OD-Characterization of all simple groups whose orders are less than 10

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 ≠ A₁₀, U₄(2), then G ≅ M; (b) If M = A₁₀, then G ≅ A₁₀ or J₂ × Z₃; (c) If M = U₄(2), then G is isomorphic to a 2-Frobenius group or U₄(2). In particular, all simple groups whose orders are less than 10⁸ but A₁₀ and U₄(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].

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)
2. Chen G Y . A new characterization of PSL₂(q). Southeast Asian Bulletin of Math, 1998, 22(3): 257–263
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)
4. Conway J H, Curtis R T, Norton S P, et al.. Atlas of Finite Groups. London/New York: Clarendon Press(Oxford), 1985
5. Gorenstein D . FiniteGroups. New York: Harper and Row, 1980
6. Iranmanesh A, Alavi S H, Khosravi B . A characterization of PSL₃(q) where q is an odd prime power. Journal of Pure and Applied Algebra, 2002, 170(2–3): 243–254. doi:10.1016/S0022‐4049(01)00113‐X
7. Iranmanesh A, Alavi S H, Khosravi B . A characterization of PSL₃(q) for q = 2^m. Acta MathSinica (English Ser), 2002, 18(3): 463–472. doi:10.1007/s101140200164
8. Iranmanesh A, Alavi S H, Khosravi B . A characterization of PSU₃(q) for q > 5. SoutheastAsian Bulletin of Math, 2002, 26(1): 33–44. doi:10.1007/s100120200024
9. Iranmanesh A, Khosravi B . A characterization of C₂(q) where q > 5. Comment Math Univ Carolinae, 2002, 43(1): 9–21
10. Kondratev A S . On prime graph components of finite simple groups. Math Sb, 1989, 180(6): 787–797
11. Lucido M S, Moghaddamfar A R . Groups in which all the connectedcomponents of their prime graphs are complete. Journal of Group Theory, 2004, 7(3): 373–384. doi:10.1515/jgth.2004.013
12. Moghaddamfar A R, Zokayi A R . Recognizing finite groupsthrough order and degree pattern. AlgebraColloquium, 2008, 15(3): 449–456
13. Moghaddamfar A R, Zokayi A R . OD-characterizations of certainfinite 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 degreesof vertices of their prime graphs. AlgebraColloquium, 2005, 12(3): 431–442
15. Passman D . PermutationGroups. New York: Benjamin Inc, 1968
16. Shi W J . A characteristic property of A₈. Acta Math Sinica(New Ser), 1987, 3(1): 92–96
17. Shi W J . A new characterization of some simple groups of Lie type. Contemporary Math, 1989, 82: 171–180
18. Shi W J . A new characterization of the sporadic simple groups.In: Group Theory, Proceeding of the 1987 SingaporeGroup Theory Conference, Berlin/NewYork: Walter de Gruyter, 1989, 531–540
19. Shi W J . Pure quantitative characterization of finite simple groups(I). Progress in Natural Science, 1994, 4(3): 316–326
20. Shi W J, Bi J X . A characteristic propertyfor each finite projective special linear group. In: Kovacs L G, ed. Groups-Canberra1989. Lecture Notes in Mathematics, Vol 1456. Berlin: Springer, 1990, 171–180
21. Shi W J, Bi J X . A characterization of Suzuki-Reegroups. Science in China, Ser A, 1991, 34(1): 14–19
22. Shi W J, Bi J X . A new characterization ofthe alternating groups. Southeast AsianBulletin of Math, 1992, 16(1): 81–90
23. Shi W J, Lipschutz S . A classification of finitesimple 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. doi:10.1016/0021‐8693(81)90218‐0
25. Zhang L C, Shi W J . OD-characterization of simple K₄-groups. Algebra Colloquium (in press)