Central factor: Difference between revisions

Latest revision as of 02:09, 20 August 2021

Terminological note

The term central factor used here refers to a subgroup that can occur as a factor in a central product. The term central factor is also used in another, completely different sense: a group that occurs as a quotient by a central subgroup in another group, or in some cases, as a quotient by the whole center (in which case, it would be isomorphic to the inner automorphism group). These other senses of the word are very different, and you can learn more about these senses at the central extension page.

Definition

Equivalent definitions in tabular format

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

No.	Shorthand	A subgroup of a group is termed a central factor if ...	A subgroup $H$ of a group $G$ is termed a central factor of $G$ if ...
1	inner automorphisms to inner automorphisms	every inner automorphism of the group restricts to an inner automorphism of the subgroup.	given any $g$ in $G$ , there is a $h$ in $H$ such that, for all $x$ in $H$ , $gxg^{-1}=hxh^{-1}$ .
2	product with centralizer is whole group	the product of the subgroup and its centralizer is the whole group.	$HC_{G}(H)=G$ where $C_{G}(H)$ denotes the centralizer of $H$ in $G$ .
3	normal and quotient maps trivially to outer automorphism group	it is a normal subgroup and the induced map from the quotient group to the outer automorphism group is trivial.	$H$ is a normal subgroup of $G$ and the map $G/H\to \operatorname {Out} (H)$ induced by the action of $G$ on $H$ by conjugation is a trivial homomorphism.
4	factor in central product	it can be expressed as one of the factor subgroups in an expression of the whole group as an internal central product of two groups.	there exists a subgroup $K$ of $G$ such that $G$ is an internal central product of $H$ and $K$ .

This definition is presented using a tabular format. |View all pages with definitions in tabular format

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

Examples

Extreme examples

The trivial subgroup in any group is a central factor.
Every group is a central factor in itself.

Subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Subgroups dissatisfying the property

Here are some examples of subgroups in basic/important groups not satisfying the property:

Here are some some examples of subgroups in relatively less basic/important groups not satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:

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 subgroup 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
Locally inner automorphism-balanced subgroup	every inner automorphism restricts to a locally inner automorphism			\|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 quantifier rank 3 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

Function restriction expression	$H$ is a central factor of $G$ if ...	This means that characteristicity is ...	Additional comments
inner automorphism $\to$ inner automorphism	every inner automorphism of $G$ restricts to a inner automorphism of $H$	the balanced subgroup property for inner automorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true

In particular, this means that the property of being a central factor is a left-inner subgroup property.

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 $G$ .

