Finitely generated abelian group: Difference between revisions
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{{group property conjunction|finitely generated group| | {{group property conjunction|finitely generated group|abelian group}} | ||
==Definition== | ==Definition== | ||
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A '''finitely generated | A '''finitely generated abelian group''' is a group satisfying the following equivalent conditions: | ||
# It is [[finitely generated group|finitely generated]] and [[ | # It is [[finitely generated group|finitely generated]] and [[abelian group|abelian]]. | ||
# It is a finitely generated module over <math>\mathbb{Z}</math>, the [[ring of integers]] | # It is a finitely generated module over <math>\mathbb{Z}</math>, the [[ring of integers]]. | ||
# It is | # It is isomorphic to an [[defining ingredient::external direct product]] of finitely many [[cyclic group]]s. | ||
===Equivalence of definitions=== | ===Equivalence of definitions=== | ||
{{further|[[structure theorem for finitely generated | {{further|[[structure theorem for finitely generated abelian groups]]}} | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::Finite abelian group]] || [[finite group|finite]] and [[abelian group|abelian]] || follows from [[finite implies finitely generated]] || the [[group of integers]] is a finitely generated abelian group that is not finite || {{intermediate notions short|finitely generated abelian group|finite abelian group}} | |||
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| [[Weaker than::Cyclic group]] || generated by one element || [[cyclic implies abelian]] || [[abelian not implies cyclic]], even in the finite case || {{intermediate notions short|finitely generated abelian group|cyclic group}} | |||
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| [[Weaker than::Finitely generated free abelian group]] || direct product of finitely many copies of the [[group of integers]] || || || {{intermediate notions short|finitely generated abelian group|finitely generated free abelian group}} | |||
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===Weaker properties=== | ===Weaker properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::Finitely presentable group]] || has a finite [[presentation of a group|presentation]] || follows from [[structure theorem for finitely generated abelian groups]] || || {{intermediate notions short|finitely presentable group|finitely generated abelian group}} | |||
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| [[Stronger than::Slender group]] || every subgroup is finitely generated || [[finitely generated abelian implies slender]] (equivalently, ring of integers is Noetherian) || || {{intermediate notions short|slender group|finitely generated abelian group}} | |||
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| [[Stronger than::Residually finite group]] || every non-identity element is outside a [[normal subgroup of finite index]] || [[finitely generated abelian implies residually finite]] || || {{intermediate notions short|residually finite group\finitely generated abelian group}} | |||
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| [[Stronger than::Finitely generated residually finite group]] || both a [[finitely generated group]] and a [[residually finite group]] || [[finitely generated abelian implies residually finite]] || || {{intermediate notions short|finitely generated residually finite group|finitely generated abelian group}} | |||
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| [[Stronger than::Hopfian group]] || every [[surjective endomorphism]] is an [[automorphism]] || [[finitely generated abelian implies Hopfian]] || || {{intermediate notions short|Hopfian group|finitely generated abelian group}} | |||
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| [[Stronger than::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) || {{intermediate notions short|finitely generated nilpotent group|finitely generated abelian group}} | |||
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==Metaproperties== | ==Metaproperties== | ||
Revision as of 21:35, 19 May 2010
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:
- It is finitely generated and abelian.
- It is a finitely generated module over , the ring of integers.
- 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
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.