Retract: Difference between revisions

Revision as of 17:01, 15 February 2013

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

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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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]

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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
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
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.			\|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

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

@@ Line 74: / Line 74: @@
 | [[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-invariant 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. || || || {{intermediate notions short|local divisibility-invariant subgroup|retract}}
+| [[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. || || || {{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-invariant) || (via local divisibility-invariant) || {{intermediate notions short|local powering-invariant 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-invariant 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-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-invariant) || (via local divisibility-invariant) || {{intermediate notions short|local powering-invariant 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}}
 |}