C-closed subgroup

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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		|FULL LIST, MORE INFO
powering-invariant subgroup	powered for all primes that power the whole group	(via local powering-invariant)		|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			|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.
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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

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^{SpringerlinkMore info}

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