PDF
(183KB)
Abstract
Let G be a finite group, and let πe(G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G ≅ M if and only if |G| = |M| and πe(G) = πe(M). In this short paper, we prove that if G is a finite group, then G ≅ M if and only if |G| = |M| and πe(G) = πe(M), where M = Dn(2) and n is even.
Keywords
Finite group
/
simple group
/
order of element
/
prime graph
Cite this article
Download citation ▾
Lingli Wang.
Characterization of finite simple group Dn(2).
Front. Math. China, 2009, 5(1): 179-190 DOI:10.1007/s11464-009-0053-1
| [1] |
Cao H. P., Shi W. J. Pure quantitative characterization of finite projective special unitary groups. Science in China, Ser A, 2002, 45(6): 761-772.
|
| [2] |
Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A. Atlas of Finite Groups, 1985, Oxford: Clarendon Press.
|
| [3] |
Mazurov V D, Khukhro E I. Unsolved Problems in Group Theory. The Kourovka Notebook. Novosibirsk, 1998, 87
|
| [4] |
Shi W. J. A new characterization of the sporadic simple groups. Group Theory-Porc Singapore Group Theory Conf, 1987, 1989, Berlin-New York: Walter de Gruyter, 531-540.
|
| [5] |
Shi W. J. A new characterization of some simple groups of Lie type. Contemporary Math, 1989, 82: 171-180.
|
| [6] |
Shi W. J. Pure quantitative characterization of finite simple groups (I). Progress in Natural Science, 1994, 4(3): 316-326.
|
| [7] |
Shi W. J., Bi J. X. A characteristic property for each finite projective special linear group. Lecture Notes in Math, 1990, 1456: 171-180.
|
| [8] |
Shi W. J., Bi J. X. A characterization of Suzuki-Ree groups. Science in China, Ser A, 1991, 34(1): 14-19.
|
| [9] |
Shi W. J., Bi J. X. A characterization of the alternating groups. Southeast Asian Bulletin of Mathematics, 1992, 16(1): 81-90.
|
| [10] |
Shi W. J., Tang C. Y. A characterization of some orthogonal groups. Progress in Natural Science, 1997, 7: 155-162.
|
| [11] |
Vasil’ev A. V. On connection between the structure of a finite group and the properties of its prime graph. Siberian Mathematical Journal, 2005, 46(3): 396-404.
|
| [12] |
Vasil’ev A. V., Vdovin E. P. An adjacency criterion for two vertices of the prime graph of a finite simple group. Algebra and Logic, 2005, 44: 381-406.
|
| [13] |
Williams J. S. Prime graph components of finite groups. J Algebra, 1981, 69: 487-513.
|
| [14] |
Xu M. C., Shi W. J. Pure quantitative characterization of finite simple groups 2Dn(q) and Dl(q) (l odd). Algebra Coll, 2003, 3: 427-443.
|
| [15] |
Zsigmondy K. Zur Theorie der Potenzreste. Monatsh Mathematical Physics, 1892, 3: 265-284.
|
