To prove , if Aut$ (G)$ is trivial then $x^2=e , \forall x \in G$

Question

If for a group $G$ the only automorphism is the identity automorphism , then how do we prove that $x^2=e ,\forall x \in G $ ? I have only been able to prove that $G$ is abelian ; Please Help .

Answer 1

Since $G$ is abelian, $x \mapsto x^{-1}$ is an automorphism. By hypothesis it has to be equal to the identity, so $x^{-1} = x \implies x^2 = e$.

Answer 2

If $G$ is abelian, consider the map $x \to x^{-1}$.