Retraction

This article defines a notion of an idempotent (one that equals its square) in a certain context

Definition

Symbol-free definition

A retraction is an idempotent endomorphism from a group to itself.

Definition with symbols

A retraction of a group $G$ is an endomorphism $f:G\to G$ such that $f^{2}=f$, in other words, $f(f(g))=f(g)$ for every $g\in G$. The image of $f$ is termed a retract, and the retraction can also be viewed as a map from $G$ to the subgroup which is this image.

Relation with other properties

Stronger properties

• Projection on a direct factor

Weaker properties

• Endomorphism