# Difference between revisions of "Central factor"

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[[importance rank::2| ]] | [[importance rank::2| ]] | ||

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==Terminological note== | ==Terminological note== | ||

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==Definition== | ==Definition== | ||

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+ | ===Equivalent definitions in tabular format=== | ||

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{{quick phrase|[[quick phrase::factor in central product]], [[quick phrase::product with centralizer is whole group]], [[quick phrase::quotient action by outer automorphisms is trivial]], [[quick phrase::every inner automorphism restricts to an inner automorphism]]}} | {{quick phrase|[[quick phrase::factor in central product]], [[quick phrase::product with centralizer is whole group]], [[quick phrase::quotient action by outer automorphisms is trivial]], [[quick phrase::every inner automorphism restricts to an inner automorphism]]}} | ||

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{| class="sortable" border="1" | {| class="sortable" border="1" | ||

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| 4 || factor in central product || it can be expressed as one of the factor subgroups in an expression of the whole group as an [[defining ingredient::internal central product]] of two groups. || there exists a subgroup <math>K</math> of <math>G</math> such that <math>G</math> is an internal central product of <math>H</math> and <math>K</math>. | | 4 || factor in central product || it can be expressed as one of the factor subgroups in an expression of the whole group as an [[defining ingredient::internal central product]] of two groups. || there exists a subgroup <math>K</math> of <math>G</math> such that <math>G</math> is an internal central product of <math>H</math> and <math>K</math>. | ||

|} | |} | ||

+ | |||

+ | {{tabular definition format}} | ||

{{stdnonbasicdef}} | {{stdnonbasicdef}} | ||

{{subgroup property}} | {{subgroup property}} | ||

{{variationof|normality}} | {{variationof|normality}} | ||

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+ | ==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=== | ||

+ | |||

+ | {{subgroups satisfying property sorted by importance rank}} | ||

+ | |||

+ | ===Subgroups dissatisfying the property=== | ||

+ | |||

+ | {{subgroups dissatisfying property sorted by importance rank}} | ||

==Metaproperties== | ==Metaproperties== | ||

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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | ||

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::abelian-tautological subgroup property]] || Yes || || If <math>G</math> is abelian and <math>H \le G</math>, then <math>H</math> is a central factor of <math>G</math> |

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::transitive subgroup property]] || Yes || [[central factor is transitive]] || If <math>H \le K \le G</math>, <math>H</math> is a central factor of <math>K</math> and <math>K</math> is a central factor of <math>G</math>, then <math>H</math> is a central factor of <math>G</math>. |

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::trim subgroup property]] || Yes || || For any group <math>G</math>, both <math>G</math> and the trivial subgroup of <math>G</math> are central factors. |

|- | |- | ||

− | | [[dissatisfies metaproperty:: | + | | [[dissatisfies metaproperty::finite-intersection-closed subgroup property]] || No || [[central factor is not finite-intersection-closed]] || It is possible to have a group <math>G</math> and central factors <math>H_1,H_2</math> such that the intersection <math>H_1 \cap H_2</math> is not a central factor. |

|- | |- | ||

− | | [[dissatisfies metaproperty:: | + | | [[dissatisfies metaproperty::finite-join-closed subgroup property]] || No || [[central factor is not finite-join-closed]] || It is possible to have a group <math>G</math> and central factors <math>H_1,H_2</math> such that the [[join of subgroups|join]] <math>\langle H_1, H_2 \rangle</math> (which in this case is also the [[product of subgroups]]) is not a central factor. |

|- | |- | ||

− | | [[dissatisfies metaproperty:: | + | | [[dissatisfies metaproperty::quotient-transitive subgroup property]] || No || [[central factor is not quotient-transitive]] || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is a central factor of <math>G</math> and <math>K/H</math> is a central factor of <math>G/H</math> but <math>K</math> is not a central factor of <math>G</math>. |

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[central factor satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> and <math>H</math> is a central factor of <math>G</math>, then <math>H</math> is a central factor of <math>K</math>. |

|- | |- | ||

− | | [[dissatisfies metaproperty:: | + | | [[dissatisfies metaproperty::transfer condition]] || No || [[central factor does not satisfy transfer condition]] || It is possible to have a group <math>G</math> and subgroups <math>H,K</math> such that <math>H</math> is a central factor of <math>G</math> but <math>H \cap K</math> is not a central factor of <math>K</math>. |

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::image condition]] || Yes || [[central factor satisfies image condition]] || If <math>H</math> is a central factor of <math>G</math> and <math>\varphi:G \to K</math> is a surjective homomorphism, then <math>\varphi(H)</math> is a central factor of <math>K</math>. |

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::direct product-closed subgroup property]] || Yes || [[central factor is direct product-closed]] || If <math>H_i \le G_i, i \in I</math> are all central factors, then the direct product of <math>H_i</math>s is a central factor of the direct product of <math>G_i</math>s |

