I know that for a commutative group $G$ the map $f(x)=x^2$ is a homomorphism from $G$ to $G$. My question : When this map $f$ will be a automorphism on $G$, where $G$ is a commutative group.
In other words, when the map $f:G\rightarrow G$ defined by $f(x)=x^2$ is an onto map for a commutative group $G$?
If $G$ is finite it is equivalent to say that $f$ is injective, i.e. there are no elements of order $2$.
If $G$ is finitely generated abelian then it must be finite (bt the structure theorem on these groups) and the previous statement applies.
In general I don't know of any characterization. For example, take a field $K$ (here the group is $K$ with addition). The situation depends a lot on $K$ (consider $K=\mathbb{Q},\mathbb{F}_p,\mathbb{R},\mathbb{C}$).