Direct factor

This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
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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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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 $H K = G$ , or equivalently, if $G$ is the internal direct product of $H$ and $K$ .

Relation with other properties

Stronger properties

Weaker properties

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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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 $N K$ .

The property of being a direct factor is a right transiter, though not the right transiter, of the property of normality.

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. Further information: 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. Further information: 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, hence the property of being a direct factor satisfies the 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).
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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.

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
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Let $H \leq K \leq G$ 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 $G$ .