Nilpotent-quotient subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup of a group is termed a nilpotent-quotient subgroup or nilpotent-quotient normal subgroup if it is a normal subgroup and the quotient group is a nilpotent group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian-quotient subgroup
upward-closed normal subgroup

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup satisfying the subgroup-to-quotient powering-invariance implication if the whole group and subgroup are powered over a prime, so is the quotient group. nilpotent-quotient implies subgroup-to-quotient powering-invariance implication |FULL LIST, MORE INFO
normal subgroup |FULL LIST, MORE INFO

Facts

The intersection of all nilpotent-quotient normal subgroups is termed the nilpotent residual, and this is also described as the \omega^{th} term of the transfinite lower central series It is trivial if and only if the group is a residually nilpotent group. In a finite group, the nilpotent residual is itself a nilpotent-quotient normal subgroup.