Group in which every subgroup is finitely presented
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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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 | |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 |