Retract: Difference between revisions

Latest revision as of 03:55, 9 March 2020

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 $σ$ of $G$ such that $σ^{2} = σ$ and the image of $σ$ 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 $N H = G$ 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 $φ : H \to K$ to any group $K$ , there exists a homomorphism $θ : G \to K$ such that $θ \|_{H} = φ$ .

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]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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]

VIEW: Definitions built on this | Facts about this: (facts closely related to Retract, all facts related to Retract) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View other semistandard definitions in group theory

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	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			\|FULL LIST, MORE INFO
lattice-complemented subgroup	there is a lattice complement to it			\|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)	\|FULL LIST, MORE INFO
central factor of normalizer		retract implies WC		\|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 $n^{t h}$ root in the whole group, it has a $n^{t h}$ root in the subgroup.	(via verbally closed)	(via verbally closed)	\|FULL LIST, MORE INFO
local powering-invariant subgroup	if an element in the subgroup has a unique $n^{t h}$ root in the whole group, that root is in the subgroup.	(via local divisibility-closed)	(via local divisibility-closed)	\|FULL LIST, MORE INFO
divisibility-closed subgroup	if every element in the subgroup has a $n^{t h}$ root in the whole group, every element has a $n^{t h}$ root in the subgroup.			\|FULL LIST, MORE INFO
powering-invariant subgroup	if every element has a unique $n^{t h}$ root in the group, every element of the subgroup has a unique $n^{t h}$ root in the subgroup.	(via local divisibility-closed)	(via local divisibility-closed)	\|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	\|FULL LIST, MORE INFO

Metaproperties

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)

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	retract is transitive	If $H \leq K \leq G$ are groups such that $H$ is a retract of $K$ and $K$ is a retract of $G$ , then $H$ is a retract of $G$ .
trim subgroup property	Yes		In any group, the whole group and the trivial subgroup are retracts.
finite-intersection-closed subgroup property	No		It is possible to have a group $G$ and subgroups $H$ and $K$ of $G$ such that both $H$ and $K$ are retracts but $H \cap K$ is not a retract of $G$ .

