Central factor

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 $K$.

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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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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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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:

	Group part	Subgroup part	Quotient part
Z2 in V4	Klein four-group	Cyclic group:Z2	Cyclic group:Z2

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
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Z4 in direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z4	Cyclic group:Z2

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

	Group part	Subgroup part	Quotient part
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
Q8 in central product of D8 and Z4	Central product of D8 and Z4	Quaternion group	Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Dihedral group:D8

Subgroups dissatisfying the property

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

	Group part	Subgroup part	Quotient part
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2
S2 in S3	Symmetric group:S3	Cyclic group:Z2

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

	Group part	Subgroup part	Quotient part
A3 in A4	Alternating group:A4	Cyclic group:Z3
A3 in A5	Alternating group:A5	Cyclic group:Z3
A3 in S4	Symmetric group:S4	Cyclic group:Z3
A4 in A5	Alternating group:A5	Alternating group:A4
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
D8 in A6	Alternating group:A6	Dihedral group:D8
D8 in S4	Symmetric group:S4	Dihedral group:D8
Klein four-subgroup of alternating group:A5	Alternating group:A5	Klein four-group
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-normal Klein four-subgroups of symmetric group:S4	Symmetric group:S4	Klein four-group
Non-normal subgroups of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2
S2 in S4	Symmetric group:S4	Cyclic group:Z2
Subgroup generated by double transposition in symmetric group:S4	Symmetric group:S4	Cyclic group:Z2
Twisted S3 in A5	Alternating group:A5	Symmetric group:S3

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

	Group part	Subgroup part	Quotient part
2-Sylow subgroup of general linear group:GL(2,3)	General linear group:GL(2,3)	Semidihedral group:SD16
Non-normal subgroups of M16	M16	Cyclic group:Z2
SL(2,3) in GL(2,3)	General linear group:GL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2

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 $H \le G$, 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:

• 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)	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
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)	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

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 $G$.