Groupprops, The Group Properties Wiki (pre-alpha)

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 1 Definition 1.1 Symbol-free definition 1.2 Equivalence of definitions 2 Examples 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties

Definition

Symbol-free definition

A finite abelian group is a group satisfying the following equivalent conditions:

1. It is both finite and abelian.
2. It is isomorphic to a direct product of finitely many finite cyclic groups.
3. It is isomorphic to a direct product of abelian groups of prime power order.
4. 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
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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
Odd-order abelian group

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finitely generated abelian group
Periodic abelian group
Finite nilpotent group				click here
Finite group that is 1-isomorphic to an abelian group				click here
Finite group that is order statistics-equivalent to an abelian group				click here