# Group in which every finite subgroup is nilpotent

From Groupprops

## Definition

A **group in which every finite subgroup is nilpotent** is a group satisfying the property that every finite subgroup is a nilpotent group, and in particular, a finite nilpotent group.

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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Relation with other properties

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

p-group | there is a prime number such that the order of every element is a power of . | uses that prime power order implies nilpotent | |FULL LIST, MORE INFO | |

group in which every finite subgroup is cyclic | |FULL LIST, MORE INFO | |||

group in which every finite subgroup is abelian | |FULL LIST, MORE INFO | |||

locally nilpotent group | every finitely generated subgroup is nilpotent | follows from finite implies finitely generated | |FULL LIST, MORE INFO | |

group in which elements of coprime finite orders commute | |FULL LIST, MORE INFO | |||

group in which order of commutator divides order of element | |FULL LIST, MORE INFO |