A new characterization of Projective Special Unitary groups $\mathbf{ U_3 (3^n )}$ by the order of group and the number of elements with the same order

Document Type : Research Paper

Authors

1 Department of Mathematics, Persian Gulf University, Bushehr, Iran

2 Department of Mathematics, Tarbiat Modares University, Tehran, Iran

10.29252/as.2022.2675

Abstract

In this paper, we prove that projective special unitary groups $U_3 (3^n)$, where $ 3^{2n}-3^{n}+1$ is a prime number and $3^n\equiv\pm2(\mod 5)$, can be uniquely determined by the order of group and the number of elements with the same order.

Keywords


[1] G. Y., Chen, On the structure of Frobenius groups and 2-Frobenius groups, J. Southwest China Normal University, 20 No. 5 (1995) 485-487.
[2] J. H., Conway, R. T., Curtis, S. P., Norton, R. A., Parker and R. A., Wilson Atlas of Finite Groups, Clarendon Press, New York, 1985.
[3] B., Ebrahimzadeh, A., Iranmanesh, and H. Parvizi Mosaed, A new characterization of Ree group 2G2(q) by the order of group and number of elements with same order, Int. J. Group Theory, 6 No. 4 (2017) 1-6.
[4] B. Ebrahimzadeh and R. Mohammadyari, A new characterization of Suzuki groups, Arch. Math., 55 No. 1 (2019) 17-21.
[5] B. Ebrahimzadeh and R. Mohammadyari, A new characterization of symplectic groups C2(3n), Acta Comment. Univ. Tartu. Math., 23 No. 1 (2019) 117-124.
[6] B. Ebrahimzadeh, A new characterization of projective special linear group l3(q), Algebra Discrete Math., 31 No. 2 (2021) 212-218.
[7] G. Frobenius, Verallgemeinerung des sylowschen satze, Berliner sitz., (1895) 981-983.
[8] D. Gorenstein, Finite groups, Harper and Row, New York, 1980.
[9] A. Khalili Asboei and A. Iranmanesh, Characterization of the linear groups L2(p), Czechoslov. Math. J., 64 No. 139 (2014) 459-464.
[10] A. Khalili Asboei and A. Iranmanesh, A characterization of Symmetric group Sr, where r is prime number, Ann. Math. Inform., 40 (2012) 13-23.
[11] A. Khalili Asboei, S. S. S. Amiri, A. Iranmanesh and A. Tehranian, A new characterization of sporadic simple groups by NSE and order, J. Algebra Appl., 12 No. 2 (2013) 1250158.
[12] M. Khatami,B. Khosravi and Z. Akhlaghi, A new characterization for some linear groups, Monatsh. Math., 163 (2011) 39-50.
[13] A. Khosravi and B. Khosravi, A new characterization of some alternating and symmetric groups (II), Houston J. Math., 30 No. 4 (2004) 465-478.
[14] A. S. Kondrat'ev, Prime graph components of finite simple groups, Math. USSR-Sb., 67 No. 1 (1990) 235-247.
[15] V. D. Mazurov and E. I. Khukhro, Unsolved problems in group theory, The Kourovka Notebook, 16 ed., Inst. Mat. Sibirsk. Otdel. Akad. Novosibirsk, 2006.
[16] H. Parvizi Mosaed, A. Iranmanesh and A. Tehranian, Characterization of suzuki group by nse and order of group, Bull. Korean Math. Soc., 53 No. 3 (2016) 651-656.
[17] C. G. Shao, W. J. Shi and Q. H. Jiang, Characterization of simple K4-groups, Front. Math. China, 3 No. 3 (2008) 355-370.
[18] C. G. Shao, W. J. Shi and Q. H. Jiang, A new characterization of some linear groups by nse, J. Algebra Appl., 13 No. 2 (2014) 1350094.
[19] W. J. Shi, Pure quantitative characterization of each finite simple groups, Front. Math. China, 2 No. 1 (2007) 123-125.
[20] W. J. Shi, A new characterization of the sporadic simple groups, Group Theory-Porc. Singapore Group Theory Conf., (1989) 531-540.
[21] W. J. Shi, A characterization of U3(2n) by their element orders, J. Southwest China Normal University, 25 No. 4 (2000) 353-360.
[22] A. V. Vasilev, M. A. Grechkoseeva and V. D. Mazurov, Characterization of finite simple groups by spectrum and order, J. Algebra and Logic, 48 No. 6 (2009) 385-409.
[23] J. S. Williams, Prime graph components of finite groups, J. Algebra, 69 No. 2 (1981) 487-513.
[24] A. V. Zavarnitsine, Recognition of the simple groups L3(q) by element orders, J. Group Theory, 7 No. 1 (2004) 81-97.