# Group in which every subgroup is finitely presented

From Groupprops

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

## Contents

## Definition

A **group in which every subgroup is finitely presented** is defined as a group in which every subgroup is a finitely presented group.

## Formalisms

### In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: finitely presented group

View other properties obtained by applying the hereditarily operator

## Relation with other properties

### Stronger properties

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

Finitely generated abelian group | both finitely generated and abelian | Finitely generated nilpotent group|FULL LIST, MORE INFO | ||

Finitely generated nilpotent group | both finitely generated and nilpotent | |FULL LIST, MORE INFO | ||

Polycyclic group | has a subnormal series where all the successive quotients are cyclic groups | |FULL LIST, MORE INFO |

### Weaker properties

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

Noetherian group | every subgroup is finitely generated | |FULL LIST, MORE INFO | ||

Finitely presented group | has a finite presentation | |FULL LIST, MORE INFO | ||

Finitely generated group | |FULL LIST, MORE INFO |