%0 Journal Article
%T Gow-Tamburini type generation of the special linear group for some special rings.
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Afre, Naresh Vasant
%A Garge, Anuradha S.
%D 2024
%\ 06/01/2024
%V 13
%N 2
%P 123-132
%! Gow-Tamburini type generation of the special linear group for some special rings.
%K Quadratic extensions
%K ring of integers of number fields
%K special linear group
%K Elementary subgroup
%R 10.22108/ijgt.2023.134366.1800
%X 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.
%U https://ijgt.ui.ac.ir/article_27604_4f939f1e06e3ae85b351097196d61860.pdf