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A characterization of the finite simple group L16(2) by its prime graph

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Abstract

In this paper as the main result we prove that the projective special linear group L 16(2) is uniquely determined by its prime graph. In fact we give a positive answer to an open problem arose in Zavarnitsin (Algebra Logic 43(4), 220–231, 2006) and we obtain a first example of a finite group with connected prime graph which is uniquely determined by its prime graph.

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References

  1. Aschbacher M. and Seitz G.M. (1976). Involutions in Chevalley groups over fields of even order. Nagoya Math. J. 63: 1–91

    MATH  MathSciNet  Google Scholar 

  2. Conway J.H., Curtis R.T., Norton S.P., Parker R.A. and Wilson R.A. (1985). Atlas of Finite Groups. Oxford University Press, Oxford

    MATH  Google Scholar 

  3. Darafsheh M.R. and Moghaddamfar A.R. (2001). A characterization of the groups PSL 5(2), PSL 6(2) and PSL 7(2). Comm. Algebra 29(1): 465–475

    Article  MATH  MathSciNet  Google Scholar 

  4. Darafsheh M.R. and Moghaddamfar A.R. (2003). Corrigendum to: “A characterization of the groups PSL 5(2), PSL 6(2) and PSL 7(2)”. Comm. Algebra 32(9): 4651–4653

    MathSciNet  Google Scholar 

  5. Darafsheh M.R. and Moghaddamfar A.R. (2000). A characterization of groups related to the linear groups PSL(n, 2), n = 5, 6, 7, 8. Pure Math. Appl. 11(4): 629–637

    MathSciNet  Google Scholar 

  6. Gorenstein D. (1968). Finite Groups. Harper and Row, New York

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Iiyori N. and Yamaki H. (1993). Prime graph components of the simple groups of Lie type over the field of even characteristic. J. Algebra 155(2): 335–343

    Article  MATH  MathSciNet  Google Scholar 

  9. Khosravi A. and Khosravi B. (2007). Quasirecognition by prime graph of the simple group 2 G 2(q). Siberian Math. J. 48(3): 570–577

    Article  MathSciNet  Google Scholar 

  10. Khosravi B., Khosravi B. and Khosravi B. (2007). Groups with the same prime graph as a CIT simple group. Houston J. Math. 33(4): 967–977

    MATH  MathSciNet  Google Scholar 

  11. Khosravi B., Khosravi B. and Khosravi B. (2007). On the prime graph of PSL(2, p) where p > 3 is a prime number. Acta. Math. Hungarica 116(4): 295–307

    Article  MathSciNet  MATH  Google Scholar 

  12. Khosravi B. and Salehi Amiri S. (2006). On the prime graph of L 2(q) where q = p α < 100. Quasigroups and related systems 14: 179–190

    MATH  MathSciNet  Google Scholar 

  13. Kondrat’ev, A.S.: On prime graph components for finite simple groups, (Russian) Mat. Sb. 180(6), 787–797 (1989), 894; translation in Math. USSR-SB., 67(1), 235–247 (1990)

  14. Lucido M.S. (1999). The diameter of the prime graph of a finite group. J. Group Theory 2(2): 157–172

    Article  MATH  MathSciNet  Google Scholar 

  15. Lucido M.S. and Moghaddamfar A.R. (2006). Recognition by spectrum of some linear groups over the binary field. Siberian Math. J. 47(1): 86–96

    Article  MathSciNet  Google Scholar 

  16. Mazurov V.D. (1997). Characterizations of finite groups by sets of orders of their elements. Algebra Logic 36(1): 23–32

    Article  MathSciNet  Google Scholar 

  17. Moghaddamfar A.R. (2006). On spectrum of linear groups over the binary field and recognizability of L 12(2). Int. J. Algebra Comput. 16(2): 341–349

    Article  MATH  MathSciNet  Google Scholar 

  18. Moghaddamfar A.R., Zokayi A.R. and Khademi M. (2005). A characterization of the finite simple group L 11(2) by its element orders. Taiwanese J. Math. 9(3): 445–455

    MATH  MathSciNet  Google Scholar 

  19. Shi W.J. (1984). A characteristic property of PSL 2(7). J. Aust. Math. Soc. Ser. A 36(3): 354–356

    Article  MATH  Google Scholar 

  20. Shi W.J. (1987). A characteristic property of A 8. Acta Math. Sin. New Ser. Ser. A 3(1): 92–96

    MATH  Google Scholar 

  21. Shi W.J., Wang L.H. and Wang S.H. (2003). The pure quantitative characterization of linear groups over the binary field (Chinese). Chin. Ann. Math. Ser. A 24(6): 675–682

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Zavarnitsin, A.V.: Element orders in covering of the groups L n (q) and recognition of the alternating group A 16, (in Russian). NIIDMI Novosibrisk (2000)

  25. Zavarnitsin A.V. (2006). Recognition of finite groups by the prime graph. Algebra Logic 43(4): 220–231

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Correspondence to Behrooz Khosravi.

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This research was in part supported by a grant from IPM (No. 86200023).

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Khosravi, B., Khosravi, B. & Khosravi, B. A characterization of the finite simple group L16(2) by its prime graph. manuscripta math. 126, 49–58 (2008). https://doi.org/10.1007/s00229-007-0160-9

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  • DOI: https://doi.org/10.1007/s00229-007-0160-9

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