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Special linear group of degree two has a class-inverting automorphism

Statement

Let k be a field and G := SL(2,k) be the Special linear group (?) of degree two over k. Then, G is a Group having a class-inverting automorphism (?). In other words, there is an automorphism \sigma of G that is a Class-inverting automorphism (?): it sends every element into the conjugacy class of its inverse.

We can choose the following to be the class-inverting automorphism: conjugation in GL(2,k) by the matrix \begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}. Alternatively, we can choose conjugation in GL(2,k) by the matrix \begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}.

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