Finite abelian group: Difference between revisions
(New page: {{group property conjunction|finite group|Abelian group}} ==Definition== ===Symbol-free definition=== A '''finite Abelian group''' is a group satisfying the following equivalent con...) |
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{{group property conjunction|finite group| | {{group property conjunction|finite group|abelian group}} | ||
==Definition== | ==Definition== | ||
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A '''finite | A '''finite abelian group''' is a [[group]] satisfying the following equivalent conditions: | ||
# It is both [[finite group|finite]] and [[ | # It is both [[finite group|finite]] and [[abelian group|abelian]]. | ||
# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s. | # It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s. | ||
# It is isomorphic to a direct product of [[Abelian group of prime power order|Abelian groups of prime power order]]. | # It is isomorphic to a direct product of [[Abelian group of prime power order|Abelian groups of prime power order]]. | ||
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===Equivalence of definitions=== | ===Equivalence of definitions=== | ||
{{proofat|[[Structure theorem for finitely generated | {{proofat|[[Structure theorem for finitely generated abelian groups]]}} | ||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Weaker than::Abelian group of prime power order]] | * [[Weaker than::Abelian group of prime power order]] | ||
* [[Weaker than::Finite cyclic group]] | * [[Weaker than::Finite cyclic group]] | ||
* [[Weaker than::Odd-order | * [[Weaker than::Odd-order abelian group]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Stronger than::Finitely generated | * [[Stronger than::Finitely generated abelian group]] | ||
Revision as of 23:18, 21 January 2009
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and abelian group
View other group property conjunctions OR view all group properties
Definition
Symbol-free definition
A finite abelian group is a group satisfying the following equivalent conditions:
- It is both finite and abelian.
- It is isomorphic to a direct product of finitely many finite cyclic groups.
- It is isomorphic to a direct product of Abelian groups of prime power order.
- It is isomorphic to a direct product of cyclic groups of prime power order.
Equivalence of definitions
For full proof, refer: Structure theorem for finitely generated abelian groups