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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History

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.

Further information: Retractions and functors

Definition

Symbol-free definition

A subgroup of a group is termed a retract if it satisfies the following equivalent conditions:

• There is an idempotent endomorphism of the group whose image is precisely that subgroup. This idempotent endomorphism is termed the retraction
• It has a normal complement: a normal subgroup that intersects it trivially, and that together with it generates the whole group
• Any homomorphism from the subgroup to any group, extends to a homomorphism from the whole group to that group

Definition with symbols

A subgroup $H$ of a group $G$ is termed a retract if it satisfies the following equivalent conditions:

• There is an endomorphism $\sigma$ of $G$ such that ${\sigma }^2=\sigma$ and the image of $\sigma$ is precisely $H$
• There is a normal subgroup $N$ of $G$ such that $NH=G$ and $N\cap H$ is trivial
• Any homomorphism from $H$ to a group $K$, can be extended to a homomorphism from $G$ to $K$

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
View other monadic second-order subgroup properties

Relation with other properties

Stronger properties

• Direct factor: For proof of the implication, refer direct factor implies retract and for proof of its strictness (i.e. the reverse implication being false) refer retract not implies direct factor.
• Free factor: For proof of the implication, refer free factor implies retract and for proof of its strictness (i.e. the reverse implication being false) refer retract not implies free factor.
• Regular retract

Weaker properties

• Conjugation-invariantly permutably complemented subgroup
• Permutably complemented subgroup
• Subset-conjugacy-closed subgroup: For proof of the implication, refer retract implies subset-conjugacy-closed and for proof of its strictness (i.e. the reverse implication being false) refer subset-conjugacy-closed not implies retract.
• Conjugacy-closed subgroup: For full proof, refer: retract implies conjugacy-closed
• WC-subgroup: For full proof, refer: Retract implies WC
• c-normal subgroup

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.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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 $K$.

For full proof, refer: Retract is transitive

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).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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.

Testing

This article is about a GAP function.