Cube map

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

Definition with symbols
The cube map on a group $$G$$ is the map sending each $$x$$ in $$G$$ to $$x^3$$.

Endomorphism, surjective endomorphism, and automorphism

 * Cube map is surjective endomorphism implies abelian
 * Cube map is endomorphism iff abelian (if order is not a multiple of 3)

Image
Elements that lie in the image of the cube map are termed cube elements. When the order of the group is not a multiple of $$3$$, then all elements are cube elements.