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Afre, Naresh, Garge, Anuradha. (1403). Gow-Tamburini type generation of the special linear group for some special rings.. , 13(2), 123-132. doi: 10.22108/ijgt.2023.134366.1800
Naresh Vasant Afre; Anuradha S. Garge. "Gow-Tamburini type generation of the special linear group for some special rings.". , 13, 2, 1403, 123-132. doi: 10.22108/ijgt.2023.134366.1800
Afre, Naresh, Garge, Anuradha. (1403). 'Gow-Tamburini type generation of the special linear group for some special rings.', , 13(2), pp. 123-132. doi: 10.22108/ijgt.2023.134366.1800
Afre, Naresh, Garge, Anuradha. Gow-Tamburini type generation of the special linear group for some special rings.. , 1403; 13(2): 123-132. doi: 10.22108/ijgt.2023.134366.1800

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

International Journal of Group Theory
مقاله 2، دوره 13، شماره 2، شهریور 2024، صفحه 123-132    اصل مقاله (417.89 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2023.134366.1800
نویسندگان
Naresh Vasant Afre1؛ Anuradha S. Garge* 2
1Department of Mathematics, University of Mumbai, Mumbai, India
2Department of Mathematics, University Mumbai, Kalina Campus, Mumbai, India
چکیده
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.
کلیدواژه‌ها
Quadratic extensions؛ ring of integers of number fields؛ Special linear group؛ Elementary subgroup
مراجع

[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 Études Sci. Publ. Math., no. 33 (1967) 59–137.

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