Finitely presented group

From Groupprops
Jump to: navigation, search

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
VIEW: Definitions built on this | Facts about this: (facts closely related to Finitely presented group, all facts related to Finitely presented group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki
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
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
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