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

From Groupprops

## Contents

## Statement

Suppose is a field and is a natural number. Let be the general linear group and be the Projective general linear group (?), i.e., the quotient group . Then, the transpose-inverse map on induces an automorphism on 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

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

## Proof

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