Gow-Tamburini type generation of the special linear group for some special rings.

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Mumbai, Mumbai, India

2 Department of Mathematics, University Mumbai, Kalina Campus, Mumbai, India

Abstract

Let $R$ be a commutative ring with unity and let $n\geq 3$ be an integer. Let $SL_n(R)$ and $E_n(R)$ denote respectively the special linear group and elementary subgroup of the general linear group $GL_n(R).$ A result of Hurwitz says that the special linear group of size atleast 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 atleast three over the ring of integers is not a finite simple group, we expect that it has more than two generators. In the special case, where $R$ is the ring of integers of an algebraic number field which is not totally imaginary, we provide for $E_n(R)$ (and hence $SL_n(R)$) a set of Gow-Tamburini matrix generators, depending on the minimal number of generators of $R$ as a $Z$-module.

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References

[1]T. Y. Lam, Serre’s problem on projective modules, Springer Monographs in Mathematics, Springer-Verlag, Berlin,
2006.
[2]N. V. Afre and A. S. Garge, Gow-Tamburini type generation of SL3 (R) over the rings of integers of imaginary
quadratic number fields of class number one, Proc. Indian Acad. Sci. Math. Sci., 132 (2022) 6 pp.
[3]R. Gow and M. C. Tamburini, Generation of SL(n, Z) by a Jordan unipotent matrix and its transpose, Linear
Algebra Appl., 181 (1993) 63–71.
[4]H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2),
Inst. Hautes EĢtudes Sci. Publ. Math., no. 33 (1967) 59–137.

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History

  • Receive Date: 02 August 2022
  • Revise Date: 29 April 2023
  • Accept Date: 08 June 2023
  • Published Online: 01 June 2024

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