# Cube map

From Groupprops

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

## Contents

## 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 is the map sending each in to .

## 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 , then all elements are cube elements.