Direct factor

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Definition

QUICK PHRASES: factor in internal direct product, normal with normal complement, has centralizing complement

Definition in tabular form

A direct factor of a group is defined in the following equivalent ways:

No.	Shorthand	A subgroup of a group is a direct factor if ...	A subgroup $H$ of a group $G$ is a direct factor of $G$ if ...
1	factor in internal direct product	its internal direct product with some subgroup is the whole group	there is a subgroup $K$ of $G$ such that $G$ is the internal direct product of $H$ and $K$
2	normal with normal complement	it is a normal subgroup with a normal complement, i.e., it is both a normal subgroup and a retract	$H$ is normal and there is a normal subgroup $K$ of $G$ such that the product $H K = G$ and $H \cap K$ is trivial.
3	has centralizing complement	there is a subgroup centralizing it, intersecting it trivially, and whose product with it is the whole group	there is a subgroup $K$ of $G$ such that $K \le C_G(H)$ (where $C G (H)$ is the centralizer in $G$ of $H$ ), $H \cap K$ is trivial, and $H K = G$ .

Equivalence of definitions

The equivalence of definitions follows largely from the equivalence of internal and external direct product.

This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]
The article text may, however, contain more than just the basic definition
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This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
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This is a variation of normal subgroup
Find other variations of normal subgroup | Read a survey article on varying normal subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and retract
View other subgroup property conjunctions | view all subgroup properties

Examples

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Extreme examples

Every group is the internal direct product of itself and the trivial subgroup. Thus:

The trivial subgroup is a direct factor of the whole group.
Every group is a direct factor of itself.

High occurrence examples

In a finite nilpotent group, all the Sylow subgroups are direct factors. In particular, a finite nilpotent group is the direct product of its Sylow subgroups. Further information: equivalence of definitions of finite nilpotent group
In a vector space, any vector subspace is a direct factor, because the complementary subspace can be taken as the complement for an internal direct product.

Relationship with external direct product and restricted external direct product

If a group $G$ arises as the external direct product of finitely or infinitely many groups $G_i, i \in I$ , then for any subset $J \subseteq I$ , the subset of $G$ arising as those elements where all coordinates outside of $J$ are trivial is a direct factor of $G$ . The complementary factor can be taken as the subgroup of $G$ where all coordinates in $J$ are trivial.
A similar observation holds for the restricted external direct product.

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

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Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
Fully invariant direct factor	direct factor and a fully invariant subgroup	click here
Characteristic direct factor	direct factor and a characteristic subgroup
Abelian direct factor	direct factor and an abelian group

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Central factor	product with centralizer is whole group	direct factor implies central factor	central factor not implies direct factor	click here
Complemented normal subgroup	normal subgroup with a (not necessarily normal) complement		complemented normal not implies direct factor	click here
Retract	subgroup with a normal complement	direct factor implies retract	retract not implies direct factor
Normal subgroup	invariant under all inner automorphisms	direct factor implies normal	normal not implies direct factor	click here
Permutably complemented subgroup	there exists a permutable complement: a subgroup intersecting it trivially and such that their product is the whole group			click here
Lattice-complemented subgroup	there exists a lattice complement: a subgroup whose intersection with it is trivial and join with it is the whole group			click here

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Related group properties

Group property	Definition in terms of direct factor
Directly indecomposable group	nontrivial group with no proper nontrivial direct factor
Complete group	it is a direct factor of any bigger group in which it is a normal subgroup
Group in which every normal subgroup is a direct factor	every normal subgroup is a direct factor

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
Transitive subgroup property	Yes	direct factor is transitive	If $M \le H \le G$ , with $M$ a direct factor of $H$ and $H$ a direct factor of $G$ , then $M$ is a direct factor of $G$ .
Finite-intersection-closed subgroup property	No	direct factor is not finite-intersection-closed	We can have $H, M$ direct factors of $G$ but $H \cap M$ not a direct factor of $G$ .
Finite-join-closed subgroup property	No	direct factor is not finite-join-closed	We can have $H, M$ direct factors of $G$ but $\langle H, M \rangle$ not a direct factor of $G$ .
Intermediate subgroup condition	Yes	direct factor satisfies intermediate subgroup condition	If $H \le M \le G$ with $H$ a direct factor of $G$ , then $H$ is a direct factor of $M$ .
Trim subgroup property	Yes		The whole group and the trivial subgroup are direct factors
Image condition	No	direct factor does not satisfy image condition	We can have a surjective homomorphism $\varphi:G \to L$ and a direct factor $H$ of $G$ such that $\varphi(H)$ is not a direct factor of $L$ .
Quotient-transitive subgroup property	Yes	direct factor is quotient-transitive	If $H \le M \le G$ with $H$ a direct factor of $G$ and $M / H$ a direct factor of $G / H$ , then $M$ is a direct factor of $G$ .
upper join-closed subgroup property	No	direct factor is not upper join-closed	We can have $H \le G$ and $M 1, M 2$ intermediate subgroups such that $H$ is a direct factor in each but not in $\langle M_1, M_2 \rangle$ .

