This question already has an answer here:
Let $G$ be a group. Show that if every non-identity element in $G$ has order $2$ then $G$ is abelian.
Let $a,b $ be non-identity elements in $G$. Since $|a|=|b|=2$ , that means $ab=babaab$ $=$ $ba$.
Is the proof correct? How can I improve it?