Pure quantitative characterization of finite simple groups

Abstract

Let G be a finite group, and let π_e(G) be the set of all element orders of G. In this short paper we prove that π_e(B_n(q)) ≠ π_e(C_n(q)) for all odd q.

Keywords

finite group / element orders / Lie type simple group

References

[1] Shi W. J. A new characterization of the sporadic simple groups. Group Theory, Proc of the 1987 Singapore Conf., 1989, Berlin: Walter de Gruyter, 531-540.

[2] Shi W. J., Bi J. X. A new characterization of the alternating groups. Southeast Asian Bull Math, 1992, 16: 81-90.

[3] Shi W. J., Bi J. X. A characteristic property for each finite projective special linear group. Lecture Notes in Math, Vol 1456, 1990, Berlin: Springer, 171-180.

[4] Cao H. P., Shi W. J. Pure quantitative characterization of finite projective special unitary groups. Science in China, Ser A, 2002, 45: 761-772.

[5] Xu M. C., Shi W. J. Pure quantitative characterization of finite simple groups ²D_n(q) and D_l(q) (l odd). Algebra Colloquium, 2003, 10: 427-443.

[6] Wang L L, Shi W J. A characterization of the finite simple group D_n(q). manuscript

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

[8] Shi W. J. Pure quantitative characterization of finite simple groups (I). Progress in Natural Science, 1994, 4: 316-326.

[9] Zsigmondy K. Zur Theorie der Potenzreste. Monatsh Mathematical Physics, 1892, 3: 265-284.

[10] Hestenes M. D. Singer groups. Can J Math, 1970, 22: 492-513.

[11] Williams J. S. Prime graph components of finite groups. J Algebra, 1981, 69(2): 487-513.

[12] Kleidman P., Liebeck M. The Subgroup Structure of the Finite Classical Groups, 1990, Cambridge: Cambridge University Press.

[13] Feit W., Seitz G. M. On finite rational groups and related topics. Illinois J Math, 1988, 33: 103-131.