The $(p,q,r)$-generations of the symplectic group $Sp(6,2)$

Document Type : Research Paper

Authors

School of Mathematical and Computer Sciences University of Limpopo (Turfloop) Sovenga 0727, South Africa.

10.29252/as.2021.1975

Abstract

A finite group $G$ is called \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 29, 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 a $(p,q,r)$-generated. In this paper we establish all the $(p,q,r)$-generations of the symplectic group $Sp(6,2).$ GAP 20 and the Atlas of finite group representations 33 are used in our computations.

Keywords


[1] F. Ali, M. A. F. Ibrahim and A. Woldar, (3, q, r)-Generations of Fischer sporadic group $Fi_{24}^{'}$, J. of Group Theory, 22 (2019) 453-489.
[2] F. Ali, M. A. F. Ibrahim and A. Woldar, On (2, 3)-generations of Fischer largest sporadic simple group
$Fi_{24}^{'}$, C. R. Acad. Sci. Paris, Ser. I, 357 (2019) 401-412.
[3] A. R. Ashraffi, Generating pairs for the Held group $He$, J. Appl. Math. and Computing, 10 (2002) 167-174.
[4] A. R. Ashraffi, (p,q,r)-generations of the sporadic group HN, Taiwanese J. Math, 10 No. 3 (2006) 613-629.
[5] A. B. M. Basheer and T. T. Seretlo, The (p; q; r)-generations of the alternating group $A_{10}$. Quaestiones Mathematicae, 43 No. 3 (2020) 395-408.
[6] A. B. M. Basheer and T. T. Seretlo, On two generation methods for the simple linear group $PSL(3,5)$ Khayyam J. Math., 5 No. 1 (2019) 125-139.
[7] A. B. M. Basheer and J. Moori, A survey on some methods of generating finite simple groups. London Mathematical Society Lecture Note Series, published as a chapter of the book of Groups St Andrews 2017 by Cambridge University Press, 455 (2019) 106-118. 
[8] A. B. M. Basheer and T. T. Seretlo, (p,q,r)-generations of the Mathieu group $ M_{22}$ Southeast Asian Bull. Math, (accepted) (2020).
[9] A. B. M. Basheer, The ranks of the classes of $A_{10}$. Bulletin of the Iranian Mathematical Society, 43 No. 7 (2017) 2125-2135.
[10] A. B. M. Basheer and J. Moori, On the ranks of finite simple groups, Khayyam J. Math., 2 No. 1 (2016) 18-24.
[11] A. B. M. Basheer, J. Motalane and T. T. Seretlo, The (p; q; r)-generations of the alternating group $A_{11}$. Accepted by the Khayyam Journal of Mathematics.
[12] A. B. M. Basheer, J. Motalane and T. T. Seretlo, The (p,q,r)-generations of the Mathieu group $ M_{23}$, submitted.
[13] M. D. E. Conder, Some results on quotients of triangle groups, Bull. Austr Math Soc., 30 (1984) 167-174.
[14] J. H. Conway et al., Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
[15] M. R. Darafsheh, A. R. Ashrafi, Generating pairs for the sporadic group Ru, J. Appl. Math. Comput., 12 No. 1-2 (2003) 143-154.
[16] M. R. Darafsheh, A. R. Ashrafi and G.A. Moghani, (p,q,r)-generations and nX-complementary generations of the sporadic group Ly, Kumamoto J. Math., 16 (2003) 13-25.
[17] M. R. Darafsheh, A. R. Ashrafi and G.A. Moghani, (p; q; r)-generations of the Conway group $Co_{1}$, for odd p, Kumamoto J. Math., 14 (2001) 1-20. 
[18] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, (p,q,r)-generations of the sporadic group O'N, Groups St. Andrews 2001 in Oxford. Vol. I
[19] S. Ganief, 2-Generations of the Sporadic Simple Groups, PhD thesis, University of KwaZulu-Natal, Pieter-maritzburg, 1997.
[20] The GAP Group, GAP -Groups, Algorithms, and Programming, Version 4.10.2; 2019. (http://www.gap-system.org) 
[21] S. Ganief and J. Moori, 2-generations of the fourth Janko group $J_4$, J. Algebra, 212 No. 1 (1999) 305-322.
[22] S. Ganief and J. Moori, 2-generations of the smallest Fischer group $Fi_{22}$, Nova J. Math. Game Theory Algebra, 6 No. 2-3 (1997) 127-145.
[23] S. Ganief and J. Moori, Generating pairs for the Conway groups $Co_2$ and $Co_3$, J. Group Theory, 1 No. 3 (1998) 237-256.
[24] S. Ganeif and J. Moori, (p,q,r)-generations and nX-complementary generations of the sporadic groups HS and McL., J. Algebra, 188 No. 2 (1997) 531-546.
[25] S. Ganief and J. Moori, (p,q,r)-generations of the smallest Conway group $Co_3$, J. Algebra, 188 No. 2 (1997) 516-530.
[26] L. Di Martino, M. Tamburini and A. Zalesskii, On Hurwitz groups of low rank, J. Algebra, 28 (2000) 5383-5404.
[27] L. Di Martino, M. Pellegrini and A. Zalesski, On generators and representations of the sporadic simple groups, Comm. Algebra, 42 (2014) 880-908.
[28] J. Moori, (2; 3; p)-generations for the Fischer group $F_{22}$, Comm. Algebra, 2 No. 11 (1994) 4597-4610.
[29] J. Moori, (p; q; r)-generations for the Janko groups $J_{1}$ and $J_{2}$, Nova J. Algebra and Geometry, 2 No. 3 (1993) 277-285.
[30] R. Ree, A theorem on permutations, J. Comb. Theory A, 10 (1971) 174-175.
[31] L. L. Scott, Matrices and cohomolgy, Ann. Math, 105 No. 3 (1977) 67-76.
[32] J. Ward, Generation of Simple Groups by Conjugate Involutions, PhD Thesis, University of London, 2009.
[33] R. A. Wilson et al., Atlas of finite group representations, (http://brauer.maths.qmul.ac.uk/Atlas/v3/)
[34] A. J. Wodlar, Sporadic simple groups which are Hurwitz, J. Algebra, 144 (1991) 443-450.