Transpose-inverse map

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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

Conditions under which it is and is not an inner automorphism

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

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