Transpose-inverse map is inner automorphism on special linear group of degree two
From Groupprops
Statement
Let be any commutative unital ring and
be the special linear group of degree two over
. Then, the transpose-inverse map, restricted to
, is an inner automorphism, and equals conjugation by the matrix
.
In other words, for any matrix :
.
Related facts
- Transpose-inverse map is composite of inner automorphism and division by determinant on general linear group of degree two
- Transpose-inverse map induces inner automorphism on projective general linear group of degree two
- Special unitary group of degree two equals special linear group of degree two over a finite field