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Effect of property operators

The finite-join-closure

Applying the finite-join-closure to this property gives: join of finitely many direct factors

The join-closure

Applying the join-closure to this property gives: join of direct factors

The image-potentially operator

Applying the image-potentially operator to this property gives: central factor

A subgroup $H$ of a group $G$ is a central factor if and only if there exists a surjective homomorphism of groups $\rho:K \to G$ such that $ρ - 1 (H)$ is a direct factor of $K$ . For full proof, refer: Central factor iff image-potentially direct factor

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: IsDirectFactor
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

Facts about Direct factorRDF feed

Applying operator gives	Join of finitely many direct factors +, Join of direct factors +, and Central factor +
Conjunction involving	Normal subgroup +, and Retract +
Defining ingredient	Internal direct product +, Normal subgroup +, Normal complement +, and Retract +
Dissatisfies metaproperty	Intersection-closed subgroup property +, Strongly intersection-closed subgroup property +, Join-closed subgroup property +, Finite-intersection-closed subgroup property +, Finite-join-closed subgroup property +, and Image condition +
Dissatisfies property	Upper join-closed subgroup property +
Page class	Term +
Quick phrase	factor in internal direct product +, normal with normal complement +, and has centralizing complement +
Satisfies metaproperty	Transitive subgroup property +, Intermediate subgroup condition +, Trim subgroup property +, Trivially true subgroup property +, Identity-true subgroup property +, Left-realized subgroup property +, Right-realized subgroup property +, Quotient-transitive subgroup property +, and Monadic second-order subgroup property +
Stronger than	Right-quotient-transitively central factor +, Complemented central factor +, Join-transitively central factor +, Conjugacy-closed subgroup +, AEP-subgroup +, Normal AEP-subgroup +, Intermediately AEP-subgroup +, Normal intermediately AEP-subgroup +, Subgroup in which every subgroup characteristic in the whole group is characteristic +, Normal subgroup in which every subgroup characteristic in the whole group is characteristic +, Intersection of direct factors +, Direct factor of characteristic subgroup +, Base of a wreath product +, Transitively normal subgroup +, Conjugacy-closed normal subgroup +, SCAB-subgroup +, Central factor +, Complemented normal subgroup +, Retract +, Normal subgroup +, Permutably complemented subgroup +, and Lattice-complemented subgroup +
Variation of	Normal subgroup +
Weaker than	Fully invariant direct factor +, Characteristic direct factor +, and Abelian direct factor +

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Right-quotient-transitively central factor	normal subgroup such that any subgroup containing it with the quotient a central factor of the whole quotient, is also a central factor	direct factor implies right-quotient-transitively central factor	right-quotient-transitively central factor not implies direct factor
Complemented central factor	central factor with a complement	direct factor implies complemented central factor	complemented central factor not implies direct factor
Join-transitively central factor	join with any central factor is a central factor	direct factor implies join-transitively central factor	join-transitively central factor not implies direct factor	click here
Conjugacy-closed subgroup	any two elements of the subgroup conjugate in the whole group are conjugate in the subgroup			click here
AEP-subgroup	every automorphism of subgroup extends to an automorphism of whole group	Direct factor implies AEP	AEP not implies direct factor	click here
Normal AEP-subgroup	both a normal subgroup and an AEP-subgroup			click here
Intermediately AEP-subgroup	AEP-subgroup in every intermediate subgroup			click here
Normal intermediately AEP-subgroup	normal and an intermediately AEP-subgroup
Subgroup in which every subgroup characteristic in the whole group is characteristic	every characteristic subgroup of whole group contained in the subgorup is characteristic in the subgroup	(via AEP)	(via AEP)	click here
Normal subgroup in which every subgroup characteristic in the whole group is characteristic	normal subgroup such that every characteristic subgroup of the whole group contained in it is characteristic in it	(via normal AEP)	(via normal AEP)	click here
Intersection of direct factors	intersection of direct factors
Direct factor of characteristic subgroup	direct factor of characteristic subgroup			click here
Base of a wreath product	base of an internal wreath product
Transitively normal subgroup	every normal subgroup of it is normal in the whole group			click here
Conjugacy-closed normal subgroup	conjugacy-closed and normal			Template:Intermediate notion short
SCAB-subgroup	every subgroup-conjugating automorphism of whole group restricts to a subgroup-conjugating automorphism of subgroup			click here

Direct factor

From Groupprops

Definition

Definition in tabular form

Equivalence of definitions

Contents

Examples

Extreme examples

High occurrence examples

Relationship with external direct product and restricted external direct product

Formalisms

Monadic second-order description

Relation with other properties

Stronger properties

Weaker properties

Related group properties

Metaproperties

Transitivity

Intersection-closedness

Join-closedness

Intermediate subgroup condition

Trimness

Image condition

Quotient-transitivity

Upper join-closedness

Effect of property operators

The finite-join-closure

The join-closure

The image-potentially operator

Testing

GAP code

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