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Contents 1 History 1.1 Origin of the concept 1.2 Origin of the term 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Formalisms 3.1 First-order description 3.2 Function restriction expression 3.3 In terms of the image-potentially operator 3.4 In terms of the upper join-closure operator 4 Relation with other properties 4.1 Analogues in other algebraic structures 4.2 Conjunction with other properties 4.3 Stronger properties 4.4 Weaker properties 5 Metaproperties 5.1 Tautology when whole group is abelian 5.2 Transitivity 5.3 Trimness 5.4 Intersection-closedness 5.5 Join-closedness 5.6 Quotient-transitivity 5.7 Intermediate subgroup condition 5.8 Transfer condition 5.9 Image condition 5.10 Direct product-closedness 5.11 Upper join-closedness 5.12 Centralizer-closedness 6 Effect of property operators 6.1 The join-transiter 6.2 The right quotient-transiter 6.3 The intersection-transiter

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.[SHOW MORE]

VIEW: Definitions built on this | Facts about this | Facts about definitions built on this |
VIEW RELATED: Analogues of this | Variations 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.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions[SHOW MORE]

VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions |
VIEW FACTS THAT: use, or start with, a subgroup satisfying the property| prove, or show that, a subgroup satisfies the property
RANDOM SUBGROUP PROPERTY: Hall subgroup: A subgroup of a finite group whose order and index are relatively prime. Sylow subgroups are a special case.

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

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

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 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:

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 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 such that ρ(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 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_is equals G.

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	click here
Central subgroup	contained in the center (equivalently, abelian central factor)	central subgroup implies central factor	central factor not implies central subgroup	click here
Cocentral subgroup	product with center is whole group	cocentral implies central factor	central factor not implies cocentral	click here
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
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	click here
Complemented central factor	central factor that is also a permutably complemented subgroup	obvious

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	click here
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	click here
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)	click here
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	click here
Conjugacy-closed normal subgroup	conjugacy-closed and normal	(via normal + conjugacy-closed)	conjugacy-closed normal not implies central factor
SCAB-subgroup	subgroup-conjugating automorphism of whole group restricts to subgroup-conjugating automorphism of subgroup	central factor implies SCAB	SCAB not implies central factor
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

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

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 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 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 |
ABOUT INTERMEDIATE SUBROUP CONDITION: View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Given any groups 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 such that H is a central factor of G, it is not necessary that 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 , if H is a central factor of G, ρ(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 H₁ is a central factor of G₁ and H₂ is a central factor of G₂. Then, is a central factor of . 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 , 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

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

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