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Presentations for 3-Dimensional Special Linear Groups Over Integer Rings

Marston Conder, Edmund Robertson and Peter Williams
Proceedings of the American Mathematical Society
Vol. 115, No. 1 (May, 1992), pp. 19-26
DOI: 10.2307/2159559
Stable URL: http://www.jstor.org/stable/2159559
Page Count: 8
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Presentations for 3-Dimensional Special Linear Groups Over Integer Rings
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Abstract

The following 2-generator 6-relator presentation is obtained for the 3-dimensional special linear group SL(3, Zk) for each odd integer $k > 1:$ \begin{equation*}\begin{split} \mathrm{SL}(3, \mathbb{Z}_k) = \langle x, y\mid x^3 = y^3 = (xy)^6 = (x^{-1}yx^{-1}y^{-1}xy)^2 \\ = (xy^{-1}xyxy^{-1}x^{-1}y^{-1})^k \\ =((xy^{-1}xyxy^{-1}x^{-1}y^{-1})^{(k - 1)/2}xy)^4 = 1\rangle.\end{split} \end{equation*} Alternative presentations for these groups and other groups associated with them are also given.

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