# Finitely presented group

## Definition

A group is said to be finitely presented or finitely presentable if it satisfies the following equivalent conditions:

1. It possesses a presentation with finitely many generators, and finitely many relations.
2. It is finitely generated and, for any finite generating set, it has a presentation with that generating set and finitely many relations.
3. It is finitely generated and, for any presentation with a finite number of generators, we can throw away all but finitely many relations to get a presentation with finitely many generators and finitely many relations.

### Equivalence of definitions

Further information: equivalence of definitions of finitely presented 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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## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite group underlying set is finite Finitely presented conjugacy-separable group, Finitely presented periodic group, Group with solvable conjugacy problem|FULL LIST, MORE INFO
Finitely generated free group Free group with finite generating set; equivalently, has a finite freely generating set (by definition, since a free group has no relations) (lots of counterexamples, such as nontrivial finite groups) Finitely presented conjugacy-separable group, Group with solvable conjugacy problem|FULL LIST, MORE INFO
Finitely generated abelian group finitely generated and abelian finitely gneerated abelian implies finitely presented -- follows from structure theorem for finitely generated abelian groups Finitely generated nilpotent group, Finitely presented conjugacy-separable group, Finitely presented solvable group, Group in which every subgroup is finitely presented, Group with solvable conjugacy problem, Polycyclic group|FULL LIST, MORE INFO
Finitely generated nilpotent group finitely generated and nilpotent finitely generated nilpotent implies finitely presented (also via polycyclic) Finitely presented solvable group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
Supersolvable group has a normal series with all quotients cyclic groups (via polycyclic) Polycyclic group|FULL LIST, MORE INFO
Polycyclic group has a subnormal series with all quotients cyclic groups; equivalently, Noetherian (slender) and solvable Finitely presented solvable group, Group in which every subgroup is finitely presented|FULL LIST, MORE INFO
Virtually polycyclic group has a subgroup of finite index that is a polycyclic group |FULL LIST, MORE INFO
Group in which every subgroup is finitely presented every subgroup is a finitely presented group |FULL LIST, MORE INFO
Finitely presented solvable group finitely presented and solvable |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finitely generated group has a finite generating set Finitely generated group for which all homomorphisms to any finite group can be listed in finite time|FULL LIST, MORE INFO