# 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 $HK = G$.
4 factor in internal direct product of multiple subgroups it is one of the subgroups occurring in an internal direct product decomposition of the whole group into (possibly more than two) subgroups. Note that we also allow infinitely many subgroups, in which case, the internal direct product would correspond to the restricted external direct product. there is a collection of subgroups $H_i, i \in I$ with $H$ equal to one of the $H_i$s, such that $G$ is the internal direct product of the $H_i$s.

### Equivalence of definitions

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

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki
This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
View a list of pivotal subgroup properties | View a complete list of subgroup properties[SHOW MORE]
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

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.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes direct factor is transitive If $H \le K \le G$, with $H$ a direct factor of $K$ and $K$ a direct factor of $G$, then $H$ 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 $\langle M_1, M_2 \rangle$.
lower central series condition Yes direct factor satisfies lower central series condition Suppose $H$ is a direct factor of a group $G$. Then, for any positive integer $k$, the lower central series member $\gamma_k(H)$ is a direct factor of $\gamma_k(G)$.

## 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 Characteristic direct factor|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) Complemented central factor, Join of direct factor and central subgroup, Join of finitely many direct factors, Join-transitively central factor, Right-quotient-transitively central factor, Upper join of direct factors|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) Complemented central factor, Complemented transitively normal subgroup, Right-transitively complemented normal subgroup|FULL LIST, MORE INFO
endomorphism kernel kernel of an endomorphism (via complemented normal) (via complemented normal) Complemented normal subgroup, Intermediately endomorphism kernel|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) Central factor, Characteristic subgroup of direct factor, Complemented central factor, Complemented normal subgroup, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Direct factor over central subgroup, Endomorphism kernel, Intermediately endomorphism kernel, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Normal AEP-subgroup, Normal subgroup having a 1-closed transversal, Normal subgroup in which every subgroup characteristic in the whole group is characteristic, Normal subgroup whose focal subgroup equals its derived subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup, Right-quotient-transitively central factor, SCAB-subgroup... further results|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 Complemented central factor, Complemented normal subgroup, Retract|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 Complemented central factor, Complemented normal subgroup, Permutably complemented subgroup, Retract, Right-transitively lattice-complemented subgroup|FULL LIST, MORE INFO
verbally closed subgroup image of subgroup under word map equals its intersection with image of whole group under word map (via retract) (via retract) Retract|FULL LIST, MORE INFO
local divisibility-closed subgroup if an element in the subgroup has a $n^{th}$ root in the whole group, it has a $n^{th}$ root in the subgroup. (via retract) (via retract) Retract, Verbally closed subgroup|FULL LIST, MORE INFO
local powering-invariant subgroup if an element in the subgroup has a unique $n^{th}$ root in the whole group, that root is in the subgroup. (via local divisibility-closed) (via local divisibility-closed) Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Retract, Verbally closed subgroup|FULL LIST, MORE INFO
divisibility-closed subgroup if every element in the subgroup has a $n^{th}$ root in the whole group, every element has a $n^{th}$ root in the subgroup. Retract, Verbally closed subgroup|FULL LIST, MORE INFO
powering-invariant subgroup if every element has a unique $n^{th}$ root in the group, every element of the subgroup has a unique $n^{th}$ root in the subgroup. (via local divisibility-closed) (via local divisibility-closed) Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Retract, Verbally closed subgroup|FULL LIST, MORE INFO
powering-invariant normal subgroup both powering-invariant and normal (follows from implications for powering-invariant and normal Endomorphism kernel, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
quotient-powering-invariant subgroup normal, and if every element has a unique $n^{th}$ root in the group, every element of the quotient group has a unique $n^{th}$ root in the quotient group. ([[endomorphism kernel implies quotient-powering-invariant|via endomorphism kernel) (via endomorphism kernel) Endomorphism kernel|FULL LIST, MORE INFO
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 Join of direct factor and central subgroup, Join of finitely many direct factors, Right-quotient-transitively central factor|FULL LIST, MORE INFO
conjugacy-closed subgroup any two elements of the subgroup conjugate in the whole group are conjugate in the subgroup Base of a wreath product, Central factor, Conjugacy-closed normal subgroup, Retract, Subset-conjugacy-closed 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 Base of a wreath product, Intermediately AEP-subgroup, Normal intermediately AEP-subgroup, Sectionally AEP-subgroup|FULL LIST, MORE INFO
normal AEP-subgroup both a normal subgroup and an AEP-subgroup Normal intermediately AEP-subgroup|FULL LIST, MORE INFO
intermediately AEP-subgroup AEP-subgroup in every intermediate subgroup Normal intermediately AEP-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) AEP-subgroup, Normal AEP-subgroup, Normal intermediately AEP-subgroup, Normal subgroup in which every subgroup characteristic in the whole group is characteristic|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) Normal AEP-subgroup|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 Central factor, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Right-quotient-transitively central factor, SCAB-subgroup, Upper join of direct factors|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 Central factor, Join of finitely many direct factors, Join-transitively central factor|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

## 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)

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

## Effect of property operators

Operator Meaning Result of application Proof
finite-join-closure a subgroup that can be expressed as a join of finitely many direct factors. join of finitely many direct factors by definition
join-closure a subgroup that can be expressed as a join of direct factors. join of direct factors by definition
image-potentially operator a subgroup that can arise as the image under a surjective homomorphism of a direct factor of some group. central factor 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