Document Type : Original Manuscript

Authors

1 Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395- 3697, Tehran, Iran.

2 Faculty of Mathematical Sciences, Department of Pure Mathematics, University of Shahrekord, P.O. Box 88186-34141, Shahrekord, Iran.

Abstract

In this paper, we prove that every finite group $ G $ with the same order and largest element order as 
$G_{2}(q)$, where $ q\leq 11 $ is necessarily isomorphic to the group $G_{2}(q)$.

Keywords

1. G. Y. Chen, On structure of frobenius groups and 2-frobenius groups, J. Southwest China Normal Univ., 20 (1995), 485–487.
2. G. Y. Chen, L. G. He and J. H. Xu, A new characterization of sporadic simple groups, Ital. J. Pure Appl. Math., 30 (2013), 373–392.
3. Y. H. Chen, G. Y. Chen and J. B. Li, Recognizing simple K4-groups by few
special conjugacy class sizes, Bull. Malays. Math. Sci. Soc., 38 (2015), 51–72.
4. D. Deriziotis, Conjugacy class and centralizers of semisimple elements in finite groups of lie type, Vorlesungen Fachbereich Math. Univ. Essen 1984.
5. B. Ebrahimzadeh, A. Iranmanesh, A. Tehranian and H. Parvizi Mosaed, A characterization of Suzuki groups by order and the largest element order, J. Sci. I. R. Iran. 27 (2016), 353–355.
6. L. G. He and G. Y. Chen, A new characterization of L2(q) where q = pn <125, Ital. J. Pure Appl. Math., 28 (2011), 127–136.
7. L. G. He and G. Y. Chen, A new characterization of some simple groups, J. Sichuan Normal Univ. (Natur. Sci)., 35 (2012), 590–594.
8. L. G. He and G. Y. Chen, A new characterization of simple K3-groups, Comm. Algebra, 40 (2012), 3903–3911.
9. L. G. He and G. Y. Chen, A new characterization of simple K4-groups, J. Math. Research Appl. 35 (2015), 400–406.
10. D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.
11. W. M. Kantor and A. Seress, Prime power graphs for groups of Lie type, J. Algebra, 247 (2002), 370–434.
12. J. B. Li, W. J. Shi and D. P. Yu, A characterization of some PGL(2; q) by maximum element orders, Bull. Korean Math. Soc., 52 (2015), 2025–2034.
13. J. B. Li, D. P. Yu, G. Y. Chen and W. J. Shi, A new characterization of simple
K4-groups of type L2(q) and their automorphism groups, Bull. Iran. Math. Soc., 43 (2017), 501–514.
14. J. S. Robinson Derek, A Course in the Theory of Groups, second edition, Springer Verlag, New York, 2003.
15. W. J. Shi, A new characterization of the sporadic simple groups, Group theory, Walter de Gruytern, Berlin-New York. (1989), 531– 540.
16. A. V. Vaseliev, M. A. Grechkoseeva and V. D. Mazurov, Characterization of finite simple groups by spectrume and order, Algebra Logic, 48 (2009), 385-409.
17. J. S. Williams, Prime graph componeents of finite group, J. Algebra, 69 (1981), 487–513.
18. D. P. Yu, G. Y. Chen, L.G. Zhang and W. J. Shi, A new characterization of simple K5-groups of type L3(p), Bull. Iran. Math. Soc., 45 (2019), 771–781.