Transpose-inverse map

Definition

The transpose-inverse map is an automorphism of the general linear group over any field or ring, and it is defined as the composition of the matrix transpose and the inverse map.

The map has order two.

Facts

• The fixed-point subgroup of this map is termed the orthogonal group.
• The special linear group is invariant (i.e., it is mapped to itself) under this map.
• The transpose-inverse map defines a map (by restriction) on the special linear group and also induces maps on the projective general linear group and projective special linear group.
• The semidirect product of the general linear group by the action of the transpose-inverse map gives the outer linear group.

Conditions under which it is and is not an inner automorphism

• Transpose-inverse map is inner automorphism on special linear group of degree two
• For the general linear group of degree two over a field, the transpose-inverse map is inner iff the field is field:F2 (this follows from the corresponding observation for class-preserving automorphisms, though it can also be deduced from the fact that the transpose-inverse map is a composite of a radial automorphism -- that is not inner -- with an inner automorphism; see endomorphism structure of general linear group of degree two over a finite field for details that cross-apply to arbitrary fields).
• For general and special linear groups of higher degree, the transpose-inverse map is never inner (follows from it not being class-preserving, see the section below).

Conditions under which it is and is not a class-preserving automorphism

• Transpose-inverse map is class-preserving automorphism for general linear group of degree two iff field has size two
• The transpose-inverse map is not a class-preserving automorphism for general linear groups of higher degree. This follows from the fact that general linear group of degree three or higher is not ambivalent

Conditions under which it is and is not a class-inverting automorphism

• Transpose-inverse map is class-inverting automorphism for general linear group
• Transpose-inverse map induces class-inverting automorphism on projective general linear group
• The transpose-inverse map does not induce a class-inverting automorphism on the special linear group (unless the field size is two). However, special linear group of degree two has a class-inverting automorphism (different from the transpose-inverse map).