Direct factor

From Groupprops

This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]
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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 normality
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Definition

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

Symbol-free definition

A direct factor of a group is a subgroup satisfying the following equivalent conditions:

Its internal direct product with another subgroup is the whole group.
It is a normal subgroup that has a normal complement.
There is another subgroup that centralizes it, intersects it trivially, and such that their product is the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed a direct factor if it satisfies the following equivalent conditions:

$H$ is a factor in an internal direct product giving $G$ , i.e., there is another subgroup $K$ of $G$ such that $G$ is the internal direct product of $H$ and $K$ .
$H$ is a normal subgroup of $G$ and there is a normal subgroup $K$ of $G$ such that $H K = G$ and $H \cap K$ is trivial.
There is a subgroup $K$ of $G$ such that every element of $H$ commutes with every element of $K$ , $H K = G$ , and $H \cap K$ is trivial.

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
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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

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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.
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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 $N K$ .

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.
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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 $\langle K_1, K_2 \rangle$ . 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 $ρ - 1 (H)$ 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 +
Defining ingredient	Internal direct product +, Normal subgroup +, and Normal complement +
Dissatisfies metaproperty	Intersection-closed subgroup property +, Strongly intersection-closed subgroup property +, and Join-closed subgroup property +
Page class	Term +
Quick phrase	factor in internal direct product +, normal with normal complement +, and has centralizing complement +
Satisfies metaproperty	Monadic second-order subgroup property +, 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 +, and Quotient-transitive 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 +, Permutably complemented subgroup +, Lattice-complemented subgroup +, 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 +, and Normal subgroup +
Variation of	Normal subgroup +
Weaker than	Fully invariant direct factor +, Characteristic direct factor +, and Abelian direct factor +