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

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.

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

Quadratic extensions , ring of integers of number fields , special linear group , Elementary subgroup