Square map

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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 definition
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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 G is the map sending each x in G to x^2.

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