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OD-characterization of all simple groups whose orders are less than 108

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Abstract

Let M be a simple group whose order is less than 108. In this paper, we prove that if G is a finite group with the same order and degree pattern as M, then the following statements hold: (a) If M ≠ A 10, U 4(2), then G ≅ M; (b) If M = A 10, then G ≅ A 10 or J 2 × ℤ3; (c) If M = U 4(2), then G is isomorphic to a 2-Frobenius group or U 4(2). In particular, all simple groups whose orders are less than 108 but A 10 and U 4(2) are OD-characterizable. As a consequence of this result, we can give a positive answer to a conjecture put forward by W. J. Shi and J. X. Bi in 1990 [Lecture Notes in Mathematics, Vol. 1456, 171–180].

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References

  1. Chen G Y. On structure of Frobenius and 2-Frobenius group. Journal of Southwest China Normal University, 1995, 20(5): 485–487 (in Chinese)

    Google Scholar 

  2. Chen G Y. A new characterization of PSL2(q). Southeast Asian Bulletin of Math, 1998, 22(3): 257–263

    MATH  Google Scholar 

  3. Chen Z M, Shi W J. On C pp-simple groups. Journal of Southwest China Normal University, 1993, 18(3): 249–256 (in Chinese)

    Google Scholar 

  4. Conway J H, Curtis R T, Norton S P, et al. Atlas of Finite Groups. London/New York: Clarendon Press (Oxford), 1985

    MATH  Google Scholar 

  5. Gorenstein D. Finite Groups. New York: Harper and Row, 1980

    MATH  Google Scholar 

  6. Iranmanesh A, Alavi S H, Khosravi B. A characterization of PSL 3(q) where q is an odd prime power. Journal of Pure and Applied Algebra, 2002, 170(2–3): 243–254

    Article  MATH  MathSciNet  Google Scholar 

  7. Iranmanesh A, Alavi S H, Khosravi B. A characterization of PSL 3(q) for q = 2m. Acta Math Sinica (English Ser), 2002, 18(3): 463–472

    Article  MATH  MathSciNet  Google Scholar 

  8. Iranmanesh A, Alavi S H, Khosravi B. A characterization of PSU 3(q) for q > 5. Southeast Asian Bulletin of Math, 2002, 26(1): 33–44

    Article  MATH  MathSciNet  Google Scholar 

  9. Iranmanesh A, Khosravi B. A characterization of C 2(q) where q > 5. Comment Math Univ Carolinae, 2002, 43(1): 9–21

    MATH  MathSciNet  Google Scholar 

  10. Kondratev A S. On prime graph components of finite simple groups. Math Sb, 1989, 180(6): 787–797

    Google Scholar 

  11. Lucido M S, Moghaddamfar A R. Groups in which all the connected components of their prime graphs are complete. Journal of Group Theory, 2004, 7(3): 373–384

    Article  MATH  MathSciNet  Google Scholar 

  12. Moghaddamfar A R, Zokayi A R. Recognizing finite groups through order and degree pattern. Algebra Colloquium, 2008, 15(3): 449–456

    MathSciNet  Google Scholar 

  13. Moghaddamfar A R, Zokayi A R. OD-characterizations of certain finite groups having connected prime graphs. Algebra Colloquium (in press)

  14. Moghaddamfar A R, Zokayi A R, Darafsheh M R. A characterization of finite simple groups by the degrees of vertices of their prime graphs. Algebra Colloquium, 2005, 12(3): 431–442

    MATH  MathSciNet  Google Scholar 

  15. Passman D. Permutation Groups. New York: Benjamin Inc, 1968

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Shi W J. A new characterization of some simple groups of Lie type. Contemporary Math, 1989, 82: 171–180

    Google Scholar 

  18. Shi W J. A new characterization of the sporadic simple groups. In: Group Theory, Proceeding of the 1987 Singapore Group Theory Conference, Berlin/New York: Walter de Gruyter, 1989, 531–540

    Google Scholar 

  19. Shi W J. Pure quantitative characterization of finite simple groups(I). Progress in Natural Science, 1994, 4(3): 316–326

    MathSciNet  Google Scholar 

  20. Shi W J, Bi J X. A characteristic property for each finite projective special linear group. In: Kovacs L G, ed. Groups-Canberra 1989. Lecture Notes in Mathematics, Vol 1456. Berlin: Springer, 1990, 171–180

    Google Scholar 

  21. Shi W J, Bi J X. A characterization of Suzuki-Ree groups. Science in China, Ser A, 1991, 34(1): 14–19

    MATH  MathSciNet  Google Scholar 

  22. Shi W J, Bi J X. A new characterization of the alternating groups. Southeast Asian Bulletin of Math, 1992, 16(1): 81–90

    MATH  MathSciNet  Google Scholar 

  23. Shi W J, Lipschutz S. A classification of finite simple groups. In: Proceedings of the MFI’99. 1999, 173–182

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Zhang L C, Shi W J. OD-characterization of simple K 4-groups. Algebra Colloquium (in press)

Download references

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Authors and Affiliations

  1. School of Mathematical Sciences, Suzhou University, Suzhou, 215006, China

    Liangcai Zhang & Wujie Shi

  2. College of Mathematics and Physics, Chongqing University, Chongqing, 400044, China

    Liangcai Zhang

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  1. Liangcai Zhang
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  2. Wujie Shi
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Correspondence to Wujie Shi.

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Zhang, L., Shi, W. OD-characterization of all simple groups whose orders are less than 108. Front. Math. China 3, 461–474 (2008). https://doi.org/10.1007/s11464-008-0026-9

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