Characterization of $G_2(q)$, where $2 < q \equiv 1(mod\ 3)$ by order components
Subject Areas : Group theory
1 - Department of Mathematics, Ilam Branch,
Islamic Azad University, Ilam, Iran
Keywords: prime graph, linear group, order component,
Abstract :
In this paper we will prove that the simple group$G_2(q)$, where $2 < q \equiv 1(mod3)$is recognizable by the set of its order components, also other word we prove that if $G$ is a finite group with $OC(G)=OC(G_2(q))$, then $G$ is isomorphic to $G_2(q)$.
[1] G. Y. Chen, A new characterization of sporadic simple groups, Algebra Colloq. 3, No. 1, 49-58(1996).
[2] G. Y. Chen, On Frobenius and 2-Frobenius group, Jornal of Southwest China Normal University, 20(5), 485-487(1995).(in Chinese).
[3] G. Y. Chen, A new characterization of P SL2(q), Southeast Asian Bull. Math., 22(3), 257-263(1998).
[4] G. Y. Chen, Characterization of 3D4(q), Southeast Asian Bull. Math., 25, 389-401(2001).
[5] G. Y. Chen and H.Shi, 2Dn(3)(9 ⩽ n = 2m + 1 not a prim) can be characterized by its order components, J. Appl. Math. Comput., 19(1-2), 353-362(2005).
[6] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford 1985.
[7] M.R.Darafsheh and A.Mahmiani, A quantitative characterization of the linear groups Lp+1(2), Kumamoto J. Math., 20, 33-50(2007).
[8] M.R.Darafsheh, Characterizability of the group 2Dp(3) by its order components, where p ⩾ 5 is a prime number not of the form 2m + 1, Acta Math. Sin., (Engl. Ser) 24(7), 1117-1126(2008).
[9] M.R.Darafsheh and A.Mahmiani, A characterization of the group 2Dn(2), where n = 2m + 1 ⩾ 5, J. Appl. Math. Comput., 31(1-2), 447-457(2009).
[10] M.R.Darafsheh, Characterization of the groups Dp+1(2) and Dp+1(3) using order components, J. Korean Math. Soc., 47(2), 311-329(2010).
[11] M.R.Darafsheh and M. Khademi, Characterization of the groups Dp(q) by order components, where p ⩾ 5 is a prime and q = 2, 3 or 5, (manuscript).
[12] A. Iranmanesh, S.H. Alavi and B. Khosravi, A characterization of P SL(3, q), where q is an odd prime power, J. Pure Appl. Algebra, 170(2-3), 243-254(2002).
[13] A. Iranmanesh, S.H. Alavi and B. Khosravi, A characterization of P SL(3, q) for q = 2n, Acta Math. Sin.(Engl. Ser.), 18(3), 463-472(2002).
[14] A. Iranmanesh, B. Khosravi and S.H. Alavi, A characterization of P SU(3, q) for q > 5, South Asian Bull. Math., 26(2), 33-44(2002).
[15] M. Khademi, Characterizability of finite simple groups by their order components: a summary of resoults, International Journal of Algebra, vol. 4, no.9, 413-420(2010).
[16] Behrooz Khosravi and Bahnam Khosravi, A characterization of E6(q), Algebras, Groups and Geometries, 19, 225-243(2002).
[17] Behrooz Khosravi and Bahnam Khosravi, A characterization of 2E6(q), Kumamoto J. Math., 16, 1-11(2003).
[18] A. Khosravi and B. Khosravi, A characterization of 2Dn(q), where n = 2m, Int. J. Math., Game theory and algebra, 13, 253-265(2003).
[19] A. Khosravi and B. Khosravi, A new characterization of P SL(p, q), Comm. Alg., 32, 2325-2339(2004).
[20] Bahman Khosravi, Behnam Khosravi and Behrooz Khosravi, A new characterization of P SU(p, q), Acta Math. Hungar., 107(3), 235-252(2005).
[21] A. Khosravi and B. Khosravi, r-recognizability of Bn(q) and Cn(q), where n = 2m ⩾ 4, Journal of pure and applied alg.,199, 149-165(2005).
[22] Behrooz Khosravi, Bahman Khosravi and Behnam Khosravi, Characterizability of P SL(p + 1, q) by its order components, Houston Journal of Mathematics, 32(3), 683-700(2006).
[23] A. Khosravi and B. Khosravi, Characterizability of P SU(p + 1, q) by its order components, Rocky mountain J. Math., 36(5), 1555-1575(2006).
[24] A.S.Kondratev, On prime graph components of finite simple groups, Mat. Sb. 180, No. 6, 787-797, (1989).
[25] H. Shi and G.Y. Chen, 2Dp+1(2)(5 ⩽ p ̸= 2m − 1) can be characterized by its order components, Kumamoto J. Math., 18, 1-8(2005).
[26] J.S.Williams, Prime graph components of finite groups, J. Alg. 69, No.2,487-513(1981).
[27] K.Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys.3, no. 1, 265-284 (1892).