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Presentations for $ 3$-dimensional special linear groups over integer rings


Authors: Marston Conder, Edmund Robertson and Peter Williams
Journal: Proc. Amer. Math. Soc. 115 (1992), 19-26
MSC: Primary 20F05; Secondary 20G40
DOI: https://doi.org/10.1090/S0002-9939-1992-1079696-5
MathSciNet review: 1079696
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Abstract: The following $ 2$-generator $ 6$-relator presentation is obtained for the $ 3$-dimensional special linear group $ \operatorname{SL}(3,{\mathbb{Z}_k})$ for each odd integer $ k > 1$:

$\displaystyle \operatorname{SL}(3,{\mathbb{Z}_k}) = \langle x,y\vert{x^3} = {y^... ...{(x{y^{ - 1}}xyx{y^{ - 1}}{x^{ - 1}}{y^{ - 1}})^{(k - 1)/2}}xy)^4} = 1\rangle .$

Alternative presentations for these groups and other groups associated with them are also given.

References [Enhancements On Off] (What's this?)

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  • [2] Marston Conder, A surprising isomorphism, J. Algebra 129 (1990), 494-501. MR 1040950 (91f:20037)
  • [3] Sherry M. Green, Generators and relations for the special linear group over a division ring, Proc. Amer. Math. Soc. 62 (1977), 229-232. MR 0430084 (55:3091)
  • [4] Jürgen Hurrelbrink, On presentations of $ \operatorname{SL}_n({\mathbb{Z}_s})$, Comm. Algebra 11 (1983), 937-947. MR 696479 (84i:20044)
  • [5] Jens Mennicke, Finite factor groups of the unimodular group, Ann. of Math. (2) 81 (1965), 31-37. MR 0171856 (30:2083)
  • [6] John Milnor, Introduction to algebraic $ K$-theory, Ann. of Math. Stud., no. 72, Princeton Univ. Press, Princeton, NJ, 1971. MR 0349811 (50:2304)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1079696-5
Article copyright: © Copyright 1992 American Mathematical Society

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