Finitely generated group

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Definition

A group is said to be finitely generated if it satisfies the following equivalent conditions:

  1. It has a finite generating set.
  2. Every generating set of the group has a subset that is finite and is also a generating set.
  3. The group has at least one minimal generating set and every minimal generating set of the group is finite.
  4. The minimum size of generating set of the group is finite.
  5. The group is a join of finitely many cyclic subgroups.

Equivalence of definitions

Further information: equivalence of definitions of finitely generated 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 that is pivotal (i.e., important) among existing group properties
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Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property No finite generation is not subgroup-closed It is possible to have a finitely generated group G and a subgroup H of G such that H is not finitely generated.
quotient-closed group property Yes finite generation is quotient-closed If G is a finitely generated group and H is a normal subgroup of G, then the quotient group G/H is a finitely generated group.
extension-closed group property Yes finite generation is extension-closed If G is a group and H is a normal subgroup of G such that both H and the quotient group G/H are finitely generated, then G is also finitely generated.
finite direct product-closed group property Yes (via extension-closed If G_1, G_2, \dots, G_n are all finitely generated groups, so is the external direct product G_1 \times G_2 \times \dots \times G_n.
finite index-closed group property Yes finite generation is finite index-closed If G is a finitely generated group and H is a subgroup of finite index in G, then H is also a finitely generated group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite group Finitely generated Hopfian group, Finitely generated group for which all homomorphisms to any finite group can be listed in finite time, Finitely generated periodic group, Finitely generated profinite group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Group with a finite complete rewriting system, Noetherian group|FULL LIST, MORE INFO
Noetherian group (also called slender group) every subgroup is finitely generated (by definition) finitely generated not implies Noetherian Finitely generated Hopfian group|FULL LIST, MORE INFO
Finitely presented group has a presentation with finitely many generators and finitely many relations finitely presented implies finitely generated finitely generated not implies finitely presented Finitely generated group for which all homomorphisms to any finite group can be listed in finite time|FULL LIST, MORE INFO

Conjunction with other properties

Conjunction Other component of conjunction Intermediate notions between finitely generated group and conjunction Intermediate notions between other component and conjunction Additional comments
Finitely generated free group Free group Burnside group, Finitely generated Hopfian group, Finitely generated group for which all homomorphisms to any finite group can be listed in finite time, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Group with a finite complete rewriting system, Group with polynomial-time solvable word problem, Group with solvable word problem|FULL LIST, MORE INFO |FULL LIST, MORE INFO A finitely generated free group is a group with finite freely generating set
Finitely generated abelian group abelian group Finitely generated Hopfian group, Finitely generated conjugacy-separable group, Finitely generated group for which all homomorphisms to any finite group can be listed in finite time, Finitely generated nilpotent group, Finitely generated residually finite group, Finitely generated solvable group, Finitely presented conjugacy-separable group, Finitely presented solvable group, Group in which every subgroup is finitely presented, Group with a finite complete rewriting system, Group with polynomial-time solvable word problem, Group with solvable word problem, Noetherian group, Polycyclic group|FULL LIST, MORE INFO Residually cyclic group|FULL LIST, MORE INFO turns out to be a direct product of finitely many cyclic groups by the structure theorem for finitely generated abelian groups
Finitely generated residually finite group residually finite group Finitely generated Hopfian group|FULL LIST, MORE INFO |FULL LIST, MORE INFO
Finitely generated Hopfian group Hopfian group |FULL LIST, MORE INFO |FULL LIST, MORE INFO
Finitely generated nilpotent group nilpotent group Finitely generated group for which all homomorphisms to any finite group can be listed in finite time, Finitely generated solvable group, Finitely presented solvable group, Group in which every subgroup is finitely presented, Group with a finite complete rewriting system, Group with polynomial-time solvable word problem, Group with solvable word problem, Noetherian group, Polycyclic group|FULL LIST, MORE INFO |FULL LIST, MORE INFO equivalent to abelianization being finitely generated
Finitely generated solvable group solvable group |FULL LIST, MORE INFO |FULL LIST, MORE INFO
Finitely generated periodic group periodic group |FULL LIST, MORE INFO |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Countable group cardinality of underlying set is either finite or countably infinite |FULL LIST, MORE INFO
Group with finitely many homomorphisms to any finite group for any fixed finite group, there are finitely many homomorphisms from the given group to that group |FULL LIST, MORE INFO
Group with no infinite minimal generating set any minimal generating set is finite |FULL LIST, MORE INFO
Normally finitely generated group normal closure of a finitely generated subgroup |FULL LIST, MORE INFO

Opposite properties

  • Locally finite group is a group where every finitely generated subgroup is finite. A group is locally finite and finitely generated if and only if it is finite.

Effect of property operators

The hereditarily operator

Applying the hereditarily operator to this property gives: slender group

A slender group, or Noetherian group, is a group such that all its subgroups are finitely generated.

Testing

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsFinitelyGeneratedGroup
View GAP-testable group properties