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 ![]() ![]() ![]() |
---|---|---|---|
1 | inner automorphisms to inner automorphisms | every inner automorphism of the group restricts to an inner automorphism of the subgroup. | given any ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | product with centralizer is whole group | the product of the subgroup and its centralizer is the whole group. | ![]() ![]() ![]() ![]() |
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. | ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() |
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 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:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
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:
Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:
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 ![]() ![]() ![]() ![]() | |
transitive subgroup property | Yes | central factor is transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
trim subgroup property | Yes | For any group ![]() ![]() ![]() | |
finite-intersection-closed subgroup property | No | central factor is not finite-intersection-closed | It is possible to have a group ![]() ![]() ![]() |
finite-join-closed subgroup property | No | central factor is not finite-join-closed | It is possible to have a group ![]() ![]() ![]() |
quotient-transitive subgroup property | No | central factor is not quotient-transitive | It is possible to have groups ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
intermediate subgroup condition | Yes | central factor satisfies intermediate subgroup condition | If ![]() ![]() ![]() ![]() ![]() |
transfer condition | No | central factor does not satisfy transfer condition | It is possible to have a group ![]() ![]() ![]() ![]() ![]() ![]() |
image condition | Yes | central factor satisfies image condition | If ![]() ![]() ![]() ![]() ![]() |
direct product-closed subgroup property | Yes | central factor is direct product-closed | If ![]() ![]() ![]() |
upper join-closed subgroup property | Yes | central factor is upper join-closed | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
centralizer-closed subgroup property | Yes | central factor is centralizer-closed | If ![]() ![]() |
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
Weaker properties
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 is a central factor in a group
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
Function restriction expression | ![]() ![]() |
This means that characteristicity is ... | Additional comments |
---|---|---|---|
inner automorphism ![]() |
every inner automorphism of ![]() ![]() |
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 of a group
is a central factor if and only if the following is true: there exists a group
, a direct factor
of
, and a surjective homomorphism
such that
.
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 of a group
is a central factor if and only if there exist groups
all contained in
such that
is a cocentral subgroup of each
(i.e.,
) and the join of the
s equals
.