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

## Statement

Let be a field and be the Special linear group (?) of degree two over . Then, is a Group having a class-inverting automorphism (?). In other words, there is an automorphism of 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 by the matrix . Alternatively, we can choose conjugation in by the matrix .

## Related facts

- Projective special linear group of degree two has a class-inverting automorphism
- Special linear group of degree two is ambivalent iff -1 is a square
- Projective special linear group of degree two is ambivalent iff -1 is a square
- Transpose-inverse map is class-inverting automorphism for general linear group
- Transpose-inverse map induces class-inverting automorphism on projective general linear group