# special linear group

Given a vector space $V$, the special linear group ${\operatorname{SL}}(V)$ is defined to be the subgroup of the general linear group $\operatorname{GL}(V)$ consisting of all invertible linear transformations $T:V\longrightarrow V$ in $\operatorname{GL}(V)$ that have determinant 1.

If $V=\mathbb{F}^{n}$ for some field $\mathbb{F}$, then the group ${\operatorname{SL}}(V)$ is often denoted ${\operatorname{SL}}(n,\mathbb{F})$ or ${\operatorname{SL}}_{n}(\mathbb{F})$, and if one identifies each linear transformation with its matrix with respect to the standard basis, then ${\operatorname{SL}}(n,\mathbb{F})$ consists of all $n\times n$ matrices with entries in $\mathbb{F}$ that have determinant 1.

Title special linear group SpecialLinearGroup 2013-03-22 12:25:38 2013-03-22 12:25:38 djao (24) djao (24) 7 djao (24) Definition msc 20G15 SL GeneralLinearGroup Group UnimodularMatrix