Transpose-inverse map is class-inverting automorphism for general linear group
This basically follows from the fact that every element of general linear group is conjugate to its transpose.
- General linear group implies every element is automorphic to its inverse
- Transpose-inverse map induces class-inverting automorphism for projective general linear group
- Projective general linear group implies every element is automorphic to its inverse
- Transpose-inverse map induces inner automorphism on projective general linear group of degree two
- Transpose-inverse map is inner automorphism on special linear group of degree two
By fact (1), we see that the transpose-inverse map is sending every element to an element that is the inverse of some conjugate of that element. Thus, the image of any element is in the conjugacy class of its inverse, and so the tranpose-inverse map is a class-inverting automorphism.