The 𝑛𝑋-complementary generations of the group 𝑃𝑆𝐿(3, 5)

Abstract

Let 𝐺 be a finite non-abelian simple group and 𝑛𝑋 be a non-trivial conjugacy class of elements of order 𝑛 in 𝐺. We say that 𝐺 is 𝑛𝑋-complementary generated, if for every 𝑥 ∈ 𝐺, there exists an element 𝑦 ∈ 𝑛𝑋 such that 𝐺 = ⟨𝑥, 𝑦⟩ . In this paper we study the 𝑛𝑋-complementary generations for all the non-trivial conjugacy classes of the projective special linear group 𝑃𝑆𝐿(3, 5). We approach this kind of generation using the structure constant method. GAP [The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.9.3; 2018. (http://www.gap-system.org)] is used frequently in our computations.

Keywords

• Conjugacy classes
• generation
• 𝑛𝑋-complementary generation
• structure constant

References

1. A. R. Ashrafi, 2003, nX-complementary generations of the Harada-Norton group HN, Acta Mathematica et Informatica Universitatis Ostraviensis, 11, no. 1, 3-9. google scholar
2. A. R. Ashrafi, 2004, nX-Complementary generations of the sporadic group O0N, Southeast Asian Bulletin of Mathematics, 28, no. 6,1011-1019. google scholar
3. A. R. Ashrafi and G. A. Moghani, 2006, nX-Complementary generations of the Fischer group Fİ23, J. Appl. Math. & Computing, 21, no. 1-2, 393-409. google scholar
4. A. B. M. Basheer and T. T. Seretlo, 2019, On two generation methods for the simple linear group PSL(3,5), Khayyam J. of Math., 5, no. 1, 125-139. google scholar
5. A. B. M. Basheer, M. J. Motalane and T. T. Seretlo, 2021 The (p, q, r)-generations of the alternating group A11, Khayyam J. of Math., 7, no. 2, 165-186. google scholar
6. A. B. M. Basheer and T. T. Seretlo, 2021, The (p, q, r)-generations of the Matheiu group M 22, Southeast Asian Bulletin of Mathematics, 45, no. 1, 11-28. google scholar
7. A. B. M. Basheer and T. T. Seretlo, 2020, The (p, q, r) -generations of the alternating group A10, Quaestiones Mathematicae, 43, no. 3, 395-408. google scholar
8. A. B. M. Basheer, M. J. Motalane, M. G. Sehoana and T. T. Seretlo, 2024 The nX-complem-entary generations of the classes of the Chevalley group G2(3), Mathematics Interdisciplinary Research, 9 no. 4, 443-460. google scholar

9. J. H. Conway et al., 1985, Atlas of Finite Groups, Clarendon Press, Oxford. google scholar
10. M. R. Darafsheh, A. R. Ashrafi and G.A. Moghani, 2001, (p, q, r)-generations of the Conway group C01, for odd p, Kumamoto J. Math., 14, 1 - 20. google scholar
11. M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, 2003, (p, q, r)-generations and nX-complementary generations of the sporadic group Ly, Kumamoto J. Math., 16, 13-25. google scholar
12. M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, 2004, nX-complementary generations of the sporadic group C01 , Acta Mathematica Vietnamica, 29, no. 1, 57-75. google scholar
13. M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani,2001, (p, q, r) -generations of the sporadic group O’N, London Mathematical Society Lecture Note Series, Groups St. Andrews in Oxford, Cambridge University Press, 304, 101 - 109. google scholar
14. S. Ganief, 1997, 2-Generations of the Sporadic Simple Groups, PhD Thesis, University of Natal, South Africa. google scholar
15. S. Ganief and J. Moori, 1997, 2-generations of the smallest Fischer group Fİ22 , Nova J. Math. Game Theory Algebra 6, no. 2-3, 127-145. google scholar
16. S. Ganeif and J. Moori, 1997, (p, q, r) -generations and nX-complementary generations of the sporadic groups HS and McL., J. Algebra, 188, no. 2, 531-6. google scholar
17. S. Ganief and J. Moori, 1998, Generating pairs for the Conway groups C02 and C03, J. Group Theory, 1, no. 3, 237-256. google scholar
18. S. Ganief and J. Moori, 1999, 2-generations of the fourth Janko group J4 , J. Algebra, 212, no. 3, 305-322. google scholar
19. The GAP Group, 2018, GAP - Groups, Algorithms, and Programming, Version 4.9.3; (http://www.gap-system.org) google scholar
20. A. J. Woldar, 1994, 2-Generation of the sporadic simple groups, Comm. Algebra 22, no. 2, 675-685. google scholar