Central factor

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 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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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:

• Every inner automorphism of the group restricts to an inner automorphism of the subgroup.
• The product of the subgroup and its centralizer is the whole group.

Definition with symbols

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

• Given any $g$ in $G$, there is a $h$ in $H$ such that, for all $x$ in $H$, $gxg^{-1}=hxh^{-1}$.
• $HC_{G}(H)=G$ where $C_{G}(H)$ denotes the centralizer of $H$ in $G$.

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.
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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.

Relation with other properties

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

• Direct factor
• Central subgroup
• Cocentral subgroup

Weaker properties

• Transitively normal subgroup
• CEP-subgroup
• Conjugacy-closed normal subgroup
• Conjugacy-closed subgroup
• SCAB-subgroup
• Subset-conjugacy-closed subgroup
• Normal subset-conjugacy-closed subgroup
• Weak subset-conjugacy-closed subgroup
• WNSCC-subgroup
• WC-subgroup
• Normal subgroup

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)

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 central factors need not be a central factor. For full proof, refer: Central factor is not intersection-closed

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

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.

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
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If $H$ is a central factor in $K_{1}$ and $K_{2}$, both sitting inside a group $G$, then $H$ is also a central factor inside $<K_{1},K_{2}>$. This follows from the fact that $H$ is a central factor inside $K_{i}$ if and only if $HC_{G}(H)$ contains $K_{i}$.