Direct factor

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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Definition

Symbol-free definition

A direct factor of a group is a subgroup whose internal direct product with another subgroup is the whole group. In other words, a direct factor is a normal subgroup that has a normal complement.

Definition with symbols

A subgroup $H$ of a group $G$ is termed a direct factor if it is normal there is another normal subgroup $K$ such that $H\cap K$ is trivial and $HK=G$, or equivalently, if $G$ is the internal direct product of $H$ and $K$.

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

• Fully characteristic direct factor
• Characteristic direct factor

Weaker properties

• Central factor
• Retract
• Complemented normal subgroup
• Permutably complemented subgroup
• Lattice-complemented subgroup
• Intersection of direct factors
• Direct factor of characteristic subgroup
• Base of a wreath product
• Normal subgroup: For proof of the implication, refer Direct factor implies normal and for proof of its strictness (i.e. the reverse implication being false) refer normal not implies direct factor. View a survey article comparing and contrasting these terms, at: Direct factor versus normal.

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
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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.
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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

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
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Under a quotient map, the image of a direct factor is again a direct factor. For full proof, refer: Direct factor satisfies 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

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
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GAP-codable subgroup property