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special linear group

Contents

Contents

Idea

Given a field kk and a natural number nn \in \mathbb{N}, the special linear group SL(n,k)SL(n,k) (or SL n(k)SL_n(k)) is the subgroup of the general linear group SL(n,k)GL(n,k)SL(n,k) \subset GL(n,k) consisting of those linear transformations that preserve the volume form on the vector space k nk^n. It can be canonically identified with the group of n×nn\times n matrices with entries in kk having determinant 11.

This group can be considered as a subvariety of the affine space M n×n(k)M_{n\times n}(k) of square matrices of size nn carved out by the equations saying that the determinant of a matrix is 1. This variety is an algebraic group over kk, and if kk is the field of real or complex numbers then it is a Lie group over kk.

Properties

Proposition

The special linear group SL n(𝔽)SL_n(\mathbb{F}) is a perfect group for any field 𝔽\mathbb{F} and any n1n \geq 1, except for the cases of the prime fields 𝔽 2\mathbb{F}_2 and 𝔽 3\mathbb{F}_3.

See for example here, or Lang 02, theorems XIII 8.3 and 9.2.

Examples

Remark

The first case admitted by Prop. is the binary icosahedral group (this Prop.):

SL(2,𝔽 5)2I SL(2,\mathbb{F}_5) \;\simeq\; 2I

References

Last revised on March 25, 2019 at 12:10:30. See the history of this page for a list of all contributions to it.