On the Ree groups $^2{}G_2(q)$ characterized by a size of a conjugacy class

Document Type : Research Paper

Authors

1 State Office of Education in Qaemiyeh, Fars Province, Iran.

2 Department of Mathematics, Payame Noor University, P. Box: 19395-3697, Tehran, Iran.

Abstract

One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group $^2{}G_2(q)$, where $q\pm\sqrt{3q}+1$ is a prime number can be uniquely determined by its order and one conjugacy class size.

Keywords

References

[1] N. Ahanjideh, On Thompson’s conjecture for some finite simple groups, J. Algebra, 344 (2011) 205-228.
[2] S. S. Amiri and A. K. Asboei, Characterization of some finite groups by order and length of one conjugacy class, Sib. Math. J., 57 No. 2 (2016) 185-189.
[3] A. K. Asboei, Characterization of PSL(5, q) by its orderand one conjugacy class size, Iran. J. Math. Sci. Inform., 15 No. 1 (2020) 35-40.
[4] A. K. Asboei, R. Mohammadyari, Recognizing alternating groups by their order and one conjugacy class length, J. Algebra. Appl., 15 No. 2 (2016) 1650021.
[5] A. K. Asboei and R. Mohammadyari, Characterization of the alternating groups by their order and one conjugacy class length, Czechoslovak Math. J., 66 No. 141 (2016) 63-70.
[6] A. K. Asboei, R. Mohammadyari and M. R. Darafsheh, The influence of order and conjugacy class length on the structure of finite groups, Hokkaido Math. J., 47 (2018) 25-32.
[7] A. K. Asboei, R. Mohammadyari and M. Rahimi, New characterization of some linear groups, Int. J. Industrial Mathematics, 8 No. 2 (2016) 165-170.
[8] G. Y. Chen, On the structure of Frobenius groups and 2-Frobenius groups, J. Southeast Univ. Nat. Sci. Ed., 20 No. 5 (1995) 485-487.
[9] Y. H. Chen and G. Y. Chen, Recognizing PSL(2, p) by its order and one special conjugacy class size, J. Inequal. Appl., 2012 No. 1 (2012) 1-10.
[10] Y. H, Chen, G. Y. Chen, Recognization of Alt10 and PSL(4, 4) by two special conjugacy class size, Ital. J. Pure Appl. Math., 29 (2012) 387-394.
[11] J. H. Conway, R. T. Curtis, S. P. Norton and R. A. Wilson, Atlas of Finite Groups, Clarendon, Oxford, 1985.
[12] M. R. Darafsheh, Characterizability of the Group 2Dp(3) by its order components, Where p ≥ 5 is a prime number not of the form 2m + 1, Acta Math. Sin. Engl., 24 No. 7 (2008) 1117-1126.
[13] B. Ebrahimzadeh, A. Iranmanesh and H. Parvizi Mosaed, A new characterization of ree group 2G2(q) by the order of group and the number of elements with the same order, International Journal of Group Theory, 6 No. 4 (2017) 1-6.
[14] B. Ebrahimzadeh, On projective special unitary groups PSU(5, q) characterized by a size of a conjugacy class, Comment. Math. Univ. Carol., (2022), Accepted.
[15] G. Frobenius, Verallgemeinerung des sylowschen satze, Berliner sitz, (1895) 981-983.
[16] M. Foroudi ghasemabadi, N. Ahanjideh, Characterization of the simple groups Dn(3) by prime graph and spectrum, Iran. J. Math. Sci. Inform, 7 No. 1 (2012) 91-106.
[17] D. Gorenstein, Finite Groups, Harper and Row, New York, 1980.
[18] I. L. Gorshkov, On characterization of a finite group by the set of conjugacy class sizes, Journal of algebra and its applications, 21 No. 11 (2022) 2250226.
[19] I. L. Gorshkov, On characterization of a finite group with Non-simple socle by the set of conjugacy class sizes, Bull. Iran. Math. Soc., 49 No. 3 (2023) 23.
[20] A. Iranmanesh and S. H. Alavi, A characterization of simple group PSL(5, q), Bull. Austral. Math. Soc., 65(2002) 211-222.
[21] N. Iiyori and H. Yamaki, Prime graph components of the simple groups of Lie type over the field of even characteristic, J. Algebra, 155 No. 2 (1993) 335-343.
[22] A. S. Kondrat’ev, Prime graph components of finite simple groups, Mathematics of the USSR-Sbornik, 67 No. 1 (1990) 235-247.
[23] J. B. Li, Finite Groups with Special Conjugacy Class Sizes or Generalized Permutable Subgroups, Southwest University, Chongqing, 2012.
[24] V. D. Mazurov, E. I. Khukhro, Unsolved problems in group theory. The Kourovka Notebook, no. 18, Institute of Mathematics, Russia, 2014.
[25] J. Moori, B. G. Rodrigues, A. Saedi and S. Zandi, Designs from maximal subgroups and conjugacy classes of Ree groups, Adv. Math. Commun., 14 No. 4 (2020) 603-611.
[26] G. R. Rezaeezadeh, M. R. Darafsheh, M. Bibak and M. Sajjadi, OD-characterization of Almost Simple Groups Related to D4(4), Iran. J. Math. Sci. Inform., 10 No. 1 (2015) 23-43.
[27] W. J. Shi, A characterization of U3(2n) by their element orders, J. Southwest China Normal Univ., 25 No. 4 (2000) 353-360.
[28] G. E. Wall, On the conjugacy classes in the unitary symplectic and orthogonal groups, Journal Australian Math. Soc., 3 (1963) 1-62.
[29] J. S. Williams, Prime graph components of finite groups, J. Algebra, 69 No. 2 (1961) 487-513.
[30] A. V. Zavarnitsine, Recognition of the simple groups L3(q) by element orders, J. Group Theory, 7 No. 1 (2004) 81-97.
Algebraic Structures and Their Applications
Volume 11, Issue 2
May 2024
Pages 165-171

Files

Share

How to cite

Statistics