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Keywords:
prime graph; order component; finite group; simple group
Summary:
The order of every finite group $G$ can be expressed as a product of coprime positive integers $m_1,\dots, m_t$ such that $\pi (m_i)$ is a connected component of the prime graph of $G$. The integers $m_1,\dots, m_t$ are called the order components of $G$. Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups $C_2(q)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a conjecture by J.G. Thompson and a conjecture by W. Shi and J. Be for $C_2(q)$ with $q>5$ are obtained.
References:
[1] Chen G.Y.: On Frobenius and $2$-Frobenius group. J. Southwest China Normal Univ. 20 (5) (1995), 485-487.
[1] Chen G.Y.: A new characterization of $G_2(q)$, $[q\equiv 0 ({mod} 3)]$. J. Southwest China Normal Univ. (1996), 47-51. MR 1374160
[3] Chen G.Y.: A new characterization of sporadic simple groups. Algebra Colloq. 3 1 (1996), 49-58. MR 1374160 | Zbl 0851.20011
[4] Chen G.Y.: On Thompson's conjecture. J. Algebra 185 (1996), 184-193. MR 1409982 | Zbl 0861.20018
[5] Chen G.Y.: Further reflections on Thompson's conjecture. J. Algebra 218 (1999), 276-285. MR 1704687 | Zbl 0931.20020
[6] Chen G.Y.: A new characterization of Suzuki-Ree groups. Sci. in China (Ser. A) 27 (5) (1997), 430-433. MR 1481680
[7] Chen G.Y.: A new characterization of $E_8(q)$. J. Southwest China Normal Univ. 21 (3) (1996), 215-21. MR 1374160
[8] Chen G.Y.: A new characterization of $PSL_2(q)$. Southeast Asian Bull. Math. 22 (1998), 257-263. MR 1684163
[9] Gruenberg K.W., Roggenkamp K.W.: Decomposition of the augmentation ideal and of the relation modules of a finite group. Proc. London Math. Soc. 31 (1975), 146-166. MR 0374247 | Zbl 0313.20004
[10] Iranmanesh A., Alavi S.H.: A new characterization of $A_p$ where $p$ and $p-2$ are primes. Korean J. Comput. & Appl. Math. 8 3 (2001), 665-673. MR 1848926 | Zbl 1013.20010
[11] Iranmanesh A., Alavi S.H., Khosravi B.: A characterization of $PSL(3,q)$ where $q$ is an odd prime power. J. Pure Appl. Algebra, to appear. MR 1904845
[12] Iranmanesh A., Khosravi B.: A characterization of $F_4(q)$ where $q$ is even. Far East J. Math. Sci. 6 (2) (2000), 853-859. MR 1808700
[13] Kondtrat'ev A.S.: Prime graph components of finite groups. Math. USSR-sb. 67 (1) (1990), 235-247. MR 1015040
[14] Shi W.: A new characterization of the sporadic simple groups. Group Theory, Proceeding of the 1987 Singapore Group Theory Conference, Walter de Gruyter, Berlin, New York, 1989, pp. 531-540. MR 0981868 | Zbl 0657.20017
[15] Shi W.: Pure quantitative characterization of finite simple groups I. Progress in Natural Science 4 (3) (1994), 316-326. MR 1402664
[16] Shi W., Jianxing Bi: A new characterization of some simple groups of Lie type. Contemporary Math. 82 (1989), 171-180. MR 0982286
[17] Shi W., Jianxing Bi: A characteristic property for each finite projective special linear group. Lecture Notes in Mathematics 1456 (1990), 171-180. MR 1092230 | Zbl 0718.20009
[18] Shi W., Jianxing Bi: A new characterization of the alternating groups. Southeast Asian Bull. Math. 17 (1) (1992), 81-90. MR 1173612 | Zbl 0790.20030
[19] Steinberg R.: Automorphism of finite linear groups. Canad. J. Math. 12 (1960), 606-615. MR 0121427
[20] Williams J.S.: Prime graph components of finite groups. J. Algebra 69 (1981), 487-513. MR 0617092 | Zbl 0471.20013

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