Definition
A group is said to be finitely presented or finitely presentable if it satisfies the following equivalent conditions:
- It possesses a presentation with finitely many generators, and finitely many relations.
- It is finitely generated and, for any finite generating set, it has a presentation with that generating set and finitely many relations.
- 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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of finite group|Find other variations of finite group |
Relation with other properties
Stronger properties
Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
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Finite group |
underlying set is finite |
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Finitely presented conjugacy-separable group, Finitely presented periodic group, Group with solvable conjugacy problem|FULL LIST, MORE INFO
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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
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Finitely generated abelian group |
finitely generated and abelian |
finitely gneerated abelian implies finitely presented -- follows from structure theorem for finitely generated abelian groups |
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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
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Finitely generated nilpotent group |
finitely generated and nilpotent |
finitely generated nilpotent implies finitely presented (also via polycyclic) |
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Finitely presented solvable group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
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Supersolvable group |
has a normal series with all quotients cyclic groups |
(via polycyclic) |
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Polycyclic group|FULL LIST, MORE INFO
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Polycyclic group |
has a subnormal series with all quotients cyclic groups; equivalently, Noetherian (slender) and solvable |
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Finitely presented solvable group, Group in which every subgroup is finitely presented|FULL LIST, MORE INFO
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Virtually polycyclic group |
has a subgroup of finite index that is a polycyclic group |
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|FULL LIST, MORE INFO
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Group in which every subgroup is finitely presented |
every subgroup is a finitely presented group |
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|FULL LIST, MORE INFO
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Finitely presented solvable group |
finitely presented and solvable |
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|FULL LIST, MORE INFO
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Weaker properties
Conjunction with other properties
Conjunction |
Other component of conjunction |
Intermediate notions between finitely presented group and conjunction |
Intermediate notions between other component and conjunction
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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
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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
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Finitely presented solvable group |
solvable group |
|FULL LIST, MORE INFO |
Finitely generated solvable group|FULL LIST, MORE INFO
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Finitely generated free group |
free group |
Finitely presented conjugacy-separable group, Group with solvable conjugacy problem|FULL LIST, MORE INFO |
|FULL LIST, MORE INFO
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Finitely presented Hopfian group |
Hopfian group |
|FULL LIST, MORE INFO |
|FULL LIST, MORE INFO
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Finitely presented residually finite group |
residually finite group |
|FULL LIST, MORE INFO |
Finitely generated residually finite group|FULL LIST, MORE INFO
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