Finite abelian group
From Groupprops
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
Contents
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
Examples
VIEW: groups satisfying this property | groups dissatisfying property finite group | groups dissatisfying property abelian group
VIEW: Related group property satisfactions | Related group property dissatisfactions
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | Yes | follows from abelianness is subgroup-closed | If ![]() ![]() ![]() ![]() |
quotient-closed group property | Yes | follows from abelianness is quotient-closed | If ![]() ![]() ![]() ![]() |
finite direct product-closed group property Yes | follows from abelianness is direct product-closed | If ![]() ![]() | |
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 ![]() ![]() ![]() |
Relation with other properties
Stronger 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 |