Retract

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Definition

Equivalent definitions in tabular format

No. Shorthand A subgroup of a group is termed a retract if ... A subgroup H of a group G is termed a retract of G 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 \sigma of G such that \sigma^2 = \sigma and the image of \sigma is precisely H.
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 N of G such that NH = G and  N \cap H 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 \varphi:H \to K to any group K, there exists a homomorphism \theta:G \to K such that \theta|_H = \varphi.
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]

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

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

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

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
endomorphism image image under an endomorphism (not necessarily idempotent) of the whole group) |FULL LIST, MORE INFO
conjugation-invariantly permutably complemented subgroup there is a permutable complement to it that is also a permutable complement to all its conjugate subgroups |FULL LIST, MORE INFO
permutably complemented subgroup there is a permutable complement to it Strongly permutably complemented subgroup|FULL LIST, MORE INFO
lattice-complemented subgroup there is a lattice complement to it Permutably complemented subgroup, Right-transitively lattice-complemented subgroup, Strongly permutably complemented subgroup|FULL LIST, MORE INFO
subset-conjugacy-closed subgroup any conjugation between two subsets in the whole group can also be achieved by conjugation in the subgroup retract implies subset-conjugacy-closed subset-conjugacy-closed not implies retract |FULL LIST, MORE INFO
conjugacy-closed subgroup any two elements that are conjugate in the whole group are conjugate in the subgroup retract implies conjugacy-closed (via subset-conjugacy-closed) Subset-conjugacy-closed subgroup|FULL LIST, MORE INFO
central factor of normalizer retract implies WC Subset-conjugacy-closed subgroup|FULL LIST, MORE INFO
c-normal subgroup |FULL LIST, MORE INFO
verbally closed subgroup |FULL LIST, MORE INFO
subgroup whose derived subgroup equals its intersection with whole derived subgroup Retract implies derived subgroup equals intersection with whole derived subgroup
local divisibility-closed subgroup if an element in the subgroup has a n^{th} root in the whole group, it has a n^{th} root in the subgroup. (via verbally closed) (via verbally closed) Verbally closed subgroup|FULL LIST, MORE INFO
local powering-invariant subgroup if an element in the subgroup has a unique n^{th} root in the whole group, that root is in the subgroup. (via local divisibility-closed) (via local divisibility-closed) Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Verbally closed subgroup|FULL LIST, MORE INFO
divisibility-closed subgroup if every element in the subgroup has a n^{th} root in the whole group, every element has a n^{th} root in the subgroup. Verbally closed subgroup|FULL LIST, MORE INFO
powering-invariant subgroup if every element has a unique n^{th} root in the group, every element of the subgroup has a unique n^{th} root in the subgroup. (via local divisibility-closed) (via local divisibility-closed) Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Verbally closed subgroup|FULL LIST, MORE INFO

Related properties

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

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

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsRetract
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
GAP-codable subgroup property