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

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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
Finitely related group				|FULL LIST, MORE INFO
Recursively presentable group	has a recursive presentation			|FULL LIST, MORE INFO

Conjunction with other properties

Conjunction	Other component of conjunction	Intermediate notions between finitely presented group and conjunction	Intermediate notions between other component and conjunction
Finitely generated abelian group	abelian group	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	Residually cyclic group|FULL LIST, MORE INFO
Finitely generated nilpotent group	nilpotent group	Finitely presented solvable group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
Finitely presented solvable group	solvable group	|FULL LIST, MORE INFO	Finitely generated solvable group|FULL LIST, MORE INFO
Finitely generated free group	free group	Finitely presented conjugacy-separable group, Group with solvable conjugacy problem|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
Finitely presented Hopfian group	Hopfian group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
Finitely presented residually finite group	residually finite group	|FULL LIST, MORE INFO	Finitely generated residually finite group|FULL LIST, MORE INFO