# Square map

From Groupprops

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

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Square map, all facts related to Square map) |Survey articles about this | Survey articles about definitions built on this

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |

View a complete list of semi-basic definitions on this wiki

## Definition

### 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 is the map sending each in to .

## Facts

### Endomorphism

A group is abelian if and only if the square map is an endomorphism on it. `For full proof, refer: Square map is endomorphism iff abelian`

### 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. `Further information: kth power map is bijective iff k is relatively prime to the order`