# Metanilpotent group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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## Definition

A group is termed **metanilpotent** if it has a nilpotent normal subgroup (i.e., a normal subgroup that is nilpotent as a group) such that the quotient group is also a nilpotent group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Nilpotent group | ||||

Abelian group | ||||

Metabelian group | ||||

Metacyclic group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Solvable group |