Special linear group of degree two

Definition

The special linear group of degree two over a field $k$, or more generally over a commutative unital ring $R$, is defined as the group of $2\times 2$ matrices with determinant $1$ under matrix multiplication. The group is denoted by $SL(2,R)$.

The underlying set of the group is:

$SL(2,R):=\{\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\mid a,b,c,d\in R,ad-bc=1\}$.

The group operation is given by:

$\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\left(\begin{array}{cc}a\text{'}& b\text{'}\\ c\text{'}& d\text{'}\\ \end{array}\right)=\left(\begin{array}{cc}aa\text{'}+cb\text{'}& ac\text{'}+bd\text{'}\\ ca\text{'}+db\text{'}& cb\text{'}+dd\text{'}\\ \end{array}\right)$.

The identity element is:

$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$.

The inverse map is given by:

<math>\begin{pmatrix} a & b \\ c & d \\\end{pmatrix}^{-1} = \begin{pmatrix} d & -b \\ -c & a \\\end{pmatrix}