Commutator-in-centralizer 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]

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 Commutator-in-center 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 Commutator-in-center subgroup, Normal subgroup contained in centralizer of derived subgroup, Subgroup contained in centralizer of derived 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