Cube map

Tihs article defines something that gives a well-defined function from every group to itself, that is invariant under group isomorphisms

Definition

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 $x^{3}$.

Facts

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.