Commutator-in-centralizer subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Commutator-in-centralizer subgroup, all facts related to Commutator-in-centralizer subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

A subgroup $H$ of a group $G$ is termed a commutator-in-centralizer subgroup if the commutator $[G,H]$ is contained in the centralizer $C_G(H)$, or equivalently, $[[G,H],H]$ is trivial.,

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Commutator-in-center subgroup	commutator with whole group is contained in center of subgroup			|FULL LIST, MORE INFO
Abelian normal subgroup				|FULL LIST, MORE INFO
Subgroup contained in centralizer of commutator subgroup	contained in the centralizer of the whole group's commutator subgroup			|FULL LIST, MORE INFO
Aut-abelian normal subgroup				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hereditarily 2-subnormal subgroup
Class two 2-subnormal subgroup
2-subnormal subgroup