@@ Line 1: / Line 1: @@
-{{stdnonbasicdef}}
+[[importance rank::2| ]]
-{{subgroup property}}
+==Terminological note==
-{{variationof|normality}}
-{{Abelian-tautological subgroup property}}
-==History==
-===Origin of the concept===
+The term '''central factor''' used here refers to a [[subgroup]] that can occur as a factor in a central product. The term ''central factor'' is also used in another, completely different sense: a group that occurs as a quotient by a [[central subgroup]] in another group, or in some cases, as a quotient by the whole [[center]] (in which case, it would be isomorphic to the [[inner automorphism group]]). These other senses of the word are very different, and you can learn more about these senses at the [[central extension]] page.
-The concept of central factor arose from the concept of [[central product]], which is a generalization of the direct product.
+==Definition==
-===Origin of the term===
+===Equivalent definitions in tabular format===
-The term central factor stems naturally from the term ''central product''. Its explicit use, however, is not very standard in the literature.
+{{quick phrase|[[quick phrase::factor in central product]], [[quick phrase::product with centralizer is whole group]], [[quick phrase::quotient action by outer automorphisms is trivial]], [[quick phrase::every inner automorphism restricts to an inner automorphism]]}}
-==Definition==
+{| class="sortable" border="1"
+! No. !! Shorthand !! A subgroup of a group is termed a central factor if ... !! A subgroup <math>H</math> of a group <math>G</math> is termed a central factor of <math>G</math> if ...
+|-
+| 1 || inner automorphisms to inner automorphisms || every [[defining ingredient::inner automorphism]] of the group restricts to an inner automorphism of the subgroup. || given any <math>g</math> in <math>G</math>, there is a <math>h</math> in <math>H</math> such that, for all <math>x</math> in <math>H</math>, <math>gxg^{-1} = hxh^{-1}</math>.
+|-
+| 2 || product with centralizer is whole group || the product of the subgroup and its [[defining ingredient::centralizer]] is the whole group. || <math>HC_G(H) = G</math> where <math>C_G(H)</math> denotes the [[centralizer]] of <math>H</math> in <math>G</math>.
+|-
+| 3 || normal and quotient maps trivially to outer automorphism group || it is a [[defining ingredient::normal subgroup]] and the [[defining ingredient::quotient group maps to outer automorphism group of normal subgroup|induced map from the quotient group to the outer automorphism group]] is trivial. || <math>H</math> is a [[normal subgroup]] of <math>G</math> and the map <math>G/H \to \operatorname{Out}(H)</math> [[quotient group maps to outer automorphism group of normal subgroup|induced by the action of]] <math>G</math> on <math>H</math> by conjugation is a trivial homomorphism.
+|-
+| 4 || factor in central product || it can be expressed as one of the factor subgroups in an expression of the whole group as an [[defining ingredient::internal central product]] of two groups. || there exists a subgroup <math>K</math> of <math>G</math> such that <math>G</math> is an internal central product of <math>H</math> and <math>K</math>.
+|}
-===Symbol-free definition===
+{{tabular definition format}}
-A subgroup of a group is termed a '''central factor''' if it satisfies the following equivalent conditions:
+{{stdnonbasicdef}}
+{{subgroup property}}
+{{variationof|normality}}
-* Every [[inner automorphism]] of the group restricts to an inner automorphism of the subgroup.
+==Examples==
-* The product of the subgroup and its centralizer is the whole group.
-===Definition with symbols===
+===Extreme examples===
-A subgroup <math>H</math> of a <math>G</math> is termed a central factor of <math>G</math> if it satisfies the following equivalent conditions:
+* The trivial subgroup in any group is a central factor.
+* Every group is a central factor in itself.
-* Given any <math>g</math> in <math>G</math>, there is a <math>h</math> in <math>H</math> such that, for all <math>x</math> in <math>H</math>, <math>gxg^{-1} = hxh^{-1}</math>.
+===Subgroups satisfying the property===
-* <math>HC_G(H) = G</math> where <math>C_G(H)</math> denotes the [[centralizer]] of <math>H</math> in <math>G</math>.
-==Formalisms==
+{{subgroups satisfying property sorted by importance rank}}
-{{wikilocal-section}}
+===Subgroups dissatisfying the property===
-{{first-order subgroup property}}
