View source for C-closed subgroup

{{semistddef}}
{{subgroup property}}
{{quotation|''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===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::Maximal among abelian subgroups]] || || || ||
|-
| [[Weaker than::Abelian critical subgroup]] || || || ||
|-
| [[Weaker than::c-closed self-centralizing subgroup]] || || || ||
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::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]] || {{intermediate notions short|fixed-point subgroup of a subgroup of the automorphism group|c-closed subgroup}}
|-
| [[Stronger than::local powering-invariant subgroup]] || unique <math>n^{th}</math> root (across whole group) of element of subgroup must lie in subgroup || [[fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant|via fixed-point subgroup of a subgroup of the automorphism group]] || || {{intermediate notions short|local powering-invariant subgroup|c-closed subgroup}}
|-
| [[Stronger than::powering-invariant subgroup]] || powered for all primes that power the whole group || (via local powering-invariant) || || {{intermediate notions short|powering-invariant subgroup|c-closed subgroup}}
|-
| [[Stronger than::algebraic subgroup]] || subgroup that is an intersection of [[elementary algebraic subset]]s || || || {{intermediate notions short|algebraic subgroup|c-closed subgroup}}
|-
| [[Stronger than::unconditionally closed subgroup]] || subgroup that is a [[closed subgroup]] for any [[T0 topological group]] structure on the whole group || || || {{intermediate notions short|unconditionally closed subgroup|c-closed subgroup}}
|}

==Metaproperties==

{{transitive}}

A c-closed subgroup of a c-closed subgroup is again c-closed. {{further|[[C-closedness is transitive]]}}
{{intersection-closed}}

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 subgroup property|strongly intersection-closed]].

{{further|[[Galois correspondence-closed implies strongly intersection-closed]]}}

==References==
* {{paperlink|Gaschutz54}}
* {{paperlink|AntChekCCI}}

==External links==
{{searchbox|"c+closed+subgroup"}}