# Difference between revisions of "Finitely generated abelian group"

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{{group property conjunction|finitely generated group|abelian group}} | {{group property conjunction|finitely generated group|abelian group}} | ||

− | + | [[importance rank::2| ]] | |

==Definition== | ==Definition== | ||

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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||

|- | |- | ||

− | | [[Stronger than:: | + | | [[Stronger than::Group in which every subgroup is finitely presented]] || || || || {{intermediate notions short|group in which every subgroup is finitely presented|finitely generated abelian group}} |

|- | |- | ||

− | | [[Stronger than:: | + | | [[Stronger than::Finitely presented group]] || has a finite [[presentation of a group|presentation]] || follows from [[structure theorem for finitely generated abelian groups]] || || {{intermediate notions short|finitely presented group|finitely generated abelian group}} |

|- | |- | ||

− | | [[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 | + | | [[Stronger than::Noetherian group]] (also called '''slender group''') || every subgroup is finitely generated || [[finitely generated abelian implies Noetherian]] (equivalently, ring of integers is Noetherian) || || {{intermediate notions short|Noetherian group|finitely generated abelian group}} |

+ | |- | ||

+ | | [[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}} | ||

|- | |- | ||

| [[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}} | | [[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::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}} | | [[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}} | ||

+ | |- | ||

+ | | [[Stronger than::Supersolvable group]] || has a [[normal series]] where all the successive quotients are [[cyclic group]]s || || || {{intermediate notions short|supersolvable group|finitely generated abelian group}} | ||

+ | |- | ||

+ | | [[Stronger than::Polycyclic group]] || has a [[subnormal series]] where all the successive quotients are [[cyclic group]]s || || || {{intermediate notions short|polycyclic group|finitely generated abelian group}} | ||

+ | |- | ||

+ | | [[Stronger than::Finitely generated solvable group]] || both a [[finitely generated group]] and a [[solvable group]] || || || {{intermediate notions short|finitely generated solvable group|finitely generated abelian group}} | ||

|} | |} | ||

## Latest revision as of 03:23, 17 December 2011

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

## Contents

## 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.