Abstract
A result of Hurwitz says that the special linear group of size greater than or equal to three over the ring of integers of an algebraic number field is finitely generated. A celebrated theorem in group theory states that finite simple groups are two-generated. Since the special linear group of size at least three over the ring of integers is not a finite simple group, we expect that it has more than two generators. In this paper, we provide a set of three generators for the special linear group of size three over the rings of integers of imaginary quadratic number fields of class number one. The speciality of this set of generators is that it is unbiased towards the choice of a particular simple root (from a Lie algebra point of view). This new set of generators is inspired by the work of Gow and Tamburini for the special linear group over the (ring of rational) integers.
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References
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Acknowledgements
The authors thank Professor R. P. Deore, former Head, Department of Mathematics, for his constant motivation when this work was in progress. The second author, Anuradha Garge takes this opportunity to thank the organizers, Anupam K. Singh and Shripad M. Garge of the workshop on Chevalley groups held at IISER, Pune in 2013. It was there that they learnt about the paper of Gow–Tamburini from the talks of Professor B. Sury. The first author, Naresh Afre takes this opportunity to thank the Principal of D. J. Sanghvi College for his support and motivation.
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Communicated by B Sury.
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Afre, N.V., Garge, A.S. Gow–Tamburini type generation of SL\(_{\varvec{3}}\varvec{(R)}\) over the rings of integers of imaginary quadratic number fields of class number one. Proc Math Sci 132, 26 (2022). https://doi.org/10.1007/s12044-022-00678-3
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DOI: https://doi.org/10.1007/s12044-022-00678-3