Square map

Symbol-free definition
The square map is a map from a group to itself that sends each element to its square.

Definition with symbols
The square map on a group $$G$$ is the map sending each $$x$$ in $$G$$ to $$x^2$$.

Endomorphism
A group is abelian if and only if the square map is an endomorphism on it.

Image
Elements that lie in the image of the square map are termed square elements. When the group is of odd order, then all elements are square elements.