# Cocentral subgroup

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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]

## Definition

### Symbol-free definition

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

- Its product with the center is the whole group.
- The inclusion map from the subgroup to the group induces maps between the corresponding inner automorphism groups and derived subgroup that give rise to an isoclinism.

### Definition with symbols

A subgroup of a group is termed **cocentral** if where denotes the center of .

## Relation with other properties

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Central factor | product with centralizer equals whole group | cocentral implies central factor | central factor not implies cocentral | Right-quotient-transitively central factor|FULL LIST, MORE INFO |

Weakly cocentral subgroup | product with center contains its normalizer | |FULL LIST, MORE INFO | ||

Abelian-quotient subgroup | contains the commutator subgroup; equivalently, normal subgroup with abelian quotient group | cocentral implies abelian-quotient | abelian-quotient not implies cocentral | |FULL LIST, MORE INFO |

Centralizer-dense subgroup | its centralizer equals its center | cocentral implies centralizer-dense | centralizer-dense not implies cocentral | |FULL LIST, MORE INFO |

Subgroup isoclinic to the whole group | there exists an isoclinism between the subgroup and the whole group | |FULL LIST, MORE INFO |

### Dichotomy

For a group with nontrivial center, any maximal subgroup of a group is either cocentral or contains the center of the group. This idea is of use in proving that every maximal subgroup of a nilpotent group is normal. It is also related to the fact that upper join-closure of cocentrality is central factor.

## Metaproperties

Metaproperty name | Satisfied? | Proof | Section in this article |
---|---|---|---|

Transitive subgroup property | Yes | Cocentrality is transitive | #Transitivity |

Identity-true subgroup property | Yes | Every group is cocentral in itself | #Trimness |

Trivially true subgroup property | No | Trivial subgroup is cocentral iff group is abelian | #Trimness |

Intermediate subgroup condition | Yes | Cocentrality satisfies intermediate subgroup condition | #Intermediate subgroup condition |

Upward-closed subgroup property | Yes | Cocentrality is upward-closed | #Upward-closedness |

Finite-intersection-closed subgroup property | No | Cocentrality is not finite-intersection-closed | #Intersection-closedness |

Upper join-closed subgroup property | No | Cocentrality is not upper join-closed | #Upper join-closedness |

### Trimness

The property of being cocentral is an identity-true subgroup property, that is, every group is cocentral as a subgroup of itself. However, it is not in general trivially true. In fact, the trivial subgroup is cocentral if and only if the group is abelian.

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If is cocentral in , is also cocentral in any intermediate subgroup . This follows from the fact that contains . `For full proof, refer: Cocentrality satisfies intermediate subgroup condition`

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

If , with a cocentral subgroup of and a cocentral subgroup of , then is cocentral in . `For full proof, refer: Cocentrality is transitive`

### Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group

View other upward-closed subgroup properties

If with a cocentral subgroup of , is also a cocentral subgroup of .

`For full proof, refer: cocentrality is upward-closed`

### Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.

Read an article on methods to prove that a subgroup property is not intersection-closed

It is possible to have a group and two cocentral subgroups and of such that is *not* a cocentral subgroup of .

`For full proof, refer: Cocentrality is not finite-intersection-closed`

### Upper join-closedness

NO:This subgroup property isnotupper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

It is possible to have a subgroup of a group that is a concentral subgroup in each of two intermediate subgroups and but not in the join .

`For full proof, refer: Cocentrality is not upper join-closed`

`Further information: Upper join-closure of cocentrality is central factor`

## Effect of property operators

### The upper join-closure operator

*Applying the upper join-closure operator to this property gives*: central factor