پیوندهای مفید
آمار
| تعداد نشریات | 7 |
| تعداد شماره‌ها | 405 |
| تعداد مقالات | 5,424 |
| تعداد مشاهده مقاله | 5,542,899 |
| تعداد دریافت فایل اصل مقاله | 5,027,044 |
On two generation methods for the simple linear group $PSL(3,7)$ | ||
| AUT Journal of Mathematics and Computing | ||
| مقاله 3، دوره 4، شماره 1، 2023، صفحه 27-37 اصل مقاله (412.22 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2022.21638.1095 | ||
| نویسنده | ||
| Thekiso Trevor Seretlo* | ||
| School of Mathematical Sciences, North West University, Mafikeng Branch P/B X2046, Mmabatho 2735, South Africa | ||
| چکیده | ||
| A finite group $G$ is said to be \textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = \left<x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right>.$ In [J. Moori, $(p, q, r)$-generations for the Janko groups $J_{1}$ and $J_{2}$, Nova J. Algebra and Geometry, \bf{2} (1993), no. 3, 277--285], Moori posed the question of finding all the $(p,q,r)$ triples, where $p,\ q$ and $r$ are prime numbers, such that a non-abelian finite simple group $G$ is $(p,q,r)$-generated. Also for a finite simple group $G$ and a conjugacy class $X$ of $G,$ the rank of $X$ in $G$ is defined to be the minimal number of elements of $X$ generating $G.$ In this paper we investigate these two generational problems for the group $PSL(3,7),$ where we will determine the $(p,q,r)$-generations and the ranks of the classes of $PSL(3,7).$ We approach these kind of generations 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 in our computations. | ||
| کلیدواژه‌ها | ||
| Conjugacy classes؛ $(p,q,r)$-Generation؛ Rank؛ Structure constant | ||
| مراجع | ||
|
[1] The GAP Group gap – groups, algorithms, and programming, version 4.9.3. http://www.gap-system.org. Accessed: 2018.
[2] A. R. Ashrafi, Generating pairs for the Held group He, J. Appl. Math. Comput., 10 (2002), pp. 167–174.
[3] , (p, q, r)-generation of the sporadic group HN, Taiwanese J. Math., 10 (2006), pp. 613–629.
[4] A. Basheer and T. T. Seretlo, The (p, q, r)-generations of the mathieu group M22, Southeast Asian Bull. Math., 45 (2021), pp. 11–28.
[5] A. B. M. Basheer, The ranks of the classes of A10, Bull. Iranian Math. Soc., 43 (2017), pp. 2125–2135.
[6] A. B. M. Basheer and J. Moori, On the ranks of finite simple groups, Khayyam J. Math., 2 (2016), pp. 18–24.
[7] , On the ranks of the alternating group An, Bull. Malays. Math. Sci. Soc., 42 (2019), pp. 1957–1973.
[8] A. B. M. Basheer and T. Seretlo, On two generation methods for the simple linear group P SL(3, 5), Khayyam J. Math., 5 (2019), pp. 125–139.
[9] , The (p, q, r)-generations of the alternating group A10, Quaest. Math., 43 (2020), pp. 395–408.
[10] M. D. E. Conder, Some results on quotients of triangle groups, Bull. Austral. Math. Soc., 30 (1984), pp. 73– 90.
[11] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray.
[12] M. R. Darafsheh and A. R. Ashrafi, Generating pairs for the sporadic group Ru, J. Appl. Math. Comput., 12 (2003), pp. 143–154.
[13] M. R. Darafsheh, A. R. Ashrafi, and G. A. Moghani, (p, q, r)-generations of the Conway group Co1 for odd p, Kumamoto J. Math., 14 (2001), pp. 1–20.
[14] , (p, q, r)-generations and nX-complementary generations of the sporadic group Ly, Kumamoto J. Math., 16 (2003), pp. 13–25.
[15] , (p, q, r)-generations of the sporadic group O’N, in Groups St. Andrews 2001 in Oxford. Vol. I, vol. 304 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2003, pp. 101–109.
[16] S. Ganeif and J. Moori, (p, q, r)-generations and nX-complementary generations of the sporadic groups HS and McL, J. Algebra, 188 (1997), pp. 531–546.
[17] S. Ganief and J. Moori, 2-generations of the smallest Fischer group Fi22, Nova J. Math. Game Theory Algebra, 6 (1997), pp. 127–145.
[18] , (p, q, r)-generations of the smallest Conway group Co3, J. Algebra, 188 (1997), pp. 516–530.
[19] , Generating pairs for the Conway groups Co2 and Co3, J. Group Theory, 1 (1998), pp. 237–256.
[20] , 2-generations of the fourth Janko group J4, J. Algebra, 212 (1999), pp. 305–322. [21] J. I. Hall and L. H. Soicher, Presentations of some 3-transposition groups, Comm. Algebra, 23 (1995), pp. 2517–2559.
[22] J. Moori, Generating sets for F22 and its automorphism group, J. Algebra, 159 (1993), pp. 488–499.
[23] , (p, q, r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom., 2 (1993), pp. 277–285.
[24] , (2, 3, p)-generations for the Fischer group F22, Comm. Algebra, 22 (1994), pp. 4597–4610.
[25] , Subgroups of 3-transposition groups generated by four 3-transpositions, Quaestiones Math., 17 (1994), pp. 83–94.
[26] , On the ranks of the Fischer group F22, Math. Japon., 43 (1996), pp. 365–367.
[27] I. Zisser, The covering numbers of the sporadic simple groups, Israel J. Math., 67 (1989), pp. 217–224. | ||
|
آمار تعداد مشاهده مقاله: 659 تعداد دریافت فایل اصل مقاله: 413 |
||