C-closed subgroup

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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 |
local powering-invariant subgroup unique 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 |
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: |
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: |
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

References

External links

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