# 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 term c-closed subgroup has 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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fixed-point subgroup of a subgroup of the automorphism group fixed-point subgroup of some subgroup of the automorphism group c-closed implies fixed-point subgroup of a subgroup of the automorphism group fixed-point subgroup of a subgroup of the automorphism group not implies c-closed |FULL LIST, MORE INFO
local powering-invariant subgroup unique $n^{th}$ root (across whole group) of element of subgroup must lie in subgroup via fixed-point subgroup of a subgroup of the automorphism group Fixed-point subgroup of a subgroup of the automorphism group|FULL LIST, MORE INFO
powering-invariant subgroup powered for all primes that power the whole group (via local powering-invariant) Fixed-point subgroup of a subgroup of the automorphism group, Local powering-invariant subgroup|FULL LIST, MORE INFO
algebraic subgroup subgroup that is an intersection of elementary algebraic subsets |FULL LIST, MORE INFO
unconditionally closed subgroup subgroup that is a closed subgroup for any T0 topological group structure on the whole group Algebraic subgroup|FULL LIST, MORE INFO

## 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 transitive
ABOUT 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-closed
ABOUT 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