Prove: A group $G$ is abelian if and only if the map $G\rightarrow G$ given by $x\mapsto x^{-1}$ is an automorphism.
Solution 1:
If $G$ is abelian then the map $x\to x^{-1}$ is a homomorphism: $$(ab)^{-1}=(ba)^{-1}=a^{-1}b^{-1}$$ If $x\to x^{-1}$ is a homomorphism, $G$ is abelian: $$ab=(b^{-1}a^{-1})^{-1}=((ba)^{-1})^{-1}=ba$$
$x\to x^{-1}$ is a bijection as proved in the question. So if it is a homomorphism, it is an automorphism.