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2-Recognizability by prime graph of PSL(2, p 2)

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Abstract

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p 2)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p 2))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p 2)| and Γ(G) = Γ(PSL(2, p 2)) then G ≅ PSL(2, p 2). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.

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References

  1. Vasil’ev A. V. and Grechkoseeva M. A., “On recognition of the finite simple orthogonal groups of dimension 2m, 2m + 1, and 2m + 2 over a field of characteristic 2,” Siberian Math. J., 45, No. 3, 420–432 (2004).

    Article  MathSciNet  Google Scholar 

  2. Iiyori N. and Yamaki H., “Prime graph components of the simple groups of Lie type over the field of even characteristic,” J. Algebra, 155, No. 2, 335–343 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  3. Kondrat’ev A. S., “Prime graph components for finite simple groups,” Math. USSR-Sb., 67, No. 1, 235–247 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  4. Williams J. S., “Prime graph components of finite groups,” J. Algebra, 69, No. 2, 487–513 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  5. Lucido M. S., “The diameter of the prime graph of a finite group,” J. Group Theory, 2, No. 2, 157–172 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  6. Hagie M., “The prime graph of a sporadic simple group,” Comm. Algebra, 31, No. 9, 4405–4424 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  7. Khosravi A. and Khosravi B., “Quasirecognition by prime graph of the simple group 2 G 2(q),” Siberian Math. J., 48, No. 3, 570–577 (2007).

    Article  MathSciNet  Google Scholar 

  8. Zavarnitsin A. V., “Recognition of finite groups by the prime graph,” Algebra and Logic, 43, No. 3, 220–231 (2006).

    Article  MathSciNet  Google Scholar 

  9. Khosravi B., Khosravi B., and Khosravi B., “On the prime graph of PSL(2, p) where p > 3 is a prime number,” Acta. Math. Hungar., 116, No. 4, 295–307 (2007).

    Article  MathSciNet  Google Scholar 

  10. Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford (1985).

    MATH  Google Scholar 

  11. Lucido M. S., “Prime graph components of finite almost simple groups,” Rend. Sem. Mat. Univ. Padova, 102, 1–22 (1999).

    MATH  MathSciNet  Google Scholar 

  12. Gruenberg K. W. and Roggenkamp K. W., “Decomposition of the augmentation ideal and of the relation modules of a finite group,” Proc. London Math. Soc., 31, No. 2, 148–166 (1975).

    Article  MathSciNet  Google Scholar 

  13. Huppert B., Endliche Gruppen. I, Springer-Verlag, Berlin; Heidelberg; New York (1967).

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  15. Sierpiński W., Elementary Theory of Numbers, Panstwowe Wydawnictwo Naukowe, Warsaw (1964).

    MATH  Google Scholar 

  16. Khosravi A. and Khosravi B., “A new characterization of some alternating and symmetric groups (II),” Houston J. Math., 30, No. 4, 465–478 (2004).

    MathSciNet  Google Scholar 

  17. Crescenzo P., “A Diophantine equation which arises in the theory of finite groups,” Adv. in Math., 17, 25–29 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  18. Zsigmondy K., “Zur Theorie der Potenzreste,” Monatsh. Math. Phys., Bd 3, 265–284 (1892).

    Article  MathSciNet  Google Scholar 

  19. Shi W. J. and Bi J., “A characteristic property for each finite projective special linear group,” in: Lecture Notes in Mathematics; 1456, Springer-Verlag, 1990, pp. 171–180.

  20. Shi W., “A new characterization of the sporadic simple groups,” in: Group Theory, Proceeding of the 1987 Singapore Group Theory Conference, Walter de Gruyter, Berlin; New York, 1989, pp. 531–540.

    Google Scholar 

  21. Shi W., “Pure quantitative characterization of finite simple groups (I),” Progress in Natural Science, 4, No. 3, 316–326 (1994).

    MathSciNet  Google Scholar 

  22. Shi W., “A new characterization of some simple groups of Lie type,” Contemporary Math., 82, 171–180 (1989).

    Google Scholar 

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Authors and Affiliations

  1. University for Teacher Education, Tehran, Iran

    A. Khosravi

  2. Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

    B. Khosravi

  3. Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran

    B. Khosravi

Authors
  1. A. Khosravi
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  2. B. Khosravi
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Correspondence to B. Khosravi.

Additional information

Original Russian Text Copyright © 2008 Khosravi A. and Khosravi B.

The second author was supported in part by the grant of the Institute for Studies in Theoretical Physics and Mathematics (IPM) (No. 84200024).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 4, pp. 934–944, July–August, 2008.

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Khosravi, A., Khosravi, B. 2-Recognizability by prime graph of PSL(2, p 2). Sib Math J 49, 749–757 (2008). https://doi.org/10.1007/s11202-008-0072-2

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  • DOI: https://doi.org/10.1007/s11202-008-0072-2

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