Document Type: Research Paper


Department of Mathematics, Tehran-North Branch, Islamic Azad University, Tehran, Iran


Throughout this paper, every groups are fi nite. The prime graph of a group $G$ is denoted by $\Gamma(G)$. Also $G$ is called recognizable by prime graph if for every fi nite group $H$ with $\Gamma(H) = \Gamma(G)$, we conclude that $G\cong H$. Until now, it is proved that if $k$ is an odd number and $p$ is an odd prime number, then $PGL(2,p^k)$ is recognizable by prime graph. So if $k$ is even, the recognition by prime graph of $PGL(2,p^k)$, where $p$ is an odd prime number, is an open problem. In this paper, we generalize this result and we prove that the almost simple group $PGL(2,25)$ is recognizable by prime graph.


Main Subjects

[1] Z. Akhlaghi, M. Khatami and B. Khosravi, Characterization by prime graph of PGL(2, pk) where p and k are odd, Int. J. Algebra Comp. 20 (7) (2010), 847-873.

[2] A. A. Buturlakin, Spectra of Finite Symplectic and Orthogonal Groups, Siberian Advances in Mathematics, 21 (3) (2011), 176-210.

[3] G. Y. Chen, V. D. Mazurov, W. J. Shi, A. V. Vasil’ev and A. Kh. Zhurtov, Recognition of the finite almost simple groups P GL2(q) by their spectrum, J. Group Theory 10(1) (2007), 71-85.

[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, 1985.

[5] M. A. Grechkoseeva, On element orders in covers of finite simple classical groups, J. Algebra, 339 (2011), 304-319.

[6] D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.

[7] M. Hagie, The prime graph of a sporadic simple group, Comm. Algebra 31(9) (2003), 4405-4424.

[8] G. Higman, Finite groups in which every element has prime power order, J. London Math. Soc. 32 (1957), 335–342.

[9] M. Khatami, B. Khosravi and Z. Akhlaghi, NCF-distinguishability by prime graph of P GL(2, p), where p is a prime, Rocky Mountain J. Math. (to appear).

[10] B. Khosravi, n-Recognition by prime graph of the simple group P SL(2, q), J. Algebra Appl. 7(6) (2008), 735-748.

[11] B. Khosravi, B. Khosravi and B. Khosravi, 2-Recognizability of P SL(2, p2 ) by the prime graph, Siberian Math. J. 49(4) (2008), 749–757.

[12] B. Khosravi, B. Khosravi and B. Khosravi, On the prime graph of P SL(2, p) where p > 3 is a prime number, Acta. Math. Hungarica 116(4) (2007), 295-307.

[13] R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern, Sci. China Math. 55 (4) (2012), 701-720.

[14] A. S. Kondrat’ev, Prime graph components of finite simple groups, Math. USSR-SB. 67(1) (1990), 235-247.

[15] A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups A + n (p) and A − n (p), J. Algebra Appl. 14(1) (2015), (12 pages).

[16] A. Mahmoudifar and B. Khosravi, On the characterization of alternating groups by order and prime graph, Sib. Math. J. 56(1) (2015), 125-131.

[17] A. Mahmoudifar, On finite groups with the same prime graph as the projective general linear group PGL(2, 81), (to appear).

[18] V. D. Mazurov, Characterizations of finite groups by sets of their element orders, Algebra Logic 36(1) (1997), 23-32.

[19] A. R. Moghaddamfar, W. J. Shi, The number of finite groups whose element orders is given, Beitr¨age Algebra Geom. 47(2) (2006), 463-479.

[20] D. S. Passman, Permutation groups, W. A. Bengamin, New York, 1968.

[21] A. V. Zavarnitsine, Fixed points of large prime-order elements in the equicharacteristic action of linear and unitary groups, Sib. Electron. Math. Rep. 8 (2011), 333-340.