Projective special linear group of degree two has a class-inverting automorphism

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Statement

Let k be a field and G := PSL(2,k) be the Projective 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.

Facts used

  1. Special linear group of degree two has a class-inverting automorphism
  2. Class-inverting automorphism induces class-inverting automorphism on any quotient

Proof

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