# Special linear group of degree two has a class-inverting automorphism

## Statement

Let $k$ be a field and $G := SL(2,k)$ be the Special linear group (?) of degree two over $k$. Then, $G$ is a Group having a class-inverting automorphism (?). In other words, there is an automorphism $\sigma$ of $G$ that is a Class-inverting automorphism (?): it sends every element into the conjugacy class of its inverse.

We can choose the following to be the class-inverting automorphism: conjugation in $GL(2,k)$ by the matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}$. Alternatively, we can choose conjugation in $GL(2,k)$ by the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$.