Central factor

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Tautology when whole group is abelian

This subgroup property is an abelian-tautological subgroup property: it is always true for a subgroup of an abelian group.
View a complete list of abelian-tautological subgroup properties

History

Origin of the concept

The concept of central factor arose from the concept of central product, which is a generalization of the direct product.

Origin of the term

The term central factor stems naturally from the term central product. Its explicit use, however, is not very standard in the literature.

Definition

Symbol-free definition

A subgroup of a group is termed a central factor if it satisfies the following equivalent conditions:

1. Every inner automorphism of the group restricts to an inner automorphism of the subgroup.
2. The product of the subgroup and its centralizer is the whole group.
3. The induced map from the quotient group to the outer automorphism group is trivial.

Definition with symbols

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

1. Given any $g$ in $G$, there is a $h$ in $H$ such that, for all $x$ in $H$, $gxg^{-1}=hxh^{-1}$.
2. $HC_{G}(H)=G$ where $C_{G}(H)$ denotes the centralizer of $H$ in $G$.
3. The map $G/H\to \operatorname {Out} (H)$ induced by the action of $G$ on $H$ by conjugation is a trivial homomorphism.

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)

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

The subgroup property of being a central factor has a first-order description as follows. A subgroup $H$ is a central factor in a group $G$ if and only if:

$\forall g\in G,\exists h\in H.\forall x\in H,\qquad hxh^{-1}=gxg^{-1}$

This is a Fraisse rank 2 expression.

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

The property of being a central factor arises in the function restriction formalism as the balanced subgroup property (function restriction formalism) with respect to the function property of being an inner automorphism. In other words:

Central factor = Inner automorphism $\to$ Inner automorphism

Meaning that a subgroup is a central factor if and only if every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.

In particular, thus, it is a left-inner subgroup property, that is, a property that can be expressd in the function restriction formalism with the left side being the property of being an inner automorphism.

In terms of the image-potentially operator

This property is obtained by applying the image-potentially operator to the property: direct factor
View other properties obtained by applying the image-potentially operator

Relation with other properties

Analogues in other algebraic structures

For a complete list, refer:

Analogues of central factor (generated semantically).

Conjunction with other properties

Some conjunctions with group properties:

• Central subgroup is a central factor that is also an abelian group.
• Nilpotent central factor is a central factor that is also a nilpotent group.

Some conjunctions with subgroup properties:

• Characteristic central factor: A central factor that is also a characteristic subgroup.

Stronger properties

property	quick decription	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Direct factor	factor in internal direct product	direct factor implies central factor	central factor not implies direct factor (see also list of examples)	|FULL LIST, MORE INFO
Central subgroup	contained in the center (equivalently, abelian central factor)	central subgroup implies central factor	central factor not implies central subgroup (see also list of examples)	|FULL LIST, MORE INFO
Cocentral subgroup	product with center is whole group	cocentral implies central factor	central factor not implies cocentral (see also list of examples)	|FULL LIST, MORE INFO
Join-transitively central factor	join with any central factor is a central factor	follows from the fact that trivial subgroup is central factor	central factor is not finite-join-closed (see also list of examples)	|FULL LIST, MORE INFO
Right-quotient-transitively central factor	normal subgroup such that any subgroup containing it with quotient a central factor in the quotient is a central factor	follows from the fact that trivial subgroup is central factor	central factor is not quotient-transitive (see also list of examples)	|FULL LIST, MORE INFO
Complemented central factor	central factor that is also a permutably complemented subgroup	obvious		|FULL LIST, MORE INFO

Weaker properties

property	quick decription	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Normal subgroup	invariant under inner automorphisms	central factor implies normal	normal not implies central factor (see also list of examples)	|FULL LIST, MORE INFO
Transitively normal subgroup	every normal subgroup of it is normal in the whole group	central factor implies transitively normal	transitively normal not implies central factor (see also list of examples)	|FULL LIST, MORE INFO
CEP-subgroup	every normal subgroup of it is the intersection with it of a normal subgroup of the whole group	(via transitively normal)	(via transitively normal)	|FULL LIST, MORE INFO
Conjugacy-closed subgroup	elements of the subgroup conjugate in the whole group are conjugate in the subgroup	central factor implies conjugacy-closed	conjugacy-closed normal not implies central factor	|FULL LIST, MORE INFO
Conjugacy-closed normal subgroup	conjugacy-closed and normal	(via normal + conjugacy-closed)	conjugacy-closed normal not implies central factor	|FULL LIST, MORE INFO
SCAB-subgroup	subgroup-conjugating automorphism of whole group restricts to subgroup-conjugating automorphism of subgroup	central factor implies SCAB	SCAB not implies central factor	|FULL LIST, MORE INFO
Subset-conjugacy-closed subgroup
Normal subset-conjugacy-closed subgroup
Weak subset-conjugacy-closed subgroup
WNSCC-subgroup
Central factor of normalizer

