Commutator-in-centralizer subgroup

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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