Central factor

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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]
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 partSubgroup partQuotient part
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2

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

 Group partSubgroup partQuotient part
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

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

 Group partSubgroup partQuotient part
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Dihedral group:D8

Subgroups dissatisfying the property

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

 Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
S2 in S3Symmetric group:S3Cyclic group:Z2

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

 Group partSubgroup partQuotient part
A3 in A4Alternating group:A4Cyclic group:Z3
A3 in A5Alternating group:A5Cyclic group:Z3
A3 in S4Symmetric group:S4Cyclic group:Z3
A4 in A5Alternating group:A5Alternating group:A4
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
D8 in A6Alternating group:A6Dihedral group:D8
D8 in S4Symmetric group:S4Dihedral group:D8
Klein four-subgroup of alternating group:A5Alternating group:A5Klein four-group
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-normal Klein four-subgroups of symmetric group:S4Symmetric group:S4Klein four-group
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
S2 in S4Symmetric group:S4Cyclic group:Z2
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
Twisted S3 in A5Alternating group:A5Symmetric group:S3

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

 Group partSubgroup partQuotient part
2-Sylow subgroup of general linear group:GL(2,3)General linear group:GL(2,3)Semidihedral group:SD16
Non-normal subgroups of M16M16Cyclic 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_is is a central factor of the direct product of G_is
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
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_is equals G.