Metaproperties

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)

Metaproperty name	Satisfied?	Proof	Section in this article
Transitive subgroup property	Yes	Central factor is transitive	#Transitivity
Trim subgroup property	Yes		#Trimness
Finite-intersection-closed subgroup property	No	Central factor is not finite-intersection-closed	#Intersection-closedness
Finite-join-closed subgroup property	No	Central factor is not finite-join-closed	#Join-closedness
Quotient-transitive subgroup property	No	Central factor is not quotient-transitive	#Quotient-transitivity
Intermediate subgroup condition	Yes	Central factor satisfies intermediate subgroup condition	#Intermediate subgroup condition
Transfer condition	No	Central factor does not satisfy transfer condition	#Transfer condition
Image condition	Yes	Central factor satisfies image condition	#Image condition
Direct product-closed subgroup property	Yes	Central factor is direct product-closed	#Direct products
Upper join-closed subgroup property	Yes	Central factor is upper join-closed	#Upper joins
Centralizer-closed subgroup property	Yes	Central factor is centralizer-closed	#Centralizer-closedness

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
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

Since the property of being a central factor is a balanced subgroup property with respect to the function restriction formalism, it is a t.i. subgroup property, that is, it is both transitive and identity-true.

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 property of being a central factor is trim, viz both the whole group and the trivial subgroup are central factors.

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 two central factors need not be a central factor. For full proof, refer: Central factor is not finite-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 two central factors of a group need not be a central factor. For full proof, refer: Central factor is not finite-join-closed

Quotient-transitivity

This subgroup property is not quotient-transitive: the corresponding quotient property is transitive.

Suppose $H\leq K\leq G$ are groups such that $H$ is a central factor of $G$ and $K/H$ is a central factor of $G/H$. Then, $K$ need not be a central factor of $G$. For full proof, refer: Central factor is not quotient-transitive

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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Given any groups $H\leq K\leq G$ such that $H$ is a central factor of $G$, $H$ is also a central factor of $K$. The reason is as follows:

Since the property of being a central factor is a left-inner subgroup property, and hence a Left-extensibility-stable subgroup property it satisfies the intermediate subgroup condition. For full proof, refer: Left-extensibility-stable implies intermediate subgroup condition

Transfer condition

This subgroup property does not satisfy the transfer condition

If $H,K\leq G$ such that $H$ is a central factor of $G$, it is not necessary that $H\cap K$ be a central factor of $K$. For full proof, refer: Central factor does not satisfy transfer condition

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
View other subgroup properties satisfying image condition

Under a surjective homomorphism of groups $\rho :G\to K$, if $H$ is a central factor of $G$, $\rho (H)$ is a central factor of $K$. For full proof, refer: Central factor satisfies image condition

Direct product-closedness

This subgroup property is direct product-closed: it is closed under taking arbitrary direct products of groups

Suppose $H_{1}$ is a central factor of $G_{1}$ and $H_{2}$ is a central factor of $G_{2}$. Then, $H_{1}\times H_{2}$ is a central factor of $G_{1}\times G_{2}$. This follows from the fact that the centralizer of the direct product of the subgroups equals the direct products of their individual centralizers. For full proof, refer: Central factor is direct product-closed

Upper join-closedness

This subgroup property is upper join-closed, viz., if a subgroup has the property in a collection of intermediate subgroups, it also has the property in their join
View other such properties

If $H$ is a central factor in $K_{i},i\in I$, all sitting inside a group $G$, then $H$ is also a central factor inside the join of $K_{i}$. This follows from the fact that $H$ is a central factor inside $K_{i}$ if and only if $HC_{G}(H)$ contains $K_{i}$. For full proof, refer: Central factor is upper join-closed

Centralizer-closedness

This subgroup property is centralizer-closed: the centralizer of any subgroup with this property, in the whole group, again has this property
View other centralizer-closed subgroup properties

If $H$ is a central factor of a group $G$, the centralizer $C_{G}(H)$ is also a central factor of $G$.

For full proof, refer: Central factor is centralizer-closed

Effect of property operators

The join-transiter

Applying the join-transiter to this property gives: join-transitively central factor

A join-transitively central factor is a subgroup of a group whose join with any central factor is a central factor. All central subgroups and all direct factors are join-transitively central factors. For full proof, refer: Central subgroup implies join-transitively central factor, direct factor implies join-transitively central factor

The right quotient-transiter

Applying the right quotient-transiter to this property gives: right-quotient-transitively central factor]]

The intersection-transiter

Applying the intersection-transiter to this property gives: intersection-transitively central factor