C-closed subgroup

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]

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Definition

Symbol-free definition

A subgroup of a group is termed a centralizer subgroup if it occurs as the centralizer of a nonempty subset (or equivalently, of a subgroup).

Alternative terminology for a centralizer subgroup is self-bicommutant subgroup, self-bicentralizer subgroup, and C-closed 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.

Metaproperties

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