Direct factor

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 $HK=G$ 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 $HK=G$.

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. 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 page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and retract
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Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Extreme examples

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

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

High occurrence examples

1. 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
2. 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

1. 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.
2. 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	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Fully invariant direct factor	direct factor and a fully invariant subgroup			|FULL LIST, MORE INFO
Characteristic direct factor	direct factor and a characteristic subgroup			|FULL LIST, MORE INFO
Abelian direct factor	direct factor and an abelian group			|FULL LIST, MORE INFO

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 (see also list of examples)	|FULL LIST, MORE INFO
Complemented normal subgroup	normal subgroup with a (not necessarily normal) complement		complemented normal not implies direct factor (see also list of examples)	|FULL LIST, MORE INFO
Retract	subgroup with a normal complement	direct factor implies retract	retract not implies direct factor (see also list of examples)	|FULL LIST, MORE INFO
Normal subgroup	invariant under all inner automorphisms	direct factor implies normal	normal not implies direct factor (see also list of examples)	|FULL LIST, MORE INFO
Permutably complemented subgroup	there exists a permutable complement: a subgroup intersecting it trivially and such that their product is the whole group			|FULL LIST, MORE INFO
Lattice-complemented subgroup	there exists a lattice complement: a subgroup whose intersection with it is trivial and join with it is the whole group			|FULL LIST, MORE INFO

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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	|FULL LIST, MORE INFO
Complemented central factor	central factor with a complement	direct factor implies complemented central factor	complemented central factor not implies direct factor	|FULL LIST, MORE INFO
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	|FULL LIST, MORE INFO
Conjugacy-closed subgroup	any two elements of the subgroup conjugate in the whole group are conjugate in the subgroup			|FULL LIST, MORE INFO
AEP-subgroup	every automorphism of subgroup extends to an automorphism of whole group	Direct factor implies AEP	AEP not implies direct factor	|FULL LIST, MORE INFO
Normal AEP-subgroup	both a normal subgroup and an AEP-subgroup			|FULL LIST, MORE INFO
Intermediately AEP-subgroup	AEP-subgroup in every intermediate subgroup			|FULL LIST, MORE INFO
Normal intermediately AEP-subgroup	normal and an intermediately AEP-subgroup			|FULL LIST, MORE INFO
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)	|FULL LIST, MORE INFO
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)	|FULL LIST, MORE INFO
Intersection of direct factors	intersection of direct factors			|FULL LIST, MORE INFO
Direct factor of characteristic subgroup	direct factor of characteristic subgroup			Direct factor of fully invariant subgroup|FULL LIST, MORE INFO
Base of a wreath product	base of an internal wreath product			|FULL LIST, MORE INFO
Transitively normal subgroup	every normal subgroup of it is normal in the whole group			|FULL LIST, MORE INFO
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			|FULL LIST, MORE INFO

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 $⟨H,M⟩$ 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 $\phi :G\to L$ and a direct factor $H$ of $G$ such that $\phi (H)$ 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 $⟨M_1,M_2⟩$.

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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

A direct factor of a direct factor is a direct factor. In fact, the normal complement is the product of the two normal complements.

In symbols, if $H$ is a direct factor of $G$ with complement $K$ and $M$ is a direct factor of $H$ with complement $N$ then $M$ is a direct factor of $G$ with complement $NK$.

For full proof, refer: Direct factor is transitive

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

An intersection of direct factors need not be a direct factor. A counterexample can be found even for Abelian $p$-groups. For full proof, refer: direct factor is not intersection-closed

Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not join-closed

A join of direct factors need not be a direct factor. A counterexample can be found even for Abelian $p$-groups. For full proof, refer: Direct factor is not join-closed

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

A direct factor of a group is also a direct factor of any intermediate subgroup. For full proof, refer: Direct factor satisfies intermediate subgroup condition

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 whole group is a direct product of itself with the trivial subgroup. Hence, the trivial subgroup and the whole group are direct factors.

Image condition

This subgroup property does not satisfy the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property need not satisfies the property

Under a quotient map, the image of a direct factor need not be a direct factor. For full proof, refer: Direct factor does not satisfy image condition

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

Let $H\le K\le G$ be groups, such that $H$ is a direct factor of $G$ and $K/H$ is a direct factor of $G/H$. Then, $K$ is also a direct factor of $G$. For full proof, refer: Direct factor is quotient-transitive

Upper join-closedness

NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

We can have a subgroup $H\le G$ and intermediate subgroups $K_1,K_2$ containing $H$ such that $H$ is a direct factor in both $K_1$ and $K_2$ but not in $⟨K_1,K_2⟩$. For full proof, refer: Direct factor is not upper join-closed

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 ${\rho }^{-1}(H)$ 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

GAP-codable subgroup property