Finite abelian group: Difference between revisions

Revision as of 23:18, 31 May 2010

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

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

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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finitely generated abelian group				\|FULL LIST, MORE INFO
Periodic abelian group				\|FULL LIST, MORE INFO
Finite nilpotent group				\|FULL LIST, MORE INFO
Finite group that is 1-isomorphic to an abelian group				\|FULL LIST, MORE INFO
Finite group that is order statistics-equivalent to an abelian group				\|FULL LIST, MORE INFO

@@ Line 24: / Line 24: @@
 ===Stronger properties===
-* [[Weaker than::Abelian group of prime power order]]
+{| class="sortable" border="1"
-* [[Weaker than::Finite cyclic group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Odd-order abelian group]]
+|-
+| [[Weaker than::Abelian group of prime power order]] || || || ||
+|-
+| [[Weaker than::Finite cyclic group]] || || || || {{intermediate notions short|finite abelian group|finite cyclic group}}
+|-
+| [[Weaker than::Odd-order abelian group]] || || || || {{intermediate notions short|finite abelian group|odd-order abelian group}}
+|}
 ===Weaker properties===
-* [[Stronger than::Finitely generated abelian group]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Finitely generated abelian group]] || || || || {{intermediate notions short|finitely generated abelian group|finite abelian group}}
+|-
+| [[Stronger than::Periodic abelian group]] || || || || {{intermediate notions short|periodic abelian group|finite abelian group}}
+|-
+| [[Stronger than::Finite nilpotent group]] || || || || {{intermediate notions short|finite nilpotent group|finite abelian group}}
+|-
+| [[Stronger than::Finite group that is 1-isomorphic to an abelian group]] || || || || {{intermediate notions short|finite group that is 1-isomorphic to an abelian group|finite abelian group}}
+|-
+| [[Stronger than::Finite group that is order statistics-equivalent to an abelian group]] || || || || {{intermediate notions short|finite group that is order statistics-equivalent to an abelian group|finite abelian group}}
+|}