# 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

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

A subgroup of a group is termed a **central factor** if it satisfies the following equivalent conditions:

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 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

## 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 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

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 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 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 .