Central factor

Definition

QUICK PHRASES: factor in central product, product with centralizer is whole group, quotient action by outer automorphisms is trivial, every inner automorphism restricts to an inner automorphism

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.

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.
VIEW: Definitions built on this | Facts about this: (facts closely related to Central factor, all facts related to Central factor) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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

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	Statement with symbols
Abelian-tautological group property	Yes		If $G$ is abelian and $H\leq G$, then $H$ is a central factor of $G$
Transitive subgroup property	Yes	Central factor is transitive	If $H\leq K\leq G$, $H$ is a central factor of $K$ and $K$ is a central factor of $G$, then $H$ is a central factor of $G$.
Trim subgroup property	Yes		For any group $G$, both $G$ and the trivial subgroup of $G$ are central factors.
Finite-intersection-closed subgroup property	No	Central factor is not finite-intersection-closed	It is possible to have a group $G$ and central factors $H_{1},H_{2}$ such that the intersection $H_{1}\cap H_{2}$ is not a central factor.
Finite-join-closed subgroup property	No	Central factor is not finite-join-closed	It is possible to have a group $G$ and central factors $H_{1},H_{2}$ such that the join $\langle H_{1},H_{2}\rangle$ (which in this case is also the product of subgroups) is not a central factor.
Quotient-transitive subgroup property	No	Central factor is not quotient-transitive	It is possible to have groups $H\leq K\leq G$ such that $H$ is a central factor of $G$ and $K/H$ is a central factor of $G/H$ but $K$ is not a central factor of $G$.
Intermediate subgroup condition	Yes	Central factor satisfies intermediate subgroup condition	If $H\leq K\leq G$ and $H$ is a central factor of $G$, then $H$ is a central factor of $K$.
Transfer condition	No	Central factor does not satisfy transfer condition	It is possible to have a group $G$ and subgroups $H,K$ such that $H$ is a central factor of $G$ but $H\cap K$ is not a central factor of $K$.
Image condition	Yes	Central factor satisfies image condition	If $H$ is a central factor of $G$ and $\varphi :G\to K$ is a surjective homomorphism, then $\varphi (H)$ is a central factor of $K$.
Direct product-closed subgroup property	Yes	Central factor is direct product-closed	If $H_{i}\leq G_{i},i\in I$ are all central factors, then the direct product of $H_{i}$s is a central factor of the direct product of $G_{i}$s
Upper join-closed subgroup property	Yes	Central factor is upper join-closed	If $H\leq G$ and $K_{i},i\in I$ are subgroups of $G$ containing $H$ such that $H$ is a central factor of each $K_{i}$, then $H$ is also a central factor of the join of subgroups $\langle K_{i}\rangle _{i\in I}$.
Centralizer-closed subgroup property	Yes	Central factor is centralizer-closed	If $H\leq G$ is a central factor, the centralizer $C_{G}(H)$ is also a central factor.

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	Meaning	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	Meaning	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

Effect of property operators

Operator	Meaning	Result of application	Proof
join-transiter	join with any central factor is a central factor	join-transitively central factor	by definition
right quotient-transiter	Any subgroup containing it so that the quotient is a central factor in the quotient of the whole group, is a central factor	right-quotient-transitively central factor	by definition
intersection-transiter	Intersection with any central factor is a central factor	intersection-transitively central factor	by definition

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

A subgroup $H$ of a group $G$ is a central factor if and only if the following is true: there exists a group $K$, a direct factor $L$ of $K$, and a surjective homomorphism $\rho :K\to G$ such that $\rho (L)=H$.

In terms of the upper join-closure operator

This property is obtained by applying the upper join-closure operator to the property: cocentral subgroup
View other properties obtained by applying the upper join-closure operator

A subgroup $H$ of a group $G$ is a central factor if and only if there exist groups $K_{i},i\in I$ all contained in $G$ such that $H$ is a cocentral subgroup of each $K_{i}$ (i.e., $HZ(K_{i})=K_{i}$) and the join of the $K_{i}$s equals $G$.