Normal subgroup of nilpotent group
This article describes a property that arises as the conjunction of a subgroup property: normal 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 normal subgroup of nilpotent group is used for a subgroup of a group where the whole group is a nilpotent group and the subgroup is a normal subgroup.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| subgroup of abelian group | ||||
| normal subgroup of group of prime power order | ||||
| characteristic subgroup of nilpotent group |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| normal subgroup | |FULL LIST, MORE INFO | |||
| normal subgroup contained in hypercenter | |FULL LIST, MORE INFO | |||
| nilpotent-quotient subgroup | normal subgroup such that the quotient group is a nilpotent group. | |FULL LIST, MORE INFO | ||
| normal subgroup satisfying the subgroup-to-quotient powering-invariance implication | if the whole group and the normal subgroup are powered over a prime, so is the quotient group. | two proofs: via normal subgroup contained in the hypercenter via nilpotent-quotient |
|FULL LIST, MORE INFO | |
| subgroup of nilpotent group | |FULL LIST, MORE INFO |