# The Validity of a Thompson’s Problem for PSL(4, 7)

Document Type : Original Article

Authors

1 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)

2 Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran

Abstract

Let \$pi_e(G)\$ be the set of elements orders of \$ G\$. Also let \$ s_n\$ be the number of elements of order \$n\$ in \$G \$ and \${rm nse}(G)= lbrace s_nmid nin pi_e(G) rbrace \$.
In this paper we prove that if \$ G\$ is a group such that \${rm nse}(G)= {rm nse}(rm PSL(4,7)) \$, \$19bigvert|G|\$ and \$19^2nmid|G|\$, then \$ Gcong rm PSL(4,7)\$. As a consequence of this result it follows that Thompson's problem is satisfied for the simple group \$rm PSL(4,7)\$.

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#### References

[1] A. K. Asboei, S. S. S. Amiri, A. Iranmanesh, A. Tehranian, A characterization of sporadic simple groups by nse and order, J. Algebra Appl, 12(2) (2013) 1250158(3 pages).
[2] A. K. Asboei, S. S. S. Amiri, A. Iranmanesh, A new characterization of PSL(2, q) for some q, Ukrainian Mathematical Journal, 67 (9) (2016) 1297–1305.
[3] Chen .Deqin, A characterization of PSU(3, 4) by nse, International Journal of Algebra and Statistics, 2(1) (2013) 51–56.
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Maximal subgroups and ordinary characters for simple groups, Clarendon Press New York, 1985.
[5] J. D. Frobenius, Verallgemeinerung des Sylowschen Satse, Berliner Sitz, (1895) 981–993.
[6] M. Hall, The Theory of Groups, Macmillan, New York 1959.
[7] A. Jafarzadeh, A. Iranmanesh, On simple Kn-groups for n = 5, 6, London Mathematical Society Lecture Note Series, 340(2) (2005) 517-526.
[8] M. Khatami, B. Khosravi, Z. Akhlaghi, A new characterization for some linear groups, Monatsh. Math, 163 (2009) 39-50.
[9] S. Liu, A characterization of PSL(3, 4), Science Asia, 39 (2013) 436–439.
[10] Leila. Mousavi, Bijan. Taeri, A characterization of L2(81) by nse, International Journal of Group Theory, 1(5) (2016) 29-35.
[11] C. G. Shao, W. J. Shi, Q. H. Jiang, A new characterization of simple K3-groups, Adv. Math, 38 (2009) 327-330.
[12] C. G. Shao, Q. Jiang, New characterization of simple linear groups by nse, J. Algebra Appl, 13, 1350094 (2014). [13] C. Shao, W. Shi, Q. Jiang, A new characterization of simple K4-groups, Front. Math. China, 3 (2008) 355-370.
[14] R. Shen, C. Shao, Q. Jiang, W. Shi, V. Mazurov, A new characterization of A5, Monatsh. Math, 160 (2010) 337-341. [15] W.J. Shi, On simple K4-groups, Chine.Sci. Bull, 36 (1991) 1281-1283.
[16] The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.8.3 (2016) http://www.gap-system.org
[17] Yong. Yang, Shitian. Liu, A characterization of some linear groups by nse, An. Stiint. Univ. Al. I. Cuza Iasi Mat. (N.S.), vol. 2 (2016).

### History

• Receive Date: 21 April 2019
• Revise Date: 11 June 2019
• Accept Date: 12 June 2019