# C-closed subgroup

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[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]

The termc-closed subgrouphas also been used in the past for conjugacy-closed subgroup

## Definition

### Symbol-free definition

A subgroup of a group is termed a **c-closed subgroup** if it satisfies the equivalent conditions:

- It equals the centralizer of its centralizer.
- It occurs as the centralizer of some subset of the group
- It occurs as the centralizer of some subgroup

Alternative terminology for a c-closed subgroup is **centralizer subgroup**, **self-bicommutant subgroup**, **self-bicentralizer subgroup**.

## Formalisms

The property of being a centralizer subgroup is a Galois correspondence-closed subgroup property, with respect to the Galois correspondence induced by relation of commuting.

## Relation with other properties

### Stronger properties

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

Maximal among abelian subgroups | ||||

Abelian critical subgroup | ||||

c-closed self-centralizing subgroup |

### Weaker properties

## Metaproperties

### 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

A c-closed subgroup of a c-closed subgroup is again c-closed. `Further information: C-closedness is transitive`

### Intersection-closedness

YES:This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closedABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of centralizer subgroups is a centralizer subgroup; this follows from general facts about Galois correspondences. In fact, even an empty intersection of centralizer subgroups is a centralizer subgroups, so the property is actually strongly intersection-closed.

`Further information: Galois correspondence-closed implies strongly intersection-closed`

## References

- Paper:Gaschutz54
^{More info} -
*Finite solvable groups with C-closed invariant subgroups*by V. A. Antonov and S. G. Chekanov,*Mathematicheski Zametski*2007 (original in Russian): This paper studies finite solvable groups in which all normal subgroups are c-closed^{Springerlink}^{More info}

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