 

  27. Suprunenko, D.A.: Matrix Groups, Translations of Mathematical Monographs, vol. 45. American Mathematical Society, Providence (1976)

    Google Scholar 

  28. Suzuki, M.: On the prime graph of a finite simple group—an application of the method of Feit–Thompson–Bender–Glauberman. Groups and combinatorics. In: Memory of Michio Suzuki, Advanced Studies in Pure Mathematics, pp. 41–207. Mathematical Society of Japan, Tokyo (2001)

  29. Tyshkevich, R.I., Chernyak, A.A.: Canonical partition of a graph defined by the degrees of its vertices. Isv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 5, 14–26 (1979). (in Russian)

    MATH  Google Scholar 

  30. Vasil’ev, A.V.: On connection between the structure of a finite group and the properties of its prime graph. Sib. Math. J. 46(3), 396–404 (2005)

    Article  MathSciNet  Google Scholar 

  31. Vasil’ev, A.V.: On finite groups isospectral to simple classical groups. J. Algebra 423, 318–374 (2015)

    Article  MathSciNet  Google Scholar 

  32. Vasil’ev, A.V., Vdovin, E.P.: An adjacency criterion in the prime graph of a finite simple group. Algebra Logic 44(6), 381–406 (2005)

    Article  MathSciNet  Google Scholar 

  33. Vasil’ev, A.V., Vdovin, E.P.: Cocliques of maximal size in the prime graph of a finite simple group. Algebra Logic 50(4), 291–322 (2011)

    Article  MathSciNet  Google Scholar 

  34. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall Inc, Upper Saddle River (2001)

    Google Scholar 

  35. Williams, J.S.: Prime graph components of finite groups. J. Algebra 69(2), 487–513 (1981)

    Article  MathSciNet  Google Scholar 

  36. Zavarnitsin, A.V.: On the recognition of finite groups by the prime graph. Algebra Logic 45(4), 220–231 (2006)

    Article  MathSciNet  Google Scholar 

  37. Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892)

    Article  MathSciNet  Google Scholar 

  38. Zvezdina, M.A.: On nonabelian simple groups having the same prime graph as an alternating group. Sib. Math. J. 54(1), 47–55 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The part of this work was done while the third author had a visiting position at the Department of Mathematical Sciences, Kent State University, USA. He would like to thank the hospitality of the Department of Mathematical Sciences of KSU. The authors are thankful to the referees for carefully reading the paper and their suggestions for improvements. A. V. Vasil’ev was supported by the program of fundamental scientific researches of the SB RAS No. I.1.1., Project No. 0314-2019-0001. M. A. Zvezdina was supported by RFBR according to the research Project No. 18-31-20011.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Mark L. Lewis

  2. Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box 16765–3381, Tehran, Iran

    J. Mirzajani & A. R. Moghaddamfar

  3. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. V. Vasil’ev & M. A. Zvezdina

  4. Novosibirsk State University, Novosibirsk, Russia

    A. V. Vasil’ev & M. A. Zvezdina

Authors
  1. Mark L. Lewis
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  2. J. Mirzajani
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  3. A. R. Moghaddamfar
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  4. A. V. Vasil’ev
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  5. M. A. Zvezdina
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Correspondence to A. R. Moghaddamfar.

Additional information

Communicated by Rosihan M. Ali.

Dedicated to Professor Victor Danilovich Mazurov on the occasion of his 75th birthday.

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Lewis, M.L., Mirzajani, J., Moghaddamfar, A.R. et al. Simple Groups Whose Gruenberg–Kegel Graph or Solvable Graph is Split. Bull. Malays. Math. Sci. Soc. 43, 2523–2547 (2020). https://doi.org/10.1007/s40840-019-00815-8

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  • DOI: https://doi.org/10.1007/s40840-019-00815-8

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