Direct factor

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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 $K\leq C_{G}(H)$ (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.

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

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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 $H\leq K\leq G$, 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 $H\leq M\leq G$ 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 $H\leq M\leq G$ 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 $H\leq G$ 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			|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
endomorphism kernel	kernel of an endomorphism	(via complemented normal)	(via complemented normal)	|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
verbally closed subgroup	image of subgroup under word map equals its intersection with image of whole group under word map	(via retract)	(via 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)	|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)	|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.			|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)	|FULL LIST, MORE INFO
powering-invariant normal subgroup	both powering-invariant and normal	(follows from implications for powering-invariant and normal		|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)	|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	|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

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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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
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GAP-codable subgroup property