Finitely generated abelian group

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and abelian group
View other group property conjunctions OR view all group properties

Definition

Symbol-free definition

A finitely generated abelian group is a group satisfying the following equivalent conditions:

  1. It is finitely generated and abelian.
  2. It is a finitely generated module over \mathbb{Z}, the ring of integers.
  3. It is isomorphic to an external direct product of finitely many cyclic groups.

Equivalence of definitions

Further information: structure theorem for finitely generated abelian groups

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite abelian group finite and abelian follows from finite implies finitely generated the group of integers is a finitely generated abelian group that is not finite |FULL LIST, MORE INFO
Cyclic group generated by one element cyclic implies abelian abelian not implies cyclic, even in the finite case |FULL LIST, MORE INFO
Finitely generated free abelian group direct product of finitely many copies of the group of integers |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Group in which every subgroup is finitely presented Finitely generated nilpotent group|FULL LIST, MORE INFO
Finitely presented group has a finite presentation 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
Noetherian group (also called slender group) every subgroup is finitely generated finitely generated abelian implies Noetherian (equivalently, ring of integers is Noetherian) Finitely generated nilpotent group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
Residually finite group every non-identity element is outside a normal subgroup of finite index finitely generated abelian implies residually finite Finitely generated conjugacy-separable group, Finitely generated residually finite group, Finitely presented conjugacy-separable group|FULL LIST, MORE INFO
Finitely generated residually finite group both a finitely generated group and a residually finite group finitely generated abelian implies residually finite Finitely generated conjugacy-separable group, Finitely presented conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO
Hopfian group every surjective endomorphism is an automorphism finitely generated abelian implies Hopfian Finitely generated Hopfian group, Finitely generated conjugacy-separable group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Finitely presented residually finite group, Noetherian group|FULL LIST, MORE INFO
Finitely generated nilpotent group both a finitely generated group and a nilpotent group abelian implies nilpotent nilpotent not implies abelian (counterexamples are there in the finite case) |FULL LIST, MORE INFO
Supersolvable group has a normal series where all the successive quotients are cyclic groups Finitely generated nilpotent group|FULL LIST, MORE INFO
Polycyclic group has a subnormal series where all the successive quotients are cyclic groups Finitely generated nilpotent group, Supersolvable group|FULL LIST, MORE INFO
Finitely generated solvable group both a finitely generated group and a solvable group Finitely generated nilpotent group, Finitely presented solvable group, Polycyclic group, Supersolvable group|FULL LIST, MORE INFO

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of a finitely generated Abelian group is finitely generated Abelian. This is related to the fact that since the ring of integers is Noetherian, any submodule of a finitely generated module over it is finitely generated.

In general, subgroups of finitely generated groups are not finitely generated; those finitely generated groups for which every subgroup is finitely generated, are termed sliender groups or Noetherian groups.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a finitely generated Abelian group is finitely generated Abelian. In fact, both the property of being finitely gneerated and the property of being Abelian are preserved on passing to quotients.

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A finite direct product of finitely generated Abelian groups is finitely generated Abelian. In fact, both the property of being finitely generated and the property of being Abelian are closed upon taking finite direct products.