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
- 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.
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
|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|
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
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
- Paper:Gaschutz54More 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-closedSpringerlinkMore info
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