Transpose-inverse map is class-inverting automorphism for general linear group

Statement

Suppose $F$ is a field and $n$ is a natural number. Let $G=GL(n,F)$ Then, the transpose-inverse map $A\mapsto (A^{T})^{-1}$ is a class-inverting automorphism of $G$: it sends every element of $G$ to an element in the conjugacy class of its inverse.

This basically follows from the fact that every element of general linear group is conjugate to its transpose.

Related facts

• 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

Facts used

1. Every element of general linear group is conjugate to its transpose

Proof

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.