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
Definition
A subgroup of a group is termed a commutator-in-center subgroup if it satisfies the following equivalent conditions:
- where denotes the commutator of two subgroups and denotes the center of .
- is a normal subgroup of and is trivial, i.e., is a commutator-in-centralizer subgroup.
![{\displaystyle [G,H]\leq Z(H)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0840114113c889a812af4e57f58e0dbb455610cd)
![{\displaystyle [G,H]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62776d62a86ef1ed138d4e4f7a49ac80b9572a9c)
![{\displaystyle [[G,H],H]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3f8a9ebc03a9b6042eaf45c6d19f50f3c3cba5)
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|FULL LIST, MORE INFO | ||
| Abelian normal subgroup | abelian and normal in whole 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) | |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) | |FULL LIST, MORE INFO | |
| Normal subgroup | ||||
| 2-subnormal subgroup |
Related group properties
A group has the property that for every group containing as a normal subgroup, is a commutator-in-center subgroup of , if and only if is a group whose inner automorphism group is central in automorphism group.