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

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].

Keywords

Simple group / prime graph / degree of a vertex / degree pattern

References

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

[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.

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

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

[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.

[7] Iranmanesh A., Alavi S. H., Khosravi B. A characterization of PSL₃(q) for q = 2^m. Acta Math Sinica (English Ser), 2002, 18(3): 463-472.

[8] Iranmanesh A., Alavi S. H., Khosravi B. A characterization of PSU₃(q) for q > 5. Southeast Asian Bulletin of Math, 2002, 26(1): 33-44.

[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 connected components of their prime graphs are complete. Journal of Group Theory, 2004, 7(3): 373-384.

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

[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.

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

[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. Group Theory, Proceeding of the 1987 Singapore Group Theory Conference, 1989, Berlin/New York: Walter de Gruyter, 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. Kovacs L. G. A characteristic property for each finite projective special linear group. Groups-Canberra 1989, 1990, Berlin: Springer, 171-180.

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

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

[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.

[25] Zhang L C, Shi W J. OD-characterization of simple K₄-groups. Algebra Colloquium (in press)