+{{subgroups dissatisfying property sorted by importance rank}}
-The subgroup property of being a central factor has a first-order description as follows. A subgroup <math>H</math> is a central factor in a group <math>G</math> if and only if:
+==Metaproperties==
-<math>\forall g \in G, \exists h \in H . \forall x \in H, \qquad  hxh^{-1} = gxg^{-1}</math>
+{{wikilocal-section}}
-This is a Fraisse rank 2 expression.
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::abelian-tautological subgroup property]] || Yes || || If <math>G</math> is abelian and <math>H \le G</math>, then <math>H</math> is a central factor of <math>G</math>
+|-
+| [[satisfies metaproperty::transitive subgroup property]] || Yes || [[central factor is transitive]] || If <math>H \le K \le G</math>, <math>H</math> is a central factor of <math>K</math> and <math>K</math> is a central factor of <math>G</math>, then <math>H</math> is a central factor of <math>G</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || || For any group <math>G</math>, both <math>G</math> and the trivial subgroup of <math>G</math> are central factors.
+|-
+| [[dissatisfies metaproperty::finite-intersection-closed subgroup property]] || No || [[central factor is not finite-intersection-closed]] || It is possible to have a group <math>G</math> and central factors <math>H_1,H_2</math> such that the intersection <math>H_1 \cap H_2</math> is not a central factor.
+|-
+| [[dissatisfies metaproperty::finite-join-closed subgroup property]] || No || [[central factor is not finite-join-closed]] || It is possible to have a group <math>G</math> and central factors <math>H_1,H_2</math> such that the [[join of subgroups|join]] <math>\langle H_1, H_2 \rangle</math> (which in this case is also the [[product of subgroups]]) is not a central factor.
+|-
+| [[dissatisfies metaproperty::quotient-transitive subgroup property]] || No || [[central factor is not quotient-transitive]] || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is a central factor of <math>G</math> and <math>K/H</math> is a central factor of <math>G/H</math> but <math>K</math> is not a central factor of <math>G</math>.
+|-
+| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[central factor satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> and <math>H</math> is a central factor of <math>G</math>, then <math>H</math> is a central factor of <math>K</math>.
+|-
+| [[dissatisfies metaproperty::transfer condition]] || No || [[central factor does not satisfy transfer condition]] || It is possible to have a group <math>G</math> and subgroups <math>H,K</math> such that <math>H</math> is a central factor of <math>G</math> but <math>H \cap K</math> is not a central factor of <math>K</math>.
+|-
+| [[satisfies metaproperty::image condition]] || Yes || [[central factor satisfies image condition]] || If <math>H</math> is a central factor of <math>G</math> and <math>\varphi:G \to K</math> is a surjective homomorphism, then <math>\varphi(H)</math> is a central factor of <math>K</math>.
+|-
+| [[satisfies metaproperty::direct product-closed subgroup property]] || Yes || [[central factor is direct product-closed]] || If <math>H_i \le G_i, i \in I</math> are all central factors, then the direct product of <math>H_i</math>s is a central factor of the direct product of <math>G_i</math>s
+|-
+| [[satisfies metaproperty::upper join-closed subgroup property]] || Yes || [[central factor is upper join-closed]] || If <math>H \le G</math> and <math>K_i, i \in I</math> are subgroups of <math>G</math> containing <math>H</math> such that <math>H</math> is a central factor of each <math>K_i</math>, then <math>H</math> is also a central factor of the [[join of subgroups]] <math>\langle K_i \rangle _{i \in I}</math>.
+|-
+| [[satisfies metaproperty::centralizer-closed subgroup property]] || Yes || [[central factor is centralizer-closed]] || If <math>H \le G</math> is a central factor, the [[centralizer]] <math>C_G(H)</math> is also a central factor.
+|}
-{{frexp}}
+==Relation with other properties==
-The property of being a central factor arises in the [[function restriction formalism]] as the {{balanced subgroup property}} with respect to the function property of being an [[inner automorphism]]. In other words:
+===Analogues in other algebraic structures===
-Central factor = Inner automorphism <math>\to</math> Inner automorphism
+For a complete list, refer:
-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.
