Commutator-in-center 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 commutator-in-centralizer subgroup
View other subgroup property conjunctions | view all subgroup properties

Contents

Definition

A subgroup $H$ of a group $G$ is termed a commutator-in-center subgroup if it satisfies the following equivalent conditions:

1. $[G,H] \le Z(H)$ where $[G,H]$ denotes the commutator of two subgroups and $Z(H)$ denotes the center of $H$.
2. $H$ is a normal subgroup of $G$ and $[[G,H],H]$ is trivial, i.e., $H$ is a commutator-in-centralizer subgroup.

A group has this property as a subgroup of itself if and only if it has nilpotency class two.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Central subgroup	contained in center of whole group			Abelian normal subgroup|FULL LIST, MORE INFO
Cyclic normal subgroup	cyclic and normal in whole group			Abelian normal subgroup, Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose automorphism group is abelian, Normal subgroup whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO
Abelian normal subgroup	abelian and normal in whole group			Normal subgroup whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO
Normal subgroup contained in centralizer of commutator subgroup	normal and contained in the group's centralizer of commutator subgroup			|FULL LIST, MORE INFO
Normal subgroup whose automorphism group is abelian	normal subgroup that is a group whose automorphism group is abelian: its automorphism group is abelian	(via normal subgroup contained in centralizer of commutator subgroup)		Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO
Normal subgroup whose inner automorphism group is central in automorphism group	normal subgroup and group whose inner automorphism group is central in automorphism group
Critical subgroup

Weaker properties

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

Related group properties

A group $H$ has the property that for every group $G$ containing $H$ as a normal subgroup, $H$ is a commutator-in-center subgroup of $G$, if and only if $H$ is a group whose inner automorphism group is central in automorphism group.