Finite abelian group: Difference between revisions

Latest revision as of 03:59, 16 April 2017

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

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	follows from abelianness is subgroup-closed	If $G$ is a finite abelian group and $H$ is a subgroup of $G$ , then $H$ is also a finite abelian group.
quotient-closed group property	Yes	follows from abelianness is quotient-closed	If $G$ is a finite abelian group and $H$ is a normal subgroup of $G$ , then the quotient group $G / H$ is also a finite abelian group.
finite direct product-closed group property Yes	follows from abelianness is direct product-closed	If $G_{1}, G_{2}, \dots, G_{n}$ are finite abelian groups, so is the external direct product $G_{1} \times G_{2} \times \dots \times G_{n}$ .
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 $G_{1}, G_{2}$ with isomorphic lattices of subgroups such that $G_{1}$ is finite abelian and $G_{2}$ is not.

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 20: / Line 20: @@
 {{group property conjunction see examples|finite group|abelian group}}
+==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::subgroup-closed group property]] || Yes || follows from [[abelianness is subgroup-closed]] || If <math>G</math> is a finite abelian group and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> is also a finite abelian group.
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || follows from [[abelianness is quotient-closed]] || If <math>G</math> is a finite abelian group and <math>H</math> is a normal subgroup of <math>G</math>, then the quotient group <math>G/H</math> is also a finite abelian group.
+|-
+| [[satisfies metaproperty::finite direct product-closed group property]] Yes || follows from [[abelianness is direct product-closed]] || If <math>G_1, G_2, \dots, G_n</math> are finite abelian groups, so is the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math>.
+|-
+| [[satisfies metaproperty::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 <math>G_1, G_2</math> with isomorphic lattices of subgroups such that <math>G_1</math> is finite abelian and <math>G_2</math> is not.
+|}
 ==Relation with other properties==
@@ Line 27: / Line 40: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Weaker than::Abelian group of prime power order]] || || || ||
+| [[Weaker than::abelian group of prime power order]] || || || ||
 |-
-| [[Weaker than::Finite cyclic group]] || || || || {{intermediate notions short|finite abelian group|finite cyclic group}}
+| [[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 than::odd-order abelian group]] || || || || {{intermediate notions short|finite abelian group|odd-order abelian group}}
 |}