Recognition of the group $G_2(5)$ by the prime graph

Subject Areas : History and biography

P. Nosratpour ¹ (Department of mathematics, ILam Branch, Islamic Azad university, Ilam, Iran)
M. R. Darafsheh ² (School of Mathematics, statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran)

Received: 2012-01-01 Accepted : 2012-01-01 Published : 2012-06-01

Keywords: prime graph, recognition, linear group,

Abstract :

Let $G$ be a finite group. The prime graph of $G$ is a graph $\Gamma(G)$ with vertex set $\pi(G)$, the set of all prime divisors of $|G|$, and two distinct vertices $p$ and $q$ are adjacent by anedge if $G$ has an element of order $pq$. In this paper we prove that if $\Gamma(G)=\Gamma(G_2(5))$, then $G$has a normal subgroup $N$ such that $\pi(N)\subseteq\{2,3,5\}$ and $G/N\equivG_2(5)$.

References:

[1] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford 1985.
[2] M.R.Darafsheh, Order of elements in the groups related to the general linear groups, Finite Fields Appl., 11, 738-747(2005).
[3] M.R.Darafsheh, Pure characterization of the projective special linear groups, Italian Journal of Pure and Applied Mathematics, No.23, 229-244(2008).
[4] M.R.Darafsheh and Y. Farjami, Calculating the set of elements in the nite linear groups, Journal of Discrete Mathematical Sciences, Vol.10, No.5, 637-653(2007).
[5] M.Hagie, The prime graph of a sporadic simple group, Comm.Alg., vol.31, No.9, 4405-4424(2003).
[6] G.Higman, Finite groups in which every element has prime power order, J.Landan Math. Soc., 32, 335-342(1957).
[7] B. Khosravi, Quasirecognition by prime graph of L10(2), Siberian Math.J., Vol.50, No.2, 355-359(2009).
[8] Behrooz Khosravi, Bahman Khosravi and Behnam Khosravi, A characterization of the nite simple group L16(2) by its prime graph, Manuscripta Math., 126, 49-58(2008).
[9] A.S.Kondratiev, On prime graph components for nite simple groups, Math.Sb., 180, No.6, 787-797(1989).
[10] M.S.Lusido and A.R.Moghaddamfar, Groups with complete prime graph connected components, Journal of Group theory, 31, 373-384(2004).
[11] V.D.Mazurov, M.C.Xu and H.P.Cao, Recognition of nite simple groups L3(2m) and U3(2m) by their element orders, Algebra Logika, 39, No.5, 567-585(2000).
[12] V.D.Mazurov, Recognition of nite simple groups S4(q) by their element orders, Algebra and Logic, Vol.41, No.2, 93-110(2002).
[13] V.D.Mazurov and G.Y.Chen, Recognisability of nite simple groups L4(2m) and U4(2m) by spectrum, Algebra and Logic, Vol.47, No.1, 49-55(2008).
[14] V.D.Mazurov, Characterization of nite groups by sets of element orders, Algebra and Logic, 36, No.1, 23-32(1997).
[15] D.S.Passman, Permutation groups, W.A.Benjamin Inc., New York, (1968).
[16] W.J.Shi and W.Z.Yang, A new characterization of A5 and the nite groups in which every non-identity element has prime order, J.Southwest China Teachers College(1984), 9-36(in chinese).
[17] A.V.Vasilev, On connection between the structure of a nite group and the properties of its prime graph, Siberian Math.J., 46, No.3, 396-404(2005).
[18] A.V.Vasilev and M.A.Grechkoseeva, On recognition by spectrum of nite simple linear groups over fields of characteristic 2, Siberian Math.J., Vol.46, No.4, 593-600(2005).
[19] J.S.Williams, Prime graph components of nite groups, J. Alg. 69, No.2,487-513(1981).
[20] A.V.Zavarnitsine, Recognition of nite groups by the prime graph, Algebra and Logic, Vol.45, No.4,(2006).
[21] A.V.Zavarnitsine, Finite simple groups with narrow prime spectrum, arXiv: 0810.0568v1(2008).