Finite abelian group
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and abelian group
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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
VIEW: groups satisfying this property | groups dissatisfying property finite group | groups dissatisfying property abelian group
VIEW: Related group property satisfactions | Related group property dissatisfactions
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|subgroup-closed group property||Yes||follows from abelianness is subgroup-closed||If is a finite abelian group and is a subgroup of , then is also a finite abelian group.|
|quotient-closed group property||Yes||follows from abelianness is quotient-closed||If is a finite abelian group and is a normal subgroup of , then the quotient group is also a finite abelian group.|
|finite direct product-closed group property Yes||follows from abelianness is direct product-closed||If are finite abelian groups, so is the external direct product .|
|lattice-determined group property||No||there exists an abelian group of prime power order that is lattice-isomorphic to a non-abelian group not of prime power order||There exist groups with isomorphic lattices of subgroups such that is finite abelian and is not.|
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|abelian group of prime power order|
|finite cyclic group|||FULL LIST, MORE INFO|
|odd-order abelian group|||FULL LIST, MORE INFO|