Normal subgroup contained in centralizer of derived subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup contained in centralizer of commutator subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup H of a group G is termed a normal subgroup contained in centralizer of commutator subgroup if H is a normal subgroup of G and H \le C_G([G,G]), i.e., H is contained in the centralizer of derived subgroup of G.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup whose automorphism group is abelian normal subgroup, also a group whose automorphism group is abelian (automorphism group is an abelian group) derived subgroup centralizes normal subgroup whose automorphism group is abelian
cyclic normal subgroup normal subgroup, also a cyclic group (via abelian automorphism group) (via abelian automorphism group) Normal subgroup whose automorphism group is abelian|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup contained in centralizer of commutator subgroup
commutator-in-center subgroup commutator with whole group is contained in its center
commutator-in-centralizer subgroup
hereditarily 2-subnormal subgroup
class two normal subgroup

Related group properties

A group H has the property that for any group G containing H as a normal subgroup, H is also contained in the centralizer of commutator subgroup of G, if and only if H is a group whose automorphism group is abelian.