Normal subgroup of nilpotent group

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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
Normal subgroup contained in the hypercenter|FULL LIST, MORE INFO
subgroup of nilpotent group |FULL LIST, MORE INFO