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Given a field and a natural number , the special linear group (or ) is the subgroup of the general linear group consisting of those linear transformations that preserve the volume form on the vector space . It can be canonically identified with the group of matrices with entries in having determinant .
This group can be considered as a subvariety of the affine space of square matrices of size carved out by the equations saying that the determinant of a matrix is 1. This variety is an algebraic group over , and if is the field of real or complex numbers then it is a Lie group over .
The special linear group is a perfect group for any field and any , except for the cases of the prime fields and .
See for example here, or Lang 02, theorems XIII 8.3 and 9.2.
The first case admitted by Prop. is the binary icosahedral group (this Prop.):
Last revised on May 17, 2021 at 06:38:03. See the history of this page for a list of all contributions to it.