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]

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 Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose automorphism group is abelian|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 Normal subgroup contained in centralizer of derived 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 Commutator-in-centralizer subgroup|FULL LIST, MORE INFO
Commutator-in-centralizer subgroup commutator with whole group is contained in centralizer
2-subnormal subgroup normal subgroup of normal subgroup Commutator-in-centralizer subgroup|FULL LIST, MORE INFO