A characterization of the Suzuki group Sz(2^9) by the set of the number of elements with the same order

Document Type : Original Paper


1 Alvand Institute of Higher Education, Hamedan, Iran

2 Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran


Let G be a group, and let nse(G) be the set of the number of elements with the same order in group G. In this paper, we prove that if G is a group and Sz(2^9) is the Suzuki simple group such that nse(G) = nse(Sz(2^9)), then G is isomorphic to the Suzuki simple group Sz(2^9). In other words, we prove that the simple Suzuki group Sz(2^9) is uniquely determined by its set of the number of elements with the same order. Consequently, Thompson’s problem is true for the Suzuki simple group Sz(2^9 ).


Main Subjects

[1] Alavi S.H., Daneshkhah A. and Parvizi Mosaed H., On Quantitative Structure of Small Ree Groups, Communications in Algebra, 45 (2017) 4099–4108.
[2] Alavi S.H., Daneshkhah A. and Parvizi Mosaed H., Finite groups of the same type as Suzuki groups, International Journal of Group Theory, 8 (2019) 35–42.
[3] Asgary S., and Ahanjideh N., A characterization of Sz(8) by nse, The 6th National Group Theory Conference, Golestan University, Gorgan, Iran, (2014) 50–54.
[4] Babai A. and Khatami M., NSE characterization of some Suzuki groups, Quasigroups and Related Systems, 27 (2019) 15–24.
[5] Frobenius G., Verallgemeinerung des sylowschen Satze, Berliner Sitz, (1895) 981–993.
[6] Gorenstein D., Finite Groups, New York, Harper and Row, 1980.
[7] Hall M., The Theory of Group, New York, Macmillan Co., 1959.
[8] Huppert B. and Blackburn N., Finite Groups III, Berlin, Springer Verlag, 1982.
[9] Mazurov V.D. and Khukhro E.I., Unsolved problems in group theory, The Kourovka Notebook, 16 ed. Inst. Mat. Sibirsk. Otdel. Akad. Novosibirsk, 2006.
[10] Miller G., Addition to a theorem due to Frobenius, Bulletin of the American Mathematical Society, 11 (1904) 6–7.
[11] Parvizi Mosaed H., Iranmanesh A. and Tehranian A., A characterization of the small Suzuki groups by the number of the same element order, Journal of Sciences, Islamic Republic of Iran, 26 (2015) 171–177.
[12] Shao C.G., ShiW.J. and Jiang Q.H., Characterization of simple K4-groups, Frontiers ofMathematics in China, 3 (2008) 355–370.
[13] Shen R., Shao C., Jiang Q., Shi W.J. and Mazurov V.D., A new characterization of A5, Monatshefte für Mathematik, 160 (2010) 337–341.
Volume 11, Issue 4 - Serial Number 4
December 2021
Pages 679-685
  • Receive Date: 14 June 1400
  • Revise Date: 23 July 1400
  • Accept Date: 03 August 1400
  • First Publish Date: 01 October 1400