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

@@ Line 1: / Line 1: @@
+[[importance rank::2| ]]
+==Definition==
+===Equivalent definitions in tabular format===
+{| class="sortable" border="1"
+! No. !! Shorthand !! A subgroup of a group is termed a retract if ... !! A subgroup <math>H</math> of a group <math>G</math> is termed a retract of <math>G</math> 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 <math>\sigma</math> of <math>G</math> such that <math>\sigma^2 = \sigma</math> and the image of <math>\sigma</math> is precisely <math>H</math>.
+|-
+| 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 <math>N</math> of <math>G</math> such that <math>NH = G</math> and <math> N \cap H</math> 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]] <math>\varphi:H \to K</math> to any group <math>K</math>, there exists a homomorphism <math>\theta:G \to K</math> such that <math>\theta|_H = \varphi</math>.
+|}
 {{subgroup property}}
 {{semistddef}}
@@ Line 14: / Line 28: @@
 {{further|[[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 an automorphism from the whole group to that group
-===Definition with symbols===
-A subgroup <math>H</math> of a group <math>G</math> is termed a '''retract''' if it satisfies the following equivalent conditions:
-* There is an endomorphism <math>\sigma</math> of <math>G</math> such that <math>\sigma^2 = \sigma</math> and the image of <math>\sigma</math> is precisely <math>H</math>
-* There is a normal subgroup <math>N</math> of <math>G</math> such that <math>NH = G</math> and <math> N \cap H</math> is trivial
-* Any homomorphism from <math>H</math> to a group <math>K</math>, can be extended to a homomorphism from <math>G</math> to <math>K</math>
 ==Formalisms==
@@ Line 42: / Line 39: @@
 ===Stronger properties===
-* [[Weaker than::Direct factor]]: {{proofofstrictimplicationat|[[direct factor implies retract]]|[[retract not implies direct factor]]}}
+{| class="sortable" border="1"
-* [[Weaker than::Free factor]]: {{proofofstrictimplicationat|[[free factor implies retract]]|[[retract not implies free factor]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Regular retract]]
+|-
+| [[Weaker than::direct factor]] || || [[direct factor implies retract]] || [[retract not implies direct factor]] || {{intermediate notions short|retract|direct factor}}
+|-
+| [[Weaker than::free factor]] || ||[[free factor implies retract]] ||[[retract not implies free factor]] || {{intermediate notions short|retract|free factor}}
+|-
+| [[Weaker than::regular retract]] || || || || {{intermediate notions short|retract|regular retract}}
+|}
 ===Weaker properties===
-* [[Stronger than::Conjugation-invariantly permutably complemented subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::Permutably complemented subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Subset-conjugacy-closed subgroup]]: {{proofofstrictimplicationat|[[retract implies subset-conjugacy-closed]]|[[subset-conjugacy-closed not implies retract]]}}
+|-
-* [[Stronger than::Conjugacy-closed subgroup]]: {{proofat|[[retract implies conjugacy-closed]]}}
+| [[Stronger than::endomorphism image]] || image under an [[endomorphism]] (not necessarily idempotent) of the whole group || || || {{intermediate notions short|endomorphism image|retract}}
-* [[Stronger than::WC-subgroup]]: {{proofat|[[Retract implies WC]]}}
+|-
-* [[Stronger than::c-normal subgroup]]
+| [[Stronger than::conjugation-invariantly permutably complemented subgroup]] || there is a permutable complement to it that is also a permutable complement to all its [[conjugate subgroups]] || || || {{intermediate notions short|conjugation-invariantly permutably complemented subgroup|retract}}
+|-
-==Metaproperties==
+| [[Stronger than::permutably complemented subgroup]] || there is a [[permutable complements|permutable complement]] to it || || || {{intermediate notions short|permutably complemented subgroup|retract}}
+|-
+| [[Stronger than::lattice-complemented subgroup]] || there is a [[lattice complements|lattice complement]] to it || || || {{intermediate notions short|lattice-complemented subgroup|retract}}
+|-
+| [[Stronger than::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]] || {{intermediate notions short|subset-conjugacy-closed subgroup|retract}}
+|-
+| [[Stronger than::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) || {{intermediate notions short|conjugacy-closed subgroup|retract}}
+|-
+| [[Stronger than::central factor of normalizer]] || || [[retract implies WC]] || || {{intermediate notions short|central factor of normalizer|retract}}
+|-
+| [[Stronger than::c-normal subgroup]] || || || || {{intermediate notions short|c-normal subgroup|retract}}
+|-
+| [[Stronger than::verbally closed subgroup]] || || || || {{intermediate notions short|verbally closed subgroup|retract}}
+|-
+| [[Stronger than::subgroup whose derived subgroup equals its intersection with whole derived subgroup]] || || [[Retract implies derived subgroup equals intersection with whole derived subgroup]] || ||
+|-
+| [[Stronger than::local divisibility-closed subgroup]] || if an element in the subgroup has a <math>n^{th}</math> root in the whole group, it has a <math>n^{th}</math> root in the subgroup. || (via verbally closed) || (via verbally closed) || {{intermediate notions short|local divisibility-closed subgroup|retract}}
+|-
+| [[Stronger than::local powering-invariant subgroup]] || if an element in the subgroup has a unique <math>n^{th}</math> root in the whole group, that root is in the subgroup. || (via local divisibility-closed) || (via local divisibility-closed) || {{intermediate notions short|local powering-invariant subgroup|retract}}
+|-
+| [[Stronger than::divisibility-closed subgroup]] || if every element in the subgroup has a <math>n^{th}</math> root in the whole group, every element has a <math>n^{th}</math> root in the subgroup. || || || {{intermediate notions short|local divisibility-closed subgroup|retract}}
+|-
+| [[Stronger than::powering-invariant subgroup]] || if every element has a unique <math>n^{th}</math> root in the group, every element of the subgroup has a unique <math>n^{th}</math> root in the subgroup. || (via local divisibility-closed) || (via local divisibility-closed) || {{intermediate notions short|local powering-invariant subgroup|retract}}
+|}
-{{transitive}}
+===Related properties===
-The property of being a retract is [[transitive subgroup property|transitive]]. In other words, a retract of a retract is a retract. In symbols, if <math>H</math> is a retract of <math>G</math>, and <math>G</math> is a retract of <math>K</math>, then <math>H</math> is a retract of <math>K</math>.
+{| class="sortable" border="1"
+! 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 automorphism]]s || [[retract not implies normal]] || [[normal not implies direct factor]] || [[direct factor]] is the conjunction || {{weaker than both short|retract|normal subgroup}}
+|}
-{{proofat|[[Retract is transitive]]}}
+==Metaproperties==
-{{trim}}
+{{wikilocal-section}}
-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.
-{{not intersection-closed}}
+{| class="sortable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::transitive subgroup property]] || Yes || [[retract is transitive]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is a retract of <math>K</math> and <math>K</math> is a retract of <math>G</math>, then <math>H</math> is a retract of <math>G</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || || In any group, the whole group and the trivial subgroup are retracts.
+|-
+| [[dissatisfies metaproperty::finite-intersection-closed subgroup property]] || No || || It is possible to have a group <math>G</math> and subgroups <math>H</math> and <math>K</math> of <math>G</math> such that both <math>H</math> and <math>K</math> are retracts but <math>H \cap K</math> is not a retract of <math>G</math>.
+|}
-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:
+==Testing==
-*The intersection of two retracts in a [[free group]] is a retract.
+{{GAP code for subgroup property|
+test = IsRetract|
+classreplist = RetractClassReps}}