+{{#ask: [[Analogue of::central factor]]|limit = 0|searchlabel = Analogues of central factor}} (generated semantically).
-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]].
-==Relation with other properties==
 ===Conjunction with other properties===
@@ Line 66: / Line 96: @@
 ===Stronger properties===
-* [[Weaker than::Direct factor]]
+{| class="sortable" border="1"
-* [[Weaker than::Central subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Cocentral subgroup]]
+|-
+| [[Weaker than::Direct factor]] || factor in [[internal direct product]] || [[direct factor implies central factor]] || [[central factor not implies direct factor]] {{strictness examples|central factor|direct factor}} || {{intermediate notions short|central factor|direct factor}}
+|-
+| [[Weaker than::Central subgroup]] || contained in the [[center]] (equivalently, abelian central factor) || [[central subgroup implies central factor]] || [[central factor not implies central subgroup]] {{strictness examples|central factor|central subgroup}} || {{intermediate notions short|central factor|central subgroup}}
+|-
+| [[Weaker than::Cocentral subgroup]] || product with [[center]] is whole group || [[cocentral implies central factor]] || [[central factor not implies cocentral]] {{strictness examples|central factor|cocentral subgroup}} || {{intermediate notions short|central factor|cocentral subgroup}}
+|-
+| [[Weaker than::Join-transitively central factor]] || [[join of subgroups|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]] {{strictness examples|central factor|join-transitively central factor}} || {{intermediate notions short|central factor|join-transitively central factor}}
+|-
+| [[Weaker than::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]] {{strictness examples|central factor|right-quotient-transitively central factor}} ||{{intermediate notions short|central factor|right-quotient-transitively central factor}}
+|-
+| [[Weaker than::Complemented central factor]] || central factor that is also a [[permutably complemented subgroup]] || obvious || || {{intermediate notions short|central factor|complemented central factor}}
+|}
 ===Weaker properties===
-* [[Stronger than::Transitively normal subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::CEP-subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Conjugacy-closed normal subgroup]]
+|-
-* [[Stronger than::Conjugacy-closed subgroup]]
+| [[Stronger than::Normal subgroup]] || invariant under [[inner automorphism]]s || [[central factor implies normal]] || [[normal not implies central factor]] {{strictness examples for subgroup property|normal subgroup|central factor}} || {{intermediate notions short|normal subgroup|central factor}}
-* [[Stronger than::SCAB-subgroup]]
+|-
-* [[Stronger than::Subset-conjugacy-closed subgroup]]
+| [[Stronger than::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]] {{strictness examples for subgroup property|transitively normal subgroup|central factor}} || {{intermediate notions short|transitively normal subgroup|central factor}}
-* [[Stronger than::Normal subset-conjugacy-closed subgroup]]
+|-
-* [[Stronger than::Weak subset-conjugacy-closed subgroup]]
+| [[Stronger than::Locally inner automorphism-balanced subgroup]] || every [[inner automorphism]] restricts to a [[locally inner automorphism]] || || || {{intermediate notions short|locally inner automorphism-balanced subgroup|central factor}}
-* [[Stronger than::WNSCC-subgroup]]
+|-
-* [[Stronger than::WC-subgroup]]
+| [[Stronger than::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) || {{intermediate notions short|CEP-subgroup|central factor}}
-* [[Stronger than::Normal subgroup]]
+|-
+| [[Stronger than::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]] || {{intermediate notions short|conjugacy-closed subgroup|central factor}}
+|-
+| [[Stronger than::Conjugacy-closed normal subgroup]] || [[conjugacy-closed subgroup|conjugacy-closed]] and [[normal subgroup|normal]] || (via normal + conjugacy-closed) || [[conjugacy-closed normal not implies central factor]] || {{intermediate notions short|conjugacy-closed normal subgroup|central factor}}
+|-
+| [[Stronger than::SCAB-subgroup]] || [[subgroup-conjugating automorphism]] of whole group restricts to subgroup-conjugating automorphism of subgroup || [[central factor implies SCAB]] || [[SCAB not implies central factor]] || {{intermediate notions short|SCAB-subgroup|central factor}}
+|-
