# 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 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]
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 $H \le G$, then $H$ is a central factor of $G$
Transitive subgroup property Yes Central factor is transitive If $H \le K \le G$, $H$ 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 $H \le K \le G$ 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 $H \le K \le G$ 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 $H_i \le G_i, i \in I$ 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 $H \le G$ 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 $H \le G$ 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:

Some conjunctions with subgroup properties:

### 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) Complemented central factor, Join of direct factor and central subgroup, Join of finitely many direct factors, Join-transitively central factor, Right-quotient-transitively central factor, Upper join of direct factors|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) Intersection-transitively central factor, Join of direct factor and central subgroup, Join-transitively central factor|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) Right-quotient-transitively central factor|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) Join-transitively central factor|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) Conjugacy-closed normal subgroup, Locally inner automorphism-balanced subgroup, Normal subgroup whose center is contained in the center of the whole group, Normal subgroup whose focal subgroup equals its derived subgroup, SCAB-subgroup, Transitively normal subgroup|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) Conjugacy-closed normal subgroup, Locally inner automorphism-balanced subgroup, SCAB-subgroup|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) Transitively normal subgroup|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 Conjugacy-closed normal subgroup, Subset-conjugacy-closed subgroup|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 $G$.