Subgroup contained in centralizer of derived subgroup

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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

A subgroup $H$ of a group $G$ is termed a subgroup contained in centralizer of commutator subgroup if $H$ is contained in $C_{G}[G,G]$, the centralizer of commutator subgroup of $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Central subgroup	contained in center			|FULL LIST, MORE INFO
Cyclic normal subgroup	cyclic and a normal subgroup			|FULL LIST, MORE INFO
Aut-abelian normal subgroup	normal subgroup whose automorphism group is abelian	aut-abelian normal subgroup is contained in centralizer of commutator subgroup		|FULL LIST, MORE INFO
Normal subgroup contained in centralizer of commutator subgroup	normal and contained in centralizer of commutator subgroup

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hereditarily 2-subnormal subgroup	every subgroup is 2-subnormal	centralizer of commutator subgroup is hereditarily 2-subnormal		|FULL LIST, MORE INFO
Commutator-in-centralizer subgroup	commutator with whole group is contained in centralizer
2-subnormal subgroup	normal subgroup of normal subgroup			|FULL LIST, MORE INFO