Transpose-inverse map induces class-inverting automorphism on projective general linear group

Statement

Suppose $F$ is a field and $n$ is a natural number. Let $G=GL(n,F)$ be the general linear group and $P=PGL(n,F)$ be the Projective general linear group (?), i.e., the quotient group $G/Z(G)$. Then, the transpose-inverse map on $G$ induces an automorphism on $P$ which is a class-inverting automorphism.

Related facts

• Transpose-inverse map is class-inverting automorphism for general linear group
• General linear group implies every element is automorphic to its inverse
• Projective general linear group implies every element is automorphic to its inverse
• Special linear group implies every element is automorphic to its inverse
• Projective special linear group implies every element is automorphic to its inverse

Facts used

1. Transpose-inverse map is class-inverting automorphism for general linear group
2. Class-inverting automorphism induces class-inverting automorphism on any quotient

Proof

The proof follows directly from facts (1) and (2).