+| [[Stronger than::Subset-conjugacy-closed subgroup]] || || || ||
+|-
+| [[Stronger than::Normal subset-conjugacy-closed subgroup]] || || || ||
+|-
+| [[Stronger than::Weak subset-conjugacy-closed subgroup]] || || || ||
+|-
+| [[Stronger than::WNSCC-subgroup]] || || || ||
+|-
+| [[Stronger than::Central factor of normalizer]] || || || ||
+|}
-==Metaproperties==
+==Effect of property operators==
-{{wikilocal-section}}
+{| class="sortable" border="1"
+! 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
+|}
-{{transitive}}
+==Formalisms==
-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.}}, that is, it is both [[transitive subgroup property|transitive]] and [[identity-true subgroup property|identity-true]].
+{{wikilocal-section}}
-{{trim}}
+{{first-order subgroup property}}
-The property of being a central factor is [[trim subgroup property|trim]], viz both the whole group and the trivial subgroup are central factors.
+The subgroup property of being a central factor has a first-order description as follows. A subgroup <math>H</math> is a central factor in a group <math>G</math> if and only if:
-{{not intersection-closed}}
+<math>\forall g \in G, \exists h \in H . \forall x \in H: \qquad  hxh^{-1} = gxg^{-1}</math>
-An intersection of central factors need not be a central factor. {{proofat|[[Central factor is not intersection-closed]]}}
+This is a quantifier rank 3 expression.
-{{not quotient-transitive}}
+{{frexp}}
-Suppose <math>H \le K \le G</math> are groups such that <math>H</math> is a central factor of <math>G</math> and <math>K/H</math> is a central factor of <math>G/H</math>. Then, <math>K</math> need not be a central factor of <math>G</math>. {{proofat|[[Central factor is not quotient-transitive]]}}
-{{intsubcondn}}
-Given any groups <math>H \le K \le G</math> such that <math>H</math> is a central factor of <math>G</math>, <math>H</math> is also a central factor of <math>K</math>. 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]].
-{{proofat|[[Left-extensibility-stable implies intermediate subgroup condition]]}}
-{{dirprodclosedsgp}}
-Suppose <math>H_1</math> is a central factor of <math>G_1</math> and <math>H_2</math> is a central factor of <math>G_2</math>. Then, <math>H_1 \times H_2</math> is a central factor of <math>G_1 \times G_2</math>. This follows from the fact that the centralizer of the direct product of the subgroups equals the direct products of their individual centralizers.
+{| class="sortable" border="1"
+! Function restriction expression !! <math>H</math> is a central factor of <math>G</math> if ... !! This means that characteristicity is ... !! Additional comments
+|-
+| {{balance-short|inner automorphism}}
+|}
-{{upper join-closed}}
+In particular, this means that the property of being a central factor is a [[satisfies metaproperty::left-inner subgroup property]].
-If <math>H</math> is a central factor in <math>K_1</math> and <math>K_2</math>, both sitting inside a group <math>G</math>, then <math>H</math> is also a central factor inside <math><K_1,K_2></math>. This follows from the fact that <math>H</math> is a central factor inside <math>K_i</math> if and only if <math>HC_G(H)</math> contains <math>K_i</math>.
+{{obtainedbyapplyingthe|image-potentially operator|direct factor}}
-==Effect of property operators==
+A subgroup <math>H</math> of a group <math>G</math> is a central factor if and only if the following is true: there exists a group <math>K</math>, a [[direct factor]] <math>L</math> of <math>K</math>, and a surjective homomorphism <math>\rho:K \to G</math> such that <math>\rho(L) = H</math>.
-{{applyingoperatorgives|join-transiter|join-transitively central factor}}
+{{obtainedbyapplyingthe|upper join-closure operator|cocentral subgroup}}
-A [[join-transitively central factor]] is a subgroup of a group whose join with any central factor is a central factor. All [[central subgroup]]s and all [[direct factor]]s are join-transitively central factors. {{proofat|[[Central subgroup implies join-transitively central factor]], [[direct factor implies join-transitively central factor]]}}
+A subgroup <math>H</math> of a group <math>G</math> is a central factor if and only if there exist groups <math>K_i, i \in I</math> all contained in <math>G</math> such that <math>H</math> is a [[cocentral subgroup]] of each <math>K_i</math> (i.e., <math>HZ(K_i) = K_i</math>) and the [[join of subgroups|join]] of the <math>K_i</math>s equals <math>G</math>.