A new characterization of chevalley groups $\mathbf{G_2 (q)}$

Subject Areas : Group theory

B. Ebrahimzadeh ¹ , A. R. Shabani ²

1 - State Office of Education in Qaemiyeh, Fars Province, Iran
2 - Department of Mathematics, Imam khomaini Naval Academy, Nowshahr, Iran

Received: 2022-09-05 Accepted : 2022-12-14 Published : 2022-12-01

Keywords: element order, the largest element order, the second largest element order, chevalley group,

Abstract :

‎In this paper‎, ‎we prove that chevalley groups $G_2 (q)$‎, ‎where $q\equiv \pm2($mod 5$)$ and $q^2+q+1$ is a prime number‎, ‎can be uniquely determined by the order of the group and the second largest element order‎.

References:

[1] G. Y. Chen, On the structure of Frobenius groups and 2-Frobenius groups, J. Southwest China Normal University. 20 (5) (1995), 485-487.

[2] G. Y. Chen, L. G. He, A new characterization of simple K4-group with type L2(p), Advanced in Mathematics. 43 (5) (2014), 667-670.

[3] G. Y. Chen, L. G. He, A new characterization of L2(q) where q = pn< 125, Italian J. Pure. Math. 38 (2011), 125-134.

[4] G. Y. Chen, L. G. He, J. H. Xu, A new characterization of sporadic simple groups, Italian J. Pure. Math. 30 (2013), 373-392.

[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, New York, 1985.

[6] B. Ebrahimzadeh, A. Iranmanesh, A. Tehranian, H. Parvizi Mosaed, A Characterization of the Suzuki groups by order and the largest elements order, Journal of Sciences. 27 (4) (2016), 353-355.

[7] B. Ebrahimzadeh, R. Mohammadyari, A new characterization of projective special unitary groups PSU3(3n), Discuss. Math. Gen. Algebra. Appl. 39 (2019), 35-41.

[8] B. Ebrahimzadeh, R. Mohammadyari, M. Y. Sadeghi, A new characterization of simple groups C4(q) by its order and the largestorder of elements, Acta et Commentationes Universitatis Tartuensis de Mathematica. 23 (2) (2019), 283-290.

[9] B. Ebrahimzadeh, M. Y. Sadeghi, A. Iranmanesh, A. Tehranian, A new characterization of symplectics groups PSP(8,q), Analele Stiintifice ale Universitatii Alexandru Ioan Cuza din Iasi. 66 (1) (2020), 93-99.

[10] D. Gorenstein, Finite groups, Harper and Row, New York, 1980.

[11] L. G. He, G. Y. Chen, A new characterization of L3(q) (q ≤ 8) and U3(q) (q ≤ 11), J. Southwest Univ. Natur. Sci. 27 (33) (2011), 81-87.

[12] W. M. Kantor, A. Seress, Large element orders and the characteristic of Lie-type simple groups, J. Algebra. 322 (2009), 802-832.

[13] J. Li, W. J. Shi, D. Yu, A characterization of some PGL(2,q) by maximum element orders, Bull. Korean. Math. Soc. 52 (2015), 2025-2034.

[14] W. J. Shi, A characterization of U3(2n) by their element orders, J. Southwest-China Normal Univ. 25 (4) (2000), 353-360.

[15] J. S. Williams, Prime graph components of finite groups, J. Algebra. 69 (2) (1981), 487-513.

[16] A. V. Zavarnitsine, Recognition of the simple groups L3(q) by element orders, J. Group Theory. 7 (1) (2004), 81-97.