$nX$-Complementary Generations of the Chevalley Group $G_{2}(3)$

Document Type : Original Scientific Paper

Authors

1 ‎School of Mathematical and Computer Sciences, ‎University of Limpopo (Turfloop), ‎P. Bag X1106‎, ‎Sovenga 0727‎, ‎South Africa

2 ‎School of Mathematical and Statistical Sciences, PAA Focus Area‎, ‎North-West University (Mahikeng), ‎P‎. ‎Bag X2046‎, ‎Mmabatho 2790‎, ‎South Africa

Abstract

‎A finite non-abelian group $G$ is said to be $(l,m,n)$-generated if it can be generated by two elements $x$ and $y$ such that $o(x)=l$‎, ‎$o(y)=m$ and $o(xy)=n$‎. ‎Also‎, ‎$G$ is said to be $nX$-complementary generated if given an arbitrary non-identity element $x\in G$‎, ‎there exists an element $y \in nX$ such that $G=\langle x,y\rangle$‎. ‎We studied the $(p,q,r)$-generation for the Chevalley group $G_{2}(3)$‎, ‎where $p$‎, ‎$q$ and $r$ are all the primes dividing the order of $G_{2}(3)$‎. ‎In the current paper‎, ‎we classify all the non-trivial conjugacy classes of $G_{2}(3)$ whether they are complementary generators or not‎. ‎To achieve this‎, ‎we mainly used the structure constant method together with other results applied to establish generation and non-generation of the group $G_{2}(3)$ by the $(p,q,r)$ triples‎. ‎Some particular algorithms‎, ‎as well as the (Gap) programming tool‎, ‎and the Atlas of finite groups have been exploited in our computations‎.

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References

[1] J. L. Brenner, R. M. Guralnick and J. Wiegold, Two generator groups III, Contemp. Math. 33 (1984) 82 - 89.
[2] A. R. Ashrafi, (p; q; r)-generations and nX-complementary generations of the thompson group Th, SUT J. Math. 39 (2003) 41 - 54, https://doi.org/10.55937/sut/1059541213.
[3] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, nX-complementary generations of the sporadic group Co1, Acta Math. Vietnam 29 (2004) 57 - 75.
[4] S. Ganief and J. Moori, 2-generations of the fourth Janko group J4, J. Algebra 212 (1999) 305 - 322.
[5] S. Ganief and J. Moori, 2-generations of the smallest Fischer group Fi22, Nova J. Math. Game Theory Algebra 6 (1997) 127 - 145.
[6] S. Ganief and J. Moori, Generating pairs for the Conway groups Co2 and Co3, J. Group Theory 1 (1998) 237 - 256.
[7] S. Ganief and J. Moori, (p; q; r)-generations and nX-complementary generations of the sporadic groups HS and McL, J. Algebra 188 (1997) 531 - 546.
[8] J. Moori, (p; q; r)-generations for the Janko groups J1 and J2, Nova J. Algebra and Geometry, 2 (1993), 277 - 285.
[9] J. Moori, (2; 3; p)-generations for the Fischer group F22, Commun. Algebra. 22 (1994) 4597 - 4610, https://doi.org/10.1080/00927879408825089.
[10] R. Wilson, J. Conway and S. Norton, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
[11] M. A. Al-Kadhi and F. Ali, (2; 3; t)-generations for the Conway group Co3, Int. J. Algebra 4 (2010) 1341 - 1353.
[12] A. B. M. Basheer and T. T. Seretlo, The (p; q; r)-generations of the alternating group A10, Quaest. Math. 43 (2020) 395 - 408, https://doi.org/10.2989/16073606.2019.1575925.
[13] A. B. M. Basheer and T. T. Seretlo, On two generation methods for the simple linear group PSL(3; 5), Khayyam J. Math. 5 (2019) 125 - 139, https://doi.org/10.22034/KJM.2019.81226.
[14] A. B. M. Basheer and J. Moori, A survey on some methods of generating finite simple groups, London Math. Soc. Lecture Note Ser. Cambridge University Press, Cambridge, 455 (2019) 106 - 118, https://doi.org/10.1017/9781108692397.005.
[15] A. B. M. Basheer and T. T. Seretlo, (p,q,r)-generations of the Mathieu group M22, Southeast Asian Bull. Math. 45 (2021) 11 - 28.
[16] A. B. M. Basheer, The ranks of the classes of A10, Bull. Iranian Math. Soc. 43 (2017) 2125 - 2135.
[17] A. B. M. Basheer and J. Moori, On the ranks of finite simple groups, Khayyam J. Math. 2 (2016) 18 - 24, https://doi.org/10.22034/KJM.2016.15511.
[18] A. B. M. Basheer, M. J. Motalane and T. T. Seretlo, The (p; q; r)-generations of the alternating group A11, Khayyam J. Math. 7 (2021) 165 - 186.
[19] A. B. M. Basheer, M. J. Motalane and T. T. Seretlo, The (p; q; r)-generations of the Mathieu group M23, Italian J. Pur and Applied Math., to appear.
[20] A. B. M. Basheer, M. J. Motalane and T. T. Seretlo, The (p; q; r)- generations of the sympletic group Sp(6; 2), Alg. Struc. Appl. 8 (2021) 31-49, https://doi.org/ 10.22034/AS.2021.1975.
[21] M. D. E. Conder, Some results on quotients of triangle groups, Bull. Austral. Math. Soc. 30 (1984) 73 - 90, https://doi.org/10.1017/S0004972700001738.
[22] S. Ganief, 2-Generations of the Sporadic Simple Groups, PhD Thesis, University of Natal, South Africa, 1997.
[23] S. Ganief and J. Moori, (p; q; r)-generations of the smallest Conway group Co3, J. Algebra 188 (1997) 516-530, https://doi.org/10.1006/jabr.1996.6828.
[24] A. B. M Basheer et al, The (p; q; r)-generations of the Chevalley group G2(3), submitted, 2024.
[25] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.2; 2019, (http://www.gap-system.org).
Mathematics Interdisciplinary Research
Volume 9, Issue 4
December 2024
Pages 443-460

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