Characteristic subgroup of nilpotent group
This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: nilpotent group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
Definition
The term characteristic subgroup of nilpotent group is used for a situation where the whole group is a nilpotent group and the subgroup is a characteristic subgroup.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| characteristic subgroup of abelian group | follows from abelian implies nilpotent | follows from nilpotent not implies abelian | |FULL LIST, MORE INFO | |
| characteristic direct factor of nilpotent group | |FULL LIST, MORE INFO | |||
| fully invariant subgroup of nilpotent group | |FULL LIST, MORE INFO | |||
| verbal subgroup of nilpotent group | |FULL LIST, MORE INFO | |||
| powering-invariant characteristic subgroup of nilpotent group | characteristic not implies powering-invariant in nilpotent group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| normal subgroup of nilpotent group | |FULL LIST, MORE INFO | |||
| nilpotent characteristic subgroup | |FULL LIST, MORE INFO |