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::upper join-closed subgroup property]] || Yes || [[central factor is upper join-closed]] || If <math>H \le G</math> and <math>K_i, i \in I</math> are subgroups of <math>G</math> containing <math>H</math> such that <math>H</math> is a central factor of each <math>K_i</math>, then <math>H</math> is also a central factor of the [[join of subgroups]] <math>\langle K_i \rangle _{i \in I}</math>. |

|- | |- | ||

− | | [[satisfies metaproperty:: | + | | [[satisfies metaproperty::centralizer-closed subgroup property]] || Yes || [[central factor is centralizer-closed]] || If <math>H \le G</math> is a central factor, the [[centralizer]] <math>C_G(H)</math> is also a central factor. |

|} | |} | ||

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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||

|- | |- | ||

− | | [[Stronger than::Normal subgroup]] || invariant under [[inner automorphism]]s || [[central factor implies normal]] || [[normal not implies central factor]] {{strictness examples|normal subgroup|central factor}} || {{intermediate notions short|normal subgroup|central factor}} | + | | [[Stronger than::Normal subgroup]] || invariant under [[inner automorphism]]s || [[central factor implies normal]] || [[normal not implies central factor]] {{strictness examples for subgroup property|normal subgroup|central factor}} || {{intermediate notions short|normal subgroup|central factor}} |

|- | |- | ||

− | | [[Stronger than::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]] {{strictness examples|transitively normal subgroup|central factor}} || {{intermediate notions short|transitively normal subgroup|central factor}} | + | | [[Stronger than::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]] {{strictness examples for subgroup property|transitively normal subgroup|central factor}} || {{intermediate notions short|transitively normal subgroup|central factor}} |

+ | |- | ||

+ | | [[Stronger than::Locally inner automorphism-balanced subgroup]] || every [[inner automorphism]] restricts to a [[locally inner automorphism]] || || || {{intermediate notions short|locally inner automorphism-balanced subgroup|central factor}} | ||

|- | |- | ||

| [[Stronger than::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) || {{intermediate notions short|CEP-subgroup|central factor}} | | [[Stronger than::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) || {{intermediate notions short|CEP-subgroup|central factor}} | ||

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{{frexp}} | {{frexp}} | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Function restriction expression !! <math>H</math> is a central factor of <math>G</math> if ... !! This means that characteristicity is ... !! Additional comments | |

− | + | |- | |

− | + | | {{balance-short|inner automorphism}} | |

− | + | |} | |

− | In particular, | + | In particular, this means that the property of being a central factor is a [[satisfies metaproperty::left-inner subgroup property]]. |

{{obtainedbyapplyingthe|image-potentially operator|direct factor}} | {{obtainedbyapplyingthe|image-potentially operator|direct factor}} |

## Latest revision as of 21:51, 12 January 2014

## 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 of a group is termed a central factor of if ... |
---|---|---|---|

1 | inner automorphisms to inner automorphisms | every inner automorphism of the group restricts to an inner automorphism of the subgroup. | given any in , there is a in such that, for all in , . |

2 | product with centralizer is whole group | the product of the subgroup and its centralizer is the whole group. | where denotes the centralizer of in . |

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. | is a normal subgroup of and the map induced by the action of on 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 of such that is an internal central product of and . |

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: (factscloselyrelated 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 is abelian and , then is a central factor of | |

transitive subgroup property | Yes | central factor is transitive | If , is a central factor of and is a central factor of , then is a central factor of . |

trim subgroup property | Yes | For any group , both and the trivial subgroup of are central factors. | |

finite-intersection-closed subgroup property | No | central factor is not finite-intersection-closed | It is possible to have a group and central factors such that the intersection 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 and central factors such that the join (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 such that is a central factor of and is a central factor of but is not a central factor of . |

intermediate subgroup condition | Yes | central factor satisfies intermediate subgroup condition | If and is a central factor of , then is a central factor of . |

transfer condition | No | central factor does not satisfy transfer condition | It is possible to have a group and subgroups such that is a central factor of but is not a central factor of . |

image condition | Yes | central factor satisfies image condition | If is a central factor of and is a surjective homomorphism, then is a central factor of . |

direct product-closed subgroup property | Yes | central factor is direct product-closed | If are all central factors, then the direct product of s is a central factor of the direct product of s |

upper join-closed subgroup property | Yes | central factor is upper join-closed | If and are subgroups of containing such that is a central factor of each , then is also a central factor of the join of subgroups . |

centralizer-closed subgroup property | Yes | central factor is centralizer-closed | If is a central factor, the centralizer 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

### 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 | is a central factor of if ... | This means that characteristicity is ... | Additional comments |
---|---|---|---|

inner automorphism inner automorphism | every inner automorphism of restricts to a 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 .