Retract

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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How retracts came about

Introduction of the concept

The concept of retract is fairly old, and came about in the beginning of the study of group theory. Retracts were first encountered as the right part in short exact sequences that split.

Introduction of the term

The term retract is not very standard, and the concept is often referred to without the use of this formal term. The term retract actually comes from the set-theoretic/topological equivalent notion.

Definition

Symbol-free definition

A subgroup of a group is termed a retract if there is an idempotent endomorphism of the group whose image is precisely that subgroup. This idempotent endomorphism is termed the retraction. Equivalently, a subgroup of a group is termed a retract if it has a normal complement.

Definition with symbols

A subgroup $H$ of a group $G$ is termed a retract if there is an endomorphism $\sigma$ of $G$ such that ${\sigma }^2=\sigma$ and the image of $\sigma$ is precisely $H$.

Equivalently, a subgroup $H$ of a group $G$ is a retract if there is a normal subgroup $N$ of $G$ such that $NH=G$ and $N\cap H$ is trivial.

What's a retract?

The property of being a retract is an element of the subgroup property space.

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Monadic second-order description

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups
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Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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The property of being a retract is transitive. In other words, a retract of a retract is a retract. In symbols, if $H$ is a retract of $G$, and $G$ is a retract of $K$, then $H$ is a retract of $K$.

Proof: to fill in (basically, compose the retractions)

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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The trivial subgroup is clearly a retract, the retraction being the trivial map. The improper subgroup, viz. the whole group, is also clearly a retract, the retraction map being the identity map. Thus, the property of being a retract is trim.

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed

Is the intersection of two retracts a retract? The answer is in general no, because it may even happen that an intersection of direct factors is not a retract. However, we do have some partial results:

• The intersection of two retracts in a free group is a retract.