Presentations for 3-dimensional special linear groups over integer rings

Conder, Marston; Robertson, Edmund; Williams, Peter

doi:10.1090/S0002-9939-1992-1079696-5

Presentations for $3$-dimensional special linear groups over integer rings
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by Marston Conder, Edmund Robertson and Peter Williams

Proc. Amer. Math. Soc. 115 (1992), 19-26

DOI: https://doi.org/10.1090/S0002-9939-1992-1079696-5

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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 $k > 1$: \[ \operatorname {SL}(3,{\mathbb {Z}_k}) = \langle x,y|{x^3} = {y^3} = {(xy)^6} = {({x^{ - 1}}y{x^{ - 1}}{y^{ - 1}}xy)^2} = {(x{y^{ - 1}}xyx{y^{ - 1}}{x^{ - 1}}{y^{ - 1}})^k} = {({(x{y^{ - 1}}xyx{y^{ - 1}}{x^{ - 1}}{y^{ - 1}})^{(k - 1)/2}}xy)^4} = 1\rangle .\] Alternative presentations for these groups and other groups associated with them are also given.

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