Tihs article defines something that gives a well-defined function from every group to itself, that is invariant under group isomorphisms
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 is the map sending each in to .
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)
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 , then all elements are cube elements.