Equivalent definitions in tabular format
|No.||Shorthand||A subgroup of a group is termed a retract if ...||A subgroup of a group is termed a retract of if ...|
|1||image of idempotent endomorphism||there is an idempotent endomorphism of the group whose image is precisely that subgroup. This idempotent endomorphism is termed the retraction.||there is an endomorphism of such that and the image of is precisely .|
|2||normal complement||it has a normal complement: a normal subgroup that intersects it trivially, and that together with it generates the whole group.||there is a normal subgroup of such that and is trivial.|
|3||homomorphism extension||any homomorphism from the subgroup to any group extends to a homomorphism from the whole group to that group||for any homomorphism of groups to any group , there exists a homomorphism such that .|
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]
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
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
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
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|direct factor||direct factor implies retract||retract not implies direct factor|||FULL LIST, MORE INFO|
|free factor||free factor implies retract||retract not implies free factor|||FULL LIST, MORE INFO|
|regular retract|||FULL LIST, MORE INFO|
|Property||Meaning||Proof of one non-implication||Proof of other non-implication||Notions stronger than both||Notions weaker than both|
|Normal subgroup||invariant under all inner automorphisms||retract not implies normal||normal not implies direct factor||direct factor is the conjunction||SCDIN-subgroup, Subgroup having a left transversal that is also a right transversal, Subgroup having a symmetric transversal|FULL LIST, MORE INFO|
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 is a retract of , and is a retract of , then is a retract of .
For full proof, refer: Retract is transitive
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.
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.
One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.GAP-codable subgroup property
View the GAP code for testing this subgroup property at: IsRetract
View other GAP-codable subgroup properties | View subgroup properties with in-built commands