Characterization of the Chevalley group $G_{2}(5)$ by the set of numbers of the same order elements

Document Type : Research Paper

Authors

1 Department of Mathematics, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran.

2 School of Mathematics, Statistics and computer Science, College of Science, University of Tehran, Tehran, Iran.

Abstract

Let $G$ be a group and $\omega(G)=\{o(g)|g\in G\}$ be the set of element orders of $G$. Let $k\in\omega(G)$ and $s_{k}=|\{g\in G|o(g)=k\}|$. Let $nse(G)=\{s_{k}|k\in\omega(G)\}.$ In this paper, we prove that if $G$ is a group and $G_{2}(5)$ is the Chevalley simple group of type $G_{2}$ over $GF(5)$ such that $nse(G)=nse(G_{2}(5))$, then $G\cong G_{2}(5)$.

Keywords

Main Subjects

References

  1. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of finite groups. Maximal
    subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray,
    Oxford University Press, Eynsham, 1985.
    2. G. Frobenius, Verallgemeinerung des sylowschen satze, Berliner Sitz, (1895) 981–993.
    3. M. Jr. Hall, The theory of groups, The Macmillan Co., New York, N. Y., 1959.
    4. M. Herzog, On finite simple groups of order divisible by three primes only, J. Algebra, 10 (1968) 383–388.
    5. A. Jafarzadeh and A. Iranmanesh, On simple Kn -groups for n = 5, 6, In Groups St. Andrews 2005 in
    Oxford, 2, Cambridge Univ. Press, Cambridge, (2007) 517–526.
    6. M. Jahandideh and M. R. Darafsheh, N SE characterization of the Chevalley group G2 (4), Arab J. Math.,
    7 (2018) 21–26.
    7. A. R. Khalili Asboei, S. S. S. Amiri, A. Iranmanesh and A. Tehranian, A characterization of sporadic simple
    groups by N SE and order, J. Algebra Appl., 12 (2013) 125–158.
    8. M. Khatami, B. Khosravi and Z. Akhlaghi, A new characterization for some linear groups, Monatsh. Math.,
    163 (2011) 39–50.
    9. Sh. Liu, A characterization of L3 (4), Sci. Asia, 39 (2013) 436–439.
    10. Sh. Liu, NSE characterization of projective special group L5 (2), Rend. Semin. Mat. Univ. Padova, 132
    (2014) 123–132.
    11. Sh. Liu,A characterization of projective special unitary group U3 (5) by nse, Arab J. Math. Sci., 20 (2014)
    133–140.
    12. G. A. Miller, Addition to a theorem due to Frobenius, Bull. Amer. Math. Soc., 11 (1904) 6–7.
    13. L. Mousavi and B. Taeri, A characterization of L2 (81) by nse, Int. J. Group Theory., 1 (2016) 2935.
    14. C. Shao and Q. Jiang, A new characterization of some linear groups by nse, J. Algebra Appl., 13 (2014) pp.
    9.
    15. C. Shao, W. Shi and W. Q. Jiang, Characterization of simple K4 -groups, Front. Math. China, 3 (2008)
    355–370.
    16. R. Shen, C. Shao, Q. Jiang, W. Shi and V. Mazurov, A new characterization of A5 , Monatsh. Math., 160
    (2010) 337–341.
    17. W. J. Shi, A new characterization of the sporadic simple groups, Group theory (Singapore, 1987), 531–540,
    de Gruyter, Berlin, 1989.

18. W. J. Shi, On simple K4 -group, Chin. Sci. Bul., 36 (1991) 1281–1283.

Files

Share

How to cite

Statistics